Direct-Sequence Communication Systems 

by Don Torrieri 

ARL-TR-3175 March 2004 

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Army Research Laboratory 

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ARL-TR-3175 March 2004 

Direct-Sequence Communication Systems 

Don Torrieri 

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This report presents a comprehensive review of the state of the art of direct-sequence communication systems. Although it is 

largely self-contained mathematically, the report presumes a thorough understanding of modern digital communications. To 

limit the report to approximately 200 pages, only the most vital aspects of the theory are emphasized, but the cited references

provide many details and minor topics. 

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Direct sequence, spread spectrum, code-division multiple access 

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Contents

Preface xii

1. Definitions and Concepts 1

2. Spreading Sequences and Waveforms 4

2.1 Random Binary Sequence . . . . . . . . . . . . . . . ............. 4

2.2 Shift-Register Sequences . . . . . . . . . . . .................. 6

2.3 Periodic Autocorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Polynomials over the Binary Field . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Long Nonlinear Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Systems with Coherent PSK and Random Spreading Sequences 23

3.1 Tone Interference at Carrier Frequency . . . . . . . . . . . . . . . . . . . . . 26

3.2 General Tone Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Gaussian Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4. Quaternary Systems with Random Spreading Sequences 32

5. Pulsed Interference 38

6. Despreading with Matched Filters 47

6.1 Matched Filters ..................... . . . . . . . . . . . . . 48

6.2 Noncoherent Systems.................. . . . . . . . . . . . . . 52

6.3 Multipath-Resistant Coherent System . . . . . . . . . . . . . . . . . . . . . . 56

7. Code Synchronization 60

iii
7.1 Acquisition with Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Serial-Search Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.3 Acquisition Correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.4 Code Tracking ...................... . . . . . . . . . . . . . 88

8. Rejection of Narrowband Interference 93

8.1 Time-Domain Adaptive Filtering . . . . . . . . . . . . . . . . . . . . . . . . 94

8.2 Transform-Domain Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.3 Nonlinear Filtering ................... . . . . . . . . . . . . . 99

9. Detection of Direct-Sequence Signals 105

9.1 Ideal Detection ........................... . . . . . . . 106

9.2 Wideband Radiometer ....................... . . . . . . . 109

10.Direct-Sequence Code-Division Multiple Access 116

10.1 Orthogonal Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

10.2 Sequences with Small Cross-Correlations . . . . . . . . . . . . . . . . . . . . 119

10.3 Symbol Error Probability for Direct-Sequence Systems . . . . . . . . . . . . 124

10.4 Complex-Valued Quaternary Sequences . . . . . . . . . . . . . . . . . . . . . 125

10.5 Direct-Sequence Systems with PSK and Random Sequences . . . . . . . . . 129

10.6 Quadriphase Direct-Sequence Systems with Random Sequences . . . . . . . . 137

10.7 Wideband Direct-Sequence CDMA . . . . . . . . . . . . . . . . . . . . . . . 140

10.8 Cellular Networks and Power Control . . . . . . . . . . . . . . . . . . . . . . 149

11.Multiuser Detectors 152

11.1 Optimum Detectors ......................... . . . . . . . 153

iv
11.2 Decorrelating detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

11.3 MMSE Detector . . . . . ...................... . . . . . . . 160

11.4 Interference Cancellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

11.5 Successive Interference Canceller . . . . . . . . . . . . . . . . . . . . . . . . . 162

11.6 Parallel Interference Canceller . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Appendices 165

A. Inequalities 165

A.1 Jensen’s Inequality ................... . . . . . . . . . . . . . 165

A.2 Chebyshev’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

B. Probability Distributions 169

B.1 Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.2 Central Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 171

B.3 Rice Distribution .......................... . . . . . . . 171

B.4 Rayleigh Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

B.5 Sum of Independent, Exponentially Distributed Random Variables . . . . . . 174

C. Signal Representations 177

C.1 Bandpass Signals .......................... . . . . . . . 177

C.2 Stationary Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . 179

C.3 Sampling Theorems ......................... . . . . . . . 185

D. Adaptive Filters 189

References 192

v
Distribution 197

List of Figures

1 Examples of (a) data modulation and (b) spreading waveform . . . . . . . . 2

2 Functional block diagram of direct-sequence systemn with PSK or DPSK: (a)

transmitter and (b) receiver . . . . . . . . . . . . . . ............. 3

3 Spectra of desired signal and interference: (a) wideband-filter output and (b)

demodulator input ................... ............. 3

4 Sample function of a random binary sequence . . . . . . . . . . . . . . . . . 4

5 General feedback shift register with m stages . . . . . . . . . . . . . . . . . . 6

6 (a) Three-stage linear feedback shift register and (b) contents after successive

shifts . . . . . . . . . . . . . . . . ........................ 7

7 Linear feedback shift register: (a) standard representation and (b) high-speed

form . . . . . . . . . . . . . . . . . . . . . . .................. 9

8 (a) Nonmaximal linear feedback shift register and (b) state diagrams . . . . 12

9 Autocorrelations of maximal sequence and random binary sequence . . . . . 14

10 Power spectral density of maximal sequence . . . . . . . . . . . . . . . . . . 14

11 Linear generator of binary sequence with period N . . . . . . . . . . . . . . 21

12 (a) Nonlinear generator and (b) its linear equivalent . . . . . . . . . . . . . . 22

13 Nonlinear generator that uses a multiplexer . . . . . . . . . . . . . . . . . . 23

14 Basic elements of correlator for direct-sequence signal with coherent PSK . . 24

15 Symbol error probability of binary direct-sequence system with tone interference at carrier frequency and G = 17 dB . . . . . . . . . . . . . . . . . . . . 30

16 Symbol error probability for direct-sequence system with PSK, rectangular

and sinusoidal chip waveforms, G = 17 dB, Es/N0 = 14 dB, and GS/I = 10

dB in the presence of tone interference . . . . . . . . . . . . . . . . . . . . . 30

vi
17 Receiver for direct-sequence signal with dual quaternary modulation; CMF

= chip-matched filter; SSG = spreading sequence generator. Delay = 0 for

QPSK; delay = Tc/2 for OQPSK and MSK . . . . . . . . . . . . . . . . . . . 33

18 Symbol error probability for quaternary and binary direct-sequence systems

with G = 17 dB, Es/N0 = 14 dB, and GS/I = 10 dB in the presence of tone

interference . . . . . . . ...................... . . . . . . . 35

19 Receiver for direct-sequence signal with balanced quaternary modulation (delay = 0 for QPSK and delay = Tc/2 for OQPSK and MSK); CMF = chipmatched filter; SSG = spreading sequence generator . . . . . . . . . . . . . . 36

20 Symbol error probability for direct-sequence systems with balanced QPSK

and dual quaternary modulations, rectangular and sinusoidal chip waveforms,

G = 17 dB, Es/N0 = 14 dB, and GS/I = 10 dB in the presence of tone

interference . . . . . . . ...................... . . . . . . . 38

21 Worst-case performance against pulsed interference for convolutional codes of

constraint length K, rate r, Eb/N0 = 20 dB, and hard decisions . . . . . . . . 41

22 Worst-case performance against pulsed interference for convolutional codes of

constraint length K, rate r, Eb/N0 = 20 dB and maximum-likelihood (ML)

and AGC metrics . . . . ...................... . . . . . . . 43

23 Performance against pulsed interference for convolutional code with whitenoise metric, and K = 7, r = 1/2, and Eb/N0 = 20 dB . . . . . . . . . . . . . 45

24 Performance against pulsed interference for convolutional code with erasures,

K = 7, r = 1/2, and Eb/N0 = 20 dB . . . . . . . . . . . . . . . . . . . . . . . 46

25 Worst-case performance against pulsed interference for convolutional codes

witrh ideal erasure decoding, constraint length K, rate r, and Eb/N0 = 20 dB 47

26 Matched filter that uses a SAW transversal filter . . . . . . . . . . . . . . . . 50

27 SAW elastic convolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

28 Chip configurations within convolver at time instants t = 4Tc, 5Tc, and 6Tc

when t0 = 0, L/v = Tc, and T = 4Tc . . . . . . . . . . . . . . . . . . . . . . . 53

29 Direct-sequence system with binary code-shift keying: (a) transmitter and (b)

receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

30 Receiver for direct-sequence system with differential phase-shift keying . . . 56

vii
31 Recirculation loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

32 Coherent decision-directed demodulator . . . . . . . . . . . . . . . . . . . . . 59

33 Conceptual waveforms of demodulator: (a) matched-filter output, (b) recirculation loop input or output, and (c) baseband integrator input . . . . . . . 60

34 Digital matched filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

35 Configuration of a serial-search acquisition system enabled by a matched filter 65

36 Serial-search acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . 66

37 Flow graph of multiple-dwell system with consecutive-count strategy . . . . . 67

38 Flow graph of multiple-dwell system with up-down strategy . . . . . . . . . . 67

39 Trajectories of search positions: (a) uniform search and (b) broken-center Z

search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

40 Trajectories of expanding-window search positions: (a) broken-center and (b)

continuous-center search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

41 Trajectories of alternating search positions: (a) uniform search and (b) nonuniform search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

42 Circular state diagram for serial-search acquisition . . . . . . . . . . . . . . . 77

43 Subsidiary state diagram for determination of H0(z) for consecutive-count

double-dwell system ................... . . . . . . . . . . . . . 78

44 Subsidiary state diagram for calculation of HD(z) and HM(z) for consecutivecount double-dwell system with two-state collective state . . . . . . . . . . . 79

45 Noncoherent correlator for acquisition system. CMF = chip matched filter.

SSG = spreading sequence generator . . . . . . . . . . . . . . . . . . . . . . 81

46 NMAT versus Ec/N0 for single-dwell system in presence of fast Rayleigh fading

or no fading. Values of Pf and M are optimized . . . . . . . . . . . . . . . . 86

47 NMAT versus Ec/N0 for double-dwell systems in presence of fast Rayleigh

fading. Step size is ∆ = 1/2. Values of PF1, PF2, M1, and M2 are optimizedn

brackets) appears on the contents list . . . . . . . . . . . . . . . . . . . . . . 87

48 Delay-locked loop .................... . . . . . . . . . . . . . 89

viii
49 Discriminator characteristic of delay-locked loop for δ = 1/2 . . . . . . . . . 91

50 Tau-dither loop ..................... . . . . . . . . . . . . . 92

51 Direct-sequence receiver with processor for rejecting narrowband interference 94

52 (a) Processor using adaptive filter and (b) two-sided adaptive transversal filter 95

53 Processor with decision-directed adaptive filter . . . . . . . . . . . . . . . . . 96

54 Transform-domain processor . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

55 Adaptive ACM filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

56 Radiometers: (a) passband, (b) baseband with integration, and (c) baseband

with sampling at rate 1/W and summation . . . . . . . . . . . . . . . . . . . 110

57 Probability of detection versus E/N0 for wideband radiometer with PF = 10−3

and various values of TW. Solid curves are the Nˆ0 = N0; dashed curve is for

Nˆ0 = 1.001N0 ............................ . . . . . . . 114

58 Energy-to-noise-density ratio versus TW for wideband radiometer with PD =

0.99 and various values of PF and h . . . . . . . . . . . . . . . . . . . . . . . 116

59 Gold sequence generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

60 (a) Feedback shift register for a quaternary sequence and (b) contents after

successive shifts . . . . . ...................... . . . . . . . 127

61 Product of quaternary data and spreading sequences . . . . . . . . . . . . . 128

62 Receiver for direct-sequence system with complex quaternary spreading sequences. CMF is chip-matched filter . . . . . . . . . . . . . . . . . . . . . . 129

63 Symbol error probability of direct-sequence system with PSK in presence of

single multiple-access interference signal and Es/N0 = 15 dB . . . . . . . . . 135

64 Symbol error probability of direct-sequence system with PSK in presence of

K − 1 equal-power multiple-access interference signals and Es/N0 = 15 dB . 135

65 Symbol error probability of direct-sequence system with PSK in presence of

3 equal-power multiple-access interference signals and Es/N0 = 15 dB . . . . 137

66 Symbol error probability of quadriphase direct-sequence system in presence

of 3 equal-power multiple-access interference signals and Es/N0 = 15 dB . . . 139

ix
67 Multicarrier direct-sequence CDMA system: (a) transmitter and (b) receiver 142

68 Symbol error probability for multicarrier systems with L carriers . . . . . . . 145

69 Symbol error probability for single-carrier systems and L ≤ 4 multipath components with different multipath intensity vectors . . . . . . . . . . . . . . . 147

70 Information-bit error probability for multicarrier system with L = 32 and

for single-carrier systems with Γ1 = (.505, .495, 0) and Γ2 = (.995, .990, 0).

Error-correcting code is BCH(63, 36) . . . . . . . . . . . . . . . . . . . . . . 149

71 Geometry of cellular network with base station at center of each hexagon.

Two concentric tiers of cells surrounding a central cell are shown . . . . . . . 150

72 Architecture of decorrelating detector and MMSE detecter. Filter bank comprises parallel correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

73 Successive interference canceller with K detector-generators to produce signal

estimates for subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

74 Structure of detector-generator for signal i . . . . . . . . . . . . . . . . . . . 162

75 Second canceller of multistage canceller using successive interference cancellers 163

76 Parallel interference canceller for two signals . . . . . . . . . . . . . . . . . . 163

77 Multistage parallel interference canceller for two signals. D=delay . . . . . . 163

x
List of Tables

1 Primitive Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

xi
Preface

This report presents a comprehensive review of the state of the art of direct-sequence

communication systems. Although it is largely self-contained mathematically, the report

presumes a thorough understanding of modern digital communications. To limit the report

to approximately 200 pages, only the most vital aspects of the theory are emphasized, but

the cited references provide many details and minor topics. Sections 1 and 2 cover the basic

definitions and concepts and the fundamental properties of spreading sequences. Sections 3

and 4 derive the symbol error probabilities of coherent systems in the presence of tone or

Gaussian interference. Methods of suppressing pulsed interference are analyzed in Section

5. In Section 6, the role of matched filters in direct-sequence systems is examined. Code

synchronization, which is the most significant issue in the design of direct-sequence

systems, is analyzed in Section 7. Several supplementary methods of rejecting narrowband

interference are presented in Section 8. The detection of the existence of a direct-sequence

signal by both an ideal receiver and the more practical wideband radiometer are analyzed

in Section 9. Section 10 explains and develops the theory of direct-sequence code-division

multiple access, which is the dominant technology of third and fourth generation cellular

mobile communications. Multiuser detectors, which have current and potential applications

in cellular receivers, are discussed in Section 11. The four appendices provide extensive

derivations of mathematical results that are used in the main text.

xii
1. Definitions and Concepts

A spread-spectrum signal is a signal that has an extra modulation that expands the signal

bandwidth beyond what is required by the underlying data modulation. Spread-spectrum

communication systems [1], [2], [3] are useful for suppressing interference, making

interception difficult, accommodating fading and multipath channels, and providing a

multiple-access capability. The most practical and dominant methods of spread-spectrum

communications are direct-sequence modulation and frequency hopping of digital

communications.

A direct-sequence signal is a spread-spectrum signal generated by the direct mixing of the

data with a spreading waveform before the final carrier modulation. Ideally, a

direct-sequence signal with binary phase-shift keying (PSK) or differential PSK (DPSK)

data modulation can be represented by

s(t) = Ad(t)p(t) cos(2πfct + θ) (1-1)

where A is the signal amplitude, d(t) is the data modulation, p(t) is the spreading

waveform, fc is the carrier frequency, and θ is the phase at t = 0. The data modulation is a

sequence of nonoverlapping rectangular pulses of duration Ts, each of which has an

amplitude di = +1 if the associated data symbol is a 1 and di = −1 if it is a 0. Equation

(1-1) implies that s(t) = Ap(t)cos[2πfct + θ + πd(t)], which explicitly exhibits the

phase-shift keying by the data modulation. The spreading waveform has the form

p (t) = X∞

i=−∞

piψ (t − iTc) (1-2)

where each pi equals +1 or −1 and represents one chip of the spreading sequence. The chip

waveform ψ(t) is ideally confined to the interval [0, Tc] to prevent interchip interference in

the receiver. A rectangular chip waveform has ψ(t) = w(t, Tc), where

w (t, T) = 1⁄2

1, 0 ≤ t<T

0, otherwise (1-3)

Figure 1 depicts an example of d(t) and p(t) for a rectangular chip waveform.

Message privacy is provided by a direct-sequence system if a transmitted message cannot

be recovered without knowledge of the spreading sequence. To ensure message privacy,

which is assumed henceforth, the data-symbol transitions must coincide with the chip

transitions. Since the transitions coincide, the processing gain G = Ts/Tc is an integer equal

to the number of chips in a symbol interval. If W is the bandwidth of p(t) and B is the

bandwidth of d(t), the spreading due to p(t) ensures that s(t) has a bandwidth W >> B.

1
d(t)

p(t)

t

(b)

1

–1

1

–1

t

(a)

Figure 1. Examples of (a) data modulation and (b) spreading waveform.

Figure 2 is a functional or conceptual block diagram of the basic operation of a

direct-sequence system with PSK. To provide message privacy, data symbols and chips,

which are represented by digital sequences of 0’s and 1’s, are synchronized by the same

clock and then modulo-2 added in the transmitter. The adder output is converted

according to 0 → −1 and 1 → +1 before the chip and carrier modulations. Assuming that

chip and symbol synchronization has been established, the received signal passes through

the wideband filter and is multiplied by a synchronized local replica of p(t). If ψ(t) is

rectangular, then p(t) = ±1 and p2(t) = 1. Therefore, if the filtered signal is given by (1-1),

the multiplication yields the despread signal

s1(t) = p(t)s(t) = Ad(t) cos (2πfct + θ) (1-4)

at the input of the PSK demodulator. Since the despread signal is a PSK signal, a

standard coherent demodulator extracts the data symbols.

Figure 3(a) is a qualitative depiction of the relative spectra of the desired signal and

narrowband interference at the output of the wideband filter. Multiplication by the

spreading waveform produces the spectra of Figure 3(b) at the demodulator input. The

signal bandwidth is reduced to B, while the interference energy is spread over a bandwidth

exceeding W. Since the filtering action of the demodulator then removes most of the

interference spectrum that does not overlap the signal spectrum, most of the original

interference energy is eliminated. An approximate measure of the interference rejection

capability is given by the ratio W/B. Whatever the precise definition of a bandwidth, W

2
Data

symbols

Received

signal

(a)

Oscillator Spreading

sequence

generator

Chip

waveform

modulator

(b)

Synchronization

system

Data

symbols

PSK demodulator

Spreading

waveform

generator

Wideband

filter

Wideband

filter

Transmitted

signal

Ts

0

∫ Carrier Symbol

synchr.

Figure 2. Functional block diagram of direct-sequence systemn with PSK or DPSK: (a)

transmitter and (b) receiver. Spectral density

Interference

Interference

(a) (b)

Frequency Frequency

Signal

Signal

Spectral density

Figure 3. Spectra of desired signal and interference: (a) wideband-filter output and (b)

demodulator input.

and B are proportional to 1/Tc and 1/Ts, respectively, with the same proportionality

constant. Therefore,

G = Ts

Tc

= W

B (1-5)

which links the processing gain with the interference rejection illustrated in the figure.

Since its spectrum is unchanged by the despreading, white Gaussian noise is not

suppressed by a direct-sequence system.

In practical systems, the wideband filter in the transmitter is used to limit the out-of-band

radiation. This filter and the propagation channel disperse the chip waveform so that it is

no longer confined to [0, Tc]. To avoid interchip interference in the receiver, the filter might

be designed to generate a pulse that satisfies the Nyquist criterion for no intersymbol

3
interference. A convenient representation of a direct-sequence signal when the chip

waveform may extend beyond [0, Tc] is

s(t) = A X∞

i=−∞

dbi/G]piψ (t − iTc) cos (2πfct + θ) (1-6)

where bxc denotes the integer part of x. When the chip waveform is assumed to be confined

to [0, Tc], then (1-6) can be expressed by (1-1) and (1-2).

2. Spreading Sequences and Waveforms

2.1 Random Binary Sequence

A random binary sequence x(t) is a stochastic process that consists of independent,

identically distributed symbols, each of duration T. Each symbol takes the value +1 with

probability 1/2 or the value −1 with probability 1/2. Therefore, E[x(t)] = 0 for all t, where

E[ ] denotes the expected value, and

P [x (t) = i] = 1/2, i = +1, −1 (2-1)

where P[A] denotes the probability of event A. The process is wide-sense stationary if the

location of the first symbol transition or start of a new symbol after t = 0 is a random

variable uniformly distributed over the half-open interval (0, T]. A sample function of a

wide-sense-stationary random binary sequence x(t) is illustrated in Figure 4.

The autocorrelation of a stochastic process x(t) is defined as

Rx(t, τ ) = E [x(t)x(t + τ )] (2-2)

x(t)

1

–1

t

Figure 4. Sample function of a random binary sequence.

4
If x(t) is a wide-sense stationary process, then Rx (t, τ ) is a function of τ alone, and the

autocorrelation is denoted by Rx(τ ). From the theorem of total probability, the definition

of a conditional probability, and (2-1), it follows that the autocorrelation of a random

binary sequence is

Rx(t, τ ) = 1

2

P[x(t + τ )=1|x(t) = 1] − 1

2

P[x(t + τ ) = −1|x(t) = 1]

+

1

2

P[x(t + τ ) = −1|x(t) = −1] − 1

2

P[x(t + τ )=1|x(t) = −1]

(2-3)

where P[A|B] denotes the conditional probability of event A given the occurrence of event

B. From the theorem of total probability, it follows that

P[x(t + τ) = i|x(t) = i] + P[x(t + τ ) = −i|x(t) = i]=1,

i = +1, −1 (2-4)

Since both of the following probabilities are equal to the probability that x(t) and x(t + τ )

differ,

P[x(t + τ )=1|x(t) = −1] = P[x(t + τ ) = −1|x(t) = 1] (2-5)

Substitution of (2-4) and (2-5) into (2-3) yields

Rx(t, τ )=1 − 2P[x(t + τ )=1|x(t) = −1] (2-6)

If |τ | ≥ T, then x(t) and x(t + τ ) are independent random variables because t and t + τ are

in different symbol intervals. Therefore,

P[x(t + τ )=1|x(t) = −1] = P[x(t + τ ) = 1] = 1/2

and (2-6) implies that Rx(t, τ ) = 0 for |τ | ≥ T. Further simplification is possible for a

stationary random binary sequence. If |τ | < T, then x(t) and x(t + τ ) are independent only

if a symbol transition occurs in the half-open interval I0 = (t, t + τ ]. Consider any half-open

interval I1 of length T that includes I0. Exactly one transition occurs in I1. Since the first

transition for t > 0 is assumed to be uniformly distributed over (0, T], the probability that

a transition in I1 occurs in I0 is |τ |/T. If a transition occurs in I0, then x(t) and x(t + τ )

are independent and differ with probability 1/2 ; otherwise, x(t) = x(t + τ ). Consequently,

P[x(t + τ )=1|x(t) = −1] = |τ |/2T if |τ | < T. Substitution of the preceding results into

(2-6) gives the autocorrelation of the stationary random binary sequence:

Rx(t, τ ) = Rx(τ ) = Λ

3 τ

T

 ́

(2-7)

where the triangular function is defined by

Λ(t) = 1⁄2

1 − |t|, |t| ≤ 1

0, |t| > 1 (2-8)

5
Feedback Logic

Clock

1 2 3 m

Figure 5. General feedback shift register with m stages.

2.2 Shift-Register Sequences

Ideally, one would prefer a random binary sequence as the spreading sequence. However,

practical synchronization requirements in the receiver force one to use periodic binary

sequences. A shift-register sequence is a periodic binary sequence generated by combining

the outputs of feedback shift registers. A feedback shift register, which is diagrammed in

Figure 5, consists of consecutive two-state memory or storage stages and feedback logic.

Binary sequences drawn from the alphabet {0,1} are shifted through the shift register in

response to clock pulses. The content of each stage is identical to its output. The contents

of the stages are logically combined to produce the input to the first stage. The initial

contents of the stages and the feedback logic determine the successive contents of the

stages. A feedback shift register and its output are called linear when the feedback logic

consists entirely of modulo-2 adders (exclusive-OR gates). Figure 6(a) illustrates a linear

feedback shift register with three stages and an output sequence extracted from the final

stage. The input to the first stage is the modulo-2 sum of the contents of the second and

third stages. After each clock pulse, the contents of the first two stages are shifted to the

right, and the input to the first stage becomes its content. If the initial contents of the

shift-register stages are 0 0 1, the subsequent contents after successive shifts are listed in

Figure 6(b). Since the shift register returns to its initial state after 7 shifts, the periodic

output sequence extracted from the final stage has a period of 7 bits.

Let sj (i) denote the content of stage j after clock pulse i. The state of the shift register

after clock pulse i is the vector

S(i)=[s1(i) s2(i)...sm(i)], i ≥ 0 (2-9)

where S(0) is the initial state. The zero state is the state for which all the contents are 0’s.

From the definition of a shift register, it follows that

sj (i) = sj−k(i − k), i ≥ k ≥ 0, k ≤ j ≤ m (2-10)

where s0(i) denotes the input to stage 1 after clock pulse i. If ai denotes the state of bit i

of the output sequence, then ai = sm(i). The state of a feedback shift register uniquely

6
1 2 3

Clock

Output

Contents

Shift Stage 1 Stage 2 Stage 3

Initial

(a)

(b)

1

2

3

4

5

6

7

0

1

0

1

1

1

0

0

0

0

1

0

1

1

1

0

1

0

0

1

0

1

1

1

Figure 6. (a) Three-stage linear feedback shift register and (b) contents after successive

shifts.

7
determines the subsequent sequence of states and the output sequence.The period N of a

periodic sequence {ai} is defined as the smallest positive integer for which ai+N = ai, i ≥ 0.

Since the number of distinct states of an m-stage shift register is 2m, the sequence of states

and the output sequence have period N ≤ 2m.

The Galois field of two elements, which is denoted by GF(2), consists of the symbols 0 and

1 and the operations of modulo-2 addition and modulo-2 multiplication. These binary

operations are defined by

0 ⊕ 0=0, 0 ⊕ 1=1, 1 ⊕ 0=1, 1 ⊕ 1=0

0 · 0=0, 0 · 1=0, 1 · 0=0, 1 · 1 = 1 (2-11)

where ⊕ denotes modulo-2 addition. From these equations, it is easy to verify that the

field is closed under both modulo-2 addition and modulo-2 multiplication and that both

operations are associative and commutative. Since −1 is defined as that element which

when added to 1 yields 0, we have −1 = 1, and subtraction is the same as addition. From

(2-11), it follows that the additive identity element is 0, the multiplicative identity is 1, and

the multiplicative inverse of 1 is 1−1 = 1. The substitutions of all possible symbol

combinations verify the distributive laws:

a(b ⊕ c) = ab ⊕ ac, (b ⊕ c)a = ba ⊕ ca (2-12)

where a, b, and c can each equal 0 or 1. The equality of subtraction and addition implies

that if a ⊕ b = c, then a = b ⊕ c.

The input to stage 1 of a linear feedback shift register is

s0(i) = Xm

k=1

cksk(i), i ≥ 0 (2-13)

where the operations are modulo-2 and the feedback coefficient ck equals either 0 or 1,

depending on whether the output of stage k feeds a modulo-2 adder. An m-stage shift

register is defined to have cm = 1; otherwise, the final state would not contribute to the

generation of the output sequence, but would only provide a one-shift delay. For example,

Figure 6 gives c1 = 0, c2 = c3 = 1, and s0(i) = s2(i) ⊕ s3(i). A general representation of a

linear feedback shift register is shown in Figure 7(a). If ck = 1, the corresponding switch is

closed; if ck = 0, it is open.

Since the output bit ai = sm(i), (2-10) and (2-13) imply that for i ≥ m,

ai = s0(i − m) = Xm

l=1

cksk(i − m) = Xm

l=1

cksm(i − k)

8
1 2 3 m

1 2 3 m

c1 c2 c3 cm-1

cm-1 cm-2 cm-3 c1

Output

Output

(a)

(b)

Figure 7. Linear feedback shift register: (a) standard representation and (b) high-speed

form.

which indicates that each output bit satisfies the linear recurrence relation:

ai = Xm

k=1

ckai−k, i ≥ m (2-14)

The first m output bits are determined solely by the initial state:

ai = sm−i(0), 0 ≤ i ≤ m − 1 (2-15)

Figure 7(a) is not necessarily the best way to implement the feedback shift register that

produces the sequence satisfying (2-14). Figure 7(b) illustrates an implementation that

allows higher-speed operation. From this diagram, it follows that

sj (i) = sj−1(i − 1) ⊕ cm−j+1sm(i − 1), i ≥ 1, 2 ≤ j ≤ m (2-16)

s1(i) = sm(i − 1) i ≥ 1 (2-17)

Repeated application of (2-16) implies that

sm(i) = sm−1(i − 1) ⊕ c1sm(i − 1) , i ≥ 1

sm−1(i − 1) = sm−2(i − 2) ⊕ c2sm(i − 2) , i ≥ 2

.

.

. (2-18)

s2(i − m + 2) = s1(i − m + 1) ⊕ cm−1sm(i − m + 1) , i ≥ m − 1

Addition of these m equations yields

sm(i) = s1(i − m + 1) ⊕

m

X−1

k=1

cksm(i − k), i ≥ m − 1 (2-19)

9
Substituting (2-17) and then ai = sm(i) into (2-19), we obtain

ai = ai−m ⊕

m

X−1

k=1

ckai−k, i ≥ m (2-20)

Since cm = 1, (2-20) is the same as (2-14). Thus, the two implementations can produce the

same output sequence indefinitely if the first m output bits coincide. However, they require

different initial states and have different sequences of states. Successive substitutions into

the first equation of sequence (2-18) yields

sm(i) = sm−i(0) ⊕X

i

k=1

cksm(i − k) , 1 ≤ i ≤ m − 1 (2-21)

Substituting ai = sm(i), ai−k = sm(i − k), and j = m − i into (2-21) and then using binary

arithmetic, we obtain

sj (0) = am−j ⊕

m

X−j

k=1

ckam−j−k , 1 ≤ j ≤ m (2-22)

If a0, a1,...am−1 are specified, then (2-22) gives the corresponding initial state of the

high-speed shift register.

The sum of binary sequence a = (a0, a1, ···) and binary sequence b = (b0, b1, ···) is defined

to be the binary sequence a ⊕ b, each bit of which is the modulo-2 sum of the

corresponding bits of a and b. Thus, if d = a ⊕ b we can write

di = ai ⊕ bi , i ≥ 0 (2-23)

Suppose that a and b are generated by the same linear feedback shift register, but may

differ because the initial states may be different. For the sequence d = a ⊕ b, (2-23) and

the associative and distributive laws of binary fields imply that

di = Xm

k=1

ckai−k ⊕Xm

k=1

ckbi−k = Xm

k=1

(ckai−k ⊕ ckbi−k)

= Xm

k=1

ck(ai−k ⊕ bi−k) = Xm

k=1

ckdi−k (2-24)

Since the linear recurrence relation is identical, d can be generated by the same linear

feedback logic as a and b. Thus, if a and b are two output sequences of a linear feedback

shift register, then a ⊕ b is also. If a = b, then a ⊕ b is the sequence of all 0’s, which can

be generated by any linear feedback shift register.

If a linear feedback shift register reached the zero state at some time, it would always

remain in the zero state, and the output sequence would subsequently be all 0’s. Since a

10
linear m-stage feedback shift register has exactly 2m − 1 nonzero states, the period of its

output sequence cannot exceed 2m − 1. A sequence of period 2m − 1 generated by a linear

feedback shift register is called a maximal or maximal-length sequence. If a linear feedback

shift register generates a maximal sequence, then all of its nonzero output sequences are

maximal, regardless of the initial states.

Out of 2m possible states, the content of the last stage, which is the same as the output bit,

is a 0 in 2m−1 states. Among the nonzero states, the output bit is a 0 in 2m−1 − 1 states.

Therefore, in one period of a maximal sequence, the number of 0’s is exactly 2m−1 − 1,

while the number of 1’s is exactly 2m−1.

Given the binary sequence a, let a(j)=(aj , aj+1,...) denote a shifted binary sequence. If a

is a maximal sequence and j 6= 0, modulo 2m − 1, then a ⊕ a(j) is not the sequence of all

0’s. Because it is generated by the same shift register as a, a ⊕ a(j) must be a maximal

sequence and, hence, some cyclic shift of a. We conclude that the modulo-2 sum of a

maximal sequence and a cyclic shift of itself by j digits, where j 6= 0, modulo 2m − 1,

produces another cyclic shift of the original sequence. This property is succinctly written as

a ⊕ a(j) = a(k), j 6= 0 (modulo 2m − 1) (2-25)

In contrast, a non-maximal linear sequence a ⊕ a(j) is not necessarily a cyclic shift of a

and may not even have the same period. As an example, consider the linear feedback shift

register depicted in Figure 8. The possible state transitions depend on the initial state.

Thus, if the initial state is 0 1 0, then the first state diagram indicates that there are four

possible states and, hence, the output sequence has a period of four. The output sequence

is a = (0, 1, 0, 1, 0, 1,...), which implies that a(1) = (1, 0, 1, 0, 1, 0,...) and

a ⊕ a(1) = (1, 1, 1, 1, 1, 1,...); this result indicates that (2-25) is not satisfied for any k.

2.3 Periodic Autocorrelations

A binary sequence a with components ai ∈ GF(2), can be mapped into a binary antipodal

sequence p with components pi ∈ {−1, +1} by means of the transformation

pi = (−1)ai+1, i ≥ 0 (2-26)

or, alternatively, pi = (−1)ai . The periodic autocorrelation of a periodic binary sequence a

with period N is defined as

θp(j) = 1

N

X

N

i=0

pipi+j (2-27)

Substitution of (2-26) into (2-27) yields

θp(j) = 1

N

N

X−1

i=0

(−1)ai+ai+j = 1

N

N

X−1

i=0

(−1)ai⊕ai+j = Aj − Dj

N (2-28)

11
1 2 3

011

(a)

Clock

Output

(b)

100

001 110 101 010 111

Figure 8. (a) Nonmaximal linear feedback shift register and (b) state diagrams.

where Aj denotes the number of agreements in the corresponding bits of a and a(j), and

Dj denotes the number of disagreements. Equivalently, Aj is the number of 0’s in one

period of a ⊕ a(j), and Dj = N − Aj is the number of 1’s.

Consider a maximal sequence. From (2-25), it follows that Aj equals the number of 0’s in a

maximal sequence if j 6= 0, modulo N. Thus, Aj = (N − 1)/2 and, similarly,

Dj = (N + 1)/2 if j 6= 0, modulo N. Therefore,

θp(j) = 1⁄2 1, j = 0(mod N)

− 1

N , j 6= 0(mod N) (2-29)

The periodic autocorrelation of a periodic function x(t) with period T is defined as

Rx(τ ) = 1

T

Z c+T

c

x(t)x(t + τ )dt (2-30)

where τ is the relative delay variable and c is an arbitrary constant. It follows that Rx(τ )

has period T. We derive the periodic autocorrelation of p(t) assuming an ideal periodic

spreading waveform of infinite extent and a rectangular chip waveform. If the spreading

sequence has period N, then p(t) has period T = NTc. Equations (1-2) and (2-30) with

c = 0 yield the autocorrelation of p(t):

Rp (τ ) = 1

NTc

N

X−1

i=0

pi

N

X−1

l=0

pl

Z NTc

0

ψ (t − iTc)ψ (t − lTc + τ ) dt (2-31)

12
If τ = jTc, where j is an integer, then ψ(t) = w(t, Tc), (1-3), and (2-31) yield

Rp (jTc) = 1

N

N

X−1

i=0

pipi+j = θp(j) (2-32)

Any delay can be expressed in the form τ = jTc + 2, where j is an integer and 0 ≤ 2<Tc.

Therefore, (2-31) and ψ(t) = w(t, Tc) give

Rp (jTc + 2) = 1

NTc

N

X−1

i=0

pipi+j

Z NTc

0

w (t − iTc, Tc)w (t − iTc + 2, Tc) dt

+

1

NTc

N

X−1

i=0

pipi+j+1 Z NTc

0

w (t − iTc, Tc)w (t − iTc + 2 − Tc, Tc) dt

(2-33)

Using (2-32) and (1-3) in (2-33), we obtain

Rp (jTc + 2) = μ

1 − 2

Tc

¶

θp(j) + 2

Tc

θp(j + 1) (2-34)

For a maximal sequence, the substitution of (2-29) into (2-34) yields Rp(τ ) over one period:

Rp (τ ) = N + 1

N

Λ

μ τ

Tc

¶

− 1

N , |τ | ≤ NTc/2 (2-35)

where Λ() is the triangular function defined by (2-8). Since it has period NTc, the

autocorrelation can be compactly expressed as

Rp(τ ) = − 1

N +

N + 1

N

X∞

i=−∞

Λ

μτ − iNTc

Tc

¶

(2-36)

Over one period, this autocorrelation resembles that of a random binary sequence, which is

given by (2-7) with T = Tc. Both autocorrelations are shown in Figure 9.

A straightforward calculation or the use of tables gives the Fourier transform of the

triangular function:

F

1⁄2

Λ

μ t

T

¶3⁄4 =

Z ∞

−∞

Λ

μ t

T

¶

exp (−j2πf t) dt

= Tsinc2

fT (2-37)

13
– NTc 1/N NTc

Tc

maximal

random binary 1

. . . τ

. . .

. . .

. . .

Figure 9. Autocorrelations of maximal sequence and random binary sequence.

– Tc

2 0 – Tc

1

Tc

1

Tc

2

f

Sp(f)

N + 1

N2

NTc

1

Figure 10. Power spectral density of maximal sequence.

where j = √−1 and sinc x = (sin πx)/πx. Since the infinite series in (2-36) is a periodic

function of τ , it can be expressed as a complex exponential Fourier series. From (2-37) and

the fact that the Fourier transform of a complex exponential is a delta function, we obtain

F

( X∞

i=−∞

Λ

μt − iNTc

Tc

¶)

= 1

N

X∞

i=−∞

sinc2

μ i

N

¶

δ

μ

f − i

NTc

¶

(2-38)

where δ( ) is the Dirac delta function. Applying this identity to (2-36), we determine

Sp(f), the power spectral density of p(t), which is defined as the Fourier transform of Rp(τ ):

Sp(f) = N + 1

N2

X∞

i=−∞ i6=0

sinc2

μ i

N

¶

δ

μ

f − i

NTc

¶

+

1

N2 δ(f) (2-39)

This function, which consists of an infinite series of delta functions, is sketched in Figure 10.

A pseudonoise or pseudorandom sequence is a periodic binary sequence with a nearly even

balance of 0’s and 1’s and an autocorrelation that roughly resembles, over one period, the

autocorrelation of a random binary sequence. Pseudonoise sequences provide practical

spreading sequences because their autocorrelations facilitate code synchronization in the

receiver (Section 7). Other sequences have peaks that hinder synchronization.

14
To derive the power spectral density of a direct-sequence signal with a periodic spreading

sequence, it is necessary to define the average autocorrelation of x(t):

R ̄x(τ ) = limT→∞

1

2T

Z T

−T

Rx (t, τ )dt (2-40)

The limit exists and may be nonzero if x(t) has finite power and infinite duration. If x(t) is

stationary, R ̄x (τ ) = Rx (τ ). The average power spectral density S ̄x(f) is defined as the

Fourier transform of the average autocorrelation.

For the direct-sequence signal of (1-1), d(t) is modeled as a random binary sequence with

autocorrelation given by (2-7), and θ is modeled as a random variable uniformly

distributed over [0, 2π) and statistically independent of d(t). Neglecting the constraint that

the bit transitions must coincide with chip transitions, we obtain the autocorrelation of the

direct-sequence signal s(t):

Rs(t, τ ) = A2

2 p(t)p(t + τ )Λ

μ τ

Ts

¶

cos 2πfcτ (2-41)

where p(t) is the periodic spreading waveform. Substituting this equation into (2-40) and

using (2-30), we obtain

R ̄s (τ ) = A2

2

Rp (τ ) Λ

μ τ

Ts

¶

cos 2πfcτ (2-42)

where Rp(τ ) is the periodic autocorrelation of p(t). For a maximal spreading sequence, the

convolution theorem, (2-42), (2-37), and (2-39) provide the average power spectral density

of s(t):

S ̄s (f) = A2

4 [Ss1 (f − fc) + Ss1 (f + fc)] (2-43)

where the lowpass equivalent density is

Ss1 (f) = Ts

N2 sinc2

fTs +

N + 1

N2 Ts

X∞

i=−∞ i6=0

sinc2

μ i

N

¶

sinc2

μ

fTs − iTs

NTc

¶

(2-44)

For a random binary sequence, Ss(f) = S ̄s(f) is given by (2-43) with Ss1(f) = Tcsinc2fTc.

2.4 Polynomials over the Binary Field

Polynomials allow a compact description of the dependence of the output sequence of a

linear feedback shift register on its feedback coefficients and initial state. A polynomial over

the binary field GF(2) has the form

f(x) = f0 + f1x + f2x2 + ··· + fnxn (2-45)

15
where the coefficients f0, f1, ··· , fn are elements of GF(2) and the symbol x is an

indeterminate introduced for covenience in calculations. The degree of a polynomial is the

largest power of x with a nonzero coefficient. The sum of two polynomials, f(x) and g(x),

over GF(2) is another polynomial over GF(2) defined as

f(x) + g(x) =

max(

Xn1,n2)

i=0

(fi ⊕ gi) xi (2-46)

where the addition of the coefficients is modulo 2, n1 is the degree of f(x), n2 is the degree

of g(x), and max(n1, n2) denotes the larger of n1 and n2. For example,

(1 + x2 + x3) + (1 + x2 + x4) = x3 + x4 (2-47)

The product of two polynomials over GF(2) is another polynomial over GF(2) defined as

f(x)g(x) =

nX1+n2

i=0 ÃX

i

j=0

fjgi−j

!

xi (2-48)

where the inner addition is modulo 2. For example,

(1 + x2 + x3

)(1 + x2 + x4)=1+ x3 + x5 + x6 + x7 (2-49)

It is easily verified that associative, commutative, and distributive laws apply to

polynomial addition and multiplication.

The characteristic polynomial associated with a linear feedback shift register of m stages is

defined as

f(x)=1+Xm

i=1

cixi (2-50)

where cm = 1 assuming that stage m contributes to the generation of the output sequence.

The generating function associated with the output sequence is defined as

G(x) = X∞

i=0

aixi (2-51)

Substitution of (2-14) into this equation yields

G(x) =

m

X−1

i=0

aixi +X∞

i=m

Xm

k=1

ckai−kxi

=

m

X−1

i=0

aixi +Xm

k=1

ckxkX∞

i=m

ai−kxi−k

=

m

X−1

i=0

aixi +Xm

k=1

ckxk

"

G (x) +

mX−k−1

i=0

aixi

#

(2-52)

16
Combining this equation with (2-50), and defining c0 = 1, we obtain

G(x)f(x) =

m

X−1

i=0

aixi +Xm

k=1

ckxk

ÃmX−k−1

i=0

aixi

!

=

m

X−1

k=0

ckxk

ÃmX−k−1

i=0

aixi

!

=

m

X−1

k=0

mX−k−1

i=0

ckaixk+i

=

m

X−1

k=0

m

X−1

j=k

ckaj−kxj =

m

X−1

j=0

X

j

k=0

ckaj−kxj (2-53)

Therefore,

G(x) =

mP−1

i=0

xi

μP

i

k=0

ckai−k

¶

f(x) , c0 = 1 (2-54)

Thus, the generating function of the output sequence generated by a linear feedback shift

register with characteristic polynomial f(x) may be expressed in the form

G(x) = φ(x)/f(x), where the degree of φ(x) is less than the degree of f(x). The output

sequence is said to be generated by f(x). Equation (2-54) explicitly shows that the output

sequence is completely determined by the feedback coefficients ck, k = 1, 2,...,m, and the

initial state ai = sm−i(0), i = 0, 1,...,m − 1.

In Figure 6, the feedback coefficients are c1 = 0, c2 = 1, and c3 = 1, and the initial state

gives a0 = 1, a1 = 0, and a2 = 0. Therefore,

G(x) = 1 + x2

1 + x2 + x3 (2-55)

Performing the long polynomial division according to the rules of binary arithmetic yields

1 + x3 + x5 + x6 + x7 + x10 + ..., which implies the output sequence listed in the figure.

The polynomial p(x) is said to divide the polynomial b(x) if there is a polynomial h(x) such

that b(x) = h(x)p(x). A polynomial p(x) over GF(2) of degree m is called irreducible if

p(x) is not divisible by any polynomial over GF(2) of degree less than m but greater than

zero. If p(x) is irreducible over GF(2), then p(0) 6= 0, for otherwise x would divide p(x). If

p(x) has an even number of terms, then p(1) = 0 and the fundamental theorem of algebra

implies that x + 1 divides p(x) . Therefore, an irreducible polynomial over GF(2) must

have an odd number of terms, but this condition is not sufficient for irreducibility. For

example, 1 + x + x2 is irreducible, but 1 + x + x5 = (1 + x2 + x3)(1 + x + x2) is not.

If a shift-register sequence {ai} is periodic with period n, then its generating function

G(x) = φ(x)/f(x) may be expressed as

17
G (x) = g (x) + xng (x) + x2ng (x) + ···

= g (x)

X∞

i=0

xin

= g (x)

1 + xn (2-56)

where g(x) is a polynomial of degree n − 1. Therefore,

g (x) = φ (x) (1 + xn)

f (x)

Suppose that f(x) and φ(x) have no common factors, which is true if f(x) is irreducible

since φ(x) is of lower degree than f(x). Then f(x) must divide 1 + xn. Conversely, if the

characteristic polynomial f(x) divides 1 + xn, then f(x)h(x)=1+ xn for some polynomial

h(x), and

G (x) = φ (x)

f (x) = φ (x) h (x)1 + xn

which has the form of (2-56). Thus, f(x) generates a sequence of period n for all φ(x) and,

hence, all initial states.

A polynomial over GF(2) of degree m is called primitive if the smallest positive integer n

for which the polynomial divides 1 + xn is n = 2m − 1. Thus, a primitive characteristic

polynomial of degree m can generate a sequence of period 2m − 1, which is the period of a

maximal sequence generated by a characteristic polynomial of degree m. Suppose that a

primitive characteristic polynomial of positive degree m could be factored so that

f(x) = f1(x)f2(x), where f1(x) is of positive degree m1 and f2(x) is of positive degree

m − m1. A partial-fraction expansion yields

1

f (x) = a (x)f1 (x) +

b (x)

f2 (x)

Since f1(x) and f2(x) can serve as characteristic polynomials, the period of the first term in

the expansion cannot exceed 2m1 − 1 while the period of the second term cannot exceed

2m−m1 − 1. Therefore, the period of 1/f(x) cannot exceed(2m1 − 1)(2m−m1 − 1) ≤ 2m − 3 ,

which contradicts the assumption that f(x) is primitive. Thus, a primitive characteristic

polynomial must be irreducible.

Theorem. A characteristic polynomial of degree m generates a maximal sequence of period

2m − 1 if and only if it is a primitive polynomial.

Proof. To prove sufficiency, we observe that if f(x) is a primitive characteristic

polynomial, it divides 1 + xn for n = 2m − 1 so a maximal sequence of period 2m − 1 is

18
generated. If a sequence of smaller period could be generated, then the irreducible f(x)

would have to divide 1 + xn1 for n1 < n, which contradicts the assumption of a primitive

polynomial. To prove necessity, we observe that if the characteristic polynomial f(x)

generates a maximal sequence with period n = 2m − 1, then f(x) cannot divide

1 + xn1 , n1 < n, because a sequence with a smaller period would result, and such a

sequence cannot be generated by a maximal sequence generator. Since f(x) does divide

1 + xn, it must be a primitive polynomial. ¤

Primitive polynomials are difficult to find, but many have been tabulated (e.g., [4]). Those

for which m ≤ 7 and one of those of minimal coefficient weight for 8 ≤ m ≤ 25 are listed in

Table 1 as octal numbers in increasing order (e.g., 51 ↔ 101100 ↔ 1 + x2 + x3). For any

positive integer m, the number of different primitive polynomials of degree m over GF(2) is

λ(m) = φe (2m − 1)

m (2-57)

where the Euler function φe(n) is the number of positive integers that are less than and

relatively prime to the positive integer n. If n is a prime number, φe(n) = n − 1. In general,

φe(n) = n

Y

k

i=1

νi − 1

νi

≤ n − 1 (2-58)

where ν1, ν2,..., νk are the prime integers that divide n. Thus, λ(6) = φe(63)/6 = 6 and

λ(13) = φe(8191)/13 = 630.

2.5 Long Nonlinear Sequences

A long sequence or long code is a spreading sequence with a period that is much longer

than the data-symbol duration and may even exceed the message duration. A short

sequence or short code is a spreading sequence with a period that is equal to or less than

the data-symbol duration. Since short sequences are susceptible to interception and long

linear sequences are inherently susceptible to mathematical cryptanalysis [1], long

nonlinear pseudonoise sequences and programmable code generators are needed for

communications with a high level of security. However, if a modest level of security is

acceptable, short or moderate-length pseudonoise sequences are preferable for rapid

acquisition, burst communications, and multiuser detection.

The algebraic structure of linear feedback shift registers makes them susceptible to

cryptanalysis. Let

c = [c1 c2 ...cm]

T (2-59)

denote the column vector of the m feedback coefficients of an m-stage linear feedback shift

register, where T denotes the transpose. The column vector of m successive sequence bits

19
Table 1. Primitive Polynomials.

Degree Primitive Degree Primitive Degree Primitive

2 7 7 103 8 534

3 51 122 9 1201

31 163 10 1102

4 13 112 11 5004

32 172 12 32101

5 15 543 13 33002

54 523 14 30214

57 532 15 300001

37 573 16 310012

76 302 17 110004

75 323 18 1020001

6 141 313 19 7400002

551 352 20 1100004

301 742 21 50000001

361 763 22 30000002

331 712 23 14000004

741 753 24 702000001

772 25 110000002

produced by the shift register starting at bit i is

ai = [ai ai+1 ...ai+m−1]

T , (2-60)

Let A(i) denote the m × m matrix with columns consisting of the aj vectors for

i ≤ j ≤ i + m − 1:

A(i) =











ai+m−1 ai+m−2 ··· ai

ai+m ai−m−1 ··· ai+1

.

.

. .

.

. .

.

.

ai+2m−2 ai+2m−3 ··· ai+m−1











(2-61)

The linear recurrence relation (2-14) indicates that the output sequence and feedback

coefficients are related by

ai+m = A(i)c, i ≥ 0 (2-62)

If 2m consecutive sequence bits are known, then A(i) and ai+n are completely known for

some i. If A(i) is invertible, then the feedback coefficients are derived from

c = A−1

(i)ai+m , i ≥ 0 (2-63)

The output sequence of a shift register is completely determined by the feedback

coefficients and any state vector. Since any m successive sequence bits determine a state

20
1 2 N–1 N Output

Figure 11. Linear generator of binary sequence with period N.

vector, 2m successive bits provide enough information to reproduce the output sequence

unless A(i) is not invertible. In that case, a few more bits are needed.

If a binary sequence has period N, it can always be generated by a N-stage linear feedback

shift register by connecting the output of the last stage to the input of the first stage and

inserting N consecutive bits of the sequence into the output sequence, as illustrated in

Figure 11. The polynomial associated with one period of the binary sequence is

g(x) =

N

X−1

i=0

aixi (2-64)

Let gcd(g(x), 1 + xN ) denote the greatest common polynomial divisor of the polynomials

g(x) and 1 + xN . Then (2-56) implies that the generating function of the sequence may be

expressed as

G(x) = g(x)/gcd ¡

g(x), 1 + xN ¢

(1 + xN ) /gcd (g(x), 1 + xN ) (2-65)

If gcd(g(x), 1 + xN ) 6= 1, the degree of the denominator of G(x) is less than N. Therefore,

the sequence represented by G(x) can be generated by a linear feedback shift register with

fewer stages than N and with the characteristic function given by the denominator. The

appropriate initial state can be determined from the coefficients of the numerator.

The linear equivalent of the generator of a sequence is the linear shift register with the

fewest stages that produces the sequence. The number of stages in the linear equivalent is

called the linear complexity of the sequence. If the linear complexity is equal to m, then

(2-63) determines the linear equivalent after the observation of 2m consecutive sequence

bits. Security improves as the period of a sequence increases, but there are practical limits

to the number of shift-register stages. To produce sequences with a long enough period for

high security, the feedback logic in Figure 5 often must be nonlinear. Alternatively, one or

more shift-register outputs or several outputs of shift-register stages may be applied to a

nonlinear device to produce the sequence [5]. Nonlinear generators with relatively few

shift-register stages can produce sequences of enormous linear complexity. As an example,

Figure 12(a) depicts a nonlinear generator in which two stages of a linear feedback shift

register have their outputs applied to an AND gate to produce the output sequence. The

initial contents of the shift-register stages are indicated by the enclosed binary numbers.

Since the linear generator produces a maximal sequence of length 7, the output sequence

21
0 0 1 A

N

D

(a)

Output

1 0 0 0 0 0

Output

(b)

Figure 12. (a) Nonlinear generator and (b) its linear equivalent.

has period 7. The first period of the sequence is (0 0 0 0 0 1 1), from which the linear

equivalent with the initial contents shown in Figure 12 (b) is derived by evaluating (2-65).

While a large linear complexity is necessary for the cryptographic integrity of a sequence, it

is not necessarily sufficient because other statistical characteristics, such as a nearly even

distribution of 1’s and 0’s, are required. For example, a long sequence of many 0’s followed

by a single 1 has a linear complexity equal to the length of the sequence, but the sequence

is very weak. The generator of Figure 12 (a) produces a relatively large number of 0’s

because the AND gate produces a 1 only if both of its inputs are 1’s.

As another example, a nonlinear generator that uses a multiplexer is shown in Figure 13.

The outputs of various stages of feedback shift register 1 are applied to the multiplexer,

which interprets the binary number determined by these outputs as an address. The

multiplexer uses this address to select one of the stages of feedback shift register 2. The

selected stage provides the multiplexer output and, hence, one bit of the output sequence.

Suppose that register 1 has m stages and register 2 has n stages. If h stages of register 1,

where h<m, are applied to the multiplexer, then the address is one of the numbers

0, 1, ..., 2h − 1. Therefore, if n ≥ 2h, each address specifies a distinct stage of register 2. The

initial states of the two registers, the feedback connections, and which stages are used for

addressing may be parts of a variable key that provides security. The security of the

nonlinear generator is further enhanced if nonlinear feedback is used in both shift registers.

22
Shift register 1

Address

inputs

Multiplexer

Output

Shift register 2

Figure 13. Nonlinear generator that uses a multiplexer.

3. Systems with Coherent PSK and Random Spreading

Sequences

A received direct-sequence signal with PSK data modulation and ideal carrier

synchronization can be represented by (1-1) or (1-6) with θ = 0 to reflect the absence of

phase uncertainty. Assuming that the chip waveform is well approximated by a waveform

ψ(t)of duration Tc, the received signal is

s(t) = √

2Sd(t)p(t) cos 2πfct (3-1)

where S is the average power, d(t) is the data modulation, p(t) is the spreading waveform,

and fc is the carrier frequency. The data modulation is a sequence of nonoverlapping

rectangular pulses, each of which has an amplitude equal to +1 or —1. Each pulse of d(t)

represents a data symbol and has a duration of Ts. The spreading waveform has the form

p(t) = X∞

i=−∞

piψ(t − iTc) (3-2)

where pi is equal to +1 or —1 and represents one chip of a spreading sequence {pi}. It is

convenient, and entails no loss of generality, to normalize the energy content of the chip

waveform according to

1

Tc

Z Tc

0

ψ2(t)dt = 1 (3-3)

Because the transitions of a data symbol and the chips coincide on both sides of a symbol,

the processing gain, defined as

G = Ts

Tc

(3-4)

is an integer equal to the number of chips in a symbol interval.

23
A practical direct-sequence system differs from the functional diagram of Figure 2. The

transmitter needs practical devices, such as a power amplifier and a filter, to limit

out-of-band radiation. In the receiver, the radio-frequency front end includes devices for

wideband filtering and automatic gain control. These devices are assumed to have a

negligible effect on the operation of the demodulator, at least for the purposes of this

analysis. Thus, the front-end circuitry is omitted from Figure 14, which shows the

optimum demodulator in the form of a correlator for the detection of a single symbol in the

presence of white Gaussian noise. This correlator is more practical and flexible for digital

processing than the alternative one shown in Figure 2. It is a suboptimal but reasonable

approach against non-Gaussian interference. An equivalent matched-filter demodulator is

implemented with a transversal filter or tapped delay line and a stored spreading sequence.

However, the matched-filter implementation is not practical for a long sequence that

extends over many data symbols. If the chip-rate synchronization in Figure 14 is accurate,

than the demodulated sequence and the receiver-generated spreading sequence are

multiplied together, and G successive products are added in an accumulator to produce the

decision variable. The effective sampling rate of the decision variable is the symbol rate.

r(t) Chip

matched

filter

Sampler

Decision

device

Synchron.

device

Chip-rate

clock

Spreading

sequence

generator

Demodulated

sequence

cos 2πfct

∑

G–1

i=0

Figure 14. Basic elements of correlator for direct-sequence signal with coherent PSK.

The sequence generator, multiplier, and summer function as a discrete-time filter matched

to the spreading sequence.

In the subsequent analysis, perfect phase, sequence, and symbol synchronization are

assumed. The received signal is

r(t) = s(t) + i(t) + n(t) (3-5)

where i(t) is the interference, and n(t) denotes the zero-mean white Gaussian noise. The

chip matched filter has impulse response ψ(−t). Its output is sampled at the chip rate to

provide G samples per data symbol. If d(t) = d0 over [0, Ts], then (3-1) to (3-5) indicate

that the demodulated sequence corresponding to this data symbol is

Zi =

Z (i+1)Tc

iTc

r(t)ψ (t − iTc) cos 2πfct dt = Si + Ji + Ni, 0 ≤ i ≤ G − 1 (3-6)

where

Si =

Z (i+1)Tc

iTc

s(t)ψ(t − iTc) cos 2πfct dt = pid0

rS

2

Tc (3-7)

24
Ji =

Z (i+1)Tc

iTc

i(t)ψ(t − iTc) cos 2πfct dt (3-8)

Ni =

Z (i+1)Tc

iTc

n(t)ψ (t − iTc) cos 2πfct dt (3-9)

and it is assumed that fc >> 1/Tc so that the integral over a double-frequency term in

(3-7) is negligible. The input to the decision device is

V =

G

X−1

i=0

piZi = d0

rS

2

Ts + V1 + V2 (3-10)

where

V1 =

G

X−1

ν=0

piJi (3-11)

V2 =

G

X−1

v=0

piNi (3-12)

Suppose that d0 = +1 represents the logic symbol 1 and d0 = −1 represents the logic

symbol 0. The decision device produces the symbol 1 if V > 0 and the symbol 0 if V < 0.

An error occurs if V < 0 when d0 = +1 or if V > 0 when d0 = —1. The probability that

V = 0 is zero.

The white Gaussian noise has autocorrelation

Rn(τ ) = N0

2

δ(t − τ ) (3-13)

where N0/2 is the two-sided noise power spectral density. Since E[n(t)] = 0, (3-12) implies

that E[V2] = 0. A straightforward calculation using (3-9), (3-12), (3-13), the limited

duration of ψ(t), and fc >> 1/Tc yields

var (V2) = 1

4

N0Ts (3-14)

It is natural and analytically desirable to model a long spreading sequence as a random

binary sequence. The random-binary-sequence model does not seem to obscure important

exploitable characteristics of long sequences and is a reasonable approximation even for

short sequences in networks with asynchronous communications. A random binary

sequence consists of statistically independent symbols, each of which takes the value +1

with probability 1

2 or the value −1 with probability 12 . Thus, E[pi] = E[p(t)] = 0. It then

follows from (3-10) to (3-12) that E[V1] = E[V2] = 0, and the mean value of the decision

variable is

E[V ] = d0

rS

2

Ts (3-15)

25
for the direct-sequence system with coherent PSK. Since pi and pj are independent for

i 6= j;

E [pipj ]=0, i 6= j (3-16)

Therefore, the independence of pi and Jj for all i and j implies that E[piJipjJj ]=0, i 6= j,

and hence

var (V1) =

G

X−1

i=0

E £J2

i

¤ (3-17)

3.1 Tone Interference at Carrier Frequency

When tone interference has the same carrier frequency as the desired signal, a nearly exact,

closed-form equation for the symbol error probability can be derived. The tone interference

has the form

i(t) = √

2I cos (2πfct + φ) (3-18)

where I is the average power and φ is the phase relative to the desired signal. Assuming

that fc >> 1/Tc, (3-8), (3-11), (3-18) and a change of variables give

V1 =

rI

2

cos φ

G

X−1

i=0

pi

Z Tc

0

ψ (t) dt (3-19)

A rectangular chip waveform has ψ(t) = w(t, Tc), which is given by (1-3). For sinusoidal

pulses in the spreading waveform, ψ(t) = ψs(t, Tc), where

ψs(t, T) = 1⁄2√2 sin ¡ π

T t

¢

, 0 ≤ t ≤ T

0, otherwise (3-20)

Let k1 denote the number of chips in [0, Ts] for which pi = +1; the number for which

pi = −1 is G − k1. Equations (3-19), (1-3), and (3-20) yield

V1 =

rIκ

2

Tc (2k1 − G) cos φ (3-21)

where

κ =

1⁄2 1, rectangular pulses

8

π2 , sinsoidal pulses (3-22)

These equations indicate that the use of sinusoidal pulses instead of rectangular pulses

effectively reduces the interference power by a factor 8/π2 if V1 6= 0. Thus, the advantage of

sinusoidal pulses is 0.91 dB against tone interference at the carrier frequency. Equation

(3-21) indicates that tone interference at the carrier frequency would be completely

rejected if k1 = G/2 in every symbol interval.

26
In the random-binary-sequence model, pi is equally likely to be +1 or −1. Therefore, the

conditional symbol error probability given the value of φ is

Ps (φ) = X

G

k1=0

μG

k1

¶ μ1

2

¶G ∙

1

2

Ps (φ, k1, +1) + 1

2

Ps (φ, k1, −1) ̧(3-23)

where Ps(φ, k1, d0) is the conditional symbol error probability given the values of φ, k1 and

d0. Under these conditions, V1 is a constant, and V has a Gaussian distribution. Equations

(3-10) and (3-21) imply that the conditional expected value of V is

E [V |φ, k1, d0] = d0

rS

2

Ts +

rIκ

2

Tc (2k1 − G) cos φ (3-24)

The conditional variance of V is equal to the variance of V2, which is given by (3-14).

Using the Gaussian density to evaluate Ps(φ, k1, +1) and Ps(φ, k1, −1) separately and then

consolidating the results yields

Ps (φ, k1, d0) = Q

"r2Es

N0

+ d0

r2ITcκ

GN0

(2k1 − G) cos φ

#

(3-25)

where Es = STs is the energy per symbol,

Q (x) = 1

√2π

Z ∞

x

exp μ−y2

2

¶

dy = 1

2

erfc μ x

√2

¶

(3-26)

and erfc( ) is the complementary error function. Assuming that φ is uniformly distributed

over [0, 2π) and exploiting the periodicity of cos φ, we obtain the symbol error probability

Ps = 1

π

Z π

0

Ps (φ) dφ (3-27)

where Ps(φ) is given by (3-23) and (3-25).

3.2 General Tone Interference

To simplify the preceding equations for Ps and to examine the effects of tone interference

with a carrier frequency different from the desired frequency, a Gaussian approximation is

used. Consider interference due to a single tone of the form

i(t) = √

2I cos (2πf1t + θ1) (3-28)

where I,f1, and θ1 are the average power, frequency, and phase angle of the interference

signal at the receiver. The frequency f1 is assumed to be close enough to the desired

27
frequency fc that the tone is undisturbed by the initial wideband filtering that precedes the

correlator. If f1 + fc >> fd = f1 − fc so that a term involving f1 + fc is negligible, (3-28)

and (3-8) and a change of variable yield

Ji =

rI

2

Z Tc

0

ψ (t) cos (2πfdt + θ1 + i2πfdTc)dt (3-29)

For a rectangular chip waveform, evaluation of the integral and trigonometry yield

Ji =

rI

2

Tc sinc (fdTc) cos (i2πfdTc + θ2) (3-30)

where

θ2 = θ1 + πfdTc (3-31)

Substituting (3-30) into (3-17) and expanding the squared cosine, we obtain

var (V1) = 1

4

IT2

c sinc2 (fdTc)

"

G +

G

X−1

i=0

cos (i4πfdTc + 2θ2)

#

(3-32)

To evaluate the inner summation, we use the identity

Xn−1

ν=0

cos (a + νb) = cos μa +

n − 1

2

b

¶ sin (nb/2)

sin (b/2) (3-33)

which is proved by using mathematical induction and trigonometric identities. Evaluation

and simplification yield

var(V1) = 1

4

ITsTcsinc2 (fdTc)

∙

1 + sinc (2fdTs)

sinc (2fdTc)

cos 2φ

 ̧

(3-34)

where

φ = θ2 + πfd (Ts − Tc) = θ1 + πfdTs (3-35)

Given the value of φ, the Ji in (3-30) are uniformly bounded constants, and hence the

terms of V1 in (3-11) are independent and uniformly bounded. Since var(V1) → ∞ as

G → ∞, the central limit theorem [6] implies that when G is large, the conditional

distribution of V1 is approximately Gaussian. Thus, V is nearly Gaussian with mean given

by (3-15) and var(V ) = var(V1) + var(V2). Because of the symmetry of the model, the

conditional symbol error probability may be calculated by assuming d0 = 1 and evaluating

the probability that V < 0. A straightforward derivation using (3-34) indicates that the

conditional symbol error probability is well approximated by

Ps (φ) = Q

"s

2Es

N0e(φ)

#

(3-36)

28
where

N0e(φ) = N0 + ITcsinc2 (fdTc)

∙

1 + sinc (2fdTs)

sinc (2fdTc)

cos 2φ

 ̧

(3-37)

and N0e(φ)/2 can be interpreted as the equivalent two-sided power spectral density of the

interference plus noise, given the value of φ. For sinusoidal pulses, a similar derivation

yields (3-36) with

N0e(φ) = N0 + ITc

μ 8

π2

¶ μ cos πfdTc

1 − 4f 2

dT2

c

¶2 ∙

1 + sinc (2fdTs)

sinc (2fdTc)

cos 2φ

 ̧

(3-38)

To explicitly exhibit the reduction of the interference power by the factor G, we may

substitute Tc = Ts/G in (3-37) or (3-38). A comparison of these two equations confirms

that sinusoidal pulses provide a π2/8=0.91 dB advantage when fd = 0, but this advantage

decreases as |fd| increases and ultimately disappears. The preceding analysis can easily be

extended to multiple tones, but the resulting equations are complicated.

If θ1 in (3-35) is modeled as a random variable that is uniformly distributed over [0, 2π),

then the modulo-2π character of cos 2φ in (3-37) implies that its distribution is the same as

it would be if φ were uniformly distributed over [0, 2π). Therefore, we can henceforth

assign a uniform distribution for φ. The symbol error probability, which is obtained by

averaging Ps(φ) over the range of φ, is

Ps = 2

π

Z π/2

0

Q

"s

2Es

N0e (φ)

#

dφ (3-39)

where the fact that cos 2φ takes all its possible values over [0, π/2] has been used to shorten

the integration interval.

Figure 15 depicts the symbol error probability as a function of the despread

signal-to-interference ratio, GS/I, for one tone-interference signal, rectangular pulses,

fd = 0, G = 50 = 17 dB, and Es/N0 = 14 dB and 20 dB. One pair of curves are computed

using the approximate model of (3-37) and (3-39), while the other pair are derived from the

nearly exact model of (3-23), (3-25), and (3-27) with κ = 1. For the nearly exact model, Ps

depends not only on GS/I, but also on G. A comparison of the two curves indicates that

the error introduced by the Gaussian approximation is on the order of or less than 0.1 dB

when Ps ≥ 10−6. This example and others provide evidence that the Gaussian

approximation introduces insignificant error if G ≥ 50 and practical values for the other

parameters are assumed.

Figure 16 uses the approximate model to plot Ps versus the normalized frequency offset

fdTc for rectangular and sinusoidal chip waveforms, G = 17dB, Es/N0 = 14 dB, and

GS/I = 10 dB. The performance advantage of sinusoidal pulses is apparent, but their

realization in a transmitted PSK waveform is difficult because of the distortion introduced

29
8 9 10 11 12 13 14 15

Despread signal-to-interference ratio, dB

Probability of symbol error

10–3

10–4

10–5

10–6

Es/N0 = 20 dB

Nearly exact

Approximate

Es/N0 = 14 dB

Figure 15. Symbol error probability of binary direct-sequence system with tone interference at carrier frequency and G = 17 dB.

–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5

Normalized frequency offset

Probability of symbol error

10–3

10–4

10–5

10–6

rectangular

sinusoidal

Figure 16. Symbol error probability for direct-sequence system with PSK, rectangular

and sinusoidal chip waveforms, G = 17 dB, Es/N0 = 14 dB, and GS/I = 10

dB in the presence of tone interference.

30
by a nonlinear power amplifier in the transmitter when the signal does not have a constant

envelope.

3.3 Gaussian Interference

Gaussian interference is interference that approximates a zero-mean, stationary Gaussian

process. If i(t) is modeled as Gaussian interference and fc >> 1/Tc, then (3-8), a

trigonometric expansion, the dropping of a negligible double integral, and a change of

variables give

E £J2

i

¤

= 1

2

Z Tc

0

Z Tc

0

Rj (t1 − t2) ψ (t1) ψ (t2) cos [2πfc (t1 − t2)] dt1dt2 (3-40)

where Rj (t) is the autocorrelation of i(t). Since E[J2

i ] does not depend on the index i,

(3-17) gives

var (V1) = GE £J2

i

¤ (3-41)

Assuming that ψ(t) is rectangular, we change variables in (3-40) by using τ = t1 − t2 and

s = t1 + t2. The Jacobian of this transformation is 2. Evaluating one of the resulting

integrals and substituting the result into (3-41) yields

var (V1) = 1

2

Ts

Z Tc

−Tc

Rj (τ )Λ

μ τ

Tc

¶

cos 2πfcτ dτ (3-42)

The limits in this equation can be extended to ±∞ because the integrand is truncated.

Since Rj (τ )Λ

3 τ

Tc

 ́

is an even function, the cosine function may be replaced by a complex

exponential. Then the convolution theorem and the known Fourier transform of Λ(t) yield

the alternative form

var (V1) = 1

2

TsTc

Z ∞

−∞

Sj (f) sinc2 [(f − fc) Tc] df (3-43)

where Sj (f) is the power spectral density of the interference after passage through the

initial wideband filter of the receiver.

Since i(t) is a zero-mean Gaussian process, the {Ji} are zero-mean and jointly Gaussian.

Therefore, if the {pi} are given, then (V1) is conditionally zero-mean and Gaussian. Since

var(V1) does not depend on the {pi}, V1 without conditioning is a zero-mean Gaussian

random variable. The independence of the thermal noise and the interference imply that

V = V1 + V2 is a zero-mean Gaussian random variable. Thus, a standard derivation yields

the symbol error probability:

Ps = Q

Ãr2Es

N0e

!

(3-44)

31
where

N0e = N0 + 2Tc

Z ∞

−∞

Sj (f) sinc2 [(f − fc) Tc] df (3-45)

If S0

j (f) is the interference power spectral density at the input and H(f) is the transfer

function of the initial wideband filter, then Sj (f) = S0

j (f)|H(f)|

2. Suppose that the

interference has a flat spectrum over a band within the passband of the wideband filter so

that

Sj (f) = 1⁄2 I

2W1 , [f − f1| ≤ W12 , |f + f1| ≤ W12

0, otherwise . (3-46)

If fc >> 1/Tc, the integration over negative frequencies in (3-45) is negligible and

N0e = N0 +

ITc

W1

Z f1+W1/2

f1−W1/2

sinc2 [(f − fc) Tc] df (3-47)

This equation shows that f1 = fc or fd = 0 coupled with a narrow bandwidth increases the

impact of the interference power. Since the integrand is upper-bounded by unity,

N0e ≤ N0 + ITc. This upper bound is intuitively reasonable because ITc ≈ I/B = I0, where

B ≈ 1/Tc is the bandwidth of narrowband interference after the despreading, and I0 is its

power spectral density. Equation (3-44) yields

Ps ≤ Q

Ãr 2Es

N0 + ITc

!

(3-48)

This upper bound is tight if fd ≈ 0 and the Gaussian interference is narrowband. A plot

of (3-48) with the parameter values of Figure 15 indicates that roughly 2 dB more interference power is required for worst-case Gaussian interference to degrade Ps as much as tone

interference at the carrier frequency.

4. Quaternary Systems with Random Spreading Sequences

A received quaternary direct-sequence signal with ideal carrier synchronization and a chip

waveform of duration Tc can be represented by

s(t) = √

Sd1(t)p1(t) cos 2πfct + √Sd2(t + t0)p2(t + t0) sin 2πfct (4-1)

where two spreading waveforms, p1(t) and p2(t), and two data signals, d1(t) and d2(t), are

used with two quadrature carriers, and t0 is the relative delay between the in-phase and

quadrature components of the signal. For a quadriphase direct-sequence system, which uses

quadriphase-shift keying (QPSK), t0 = 0. For a direct-sequence system with offset QPSK

or minimum-shift keying (MSK), t0 = Tc/2. For offset QPSK, the chip waveforms are

32
rectangular; for MSK, they are sinusoidal. The use of MSK limits the spectral sidelobes of

the direct-sequence signal, which may interfere with other signals.

Consider the classical or dual quaternary system in which d1(t) and d2(t) are independent.

Let Ts denote the duration of the data symbols before the generation of (4-1), and let Ts1

= 2Ts denote the duration of the channel symbols, which are transmitted in pairs. Let Tc

denote the common chip duration of p1(t) and p2(t). The number of chips per channel

symbol is 2G, where G = Ts/Tc. It is assumed that the synchronization is perfect in the

receiver, which is shown in Figure 17. Consequently, if the received signal is given by (4-1),

then the upper decision variable applied to the decision device at the end of a symbol

interval during which d1(t) = d10 is

V = d10

√

S Ts +

2

X

G−1

i=0

p1iJi +

2

X

G−1

i=0

p1iNi (4-2)

where Ji and Ni are given by (3-8) and (3-9), respectively. The term representing crosstalk,

Vc =

2

X

G−1

i=0

p1i

√

S

2

Z (i+1)Tc

iTc

d2 (t + t0) p2 (t + t0) ψ (t − iTc) sin 4πfct dt (4-3)

is negligible if fc >> 1/Tc so that the sinusoid in (4-3) varies much more rapidly than the

other factors. Similarly, the lower decision variable at the end of a channel-symbol interval

during which d2(t) = d20 is

U = d20

√

STs +

2

X

G−1

i=0

p2iJ0

i +

2

X

G−1

i=0

p2iN0

i (4-4)

where

J0

i =

Z (i+1)Tc

iTc

i(t)ψ (t − iTc) sin 2πfct dt (4-5)

Decision

device

Decision

device

Parallelto-serial

converter

Output

symbols

Received

signals Sync device

cos 2πfct

sin 2πfct

π/2

CMF

CMF

Delay

Sampler

Sampler

Chip-rate clock

SSG

SSG

∑

G-1

i=0

∑

G-1

i=0

Figure 17. Receiver for direct-sequence signal with dual quaternary modulation; CMF

= chip-matched filter; SSG = spreading sequence generator. Delay = 0 for

QPSK; delay = Tc/2 for OQPSK and MSK.

33
N0

i =

Z (i+1)Tc

iTc

n(t)ψ (t − iTc) sin 2πfct dt (4-6)

Of the available desired-signal power S, half is in each of the two components of (4-1).

Since Ts1 = 2Ts, the energy per channel symbol is Es = STs, the same as for a

direct-sequence system with PSK, and

E[V ] = d10

√

S Ts, E(U) = d20

√

S Ts (4-7)

A derivation similar to the one leading to (3-14) gives the variances of the noise terms V2

and U2 in (4-2) and (4-4):

var (V2) = var (U2) = 1

2

N0Ts (4-8)

Using the tone-interference model of Section 3.2, and averaging the error probabilities for

the two parallel symbol streams, we obtain the conditional symbol error probability:

Ps (φ) = 1

2

Q

"s 2Es

N(0)

0e (φ)

#

+

1

2

Q

"s 2Es

N(1)

0e (φ)

#

(4-9)

where N(0)

0e (φ) and N(1)0e (φ) arise from the upper and lower branches of Figure 17,

respectively. For rectangular chip waveforms (QPSK and OQPSK signals),

N(l)

0e (φ) = N0 + ITcsinc2 (fdTc)

∙

1 + sinc (4fdTs)

sinc (2fdTc)

cos(2φ + lπ)

 ̧

(4-10)

and for sinusoidal chip waveforms,

N(l)

oe (φ) = N0 + ITc

μ 8

π2

¶ μ cos πfdTc

1 − 4f 2

dT2

c

¶2 ∙

1 + sinc (4fdTs)

sinc (2fdTc)

cos(2φ + lπ)

 ̧

(4-11)

where l = 0, 1, and we have used Ts1 = 2Ts and

φ = θ1 + 2πfdTs (4-12)

These equations indicate that Ps(φ) for a quaternary direct-sequence system and the worst

value of φ is usually lower than Ps(φ) for a binary direct-sequence system with the same

chip waveform and the worst value of φ. The symbol error probability is determined by

integrating Ps(φ) over the distribution of φ. For a uniform distribution, the two integrals

are equal. Using the periodicity of cos 2φ to shorten the integration interval, we obtain

Ps = 2

π

Z π/2

0

Q

"s 2Es

N(0)

0e (φ)

#

dφ (4-13)

34
The quaternary system provides a slight advantage relative to the binary system against

tone interference. Both systems provide the same Ps when fd = 0 and nearly the same Ps

when fd > 1/Ts. Figure 18 illustrates Ps versus the normalized frequency offset fdTc for

quaternary and binary systems, G = 17 dB, Es/N0 = 14 dB, and GS/I = 10 dB.

In another version of the quaternary direct-sequence system, the same data symbols are

carried by both the in-phase and quadrature components, which implies that the received

direct-sequence signal has the form given by (4-1) with d1(t) = d2(t) = d(t). Thus,

although the spreading is done by quadrature carriers, the data modulation may be

regarded as binary PSK. A receiver for this balanced quaternary system is shown in Figure

19. The synchronization system is assumed to operate perfectly in the subsequent analysis.

If fc >> 1/Tc, the crosstalk terms similar to (4-3) are negligible. If the transmitted symbol

is d10 = d20 = d0, then the input to the decision device is

V = d0

√

S Ts +

G

X−1

i=0

p1iJi +

G

X−1

i=0

p2iJ0

i +

G

X−1

i=0

p1iNi +

G

X−1

i=0

p2iN0

i (4-14)

where Ts is the duration of both a data symbol and a channel symbol. If p1(t) and p2(t),

are approximated by independent random binary sequences, then the last four terms of

(4-14) are zero-mean uncorrelated random variables. Therefore, the variance of V is equal

–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1

Normalized frequency offset

Probability of symbol error

quaternary

binary

10–3

10–4

10–5

Figure 18. Symbol error probability for quaternary and binary direct-sequence systems

with G = 17 dB, Es/N0 = 14 dB, and GS/I = 10 dB in the presence of tone

interference.

35
∑

+

+

Sync device

π/2

Received

signals Decision

device

Output

symbols

cos 2πfct

sin 2πfct

CMF

CMF

Delay

Sampler

Sampler

Chip-rate clock

SSG

SSG

∑

G-1

i=0

Figure 19. Receiver for direct-sequence signal with balanced quaternary modulation (delay = 0 for QPSK and delay = Tc/2 for OQPSK and MSK); CMF = chipmatched filter; SSG = spreading sequence generator.

to the sum of the variances of these four random variables, and

E[V ] = d0

√

STs (4-15)

It is easily verified that both types of quaternary signals provide the same performance

against Gaussian interference as direct-sequence signals with PSK.

Consider a balanced QPSK system, for which t0 = 0. If i(t) is a tone, then a

straightforward extension of the preceding analysis for general tone interference (Section

3.2) yields a Ps(φ) that is independent of φ. Therefore,

Ps = Ps (φ) = Q

Ãr2Es

N0e

!

(4-16)

where for rectangular pulses,

N0e = N0 + ITcsinc2 (fdTc) (4-17)

and for sinusoidal pulses,

N0e = N0 + ITc

μ 8

π2

¶ μ cos πfdTc

1 − 4f 2

dT2

c

¶2

(4-18)

If fd = 0, a nearly exact model similar to the one in Section 3.1 implies that the

conditional symbol error probability is

Ps (φ) = X

G

k1=0

X

G

k2=0

μG

k1

¶ μG

k2

¶ μ1

2

¶2G ∙

1

2

Ps (φ, k1, k2, +1) + 1

2

Ps (φ, k1, k2, −1) ̧

(4-19)

where k1 and k2 are the number of chips in a symbol for which p1(t) = +1 and p2(t) = +1,

respectively, and Ps(φ, k1, k2, d0) is the conditional symbol error probability given the

36
values of φ, k1, and k2 and that d(t) = d0. A derivation analogous to that of (3-25) yields

Ps (φ, k1, k2,d0) = Q

(r2Es

N0

+ d0

rITcκ

GN0

[(2k1 − G) cos φ − (2k2 − G) sin φ]

)

(4-20)

If φ is uniformly distributed over [0, 2π), then

Ps = 1

2π

Z 2π

0

Ps(φ)dφ (4-21)

Numerical comparisons of the nearly exact model with the approximate results given by

(4-16) for fd = 0 indicate that the approximate results typically introduce an insignificant

error if G ≥ 50 .

If g(x) is a convex function over an interval containing the range of a random variable X,

then Jensen’s inequality (Appendix A) states that

g (E [X]) ≤ E [g(X)] (4-22)

provided that the indicated expected values exist. Suppose that g(x) has the form

g(x) = Q

Ãr 1

a + bx!

(4-23)

Since the second derivative of g(x) is nonnegative over the interval such that

0 < a + bx ≤ 1/3, g(x) is a convex function over that interval, and Jensen’s inequality is

applicable.

The application of this result to (4-13) with X = cos 2φ and the fact that E[cos 2φ]=0

yields a lower bound identical to the right-hand side of (4-16). Thus, the balanced QPSK

system, for which d1(t) = d2(t), provides a lower symbol error probability against tone

interference than the dual quaternary or QPSK system for which d1(t) 6= d2(t). A sufficient

convexity condition for all fd is

Es ≥

3

2 (N0 + 2ITc) (4-24)

Figure 20 illustrates the performance advantage of the balanced QPSK system of Figure 19

against tone interference when fd < 1/Ts. Equations (4-9) to (4-13) and (4-16) to (4-18)

are used for the dual quaternary and the balanced QPSK systems, respectively, and G = 17

dB, Es/N0 = 14 dB, and GS/I = 10 dB. The normalized frequency offset is fdTc. The

advantage of the balanced QPSK system when fd is small exists because a tone at the

carrier frequency cannot have a phase that causes desired-signal cancellation

simultaneously in both branches of Figure 19.

37
–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5

Normalized frequency offset

Probability of symbol error

balanced, rectangular

balanced, sinusoidal

dual quaternary

10–3

10–4

10–5

10–6

-.-.-.-

Figure 20. Symbol error probability for direct-sequence systems with balanced QPSK

and dual quaternary modulations, rectangular and sinusoidal chip waveforms,

G = 17 dB, Es/N0 = 14 dB, and GS/I = 10 dB in the presence of tone

interference.

5. Pulsed Interference

Pulsed interference is interference that occurs periodically or sporadically for brief

durations. Whether it is generated unintentionally or by an opponent, pulsed interference

can cause a substantial increase in the bit error rate of a communication system relative to

the rate caused by continuous interference with the same average power. Pulsed

interference may be produced in a receiver by a signal with a variable center frequency that

sweeps over a frequency range that intersects or includes the receiver passband.

Consider a direct-sequence system with binary PSK that operates in the presence of pulsed

interference. Let μ denote either the pulse duty cycle, which is the ratio of the pulse

duration to the repetition period, or the probability of pulse occurrence if the pulses occur

randomly. During a pulse, the interference is modeled as Gaussian interference with power

I/μ, where I is the average interference power. According to (3-47),. the equivalent

noise-power spectral density may be decomposed as

N0e = N0 + I0 (5-1)

38
where the power spectral density of continuous interference (μ = 1) is

I0 = ITc

W1

Z f1+W1/2

f1−W1/2

sinc2[(f − fc) Tc] df (5-2)

In the absence of a pulse, N0e = N0, whereas N0e = N0 + I0/μ in the presence of a pulse. If

the interference pulse duration approximately equals or exceeds the channel-symbol

duration, then (3-44) implies that

Ps ∼= μQ Ãs

2Es

N0 + I0/μ!

+ (1 − μ) Q

Ãr2Es

N0

!

, 0 ≤ μ ≤ 1 (5-3)

If μ is treated as a continuous variable over [0, 1] and I0 À N0, calculus gives the value of μ

that maximizes Ps:

μ0 ∼=

(

0.7

3

Es

I0

 ́−1

, Es

I0 > 0.7

1 , Es

I0 ≤ 0.7

(5-4)

Thus, worst-case pulsed interference is more damaging than continuous interference if

Es/I0 > 0.7.

By substituting μ = μ0 into (5-3), we obtain an approximate expression for the worst-case

Ps when I0 À N0:

Ps ∼=







0.083 3

Es

I0

 ́−1

, Es

I0 > 0.7

Q

3q2Es

I0

 ́

, Es

I0 ≤ 0.7

(5-5)

This equation indicates that the worst-case Ps varies inversely, rather than exponentially,

with Es/I0 if this ratio is sufficiently large. To restore a nearly exponential dependence on

Es/I0, an error-correcting code and symbol interleaving are necessary.

Decoding metrics that are effective against white Gaussian noise are not necessarily

effective against worst-case pulsed interference. We examine the performance of five

different metrics against pulsed interference when the direct-sequence system uses PSK,

ideal symbol interleaving, a binary convolutional code, and Viterbi decoding [7]. The

results are the same when either dual or balanced QPSK is the modulation.

Let B(l) denote the total information weight of the paths at Hamming distance l from the

correct path over an unmerged segment in the trellis diagram of the convolutional code.

Let P2(l) denote the probability of an error in comparing the correct path segment with a

path segment that differs in l symbols. The information-bit error rate satisfies the union

bound [1]

Pb ≤ X∞

l=df

B(l)P2(l) (5-6)

39
where df is the minimum free distance. If r is the code rate, Eb is the energy per

information bit, Tb is the bit duration, and Gu is the processing gain of the uncoded

system, then

Es = rEb, Ts = rTb, G = rGu. (5-7)

The decrease in the processing gain is compensated by the coding gain. An upper bound on

Pb for worst-case pulsed interference is obtained by maximizing the right-hand side of (5-6)

with respect to μ, where 0 ≤ μ ≤ 1. The maximizing value of μ , which depends on the

decoding metric, is not necessarily equal to the actual worst-case μ because a bound rather

than an equality is maximized. However, the discrepancy is small when the bound is tight.

The simplest practical metric to implement is provided by hard-decision decoding. When

the correct path segment is compared to an incorrect one, correct decoding results if the

number of incorrect symbols in the decoder input is less than half the number of symbols

in which the two segments differ. If the number of symbol errors is exactly half the number

of differing symbols, then either of the two segments is chosen with equal probability.

Assuming that the deinterleaving ensures the independence of symbol errors, it follows that

P2(l) =







P

l

i=(l+1)/2

μ

l

i

¶

Pi

s (1 − Ps)

l−i

, l is odd

P

l

i=l/2+1 μ

l

i

¶

Pi

s (1 − Ps)

l−i + 1

2

μ l

l/2

¶

[Ps (1 − Ps)]l/2 , l is even

(5-8)

Since μ = μ0 approximately maximizes Ps, it also approximately maximizes the upper

bound on Pb for hard-decision decoding given by (5-6) to (5-8).

Figure 21 depicts the upper bound on Pb as a function of Eb/I0 for worst-case pulsed

interference, Eb/N0 = 20 dB, and binary convolutional codes with several constraint lengths

and rates. Tabulated values [8] of B(l) are used, and the series in (5-6) is truncated after

the first 7 terms. This truncation gives reliable results only if Pb ≤ 10−3 because the series

converges very slowly. However, the truncation error is partially offset by the error incurred

by the use of the union bound because the latter error is in the opposite direction. Figure

21 indicates the significant advantage of raising the constraint length K and reducing r at

the cost of increased implementation complexity and synchronization requirements,

respectively.

Let N0i denote the equivalent one-sided noise-power spectral density in output sample yi of

a coherent PSK demodulator. For convenience, yi is assumed to have the form of the

right-hand side of (3-10) normalized by multiplying the latter by p2/Ts. Thus, yi has

variance N0i/2. Given that code symbol i of sequence j has value xji, the conditional

probability density function of yi is determined from the Gaussian character of the

40
2 3 4 5 6 7 8

Energy-to-interference density ratio, dB

Bit error probability

K = 7

K = 9

10–2

10–3

10–4

10–5

10–6

r = 1/2

1/3

1/3

1/2

Figure 21. Worst-case performance against pulsed interference for convolutional codes of

constraint length K, rate r, Eb/N0 = 20 dB, and hard decisions.

interference and noise. For a sequence of L code symbols, the density is

f (yi|xji) = 1

√πN0i

exp "−(yi − xji)

2

N0i

#

, i = 1, 2,...,L (5-9)

From the log-likelihood function and the statistical independence of the samples, it follows

that when the values of N01, N02,...,N0L are known, the maximum-likelihood metric for

optimal decoding of the sequence is

U(j) = X

L

i=1

xjiyi

N0i

(5-10)

This metric weights each output sample yi according to the level of the equivalent noise.

Since each yi is assumed to be an independent Gaussian random variable, U(j) is a

Gaussian random variable.

Without loss of generality, let j = 1 label the correct sequence and j = 2 label an incorrect

one at distance l. We assume that there is no quantization of the sample values or that the

quantization is infinitely fine. Therefore, the probability that U(2) = U(1) is zero, and the

probability of an error in comparing a correct sequence with an incorrect one that differs in

l symbols, P2(l), is equal to probability that M0 = U(2) − U(1) > 0. The symbols that are

41
the same in both sequences are irrelevant to the calculation of P2(l) and are ignored

subsequently. Let P2(l|ν) denote the conditional probability that M0 > 0 given that an

interference pulse occurs during ν out of l differing symbols and does not occur during

l − ν symbols. Because of the interleaving, the probability that a symbol is interfered is

statistically independent of the rest of the sequence and equals μ. Thus, (5-6 ) yields

Pb ≤ X∞

l=df

B(l)

X

l

ν=0

μl

ν

¶

μν (1 − μ)

l−νP2(l/ν) (5-11)

Since M0 is a Gaussian random variable, P2(l|ν) is determined from the conditional mean

and variance. A straightforward calculation gives

P2(l|ν) = Q

Ã

−E [M0|ν]

pvar [M0|ν]

!

(5-12)

where E[M0|ν] is the conditional mean and var[M0|ν] is the conditional variance. When an

interference pulse occurs, N0i = N0 + I0; otherwise, N0i = N0. Reordering the symbols for

calculative simplicity and observing that x2i = −x1i, x2

1i = Es, and E[yi] = x1i, we obtain

E[M0|ν] = Xν

i=1

(x2i − x1i) E [yi]

N0 + I0/μ + X

l

i=ν+1

(x2i − x1i) E [yi]

N0

= Xν

i=1

−2Es

N0 + I0/μ + X

l

i=ν+1

−2Es

N0

= −2Es

∙ ν

N0 + I0/μ +

l − ν

N0

 ̧

(5-13)

Using the statistical independence of the samples and observing that

var[yi] = N0i/2, we find similarly that

var [M0|ν]=2Es

∙ ν

N0 + I0/μ +

l − ν

N0

 ̧

(5-14)

Substituting (5-13) and (5-14) into (5-12), we obtain

P2[l|ν] = Q







r2Es

N0

"

l − ν

μ

1 + μN0

I0

¶−1

#1/2







(5-15)

The substitution of this equation into (5-11) gives the upper bound on Pb for the

maximum-likelihood metric.

The upper bound on Pb versus Eb/I0 for worst-case pulsed interference, Eb/N0 = 20 dB, and

several binary convolutional codes is shown in Figure 22. Although the worst value of μ

42
0 1 2 3 4 5 6

Energy-to-interference-density ratio, dB

Bit error probability

ML, K=7

ML, K=9

AGC, K=7,r =1/2

10–2

10–3

10–4

10–5

10–6

r = 1/3 1/2 1/3 1/2

-.-.-.-

Figure 22. Worst-case performance against pulsed interference for convolutional codes of

constraint length K, rate r, Eb/N0 = 20 dB and maximum-likelihood (ML)

and AGC metrics.

varies with Eb/I0, it is found that worst-case pulsed interference causes very little

degradation relative to continuous interference. When K = 9 and r = 1/2, the

maximum-likelihood metric provides a performance that is more than 4 dB superior at

Pb = 10−5 to that provided by hard-decision decoding; when K = 9 and r = 1/3, the

advantage is approximately 2.5 dB. However, the implementation of the

maximum-likelihood metric entails knowledge of not only the presence of interference, but

also its density level. Estimates of the N0i might be based on power measurements in

adjacent frequency bands only if the interference spectral density is fairly uniform over the

desired-signal and adjacent bands. Any measurement of the power within the

desired-signal band is contaminated by the presence of the desired signal, the average

power of which is usually unknown a priori because of the fading.

Consider an automatic gain control (AGC) device that measures the average power at the

demodulator output before sampling and then weights the sampled demodulator output yi

in proportion to the inverse of the measured power to form the AGC metric. The average

power during channel-symbol i is N0iB + Es/Ts, where B is the equivalent bandwidth of

the demodulator and Ts is the channel-symbol duration. If the power measurement is

43
perfect and BTs ≈ 1, then the AGC metric is

U(j) = X

L

i=1

xjiyi

N0i + Es

(5-16)

which is a Gaussian random variable. This metric and (5-12) yield

P2(l/ν) = Q

(r2Es

N0

l(N0 + Es + I0/μ) − νI0/μ

£

l(N0 + Es + I0/μ)

2 − ν (N0 + I0/μ − E2

s /N0) I0/μ¤1/2

)

(5-17)

This equation and (5-11) give the upper bound on Pb for the AGC metric.

The upper bound on Pb versus Eb/I0 for worst-case pulsed interference, the AGC metric,

the rate-1/2 binary convolutional code with K = 7, and Eb/N0 = 20 dB is plotted in Figure

22. The figure indicates that the potential performance of the AGC metric is nearly as

good as that of the maximum-likelihood metric.

The measurement of N0iBTs + Es may be performed by a radiometer, which is a device

that measures the energy at its input. An ideal radiometer (Section 9.2) provides an

unbiased estimate of the energy received during a symbol interval. The radiometer outputs

are accurate estimates only if the standard deviation of the output is much less than its

expected value. This criterion and theoretical results for BTs = 1 indicate that the energy

measurements over a symbol interval will be unreliable if Es/N0i ≤ 10 during interference

pulses. Thus, the potential performance of the AGC metric is expected to be significantly

degraded in practice unless each interference pulse extends over many channel symbols and

its energy is measured over the corresponding interval.

The maximum-likelihood metric for continuous interference (N0i is constant for all i) is the

white-noise metric:

U(j) = X

L

i=1

xjiyi (5-18)

which is much simpler to implement than the AGC metric. For the white-noise metric,

calculations similar to the preceding ones yield

P2(l|ν) = Q

(r2Es

N0

l

μ

l + ν

I0

μN0

¶−1/2

)

(5-19)

This equation and (5-11) give the upper bound on Pb for the white-noise metric. Figure 23

illustrates the upper bound on Pb versus Eb/I0 for K = 7, r = 1/2, Eb/N0 = 20 dB, and

several values of ζ = μ/μ0. The figure demonstrates the vulnerability of soft-decision

decoding with the white-noise metric to short high-power pulses if interference power is

conserved. The high values of Pb for ζ < 1 are due to the domination of the metric by a

44
2 3 4 5 6 7 8

Energy-to-interference-density ratio, dB

Bit error probability

1.0

0.85

0.7

10–2

10–3

10–4

10–5

10–6

ζ = 0.4

Figure 23. Performance against pulsed interference for convolutional code with whitenoise metric, and K = 7, r = 1/2, and Eb/N0 = 20 dB.

few degraded symbol metrics. Consider a coherent PSK demodulator that erases its output

and, hence, a received symbol whenever an interference pulse occurs. The presence of the

pulse might be detected by examining a sequence of the demodulator outputs and

determining which ones have inordinately large magnitudes compared to the others.

Alternatively, the demodulator might decide that a pulse has occurred if an output has a

magnitude that exceeds a known upper bound for the desired signal. Consider an ideal

demodulator that unerringly detects the pulses and erases the corresponding received

symbols. Following the deinterleaving of the demodulated symbols, the decoder processes

symbols that have a probability of being erased equal to μ. The unerased symbols are

decoded by using the white-noise metric. The erasing of ν symbols causes two sequences

that differ in l symbols to be compared on the basis of l − ν symbols where 0 ≤ ν ≤ l.

Therefore, we find that

P2(l|ν) = Q

"r2Es

N0

(l − ν)

#

(5-20)

This equation and (5-11) give the upper bound on Pb for the erasures.

The upper bound on Pb is illustrated in Figure 24 for K = 7, r = 1/2, Eb/N0 = 20 dB, and

several values of ζ = μ/μ0. In this example, erasures provide no advantage over the whitenoise metric in reducing the required Eb/I0 for Pb = 10−5 if ζ > 0.85, but are increasingly

useful as ζ decreases. Consider an ideal demodulator that activates erasures only when μ is

45
2 3 4 5 6 7 8

Energy-to-interference-density ratio, dB

Bit error probability

10–2

10–3

10–4

10–5

10–6

ζ = 0.4 ζ = 0.7 0.85 1.0

Figure 24. Performance against pulsed interference for convolutional code with erasures,

K = 7, r = 1/2, and Eb/N0 = 20 dB.

small enough that the erasures are more effective than the white-noise metric. When this

condition does not occur, the white-noise metric is used. The upper bound on Pb for this

ideal erasure decoding, worst-case pulsed interference, Eb/N0 = 20 dB, and several binary

convolutional codes is illustrated in Figure 25. The required Eb/I0 at Pb = 10−5 is roughly 2

dB less than for worst-case hard-decision decoding. However, a practical demodulator will

sometimes erroneously make erasures or fail to erase, and its performance advantage may be

much more modest.

46
2 3 4 5 6 7 8

Energy-to-interference-density ratio, dB

Bit error probability

10–2

10–3

10–4

10–5

10–6

r = 1/3 1/3 1/2 1/2

K = 7

K = 9

Figure 25. Worst-case performance against pulsed interference for convolutional codes

witrh ideal erasure decoding, constraint length K, rate r, and Eb/N0 = 20 dB.

6. Despreading with Matched Filters

If the security requirements are mild enough that a short spreading sequence is adequate,

then simple direct-sequence systems can use matched filters as despreaders that provide

inherent code synchronization. When a short sequence is used, the spreading waveform

may be expressed as

p(t) = X∞

i=−∞

p1(t − iT) (6-1)

where p1(t) is one period of the spreading waveform and T is its period. If the short

spreading sequence has length N, then

p1(t) =







N

P−1

i=0

piψ (t − iTc), 0 ≤ t ≤ T

0, otherwise

(6-2)

where pi = ±1, and T = NTc.

47
6.1 Matched Filters

A filter is said to be matched to a signal x(t) that is zero outside the interval [0, T] if the

impulse response of the filter is h(t) = x(T − t). When x(t) is the input to a filter matched

to it, the filter output is

y(t) = Z ∞

−∞

x(u)h(t − u)du =

Z ∞

−∞

x(u)x(u + T − t)du

=

Z min(t,T)

max(t−T,0)

x(u)x(u + T − t)du (6-3)

The aperiodic autocorrelation of a deterministic signal with finite energy is defined as

Rx(τ ) = Z ∞

−∞

x(u)x(u + τ )du =

Z ∞

−∞

x(u)x(u − τ )du (6-4)

Therefore, the response of a matched filter to the matched signal is

y(t) = Rx(t − T) (6-5)

If this output is sampled at t = T, then y(T) = Rx(0), the signal energy.

Consider a bandpass matched filter that is matched to

x(t) = 1⁄2

p1(t) cos (2πfct + θ1), 0 ≤ t ≤ T

0, otherwise (6-6)

where p1(t) is one period of a spreading waveform and fc is the desired carrier frequency.

We evaluate the filter response to the received signal corresponding to a single data symbol:

s(t) = 1⁄2

2Ap1(t − t0) cos (2πf1t + θ), t0 ≤ t ≤ t0 + T

0, otherwise (6-7)

where t0 is a measure of the unknown arrival time, the polarity of A is determined by the

data symbol, and f1 is the received carrier frequency, which differs from fc because of

oscillator instabilities and the Doppler shift. The matched-filter output is

ys(t) = Z t

t−T

s(u)p1(u + T − t) cos [2πfc(u + T − t) + θ1] du (6-8)

If fc >> 1/T, then substituting (6-7) into (6-8) yields

ys(t) = A

Z min(t,t0+T)

max(t−T,t0)

p1 (u − t0) p1(u − t + T) cos (2πfdu + 2πfct + θ2) du (6-9)

where θ2 = θ − θ1 − 2πfcT is the phase mismatch and fd = f1 − fc. If fd << 1/T, the

carrier-frequency error is inconsequential, and

ys(t) ≈ As(t) cos (2πfct + θ2) (6-10)

48
where

As(t) = A

Z min(t,t0+T)

max(t−T,t0)

p1 (u − t0) p1(u − t + T)du (6-11)

In the absence of noise, the matched-filter output is a sinusoidal spike with a polarity

determined by A. Assuming that (3-3) is applicable, the peak magnitude, which occurs at

t = t0 + T, equals |A|T. However, if fd > 0.1/T, then (6-9) is not well-approximated by

(6-10), and the matched-filter output is significantly degraded.

If fc >> 1/T, the response of the matched filter to the interference plus noise, denoted by

N(t) = i(t) + n(t), may be expressed as

yn(t) = Z t

t−T

N(u)p1(u + T − t) cos [2πfc(u + T − t) + θ1] du

= N1(t) cos (2πfct + θ2) + N2(t) sin (2πfct + θ2) (6-12)

where

N1(t) = Z t

t−T

N(u)p1(u + T − t) cos (2πfcu + θ) du (6-13)

N2(t) = Z t

t−T

N(u)p1(u + T − t) sin (2πfcu + θ) du (6-14)

These equations exhibit the spreading of the interference spectrum.

The envelope of the matched-filter output y(t) = ys(t) + yn(t) is

E(t) = ©[As(t) + N1(t)]2 + N2

2 (t)

a1/2 (6-15)

Define 2 such that 2πfc(t0 + 2) + θ − θ1 is an integer times 2π. If fc is sufficiently large that

2 << t0 + T, then (6-10) and (6-12) imply that if y(t) is sampled at t = t0 + T + 2,

y (t0 + T + 2) = ys (t0 + T + 2) + yn (t0 + T + 2)

= As (t0 + T + 2) + N1 (t0 + T + 2) (6-16)

≈ AT + N1 (t0 + T + 2)

where As(t0 + T) = AT. If

|AT + N1 (t0 + T)| >> |N2 (t0 + T)| (6-17)

then (6-15) implies that

E (t0 + T) ≈ |AT + N1 (t0 + T)| (6-18)

A comparison of this equation with (6-16) indicates that there is relatively little

degradation in using an envelope detector after the matched filter rather than directly

49
detecting the peak magnitude of the matched-filter output. The latter procedure would be

much more difficult to implement.

A passive matched filter can be implemented as an analog or digital transversal filter that

essentially stores a replica of the underlying spreading sequence and waits for the received

sequence to align itself with the replica. Figure 26 illustrates the basic form of a

surface-acoustic-wave (SAW) transversal filter. The SAW delay line consists primarily of a

piezoelectric substrate, which serves as the acoustic propagation medium, and interdigital

transducers, which serve as the taps and the input transducer. The transversal filter is

matched to one period of the spreading waveform, the propagation delay between taps is

Tc, and fcTc is an integer. The chip matched filter following the summer is matched to

ψ(t) cos (2πfct + θ). It is easily verified that the impulse response of the transversal filter is

that of a filter matched to p1(t) cos (2πfct + θ).

A convolver is an active matched filter that produces the convolution of the received signal

with a local reference [9]. When used as a direct-sequence matched filter, a convolver uses

a recirculating, time-reversed replica of the spreading waveform as a reference waveform. In

a SAW elastic convolver, which is depicted in Figure 27, the received signal and the

reference are applied to interdigital transducers that generate acoustic waves at opposite

ends of the substrate. The acoustic waves travel in opposite directions with speed v, and

the acoustic terminations suppress reflections. The signal wave is launched at position

x = 0 and the reference wave at x = L. The signal wave travels to the right in the

substrate and has the form

F(t, x) = f

3

t − x

υ

 ́

cos h2πfc

3

t − x

v

 ́

+ θ

i

(6-19)

Input

Output

Chip

matched

filter

pN–1 pN–2

SAW delay line

∑

p0

Figure 26. Matched filter that uses a SAW transversal filter.

50
Signal

input

Acoustic

termination

Beam

compressor

Output

Reference

input

Figure 27. SAW elastic convolver.

where f(t) is the modulation at position x = 0. The reference wave travels to the left and

has the form

G(t, x) = g

μ

t +

x

υ − Lυ

¶

cos ∙2πfc

μ

t +

x

υ − Lυ

¶

+ θ1

 ̧

(6-20)

where g(t) is the modulation at position x = L. Both f(t) and g(t) are assumed to have

bandwidths much smaller than fc. The beam compressors, which consist of thin metallic

strips, focus the acoustic energy to increase the convolver’s efficiency. When the acoustic

waves overlap beneath the central electrode, a nonlinear piezoelectric effect causes a surface

charge distribution that is spatially integrated by the electrode. The convolver output has

the primary component

y(t) = Z L

0

α[F(t, x) + G(t, x)]2dx (6-21)

where α is a constant. Substituting (6-19) and (6-20) into (6-21) and using trigonometry,

we find that y(t) is the sum of a number of terms, some of which are negligible if

fcL/v >> 1. Others are slowly varying and are easily blocked by a filter. The most useful

component of the convolver output is

ys(t) = ∙

α

Z L

0

f

3

t − x

υ

 ́

g

μ

t +

x

υ − Lυ

¶

dx ̧cos (4πfct + θ2) (6-22)

where θ2 = θ + θ1 − 2πfcL/v. Changing variables, we find that the amplitude is

proportional to

As(t) = Z t

t−L/υ

f(y)g(2t − y − L/υ)dy (6-23)

where the factor 2t results from the counterpropagation of the two acoustic waves.

Suppose that an acquisition pulse is a single period of the spreading waveform. Then

f(t) = Ap1 (t − t0) and g(t) = p(T − t), where t0 is the uncertainty in the arrival time of an

acquisition pulse relative to the launching of the reference signal at x = L. The periodicity

51
of g(t) allows the time origin to be selected so that 0 ≤ t0 ≤ T. Equations (6-23) and (6-1)

and a change of variables yield

As(t) = A X∞

i=−∞

Z t−t0

t−t0−L/υ

p1(y)p1 (y + iT + t0 − 2t + L/υ) dy (6-24)

Since p1(t) = 0 unless 0 ≤ t < T,As(t) = 0 unless t0 <t<t0 + T + L/v. For every positive

integer k, let

τk = kT + t0 + L/v

2 , k = 1, 2,... (6-25)

Only one term in (6-24) can be nonzero when t = τk, and

As (τk) = A

Z τk−t0

τk−t0−L/v

p2

1(y)dy (6-26)

The maximum possible magnitude of As(τk) is produced if τk − t0 ≥ T and

τk − t0 − L/v ≤ 0; that is, if

t0 + T ≤ τk ≤ t0 +

L

v (6-27)

Since (6-25) indicates that τk+1 − τk = T/2, there is some τk that satisfies (6-27) if

L ≥

3

2

vT (6-28)

Thus, if L is large enough, then there is some k such that As(τk) = AT, and the envelope

of the convolver output at t = τk has the maximum possible magnitude |A|T. If L = 3vT/2

and t0 6= T/2, only one peak value occurs in response to the single received pulse.

As an example, let t0 = 0, L/v = 6Tc, and T = 4Tc. The chips propagating in the convolver

for three separate time instants t = 4Tc, 5Tc, and 6Tc are illustrated in Figure 28. The top

diagrams refer to the counterpropagating periodic reference signal, whereas the bottom

diagrams refer to the single received pulse of four chips. The chips are numbered

consecutively. The received pulse is completely contained within the convolver during

4Tc ≤ t ≤ 6Tc. The maximum magnitude of the output occurs at time t = 5Tc, which is the

instant of perfect alignment of the reference signal and the received chips.

6.2 Noncoherent Systems

In a noncoherent direct-sequence system with binary code-shift keying (CSK), one of two

orthogonal spreading sequences is transmitted, as shown in Figure 29(a). One sequence

represents the symbol 1, and the other represents the symbol 0. The receiver uses two

matched filters, each matched to a different sequence and followed by an envelope detector,

as shown in Figure 29(b). In the absence of noise and interference, each sequence causes

52
2 1 4 3 2 1 1 4 3 2 1 4 4 3 2 1 4 3

4 3 2 1 4 3 2 1 4 3 2 1

t = 4Tc t = 5Tc t = 6Tc

reference

signal

Figure 28. Chip configurations within convolver at time instants t = 4Tc, 5Tc, and 6Tc

when t0 = 0, L/v = Tc, and T = 4Tc.

Data bits

∑

Transmitted

signal

Oscillator

OR

A

N

D

A

N

D

Inverter

Spreading

sequence

generator

Code clock

Spreading

sequence

generator

Received

signal Output

Symbol

synchron.

Comparator

Envelope

detector 2

Envelope

detector 1

Matched

filter 2

Matched

filter 1

Wideband

filter

(a)

(b)

Figure 29. Direct-sequence system with binary code-shift keying: (a) transmitter and (b)

receiver.

53
only one envelope detector to produce a significant output. The data is recovered by

comparing the two detector outputs every symbol period. Since each of the two orthogonal

sequences has a period equal to the symbol duration, symbol or bit synchronization is

identical to code synchronization. The symbol synchronizer, which provides timing pulses

to the comparator or decision device, must lock onto the autocorrelation spikes appearing

in the envelope-detector outputs. Ideally, these spikes have a triangular shape. The symbol

synchronizer must be impervious to the autocorrelation sidelobe peaks and any

cross-correlation peaks. A simple implementation with a single threshold detector would

result in an unacceptable number of false alarms, premature detections, or missed

detections when the received signal amplitude is unknown and has a wide dynamic range.

Limiting or automatic gain control only exacerbates the problem when the signal power

level is below that of the interference plus noise. More than one threshold detector with

precedence given to the highest threshold crossed will improve the accuracy of the decision

timing or sampling instants produced by the symbol synchronizer [10]. Another approach

is to use peak detection based on a differentiator and a zero-crossing detector. Finally, a

phase-locked or feedback loop of some type could be used in the symbol synchronizer. A

preamble may be transmitted to initiate accurate synchronization so that symbols are not

incorrectly detected while synchronization is being established.

Consider the detection of a symbol represented by (6-7), where p1(t) is the CSK waveform

to which filter 1 is matched. Assuming perfect symbol synchronization, the channel symbol

is received during the interval 0 ≤ t ≤ Ts. From (6-10) to (6-15) with T = Ts and t0 = 0,

we find that the output of envelope detector 1 at t = Ts is

R1 = ¡Z2

1 + Z22

¢1/2 (6-29)

where

Z1 = ATs +

Z Ts

0

N(u)p1(u) cos (2πfcu + θ) du (6-30)

Z2 =

Z Ts

0

N(u)p1(u) sin (2πfcu + θ)du (6-31)

Similarly, if filter 2 is matched to sequence p2(t), then the output of envelope detector 2 at

t = Ts is

R2 = ¡

Z2

3 + Z24

¢1/2 (6-32)

where

Z3 =

Z Ts

0

N(u)p2(u) cos (2πfcu + θ) du (6-33)

Z4 =

Z Ts

0

N(u)p2(u) sin (2πfcu + θ) du (6-34)

and the response to the transmitted symbol at t = Ts is zero because of the orthogonality

of the sequences.

54
Suppose that the interference plus noise N(t) is modeled as zero-mean, Gaussian

interference, and the spreading sequences are modeled as deterministic and orthogonal.

Then E[Z1] = ATs and E[Zi]=0, i = 2, 3, 4. If N(t) is assumed to be wideband enough

that its autocorrelation is approximated by (3-13), then straightforward calculations using

fcTs >> 1 and the orthogonality of p1(t) and p2(t) indicate that Z1, Z2, Z3, and Z4 are all

uncorrelated with each other. The jointly Gaussian character of the random variables then

implies that they are statistically independent of each other, and hence R1 and R2 are

independent. Analogous results can be obtained when the transmitted symbol is

represented by CSK waveform p2(t). A straightforward derivation similar to the classical

one for orthogonal signals then yields the symbol error probability:

Ps = 1

2 exp μ

− Es

2N0e

¶

(6-35)

where N0e is given by (3-47). A comparison of (6-35) with (3-44) indicates that the

performance of the direct-sequence system with noncoherent binary CSK in the presence of

wideband Gaussian interference is approximately 4 dB worse than that of a direct-sequence

system with coherent binary PSK. This difference arises because binary CSK uses

orthogonal rather than antipodal signaling. A much more complicated coherent version of

Figure 29 would only recover roughly 1 dB of the disparity.

A direct-sequence system with q-ary CSK encodes each group of m binary symbols as one

of q = 2m sequences chosen to have negligible cross correlations. Suppose that bandwidth

constraints limit the chip rate of a binary CSK system to G chips per data bit. For a fixed

data-bit rate, the q-ary CSK system produces mG chips to represent each group of m bits,

which may be regarded as a single q-ary symbol. Thus, the processing gain relative to a

data symbol is mG, which indicates an enhanced ability to suppress interference. In the

presence of wideband Gaussian interference, the performance improvement of quaternary

CSK is more than 2 dB relative to binary CSK, but four filters matched to four

double-length sequences are required. When the chip rate is fixed, q-ary CSK provides a

means of increasing the data-bit or code-symbol rate without sacrificing the processing

gain.

Elimination of the lower branch in Figure 29(b) leaves a system that uses a single CSK

sequence and a minimum amount of hardware. The symbol 1 is signified by the

transmission of the sequence, whereas the symbol 0 is signified by the absence of a

transmission. Decisions are made after comparing the envelope-detector output with a

threshold. One problem with this system is that the optimal threshold is a function of the

amplitude of the received signal, which must somehow be estimated. Another problem is

the degraded performance of the symbol synchronizer when many consecutive zeros are

transmitted. Thus, a system with binary CSK is much more practical.

55
A direct-sequence system with differential phase-shift keying (DPSK) signifies the symbol 1

by the transmission of a spreading sequence without any change in the carrier phase; the

symbol 0 is signified by the transmission of the same sequence after a phase shift of π

radians in the carrier phase or multiplication of the signal by −1. A matched filter

despreads the received direct-sequence signal, as illustrated in Figure 30. The filter output

is applied to a standard DPSK demodulator that makes symbol decisions. An analysis of

this system in the presence of wideband Gaussian interference indicates that it is more

than 2 dB superior to the system with binary CSK. However, the system with DPSK is

more sensitive to Doppler shifts and is more than 1 dB inferior to a system with coherent

binary PSK.

Received

signal

Output

Symbol

synchron.

Envelope

detector

Matched

filter

Wideband

filter

DPSK

demodulator

Figure 30. Receiver for direct-sequence system with differential phase-shift keying.

6.3 Multipath-Resistant Coherent System

Carrier synchronization is essential for the coherent demodulation of a direct-sequence

signal. Prior to despreading, the signal-to-interference-plus-noise ratio (SINR) may be too

low for the received signal to serve as the input to a phase-locked loop that produces a

phase-coherent carrier. Although the despread matched-filter output has a large SINR near

the autocorrelation peak, the average SINR may be insufficient for a phase-locked loop. An

alternative approach is to use a recirculation loop to produce a synchronized carrier during

the main lobe of the matched-filter output.

A recirculation loop, is designed to reinforce a periodic input signal by positive feedback.

As illustrated in Figure 31, the feedback elements are an attenuator of gain K and a delay

line with delay Tˆ

s approximating a symbol duration Ts. The basic concept behind this

architecture is that successive signal pulses are coherently added while the interference and

noise are noncoherently added, thereby producing an output pulse with an improved SINR.

The periodic input consists of N symbol pulses such that

s0(t) = X

N

i=0

g (t − iTs) (6-36)

56
∑

K Delay Ts

Input Output

^

Figure 31. Recirculation loop.

where g(t) = 0 for t < 0 or t ≥ Ts. The figure indicates that the loop output is

s1(t) = s0(t) + Ks1

3

t − Tˆ

s

 ́

(6-37)

Substitution of this equation into itself yields

s1(t) = s0(t) + Ks0

3

t − Tˆ

s

 ́

+ K2s1

3

t − 2Tˆ

s

 ́

(6-38)

Repeating this substitution process n times leads to

s1(t) = Xn

m=0

Kms0

3

t − mTˆ

s

 ́

+ Kn+1s1

h

t − (n + 1)Tˆ

s

i

(6-39)

which indicates that s1(t) increases with n if K ≥ 1 and enough input pulses are available.

To prevent an eventual loop malfunction, K < 1 is a design requirement that is assumed

henceforth.

During the interval [nTˆ

s,(n + 1)Tˆs], n or fewer recirculations of the symbols have occurred.

Since s1(t) = 0 for t < 0, the substitution of (6-36) into (6-39) yields

s1(t) = Xn

m=0

X

N

i=0

Kmg

3

t − mTˆ

s − iTs

 ́

, nTˆ

s ≤ t < (n + 1)Tˆs (6-40)

This equation indicates that if Tˆ

s is not exactly equal to Ts, then the pulses do not add

coherently, and may combine destructively. However, since K < 1, the effect of a particular

pulse decreases as m increases and will eventually be negligible. The delay Tˆ

s is designed to

match Ts. Suppose that the design error is small enough that

N

 ̄

 ̄

 ̄

Tˆ

s − Ts

 ̄

 ̄

 ̄

<< Tˆ

s (6-41)

Since t − mTˆ

s − iTs = t − (m + i)Ts − m(Tˆs − Ts) and g(t) is time-limited, (6-41) and

n ≤ N imply that only the term in (6-40) with i = n − m contributes appreciably to the

output. Therefore,

s1(t) ≈ Xn

m=0

Kmg

h

t − nTs − m

3

Tˆ

s − Ts

 ́i , nTs ≤ t < (n + 1)Ts (6-42)

57
Let ν denote a positive integer such that Km is negligible if m > ν. Consider an input

pulse of the form

g(t) = A(t) cos 2πfct, 0 ≤ t < min 3Ts, Tˆ

s

 ́

(6-43)

which implies that each of the N pulses in (6-36) has the same initial phase. Assume that

the amplitude A(t) varies slowly enough that

A

h

t − nTs − m

3

Tˆ

s − Ts

 ́i ≈ A (t − nTs), 0 ≤ m ≤ ν (6-44)

and that the design error is small enough that

νfc

 ̄

 ̄

 ̄

Tˆ

s − Ts

 ̄

 ̄

 ̄

<< 1 (6-45)

Then (6-42) to (6-45) yield

s1(t) ≈ g (t − nTs)

Xn

m=0

Km

= g (t − nTs)

μ1 − Kn+1

1 − K

¶

, nTs ≤ t ≤ (n + 1)Ts (6-46)

This equation indicates that the average power in an output pulse during the interval

nTs ≤ t < (n + 1)Ts is approximately

Sn =

μ1 − Kn+1

1 − K

¶2

S, K < 1 (6-47)

where S is the average power in an input pulse.

Let σ2 denote the average thermal noise power at the output. If Tˆ

s is large enough that the

recirculated noise is uncorrelated with the input noise, then the noise power after n

recirculations is

σ2

n = σ2 Xn

m=0

¡

K2¢m

= σ2

μ1 − K2n+2

1 − K2

¶

, K< 1 (6-48)

The improvement in the signal-to-noise ratio due to the presence of the recirculation loop is

I (n, K) = Sn/σ2

n

S/σ2 = (1 − Kn+1) (1 + K)(1 + Kn+1) (1 − K)

≤ 1 + K

1 − K , K< 1 (6-49)

Since it was assumed that Km is negligibly small when m > ν, the maximum improvement

is nearly attained when n ≥ ν. However, the upper bound on ν for the validity of (6-45)

58
decreases as the loop phase error 2πfc|Tˆ

s − Ts| increases. Thus, K must be decreased as

the phase error increases. The phase error of a practical SAW recirculation loop may be

caused by a temperature fluctuation, a Doppler shift, oscillator instability, or an imprecise

delay-line length. Various other loop imperfections limit the achievable value of K and,

hence, the improvement provided by the recirculation loop [11]. Figure 32 illustrates a

coherent decision-directed demodulator for a direct-sequence signal with binary PSK and

the same carrier phase at the beginning of each symbol. The bandpass matched filter

removes the spreading waveform and produces compressed sinusoidal pulses, as indicated

by (6-10) and (6-11) when A is bipolar. A compressed pulse due to a direct-path signal

may be followed by one or more compressed pulses due to multipath signals, as illustrated

conceptually in Figure 33(a) for pulses corresponding to the transmitted symbols 101. Each

compressed pulse is delayed by one symbol and then mixed with the demodulator’s output

symbol. If this symbol is correct, it coincides with the same data symbol that is modulated

onto the compressed pulse. Consequently, the mixer removes the data modulation and

produces a phase-coherent reference pulse that is independent of the data symbol, as

illustrated in Figure 33(b), where the middle pulses are inverted in phase relative to the

corresponding pulses in Figure 33(a). The reference pulses are amplified by a recirculation

loop. The loop output and the matched-filter output are applied to a mixer that produces

the baseband integrator input illustrated in Figure 33(c). The length of the integration

interval is equal to a symbol duration. The integrator output is sampled and applied to a

decision device that produces the data output. Since multipath components are coherently

integrated, the demodulator provides an improved performance in a fading environment.

Even if the desired-signal multipath components are absent, the coherent decision-directed

receiver suppresses interference approximately as much as the correlator of Figure 14. The

decision-directed receiver is much simpler to implement because code acquisition and tracking systems are unnecessary, but it requires a short spreading sequence. More efficient

exploitation of multipath components is possible with rake combining (Section 10.7).

Input

Output

Decision

device

Integrator

and

sampler

Symbol

synchron.

Envelope

detector

Recirculation

loop

One-symbol

delay

Matched

filter

Figure 32. Coherent decision-directed demodulator.

59
(a)

(b)

(c)

Integration interval

Direct Multipath

Figure 33. Conceptual waveforms of demodulator: (a) matched-filter output, (b) recirculation loop input or output, and (c) baseband integrator input.

7. Code Synchronization

A spread-spectrum receiver must generate a spreading sequence that is synchronized with

the received sequence; that is, the corresponding chips must precisely or nearly coincide.

Any misalignment causes the signal amplitude at the demodulator output to fall in

accordance with the autocorrelation or partial autocorrelation function. Range uncertainty,

clock drifts, and the Doppler shift are the primary sources of synchronization errors. Code

synchronization might be obtained from separately transmitted pilot or timing signals.

However, to reduce the cost in power and overhead, most direct-sequence receivers acquire

code synchronization from the received direct-sequence signal.

In deriving the maximum-likelihood estimate of the code phase or timing offset of the

spreading sequence, several assumptions are made. Since the presence of the data

modulation impedes code synchronization, the transmitter is assumed to assist the

synchronization by transmitting the spreading sequence without any data modulation. In

nearly all applications, noncoherent code synchronization must precede carrier

synchronization because the signal energy is spread over a wide spectral band. Prior to

despreading, which requires code synchronization, it is difficult to establish a high enough

60
signal-to-noise ratio in a phase-locked loop for successful carrier tracking. The received

signal is

r(t) = s(t) + n(t) (7-1)

where s(t) is the desired signal and n(t) is the additive white Gaussian noise. For a

direct-sequence system with PSK modulation, the desired signal is

s(t) = √

2Sp(t − τ ) cos (2πfct + 2πfdt + θ) (7-2)

where S is the average power, p(t) is the spreading waveform, fc is the carrier frequency, θ

is the random carrier phase, and τ and fd are the unknown code phase and frequency

offset, respectively, that must be estimated. The frequency offset is due to a Doppler shift

or to a drift or instability in the transmitter oscillator.

The coefficients in the expansion of the observed waveform in terms of orthonormal basis

functions constitute the vector r = [r1 r2 ...rN ]. The likelihood function for the unknown τ

and fd is the conditional density function of r given the values of τ and fd. Since θ is a

random variable, the likelihood function is

Λ(r) = Eθ [f (r|τ, fd, θ)] (7-3)

where f(r|τ, fd, θ) is the conditional density function of r given the values of τ, fd, and θ,

and Eθ is the expectation with respect to θ. The maximum-likelihood estimates are those

values of τ and fd that maximize Λ(r).

The coefficients in the expansion of r(t) in terms of the orthonormal basis functions are

statistically independent. Since each coefficient is Gaussian with variance N0/2,

f (r|τ, fd, θ) = Y

N

i=1

1

√πN0

exp "−(ri − si)

2

N0

#

(7-4)

where the {si} are the coefficients of the signal s(t) when τ, fd, and θ are given.

Substituting this equation into (7-3) and eliminating factors irrelevant to the

maximum-likelihood estimation, we obtain

Λ(r) = Eθ

(

exp "

2

N0

X

N

i=1

risi − 1

N0

X

N

i=1

s2

i

#) (7-5)

Expansions in the orthonormal basis functions indicate that if N → ∞, the likelihood

function may be expressed in terms of the signal waveforms as

Λ[r(t)] = Eθ

1⁄2

exp ∙ 2

N0

Z T

0

r(t)s(t)dt − E

N0

 ̧3⁄4 (7-6)

where E is the energy in the signal waveform over the observation interval of duration T.

Assuming that E does not vary significantly over the ranges of τ and fd considered, the

61
factor involving E may be dropped from further consideration. The substitution of s(t) in

(7-2) into (7-6) then yields

Λ[r(t)] = Eθ

(

exp "

2

√

2S

N0

Z T

0

r(t)p(t − τ ) cos (2πfct + 2πfdt + θ) dt#) (7-7)

For noncoherent estimation, the received carrier phase θ is assumed to be uniformly

distributed over [0, 2π). A trigonometric expansion followed by an integration of (7-7) over

θ gives (cf. (9-13))

Λ[r(t)] = I0

Ã

2

p2SR (τ, fd)

N0

!

(7-8)

where I0( ) is the modified Bessel function of the first kind and order zero given by (B-30),

and

R (τ, fd) = ∙Z T

0

r(t)p(t − τ ) cos (2πfct + 2πfdt) dt ̧2

+

∙Z T

0

r(t)p(t − τ ) sin (2πfct + 2πfdt) dt ̧2(7-9)

Since I0(x) is a monotonically increasing function of x, (7-8) implies that R(τ, fd) is a

sufficient statistic for maximum-likelihood estimation. Ideally, the estimates are

determined by considering all possible values of τ and fd, and then choosing those values

that maximize (7-9). A device that implements (7-9) is called a noncoherent correlator.

A practical implementation of maximum-likelihood estimation or other type of estimation

is greatly facilitated by dividing synchronization into the two operations of acquisition and

tracking. Acquisition provides coarse synchronization by limiting the choices of the

estimated values to a finite number of quantized candidates. Following the acquisition,

tracking provides and maintains fine synchronization.

One method of acquisition is to use a parallel array of processors, each matched to

candidate quantized values of the timing and frequency offsets. The largest processor

output then indicates which candidates are to selected as the estimates. An alternative

method of acquisition, which is much less complex, but significantly increases the time

needed to make a decision, is to serially search over the candidate offsets. Since the

frequency offset is usually negligible or requires only a few candidate values, the remainder

of this section analyzes code synchronization in which only the timing offset τ is estimated.

Search methods rather than parallel processing are examined. Code acquisition is the

operation by which the phase of the receiver-generated sequence is brought to within a

fraction of a chip of the phase of the received sequence. After this condition is detected and

verified, the tracking system is activated. Code tracking is the operation by which

62
synchronization errors are further reduced or at least maintained within certain bounds.

Both the acquisition and tracking devices regulate the clock rate. Changes in the clock rate

adjust the phase or timing offset of the receiver-generated or local sequence relative to the

phase or timing offset of the received sequence. In a benign environment, sequential

estimation methods provide rapid acquisition [12]. Successive received chips are

demodulated and then loaded into the receiver’s code generator to establish its initial state.

The tracking system then ensures that the code generator maintains synchronization.

However, because chip demodulation is required, the usual despreading mechanism cannot

be used to suppress interference during acquisition. Since an acquisition failure completely

disables a communication system, an acquisition system must be capable of rejecting the

anticipated level of interference. To meet this requirement, matched-filter acquisition and

serial-search acquisition are the most effective techniques in general.

7.1 Acquisition with Matched Filter

Matched-filter acquisition provides potentially rapid acquisition when short programmable

sequences give adequate security. The matched filter in an acquisition system is matched to

one period of the spreading waveform (Section 6.2), which is usually transmitted without

modulation during acquisition. The sequence length or integration time of the matched

filter is limited by frequency offsets and chip-rate errors. The output envelope, which

ideally comprises triangular autocorrelation spikes, is compared with one or more

thresholds, one of which is close to the peak value of the spikes. If the data-symbol

boundaries coincide with the beginning and end of a spreading sequence, the occurrence of

a threshold crossing provides timing information used for both symbol synchronization and

acquisition. A major application of matched-filter acquisition is for burst communications,

which are short and infrequent communications that do not require a long spreading

sequence.

A digital matched filter that generates R(τ, 0) for noncoherent acquisition of a binary

spreading waveform is illustrated in Figure 34. The digital matched filter offers great

flexibility, but is limited in the bandwidth it can accommodate. The received spreading

waveform is decomposed into in-phase and quadrature baseband components, each of

which is applied to a separate branch. The outputs of each digitizer are applied to a

transversal filter. Tapped outputs of each transversal filter are multiplied by stored weights

and summed. The two sums are squared and added together to produce the final

matched-filter output.

63
∑

Input

signal Output

+

+

Transversal filter Squarer

Filter,

sampler,

digitizer

Reference

weights

Transversal filter Squarer

Filter,

sampler,

digitizer

π/2 

Oscillator

Figure 34. Digital matched filter.

A one-bit digitizer makes hard decisions on the received chips by observing the polarities of

the sample values. Each transversal filter is a shift register, and the reference weights are

sequence chips stored in shift-register stages. The transversal filter contains G successive

received spreading-sequence chips and a correlator that computes the number of received

and stored chips that match. The correlator outputs are applied to the squarers.

Matched-filter acquisition for continuous communications is useful when serial-search

acquisition with a long sequence fails or takes too long. The transmission of the short

sequence may be concealed by embedding it within the long sequence. The short sequence

may be a subsequence of the long sequence that is presumed to be ahead of the received

sequence and is stored in the programmable matched filter. Figure 35 depicts the

configuration of a matched filter for short-sequence acquisition and a serial-search system

for long-sequence acquisition. The control signal provides the short sequence that is stored

or recirculated in the matched filter. The control signal activates the matched filter when it

is needed and deactivates it otherwise. The short sequence is detected when the envelope of

the matched-filter output crosses a threshold. The threshold-detector output starts a

long-sequence generator in the serial-search system at a predetermined initial state. The

long sequence is used for verifying the acquisition and for despreading the received

direct-sequence signal. A number of matched filters in parallel can be used to expedite the

process.

7.2 Serial-Search Acquisition

Serial-search acquisition consists of a search, usually in discrete steps, of the possible time

alignments of a local sequence relative to a received spreading sequence. The timing

uncertainty region is quantized into a finite number of cells, which are search positions of

relative code phases or timing alignments. The cells are serially tested until it is determined

64
Input

To tracking system

and demodulator

Envelope

detector

Threshold

detector

Serial-search

acquisition

system

Control

Matched

filter

Figure 35. Configuration of a serial-search acquisition system enabled by a matched filter.

that a particular cell corresponds to the alignment of the two sequences to within a fraction

of a chip. The step size or separation between cells is typically one chip or a half chip.

Figure 36 depicts the principal components of a serial-search acquisition system. The

received direct-sequence signal and a local spreading sequence are applied to a noncoherent

correlator that produces the statistic (7-9). If the received and local spreading sequences

are not aligned, the sampled correlator output is low. Therefore, the threshold is not

exceeded, the cell under test is rejected, and the phase of the local sequence is retarded or

advanced, possibly by generating an extra clock pulse or by blocking one. A new cell is

then tested. If the sequences are nearly aligned, the sampled correlator output is high, the

threshold is exceeded, the search is stopped, and the two sequences run in parallel at some

fixed phase offset. Subsequent tests verify that the correct cell has been identified. If a cell

fails the verification tests, the search is resumed. If a cell passes, the two sequences are

assumed to be coarsely synchronized, demodulation begins, and the tracking system is

activated. The threshold-detector output continues to be monitored so that any subsequent

loss of synchronization activates the serial search.

There may be several cells that potentially provide a valid acquisition. However, if none of

these cells corresponds to perfect synchronization, the detected energy is reduced below its

potential peak value. If the step size is one-half of a chip, then one of the cells corresponds

to an alignment within one-fourth of a chip. On the average, the misalignment of this cell

is one-eighth of a chip, which may cause a negligible degradation. As the step size is

decreased, the average detected energy during acquisition increases, but the number of cells

to be searched increases.

The dwell time is the amount of time required for testing a cell and is approximately equal

to the length of the integration interval in the noncoherent correlator (Section 7.3). An

65
Noncoherent

correlator

To tracking system

and demodulator

From tracking

system

Search

control

Voltagecontrolled

clock

Spreading

sequence

generator

Threshold

detector

Input

Figure 36. Serial-search acquisition system.

acquisition system is called a single-dwell system if a single test determines whether a cell

is accepted as the correct one. If verification testing occurs before acceptance, the system is

called a multiple-dwell system. The dwell times may be either fixed or variable but

bounded by some maximum value. The dwell time for the initial test of a cell is usually

designed to be much shorter than the dwell times for verification testing. This approach

expedites the acquisition by quickly eliminating the bulk of the incorrect cells. In any

serial-search system, the dwell time allotted to a test is limited by the Doppler shift, which

causes the received and local chip rates to differ. As a result, an initial close alignment of

the two sequences may disappear by the end of the test unless the dwell time is much

smaller than the inverse of the maximum difference between the chip rates.

A multiple-dwell system may use a consecutive-count strategy, in which a failed test causes

a cell to be immediately rejected, or an up-down strategy, in which a failed test causes a

repetition of a previous test. Figures 37 and 38 depict the flow graphs of the

consecutive-count and up-down strategies, respectively, that require D tests to be passed

before acquisition is declared. If the threshold is not exceeded during test 1, the cell fails

the test, and the next cell is tested. If it is exceeded, the cell passes the test, the search is

stopped, and the system enters the verification mode. The same cell is tested again, but

the dwell time and the threshold may be changed. Once all the verification tests have been

passed, the code tracking is activated, and the system enters the lock mode. In the lock

mode, the lock detector continually verifies that code synchronization is maintained. If the

lock detector decides that synchronization has been lost, reacquisition begins in the search

mode.

The order in which the cells are tested is determined by the general search strategy. Figure

39(a) depicts a uniform search over the q cells of the timing uncertainty region. The broken

lines represent the discontinuous transitions of the search from the one part of the

uncertainty region to another. The broken-center Z search, illustrated in Figure 39(b), is

appropriate when a priori information makes part of the uncertainty region more likely to

66
Search mode

Vertification mode

Lock

mode

Pass Fail

Reject

Fail Fail

Test 1 Test 2 Test D

Start Pass Pass

Figure 37. Flow graph of multiple-dwell system with consecutive-count strategy.

Search mode

Lock

mode Reject

Pass

Fail Fail

Fail

Test 1 Test 2 Test D

Pass Pass Start

Verification mode

Figure 38. Flow graph of multiple-dwell system with up-down strategy.

Cell

(a) (b)

Time Time

Uncertainty

1 2 q

Uncertainty

Cell

1 2 q

Figure 39. Trajectories of search positions: (a) uniform search and (b) broken-center Z

search.

contain the correct cell than the rest of the region. A priori information may be derived

from the detection of a short preamble. If the sequences are synchronized with the time of

day, then the receiver’s estimate of the transmitter range combined with the time of day

provide the a priori information.

The acquisition time is the amount of time required for an acquisition system to locate the

correct cell and initiate the code tracking system. To derive the statistics of the acquisition

time [13], one of the q possible cells is considered the correct cell, and the other (q − 1) cells

are incorrect. The difference in timing offsets among cells is ∆Tc, where the step size ∆ is

usually either 1 or 1/2. However, it is convenient to allow the correct cell to include two or

more timing offsets or code phases. Let L denote the number of times the correct cell is

tested before it is accepted and acquisition terminates. Let C denote the number of the

67
correct cell and πj denote the probability that C = j. Let ν(L, C) denote the number of

incorrect cells tested during the acquisition process. The functional dependence is

determined by the search strategy. Let Tr(L, C) denote the total rewinding time it takes

the search to move discontinuously within the uncertainty region. Since an incorrect cell is

always ultimately rejected, there are only three types of events that occur during a serial

search. Either the nth incorrect cell is dismissed after T11(n) seconds, a correct cell is

falsely dismissed for the mth time after T12(m) seconds, or a correct cell is accepted after

T22 seconds, where the first subscript is 1 if dismissal occurs, and 2 otherwise; the second

subscript is 1 if the cell is incorrect, and 2 otherwise. Each of these decision times is a

random variable assumed to be identically distributed for each cell to which it applies. If

an incorrect cell is accepted, the receiver eventually recognizes the mistake and reinitiates

the search at the next cell. The wasted time expended in code tracking is called the penalty

time.

From the preceding definitions, it follows that the acquisition time is the random variable

given by

Ta =

ν

X

(L,C)

n=1

T11(n) + X

L−1

m=1

T12(m) + T22 + Tr(L, C) (7-10)

The most important performance measures of the serial search are the mean and variance

of Ta. Given L = i and C = j, the conditional expected value of Ta is

E [Ta|i, j] = ν(i, j)T ̄

11 + (i − 1)T ̄12 + T ̄22 + Tr(i, j) (7-11)

where T ̄

11, T ̄12, and T ̄22 are the expected values of each T11(n), T12(m), and T22, respectively.

Therefore, the mean acquisition time is

T ̄

a = T ̄22 +X∞

i=1

PL(i)

X

q

j=1

πj

£

ν(i, j)T ̄

11 + (i − 1)T ̄12 + Tr(i, j)

¤ (7-12)

where PL(i) is the probability that L = i. We assume that the test statistics are

independent and identically distributed. Therefore,

PL(i) = PD (1 − PD)

i−1 (7-13)

where PD is the probability that the correct cell is detected when it is tested during a scan

of the uncertainty region.

After calculating the conditional expected value of T2

a given that L = i and C = j, and

using the identity x2 = var(x)+ ̄x2, we obtain

T2

a = X∞

i=1

PL(i)

X

q

j=1

πj{[ν(i, j)T ̄

11 + (i − 1)T ̄12 + T ̄22 + Tr(i, j)]2

+ν(i, j)var(T11)+(i − 1)var(T12) + var(T22)} (7-14)

68
The variance of Ta is

σ2

a = T2

a − T ̄2

a (7-15)

In some applications, the serial-search acquisition must be completed within a specified

period of duration Tmax. If the acquisition time exceeds Tmax, the serial search is

terminated, and special measures are undertaken. For example, matched-filter acquisition

of a short sequence. The probability that Ta ≤ Tmax can be bounded by using Chebyshev’s

inequality (Appendix A):

P (Ta ≤ Tmax) ≥ P ¡ ̄

 ̄Ta − T ̄

a

 ̄

 ̄ ≤ Tmax − T ̄

a

¢

≥ 1 − σ2

a ¡

Tmax − T ̄

a

¢2 (7-16)

where P(A) denotes the probability of the event A.

Uniform search with uniform a priori distribution

As an important application, we consider the uniform search of Figure 39(a) and a uniform

a priori distribution for the location of the correct cell given by

πj = 1

q

, 1 ≤ j ≤ 1. (7-17)

If the cells in Figure 39(a) are labeled consecutively from left to right, then

ν(i, j)=(i − 1)(q − 1) + j − 1 (7-18)

If the rewinding time associated with each broken line is Tr, then

Tr(i, j) = Tr(i)=(i − 1)Tr (7-19)

If the uncertainty region covers an entire sequence period, then the cells at the two edges

are actually adjacent and Tr = 0.

To evaluate T ̄

a and T2

a , we substitute (7-13), (7-17), (7-18), and (7-19) into (7-12) and

(7-14) and use the following identities:

X∞

i=0

ri = 1

1 − r

, X∞

i=1

iri = r

(1 − r)2 , X∞

i=1

i

2

ri = r(1 + r)

(1 − r)3

Xn

i=1

i = n(n + 1)

2 , Xn

i=1

i

2 = n(n + 1)(2n + 1)

6 (7-20)

where 0 ≤ |r| < 1. Defining

α = (q − 1)T ̄

11 + T ̄12 + Tr (7-21)

69
we obtain

T ̄

a = (q − 1) μ2 − PD

2PD

¶

T ̄

11 +

μ1 − PD

PD

¶ ¡

T ̄

12 + Tr

¢

+ T ̄

22 (7-22)

and

T2

a = (q − 1) μ2 − PD

2PD

¶

var (T11) + μ1 − PD

PD

¶

var (T12) + var (T22)

+(2q + 1)(q + 1)

6

T ̄2

11 +

α2 (1 − PD) (2 − PD)

P2

D

+ (q + 1)α

μ1 − PD

PD

¶

+(q + 1)T ̄

11 ¡

T ̄

22 − T ̄11¢

+ 2α

μ1 − PD

PD

¶ ¡

T ̄

22 − T ̄11¢

+ ¡T ̄

22 − T ̄11¢2

(7-23)

In most applications, the number of cells to be searched is large, and simpler asymptotic

forms for the mean and variance of the acquisition time are applicable. As q → ∞, (7-22)

gives

T ̄

a → q

μ2 − PD

2PD

¶

T ̄

11, q → ∞ (7-24)

Similarly, (7-23) and (7-15) yield

σ2

a → q2

μ 1

P2

D

− 1

PD

+

1

12¶

T ̄2

11, q → ∞ (7-25)

These equations must be modified in the presence of an uncorrected Doppler shift. The

fractional change in the received chip rate of the spreading sequence is equal to the

fractional change in the carrier frequency due to the Doppler shift. If the chip rate changes

from 1/Tc to 1/Tc + δ, then the average change in the code or sequence phase during the

test of an incorrect cell is δT ̄

11. The change relative to the step size is δT ̄11/∆. The number

of cells that are actually tested in a sweep of the timing uncertainty region becomes

q(1 + δT ̄

11/∆)−1. Since incorrect cells predominate, the substitution of the latter quantity

in place of q in (7-24) and (7-25) gives approximate asymptotic expressions for T ̄

a and σ2

a

when the Doppler shift is significant.

Consecutive-count double-dwell system

For further specialization, consider the consecutive-count double-dwell system described by

Figure 37 with D = 2. Assume that the correct cell actually subsumes two consecutive cells

with detection probabilities Pa and Pb, respectively. If the test results are assumed to be

statistically independent, then

PD = Pa + (1 − Pa)Pb (7-26)

Let τ1, PF1, Pa1, and Pb1 denote the search-mode dwell time, false-alarm probability, and

successive detection probabilities, respectively. Let τ2, PF2, Pa2, and Pb2 denote the

70
verification-mode dwell time, false-alarm probability, and successive detection probabilities,

respectively. Let T ̄

p denote the mean penalty time, which is incurred by the incorrect

activation of the tracking mode. The flow graph indicates that since each cell must pass

two tests,

Pa = Pa1Pa2, Pb = Pb1Pb2 (7-27)

and

T ̄

11 = τ1 + PF1

¡

τ2 + PF2T ̄

p

¢ (7-28)

Equations (7-26) to (7-28) are all that is needed to evaluate the asymptotic values of the

mean and variance given by (7-24) and (7-25). For a more accurate evaluation of the mean

acquisition time, expressions for the conditional means T ̄

22 and T ̄12 are needed. Expressing

T ̄

22 as the conditional expectation of the correct-cell test duration given cell detection,

enumerating the possible durations and their conditional probabilities, and then

simplifying, we obtain

T ̄

22 = τ1 + τ2 + τ1

(1 − Pa) Pb

PD

+ τ2

Pa1 (1 − Pa2) Pb

PD

(7-29)

Similarly,

T ̄

12 = 2τ1 + τ2

∙

Pa1 (1 − Pa2) (1 − Pb) + (1 − Pa) Pb1 (1 − Pb2)

1 − PD

 ̧

(7-30)

Single-dwell and matched-filter systems

Results for a single-dwell system are obtained by setting

Pa2 = Pb2 = PF2 = 1, τ2 = 0, Pa = Pa1, Pb = Pb1, PF1 = PF , and τ1 = τd in (7-28) to (7-30).

We obtain

T ̄

11 = τd + PF T ̄p, T ̄22 = τd

∙

1 + (1 − Pa) Pb

PD

 ̧

, T ̄

12 = 2τd (7-31)

Thus, (7-22) yields

T ̄

a = (q − 1) (2 − PD)

¡

τd + PF T ̄

p

¢

+ 2τd (2 − Pa) + 2 (1 − PD) Tr

2PD

(7-32)

Since the single-dwell system may be regarded as a special case of the double-dwell system,

the latter can provide a better performance by appropriate setting of its additional

parameters.

The approximate mean acquisition time for a matched filter can be derived in a similar

manner. Suppose that many periods of a short spreading sequence with N chips per period

is received, and the matched-filter output is sampled m times per chip. Then the number

of cells that are tested is q = mN and Tr = 0. Each sampled output is compared to a

threshold so τd = Tc/m is the time duration associated with a test. For m = 1 or 2, it is

reasonable to regard two of the cells as the correct ones. These cells are effectively tested

71
when a signal period fills or nearly fills the matched filter. Thus, (7-26) is applicable with

Pa ≈ Pb, and (7-32) yields

T ̄

a ≈ NTc

μ2 − PD

2PD

¶

(1 + mKPF ), q >> 1 (7-33)

where K = T ̄

p/Tc. Ideally, the threshold is exceeded once per period, and the threshold

crossing provides the timing marker that initiates the tracking system.

Up-down double-dwell system

For the up-down double-dwell system with two correct cells, the flow graph of Figure 38

with D = 2 indicates that

Pa = Pa1Pa2

X∞

i=0

[Pa1 (1 − Pa2)]i = Pa1Pa2

1 − Pa1 (1 − Pa2) (7-34)

Similarly,

Pb = Pb1Pb2

1 − Pb1 (1 − Pb2) (7-35)

and PD is given by (7-26). If an incorrect cell passes the initial test but fails the verification

test, then the cell begins the testing sequence again without any memory of the previous

testing. Therefore, for an up-down double-dwell system, a recursive evaluation gives

T ̄

11 = (1 − PF1) τ1 + PF1PF2

¡

τ1 + τ2 + T ̄

p

¢

+ PF1 (1 − PF2)

¡

τ1 + τ2 + T ̄

11¢

= τ1 + PF1

¡

τ2 + PF2T ̄

p

¢

1 − PF1 (1 − PF2) (7-36)

Substitution of (7-34) to (7-36) into (7-24) to (7-26) gives the asymptotic values of the

mean and variance of the acquisition time.

From the possible durations and their conditional probabilities, we obtain

T ̄

22 = τ1 + τ2 + Pa1 (1 − Pa2) T ̄22 + (1 − Pa1) Pb1Pb2

PD

τ1

+(1 − Pa1) Pb1 (1 − Pb2) Pb

PD

¡

τ1 + T ̄0

22¢ (7-37)

where T ̄0

22 is the expected delay for the detection of the correct cell given that the testing

begins at the second correct cell. A recursive evaluation gives

T ̄0

22 = τ1 + τ2 + Pb1 (1 − Pb2) T ̄022

= τ1 + τ2

1 − Pb1 (1 − Pb2) (7-38)

72
Similarly, T ̄

12 is determined by the recursive equation

T ̄

12 = τ1 + (1 − Pa1) (1 − Pa2)

1 − PD

τ1 + Pa1 (1 − Pa2)

¡

τ2 + T ̄

12¢

+(1 − Pa1) Pb1 (1 − Pb2) (1 − Pb)

1 − PD

¡

τ1 + τ2 + T ̄0

12¢ (7-39)

with

T ̄0

12 = τ1 + Pb1 (1 − Pb2) τ2

1 − Pb1 (1 − Pb2) (7-40)

Penalty time

The lock detector that monitors the code synchronization in the lock mode performs tests

to verify the lock condition. The time that elapses before the system incorrectly leaves the

lock mode is called the holding time. It is desirable to have a large mean holding time and

a small mean penalty time, but the realization of one of these goals tends to impede the

realization of the other. As a simple example, suppose that each test has a fixed duration τ

and that code synchronization is actually maintained. A single missed detection, which

occurs with probability 1 − PDL, causes the lock detector to assume a loss of lock and to

initiate a search. Assuming the statistical independence of the lock-mode tests, the mean

holding time is

T ̄

h = X∞

i=1

iτ (1 − PDL) Pi−1

DL

= τ

1 − PDL

(7-41)

This result may also be derived by recognizing that T ̄

h = τ + PDLT ̄h because once the lock

mode is verified, the testing of the same cell is renewed without any memory of the

previous testing. If the locally generated code phase is incorrect, the penalty time expires

unless false alarms, each of which occurs with probability PF L, continue to occur every τ

seconds. A derivation similar to that of (7-41) yields the mean penalty time for a

single-dwell lock detector:

T ̄

p = τ

1 − PF L

(7-42)

A tradeoff between a high T ̄

h and a low T ̄p exists because increasing PDL tends to increase

PF L.

When a single test verifies the lock condition, the synchronization system is vulnerable to

deep fades and pulsed interference. A preferable strategy is for the lock mode to be

maintained until a number of consecutive or cumulative misses occur during a series of

tests. The performance analysis is analogous to that of serial-search acquisition.

73
Other search strategies

In a Z search, no cell is tested more than once until all cells in the timing uncertainty

region have been tested. Both strategies of Figure 39 are Z searches. A characteristic of

the Z search is that

ν(i, j)=(i − 1)(q − 1) + ν(1, j) (7-43)

where ν(1, j) is the number of incorrect cells tested when PD = 1 and, hence, L = 1. For

simplicity, we assume that q is even. For the broken-center Z search, the search begins with

cell q/2 + 1, and

ν(1, j) = 1⁄2 j − q

2 − 1, j ≥ q2 + 1

q − j, j ≤ q

2

(7-44)

whereas ν(1, j) = j − 1 for the uniform search. If the rewinding time is negligible, then

(7-12), (7-13), and (7-43) yield

T ̄

a = 1 − PD

PD

£

(q − 1)T ̄

11 + T ̄12¤

+ T ̄

22 + T ̄11ν(1) (7-45)

where ν(1) is the average number of incorrect cells tested when PD = 1:

ν(1) = X

q

j=1

ν(1, j)πj (7-46)

If C has a uniform distribution, then ν(1) and, hence, T ̄

a are the same for both strategies.

If the distribution of C is symmetrical about a pronounced central peak and PD ≈ 1, then

a uniform search gives ν(1) ≈ q/2. Since a broken-center Z search usually ends almost

immediately or after slightly more than q/2 tests,

ν(1) ≈ 0

μ1

2

¶

+ q

2

μ1

2

¶

= q

4 (7-47)

which indicates that for large values of q and PD close to unity, the broken-center Z search

reduces T ̄

a approximately by a factor of 2 relative to its value for the uniform search.

An expanding-window search attempts to exploit the information in the distribution of C

by continually retesting cells with high a priori probabilities of being the correct cell. Tests

are performed on all cells within a radius R1 from the center. If the correct cell is not

found, then tests are performed on all cells within an increased radius R2. The radius is

increased successively until the boundaries of the uncertainty region are reached. The

expanding-window search then becomes a Z search. If the rewinding time is negligible and

C is centrally peaked, then the broken-center search of Figure 40(a) is preferable to the

continuous-center search of Figure 40(b) because the latter retests cells before testing all

the cells near the center of the uncertainty region. In an equiexpanding search, the radii

have the form

Rn = nq

2N , n = 1, 2,...,N (7-48)

74
(a) (b)

Time Time

Uncertainty

Cell

1 2 q

Cell

1 2 q

Uncertainty

Figure 40. Trajectories of expanding-window search positions: (a) broken-center and (b)

continuous-center search.

where N is the number of sweeps before the search becomes a Z search. If the rewinding

time is negligible, then it can be shown [14] that the broken-center equiexpanding-window

search is optimized for PD ≤ 0.8 by choosing N = 2. For this optimized search, T ̄

a is

moderately reduced relative to its value for the broken-center Z search.

When Tr(i, j) = 0 and PD = 1, the optimal search, which is called a uniform alternating

search, tests the cells in order of decreasing a priori probability. For a symmetric,

unimodal, centrally peaked distribution of C, this optimal search has the trajectory

depicted in Figure 41(a). Once all the cells in the uncertainty region have been tested, the

search repeats the same pattern. Equations (7-43) and (7-45) are applicable. If PD ≈ 1 and

the distribution of C has a pronounced central peak, then ν(1) is small, and (7-47)

indicates that the uniform alternating search has an advantage over the broken-center

expanding-window search when q >> 4 and the rewinding time for any discontinuous

transition is much smaller than T ̄

11. However, computations show that this advantage

dissipates as PD decreases [14], which occurs because all cells are tested with the same

(a) (b)

Time Time

Uncertainty Uncertainty

Cell

1 2 q

Cell

1 2 q

Figure 41. Trajectories of alternating search positions: (a) uniform search and (b) nonuniform search.

75
frequency without accounting for the distribution of C. In the nonuniform alternating

search, illustrated in Figure 41(b), a uniform search is performed until a radius R1 is

reached. Then a second uniform search is performed within a larger radius R2. This

process continues until the boundaries of the time uncertainty region are reached and the

search becomes a uniform alternating search. Computations show that for a centrally

peaked distribution of C, the nonuniform alternating search can give a significant

improvement over the uniform alternating search if PD < 0.8, and the radii

Rn, n = 1, 2,..., are optimized [14]. However, if the radii are optimized for PD < 1, then

the nonuniform search becomes inferior to the uniform search as PD → 1.

Density function of the acquisition time

The density function of Ta is needed to accurately calculate P(Ta ≤ Tmax) and other

probabilities. However, the exact calculation of the density [15] is complicated because the

penalty time is, in general, a random variable that depends on the method of lock

detection. The density function of Ta may be decomposed as

fa(t) = PD

X∞

i=1

(1 − PD)

i−1X

q

j=1

πjfa(t|i, j) (7-49)

where fa(t|i, j) is the conditional density of Ta given that L = i and C = j. Let ∗ denote

the convolution operation, [f(t)]∗n denote the n-fold convolution of the density f(t) with

itself, [f(t)]∗0 = 1, and [f(t)]∗1 = f(t). Using this notation, we obtain

fa(t|i, j)=[f11(t)]∗ν(i,j) ∗ [f12(t)]∗(i−1) ∗ [f22(t)] (7-50)

where f11(f), f12(t), and f22(t) are the densities associated with T11, T12, and T22,

respectively. If one of the decision times is a constant, then the associated density is a

delta function.

Since the acquisition time conditioned on L = i and C = j is the sum of independent

random variables, it is reasonable to approximate fa(t|i, j) by a truncated Gaussian density

with mean

μij = ν(i, j)T ̄

11 + (i − 1)T ̄12 + T ̄22 + Tr(i) (7-51)

and variance

σ2

ij = ν(i, j)var (T11)+(i − 1)var (T12) + var (T22) (7-52)

The truncation is such that fa(t|i, j) 6= 0 only if 0 ≤ t ≤ Tmax or 0 ≤ t ≤ μij + 3σij . When

PD is large, the infinite series in (7-49) converges rapidly enough that the series can be

accurately approximated by its first few terms.

76
Alternative analysis

An alternative method of analyzing acquisition relies on transfer functions [16]. Each phase

offset of the local code defines a state of the system. Of the total number of q states, q − 1

are states that correspond to offsets (cells) that equal or exceed a chip duration. One state

is a collective state that corresponds to all phase offsets that are less than a chip duration

and, hence, cause acquisition to be terminated and code tracking to begin. The

serial-search acquisition process is represented by its circular state diagram, a segment of

which is illustrated in Figure 42. The a priori probability distribution πj , j = 1, 2,...,q,

gives the probability that the search begins in state j. The rewinding time is assumed to

be negligible.

q-2

Lock mode

q

H0(z)

H0(z) Hm(z)

H0(z)

HD(z)

H0(z) 3

2

q-1 1

Figure 42. Circular state diagram for serial-search acquisition.

The branch labels between two states are transfer functions that contain information about

the delays that may occur during the transition between the two states. Let z denote the

unit-delay variable and let the power of z denote the time delay. A single-dwell system

with dwell τ , false-alarm probability PF , and constant penalty time Tp has transfer

function H0(z) = (1 − PF ) zτ + PF zτ+Tp for all branches that do not originate in collective

state q because the transition delay is τ with probability 1 − PF and τ + Tp with

probability PF . For a multiple-dwell system, H0(z) is determined by first drawing a

subsidiary state diagram representing intermediate states and transitions that may occur

as the system progresses from one state to the next one in the original circular state

diagram. For example, Figure 43 illustrates the subsidiary state diagram for a

consecutive-count double-dwell system with false alarms PF1 and PF2 and delays τ1 and τ2

for the initial test and the verification test, respectively. Examination of all possible paths

between the initial state and the next state indicates that

H0(z) = (1 − PF1) zτ1 + PF1zτ1 £

(1 − PF2) zτ2 + PF2zτ2+TP

¤

= (1 − PF1) zτ1 + PF1 (1 − PF2) zτ1+τ2 + PF1PF2zτ1+τ2+Tp (7-53)

77
initial

state

τ2

False alarm

PF2z τ2

z

τ (1-PF2)z P 1 F1z

next

state

(1-PF1)z τ1

Tp

Figure 43. Subsidiary state diagram for determination of H0(z) for consecutive-count

double-dwell system.

Let HD(z) denote the transfer function between the collective state q and the lock mode.

Let HM(z) denote the transfer function between state q and state 1, which represents the

failure to recognize code-phase offsets that are less than a chip duration. These transfer

functions may be derived in the same manner as H0(z). For example, consider a

consecutive-count, double-dwell system with a collective state that comprises two states.

Figure 44 depicts the subsidiary state diagram representing intermediate states and

transitions that may occur as the system progresses from state q (with subsidiary states a

and b) to either the lock mode or state 1. Examination of all possible paths yields

HD(z) = Pa1Pa2zτ1+τ2 + Pa1 (1 − Pa2) Pb1Pb2z2τ1+2τ2

+ (1 − Pa1) Pb1Pb2z2τ1+τ2 (7-54)

HM(z) = (1 − Pa1) (1 − Pb1) z2τ1 + (1 − Pa1) Pb1 (1 − Pb2) z2τ1+τ2

+Pa1 (1 − Pa2) (1 − Pb1) z2τ1+τ2

+Pa1 (1 − Pa2) Pb1 (1 − Pb2) z2τ1+2τ2 (7-55)

For a single-dwell system with a collective state that comprises N states,

HD(z) = P1zτ +X

N

j=2

Pj

"

Y

j−1

i=1

(1 − Pj )

#

zjτ (7-56)

HM(z) = "

Y

N

j=1

(1 − Pj )

#

zNτ (7-57)

78
Lock mode

1

τ P 2 b2z

τ1 (1-Pb2)z

τ1

Pb1z τ1

(1-Pb1)z τ b 1

(1-Pa2)z

(1-Pa1)z

τ2

a

Pa1z

Pa2z τ2

τ2

Figure 44. Subsidiary state diagram for calculation of HD(z) and HM(z) for consecutivecount double-dwell system with two-state collective state.

H0(z) = (1 − PF ) zτ + PF zτ+Tp (7-58)

where τ is the dwell time, PF is the false-alarm probability, and Pj is the detection

probability of state j within the collective state. To calculate the statistics of the

acquisition time, we seek the generating function defined as the polynomial

H(z) = X∞

i=0

pi (τi) zτi (7-59)

where pi(τi) is the probability that the acquisition process will terminate in the lock mode

after τi seconds. If H(z) is known, then the mean acquisition time is

T ̄

a = X∞

i=0

τipi(τi) = dH(z)

dz

 ̄

 ̄

 ̄

 ̄

z=1

(7-60)

The second derivative of H(z) gives

d2H(z)

dz2

 ̄

 ̄

 ̄

 ̄

z=1

= X∞

i=0

τi (τi − 1) pi (τi) = T2

a − T ̄

a (7-61)

Therefore, the variance of the acquisition time is

σ2

a =

(

d2H(z)

dz2 +

dH(z)

dz −

∙

dH(z)

dz  ̧2

) ̄

 ̄

 ̄

 ̄

 ̄

z=1

(7-62)

To derive H(z), we observe that it may be expressed as

H(z) = X

q

j=1

πjHj (z) (7-63)

79
where Hj (z) is the transfer function from an initial state j to the lock mode. Since the

circular state diagram of Figure 42 may be traversed an indefinite number of times during

the acquisition process,

Hj (z) = Hq−j

0 (z)HD(z)

X∞

i=0

£

HM(z)Hq−1

0 (z)

¤i

= Hq−j

0 (z)HD(z)

1 − HM(z)Hq−1

0 (z) (7-64)

Substitution of this equation into (7-63) yields

H(z) = HD(z)

1 − HM(z)Hq−1

0 (z)

X

q

j=1

πjHq−j

0 (z) (7-65)

The generating function may be expressed as the polynomial in (7-59) by means of

polynomial long division.

For the uniform a priori distribution given by (7-17),

H(z) = HD(z) [1 − Hq

0 (z)]

q

£

1 − HM(z)Hq−1

0 (z)

¤

[1 − H0(z)] (7-66)

Since the progression from one state to another is inevitable until the lock mode is reached,

H0(1) = 1. Since HD(1) + HM(1) = 1, (7-65) and (7-60) yield

T ̄

a = 1

HD(1) 1⁄2

H0

D(1) + H0M(1) + (q − 1)H00(1) ∙

1 − HD(1)

2

 ̧3⁄4 (7-67)

where the prime indicates differentiation with respect to z. As an example, consider a

single-dwell system with a two-state collective state. The evaluation of (7-67) using (7-56)

to (7-58) with N = 2 yields (7-32) with Tr = 0 if we set P1 = Pa, P2 = Pb, Tp = T ̄

p, τ = τd,

and define PD by (7-26).

7.3 Acquisition Correlator

The noncoherent correlator of Figure 36 provides the approximate maximization of

V (τ ) = R(τ, 0) given by (7-9). It is assumed that chip synchronization is established by one

of the standard methods of symbol synchronization. Consequently, the test interval can be

defined with boundaries that coincide with chip boundaries, and we test code phases such

that τ = νTc, where ν is an integer. Let MTc denote the duration of the test interval,

where M is a positive integer. If the Doppler shift is not estimated, fd may be absorbed

into fc in (7-9). If the test interval begins with chip ν of the local spreading sequence, then

(3-2) and (7-9) imply that the decision variable for one test of a specific code phase νTc is

V = V 2

c + V 2s (7-68)

80
where

Vc =

M

X−1

k=0

pk−νxk, Vs =

M

X−1

k=0

pk−νyk (7-69)

xk =

Z (k+1)Tc

kTc

r(t)ψ (t − kTc) cos 2πfct dt (7-70)

yk =

Z (k+1)Tc

kTc

r(t)ψ (t − kTc) sin 2πfct dt (7-71)

The sequences {xk} and {yk} can be obtained by an in-phase and quadrature

downconversions followed by chip-matched filters sampled at times t = kTc. Thus, the

acquisition correlator has the form depicted in Figure 45. The decision variable V is

applied to a threshold detector to determine whether or not a test of a particular code

phase is passed. If a quaternary data modulation is used instead of PSK, then the only

modification necessary is that separate spreading sequence generators are assigned to the

two parallel branches of the correlator.

Sync device

π/2

Received

signals

cos 2πfct

sin 2πfct

CMF

CMF Sampler

Sampler

Chip-rate clock SSG

∑

∑

∑

M-1

M-1

k=0

k=0

+

Squarer

Squarer

To threshold

detector

+

yk

xk

Figure 45. Noncoherent correlator for acquisition system. CMF = chip matched filter.

SSG = spreading sequence generator.

The sequences {xk} and {yk} can be applied to multiple parallel inner products with

different values of ν simultaneously. This procedure allows a parallel search of various code

phases with a moderate amount of additional hardware or software. Since pk = ±1, each

inner product may be computed by either adding or subtracting each component of {xk} or

{yk}.

To analyze the performance of the acquisition correlator under fading conditions, we

assume that the received signal is

r(t) = √2Sαp(t − τ ) cos(2πfct + θ) + n(t) (7-72)

where α is the attenuation due to fading, S is the average power when α = 1, p(t) is the

spreading waveform, fc is the carrier frequency, θ is the random carrier phase, τ is the

delay due to the unknown code phase, and n(t) is the interference plus noise modeled as

81
additive white Gaussian noise. The data modulation d(t) is omitted because either it is not

transmitted during acquisition or the test duration MTc is much smaller than a symbol

duration Ts = GTc. In the latter case, the probability that a symbol transition occurs

during a test is negligible, and the squaring operations eliminate the symbol value from V .

Let ˆτ = νTc denote the delay associated with the code phase of the local spreading

sequence. The difference between ˆτ and τ may be expressed in the form ˆτ − τ = NTc + 2Tc,

where N is an integer and 0 <2< 1. For a rectangular chip waveform, (7-69), (7-70), and

(7-72), fcTc >> 1, and the definition of chip ν yield

Vc =

rS

2

αTc cos θ

M

X−1

k=0

pk−ν [(1 − 2) pk−ν+N + 2pk−ν+N+1] +

M

X−1

k=0

pk−νnk (7-73)

where

nk =

Z (k+1)Tc

kTc

n(t)ψ (t − kTc) cos 2πfct dt (7-74)

The alignment of the received and local spreading sequences is close enough for acquisition

if N = −1 or N = 0. If N 6= −1 and N 6= 0, then the cell may be considered incorrect. The

equation for Vs is the same as (7-74) except that − sin θ replaces cos θ, and nk is given by

(7-74) with sin 2πfct replacing cos 2πfct. The first term of Vc in (7-73) contributes

self-interference that may cause a false alarm. The self-interference is small if the

autocorrelation of the spreading sequence is sharply peaked. In a network of similar

systems, interfering sequences are substantially suppressed if the cross-correlations among

sequences are small, as they are if all the sequences are Gold or Kasami sequences (Section

9.2).

In the performance analysis, the spreading sequence {pk} is modeled as a random binary

sequence and 2 is modeled as a random variable. Thus, given the values of α and θ, the

self-interference varies with respect to its mean value and, hence, degrades acquisition even

when the noise term is negligible. If the variable part the self-interference is negligible,

then (7-73) can be approximated by

Vc = E[Vc] + Ngc (7-75)

where Ngc is the second term in (7-73). Since n(t) is zero-mean, white Gaussian noise, nk is

a zero-mean Gaussian random variable. Since pk−ν = ±1 and is independent of nk, the

product pk−νnk is zero-mean and Gaussian. The independence of the terms in the sum

then indicates that Ngc is a zero-mean Gaussian random variable. Similarly, we obtain the

approximation

Vs = E[Vs] + Ngs (7-76)

where Ngs is a zero-mean, Gaussian random variable. Straightforward calculations using

fcTc >> 1 indicate that Ngc and Ngs are statistically independent with the same variance:

σ2 = var (Ngc) = var (Ngs) = N0MTc

4 (7-77)

82
To determine the condition under which the self-interference is well-approximated by its

mean value, we calculate var(Vc). Given α, θ, and 2, (7-73) yields

var (Vc) = Sα2MT2

c cos2 θ

2

¡

22

2 − 22 + 1¢

+

N0MTc

4 (7-78)

where g(2) = 22 if N = 0, g(2) = (1 − 2)2 if N = −1, and g(2)=222 − 22 + 1 if N 6= 0, −1.

The first term is much smaller than the second term if Ec/N0 << 1, where Ec = Sα2Tc is

the energy per chip. This condition is satisfied with high probability in most practical

systems, especially if N0 incorporates the power spectral densities due to multiple-access

interference and multipath signals. Accordingly, we proceed with the analysis using the

approximations (7-75) and (7-76). Without these approximations, alternative

approximations and assumptions are necessary or the analysis becomes much more

complicated [17]. A common approximation is that 2 ≈ 0. If ˆτ − τ ≥ Tc, then a cell is

incorrect and (7-73) with N 6= −1, 0 implies that E[Vc] = E[Vs] = 0. If ˆτ − τ < Tc, the

values of E[Vc] and E[Vs] depend on the step size of the serial search, which is denoted by

∆. When ∆ = 1, two consecutive cells are considered correct. If ˆτ − τ is increasing, then

the cell corresponding to N = −1 occurs first and is followed by the cell corresponding to

N = 0. If 2 is assumed to be uniformly distributed over (0,1) and N1 = −1 or 0, then

(7-73) and the similar equation for Vs yield the conditional means given α and θ:

E [Vc] = rS

2

αMTc cos θ

2 , E [Vs] = −

rS

2

αMTc sin θ

2 , ∆ = 1 (7-79)

When ∆ = 1/2, the two consecutive cells with the smallest values of ˆτ − τ are considered

the two correct cells. For all the others, we assume that E[Vc] ≈ E[Vs] ≈ 0. The first

correct cell corresponds to N = −1 and 1/2 ≤ 2 < 1, whereas the second one corresponds

to N = 0 and 0 < 2 ≤ 1/2. If 2 is assumed to be uniformly distributed over the latter

intervals, then for both cells, we obtain

E [Vc] = rS

2

3αMTc cos θ

4 , E [Vs] = −

rS

2

3αMTc sin θ

4 , ∆ = 12 (7-80)

Let V1 denote the decision variable V when the correct cell is tested, and let V0 denote V

when the incorrect cell is tested. Equations (7-68), (7-75), and (7-76) and the preceding

analysis indicate that V0 is the sum of the squares of two independent, zero-mean Gaussian

random variables. The results of Appendix B then indicate that V0 has a central chi-square

distribution with two degrees of freedom and a probability density function

fc(x) = 1

2σ2 exp 3

− x

2σ2

 ́

u(x) (7-81)

where u(x)=1, x ≥ 0, and u(x)=0,x< 0, and σ2 = var(Ngc) = var(Ngs). The

false-alarm probability for a test of an incorrect cell is the probability that V0 > Vt, where

83
Vt is the threshold. The integration of (7-81) gives the false-alarm probability:

Pf = exp μ

− Vt

2σ2

¶

(7-82)

Similarly, given α and θ, V1 is the sum of the squares of two independent Gaussian random

variables with nonzero means. The results of Appendix B then indicate that V1 has a

noncentral chi-square distribution with two degrees of freedom and a probability density

function

f1(x) = 1

2σ2 exp μ

−λ + x

2σ2

¶

I0

Ã√

λx

σ2

!

u(x) (7-83)

where

λ = (E [Vc])2 + (E [Vs])2 = 9

32fSα2

M2T2

c (7-84)

and

f =

1⁄2 1, ∆ = 1/2

4/9, ∆ = 1 (7-85)

The detection probability for a test of a correct cell is the probability that V1 > Vt. The

integration of (7-83) and the substitution of (7-84) gives the detection probability

Pd = Q1

μpξα,

√Vt

σ

¶

(7-86)

where Q1( ) is the generalized Q-function defined by (B-15),

ξ = 9

8

fM Ec

N0

(7-87)

and Ec = STc is the signal energy per chip when fading is absent and α = 1.

Combining (7-86) and (7-82) yields

Pd = Q1

3pξα,

p−2 ln Pf

 ́

(7-88)

Thus, if Pf is specified, Pd is given by (7-88). The threshold needed to realize a specified

Pf is

Vt = −N0MTc

2

ln Pf (7-89)

which requires an accurate estimate of N0.

In the presence of fast Rayleigh fading, α has the Rayleigh probability density (Appendix

B-4):

fα(x)=2x exp(−x2)u(x) (7-90)

84
where E[α2] = 1 so that S remains the average signal power in (7-72). It is assumed that α

is approximately constant during a test, but independent from test-to-test. Since (7-86) is

conditioned on α, the detection probability in the presence of fast fading is

Pd =

Z ∞

0

2x exp ¡−x2¢

Q1

3pξx,p−2 ln Pf ́

dx (7-91)

To evaluate this integral, we substitute the integral definition of Q1( ) given by (B-15),

interchange the order of integration in the resulting double integral, and then use (B-33) to

evaluate one of the integrals. The remaining integration over an exponential function is

elementary. The final result is

Pd = P2/(2+ξ)

f (7-92)

For slow Rayleigh fading with a coherence time much larger than the acquisition time, it is

appropriate to use (7-86) in calculating the conditional mean acquisition time and then

integrate over the Rayleigh density to obtain the mean acquisition time.

Let C denote the number of chips in the time uncertainty region. The normalized mean

acquisition times (NMAT) is defined as T ̄

a/CTc. The normalized standard deviation (NSD)

is defined as σa/CTc.

Example 1

As an example of the application of the preceding results, consider a single-dwell system

with a uniform search and a uniform a priori correct-cell location distribution. Let

τd = MTc, where M is the number of chips per test, and T ̄

p = KTc, where K is the number

of chips in the mean penalty time. It is assumed that there are two independent correct

cells with the common detection probability Pd = Pa = Pb. If q >> 1, (7-32) and (7-26)

yield the NMAT:

T ̄

a

CTc

=

μ2 − PD

2PD

¶ q

C (M + KPF ) (7-93)

where

PD = 2Pd − P2

d (7-94)

In a single-dwell system, PF = Pf , which is given by (7-82). For step size ∆ = 1, q/C = 1;

for ∆ = 1/2, q/C = 2. In the absence of fading, (7-88) relates Pd and Pf , whereas (7-92)

relates them in the presence of fast Rayleigh fading

Figure 46 shows the NMAT as a function of Ec/N0 for fast Rayleigh fading and no fading.

At each value of Ec/N0, the values of Pf and M are selected to minimize the NMAT. The

figure indicates the advantage of ∆ = 1/2 when K ≥ 105 and the advantage of ∆ = 1 when

K ≤ 104. The large increase in the NMAT due to fast Rayleigh fading is apparent. From

(7-25), it is found that each plot of the NSD is similar to that of the corresponding NMAT.

85
500

450

400

350

300

250

200

150

100

50

0

–20 –18 –16 –14 –12 –10 –8 –6 –4

K = 104

no fading

K = 105

fast fading

K = 104

fast fading

Normalized mean acquisition time

Chip-energy-to-noise-density ratio, dB

∆ = 1/2

∆ = 1

Figure 46. NMAT versus Ec/N0 for single-dwell system in presence of fast Rayleigh fading

or no fading. Values of Pf and M are optimized.

Example 2

Consider double-dwell systems with a uniform search, a uniform a priori correct-cell

location distribution, and two independent correct cells with Pd = Pa = Pb, Pa1 = Pb1, and

Pa2 = Pb2. The test durations are τ1 = M1Tc and τ2 = M2τc. If q >> 1, the NMAT is

obtained from (7-24) and (7-94), where T ̄

11 is given by (7-28) for a consecutive-count

system and (7-36) for an up-down system. By replacing Pd with Pai and Pf with PF i, the

probabilities Pai and PF i, i = 1 or 2, are related through (7-88) with α = 1 for no fading

and (7-92) for fast Rayleigh fading.

Figure 47 shows the NMAT as a function of Ec/N0 for double-dwell systems in the presence

of fast Rayleigh fading. The step size is ∆ = 1/2, which is found to be advantageous for

the parameter values chosen. At each value of Ec/N0, the values of PF1, PF2, M1, and M2

are selected to minimize the NMAT. The figure illustrates the advantage of the up-down

system in most practical applications. From (7-25), it is found that each plot of the NSD is

similar to that of the corresponding NMAT. A comparison of Figure 47 with Figure 46

indicates that double-dwell systems are capable of significantly lowering the NMAT relative

to a single-dwell system. ¤

The existence of two consecutive correct cells can be directly exploited in joint two-cell

detection, which can be shown to provide a lower NMAT than the conventional cell-by-cell

86
–20 –18 –16 –14 –12 –10 –8 –6 –4

500

450

400

350

300

250

200

150

100

50

0

Chip-energy-to-noise-density ratio, dB

Normalized mean acquisition time

K = 104

Consecutive-count

Up-down

K = 106

Figure 47. NMAT versus Ec/N0 for double-dwell systems in presence of fast Rayleigh

fading. Step size is ∆ = 1/2. Values of PF1, PF2, M1, and M2 are optimized.

detection [18]. In the presence of frequency-selective fading with a large number of

resolvable multipath signals, the NMAT of serial-search acquisition is usually increased

because the increased self-interference is more significant than the higher number of

nonconsecutive correct cells with correct phases. However, joint two-cell detection is more

resistant to multiple-access interference and more robust against variations in the detection

threshold, the power level of the desired signal, and the number of multipath signals. The

advantage of joint two-cell detection over cell-by-cell detection is the result of the efficient

combining of the energy of two adjacent correct-phase samples. The detection threshold of

(7-89) depends on an estimate of N0, the equivalent noise-power spectral density. An

accurate estimate usually requires a long observation interval. However, in mobile

communication systems and in the presence of jamming, the instantaneous interference

power may be rapidly varying. To cope with this environment, an adaptive threshold may

be set by the instantaneous received power [19]. As a result, the mean acquisition time is

lowered relative to its value for nonadaptive schemes when Rayleigh fading or pulsed

Gaussian noise jamming is present. When a rake receiver (Section 7.8 ) is used, each finger

of the receiver must acquire the timing of a separate multipath signal. Whether matched

filtering or a serial search is used, some mechanism is needed to ensure that each finger

acquires a distinct multipath signal [20].

An alternative to acquisition tests of fixed dwell time or number of detector samples is

sequential detection, which uses only the number necessary for a reliable decision. Thus,

87
some sample sequences may allow a quick decision, while others may warrant using a large

number of samples in the evaluation of a single phase of the spreading waveform. The

sequential probability ratio test [2], [3] entails the recalculation of the likelihood ratio after

each new detector sample is produced. This ratio is compared with both upper and lower

thresholds to determine if the test is terminated and no more samples need to be extracted.

If the upper threshold is exceeded, the receiver declares acquisition and the lock mode is

entered. If the likelihood ratio drops below the lower threshold, the test fails, and another

code phase is tested. As long as the likelihood ratio lies between the two thresholds, a

decision is postponed and the ratio continues to be updated. Although the sequential

detector is capable of significantly reducing the mean acquisition time relative to detectors

that use a fixed number of detector samples, it has a number of practical limitations. Chief

among them is the computational complexity of the calculation of the likelihood ratio or

log-likelihood ratio.

7.4 Code Tracking

Coherent code-tracking loops operate at baseband following the coherent removal of the

carrier of the received signal. An impediment to their use is that the input signal-to-noise

ratio is usually too low for carrier synchronization prior to code synchronization and the

subsequent despreading of the received signal. Furthermore, coherent loops cannot easily

accommodate the effects of data modulation. Noncoherent loops operate directly on the

received signal and are unaffected by the data modulation.

To motivate the design of the noncoherent loop, one may adapt the statistic (7-9). If the

maximum-likelihood estimate ˆτ is assumed to be within the interior of its uncertainty

region and R(τ, fd) is a differentiable function of τ , then the estimate ˆτ that maximizes

R(τ, fd) may be found by setting

∂R (τ, fd)

∂τ

 ̄

 ̄

 ̄

 ̄

τ=ˆτ

= 0 (7-95)

A major problem with this approach is that R(τ, fd) given by (7-9) is not differentiable if

the chip waveform is rectangular. This problem is circumvented by using a difference

equation as an approximation of the derivative. Thus, for a positive δTc, we set

∂R (τ, fd)

∂τ

≈

R (τ + δTc, fd) − R (τ − δTc, fd)

2δTc

(7-96)

This equation implies that the solution of (7-95) may be approximately obtained by a

device that finds the ˆτ such that

R (ˆτ + δTc, fd) − R (ˆτ − δTc, fd) = 0 (7-97)

88
To derive an alternative to this equation, we assume that no noise is present, fd = 0, and

that the correct timing offset of the received signal is τ = 0. Substituting (7-2) with τ = 0

into (7-9) and using trigonometry, we obtain

R (ˆτ , 0) = S

2

∙Z T

0

p(t)p(t − τˆ)dt ̧2(7-98)

If p(t) is modeled as the spreading waveform for a random binary sequence and the interval

[0, T] includes many chips, then the integral is reasonably approximated by its expected

value, which is proportional to the autocorrelation

Rp(τ ) = Λ

μ τ

Tc

¶

(7-99)

Substituting this result into (7-97), we find that the maximum-likelihood estimate is

approximately obtained by a device that finds the ˆτ such that

R2

p (ˆτ + δTc) − R2p (ˆτ − δTc) = 0 (7-100)

The noncoherent delay-locked loop [21], which is diagrammed in Figure 48, implements an

approximate computation of the difference on the left-hand side of (7-100) and then

continually adjusts ˆτ so that this difference remains near zero. The estimate is used to

produce the synchronized local spreading sequence that is used for despreading the received

direct-sequence signal. The code generator produces three sequences, one of which is the

reference sequence used for acquisition and demodulation. The other two sequences are

advanced and delayed, respectively, by δTc relative to the reference sequence. The product

δTc is usually equal to the acquisition step size, and thus usually δ = 1/2, but other values

are plausible. The advanced and delayed sequences are multiplied by the received

direct-sequence signal in separate branches.

Reference

Error

Square-law

device

Bandpass

filter

Square-law

device

Bandpass

filter

Input

Delayed Advanced

Loop

filter

Voltagecontrolled

clock

Spreading

sequence

generator

+

Σ –

Figure 48. Delay-locked loop.

89
For the received direct-sequence signal (7-2), the signal portion of the upper-branch mixer

output is

su1(t) = Ad(t)p(t)p (t + δTc − 2Tc) cos (2πfct + θ) (7-101)

where A = √

2S and 2Tc is the delay of the reference sequence relative to the received

sequence. Although 2 is a function of time because of the loop dynamics, the time

dependence is suppressed for notational convenience. Since each bandpass filter has a

bandwidth on the order of 1/Ts, where Ts is the duration of each symbol, d(t) is not

significantly distorted by the filtering. Nearly all spectral components except the slowly

varying expected value of p(t)p(t + δTc − 2Tc) are blocked by the upper-branch bandpass

filter. Since this expected value is the autocorrelation of the spreading sequence, the filter

output is

su2(t) ≈ Ad(t)Rp (δTc − 2Tc) cos (2πfct + θ) (7-102)

Any double-frequency component produced by the square-law device is ultimately

suppressed by the loop filter and thus is ignored. Since d2(t) = 1, the data modulation is

removed, and the upper-branch output is

su3(t) ≈

A2

2 R2

p (δTc − 2Tc) (7-103)

Similarly, the output of the lower branch is

sl3(t) ≈

A2

2

R2

p (−δTc − 2Tc) (7-104)

The difference between the outputs of the two branches is the error signal:

se(t) ≈

A2

2

£

R2

p (δTc − 2Tc) − R2p (−δTc − 2Tc)

¤ (7-105)

Since Rp(τ ) is an even function, the error signal is proportional to the left-hand side of

(7-100).

The substitution of (7-99) and (2-8) into (7-105) yields

se(t) ≈

A2

2

S(2, δ) (7-106)

where S(2, δ) is the discriminator characteristic or S-curve of the tracking loop. For

0 ≤ δ ≤ 1/2,

S(2, δ) =







42(1 − δ), 0 ≤ 2 ≤ δ

4δ(1 − 2), δ ≤ 2 ≤ 1 − δ

1+(2 − δ)(2 − δ − 2), 1 − δ ≤ 2 ≤ 1 + δ

0, 1 + δ ≤ 2

(7-107)

90
For 1/2 ≤ δ ≤ 1,

S(2, δ) =







42(1 − δ), 0 ≤ 2 ≤ 1 − δ

1+(2 − δ)(2 − δ + 2), 1 − δ ≤ 2 ≤ δ

1+(2 − δ)(2 − δ − 2), δ ≤ 2 ≤ 1 + δ

0, 1 + δ ≤ 2

(7-108)

In both cases,

S(−2, δ) = −S(2, δ) (7-109)

Figure 49 illustrates the discriminator characteristic for δ = 1/2. The filtered error signal is

applied to the voltage-controlled clock. Changes in the clock frequency cause the reference

sequence to converge toward alignment with the received spreading sequence. When

0 < 2(t) < 1 + δ, the reference sequence is delayed relative to the received sequence. As

shown in Figure 49, S(2, δ) is positive, so the clock rate is increased, and 2(t) decreases.

The figure indicates that se(t) → 0 as 2(t) → 0. Similarly, when 2(t) < 0, we find that

se(t) → 0 as 2(t) → 0. Thus, the delay-locked loop tracks the received code timing once the

acquisition system has finished the coarse alignment.

-1.5 0.5 1.5

ε

S(ε, d)

1

-1

Figure 49. Discriminator characteristic of delay-locked loop for δ = 1/2.

The discriminator characteristic of code-tracking loops differs from that of phase-locked

loops in that it is nonzero only within a finite range of 2. Outside that range, code tracking

cannot be sustained, the synchronization system loses lock, and a reacquisition search is

initiated by the lock detector. Tracking resumes once the acquisition system reduces 2 to

within the range for which the discriminator characteristic is nonzero.

When short spreading sequences are used in a synchronous direct-sequence network, the

reduced randomness in the multiple-access interference (Section 10) may cause increased

91
tracking jitter or even an offset in the discriminator characteristic [22]. For orthogonal

sequences, the interference is zero when synchronization exists, but becomes large when

there is a code-phase error in the local spreading sequence. In the presence of a tracking

error, the delay-locked-loop arm with the larger offset relative to the correct code phase

receives relatively more noise power than the other arm. This disparity reduces the slope of

the discriminator characteristic and, hence, degrades the tracking performance. Moreover,

because of the nonsymmetric character of the crosscorrelations among the spreading

sequences, the discriminator characteristic may be biased in one direction, which will cause

a tracking offset.

The main components of a noncoherent tau-dither loop are depicted in Figure 50. The

dither generator controls a switch that alternately passes an advanced or delayed local

sequence. Let D(t) denote the dither signal, a square wave that alternates between +1 and

−1. In the absence of noise, the output of the switch can be represented by

s1(t) = ∙

1 + D(t)

2

 ̧

p (t + δTc − 2Tc) + ∙

1 − D(t)

2

 ̧

p (t − δTc − 2Tc) (7-110)

where the two factors within brackets are orthogonal functions of time and alternate

between +1 and 0. Only one of the factors is nonzero at any instant. The received

direct-sequence signal is multiplied by s1(t), filtered, and then applied to a square-law

device. If the bandpass filter has a sufficiently narrow bandwidth, then a derivation similar

to that of (7-103) indicates that the device output is

s2(t) ≈

A2

2

∙

1 + D(t)

2

 ̧

R2

p (δTc − 2Tc) + A2

2

∙

1 − D(t)

2

 ̧

R2

p (−δTc − 2Tc) (7-111)

Spreading

sequence

generator

D(t) D(t)

Input

Advanced Delayed

Reference

Switch

Square-law

device

Bandpass

filter

Dither

generator

Loop

filter

Voltagecontrolled

clock

Figure 50. Tau-dither loop.

92
Since D(t)[1 + D(t)] = 1 + D(t) and D(t)[1 − D(t)] = −[1 − D(t)], the input to the loop

filter is

s3(t) ≈

A2

2

∙

1 + D(t)

2

 ̧

R2

p (δTc − 2Tc) − A2

2

∙

1 − D(t)

2

 ̧

R2

p (−δTc − 2Tc) (7-112)

which is a rectangular wave if the time variation of 2 is ignored. Since the loop filter has a

narrow bandwidth relative to that of D(t), its output is approximately the direct-current

component of s3(t), which is the average value of s3(t). Averaging the two terms of (7-112),

we obtain the filter output:

s4(t) ≈

A2

4

£

R2

p (δTc − 2Tc) − R2p (δTc − 2Tc)

¤ (7-113)

The substitution of (7-99) yields the clock input:

s4(t) = A2

4

S(2, δ) (7-114)

where the discriminator characteristic is given by (7-107) to (7-109). Thus, the tau-dither

loop can track the code timing in a manner similar to that of the delay-locked loop. The

tau-dither loop requires less hardware than the delay-locked loop and eliminates the need

to balance the gains and delays in the two branches of the delay-locked loop. However, a

detailed analysis indicates that the tau-dither loop provides less accurate code tracking [2],

[3].

In the presence of frequency-selective fading, the discriminator characteristics of tracking

loops are severely distorted. Much better performance is potentially available from a noncoherent tracking loop with diversity and multipath-interference cancellation [23], but a large

increase in implementation complexity is required.

8. Rejection of Narrowband Interference

Narrowband interference presents a crucial problem for spread-spectrum overlay systems,

which are systems that have been assigned a spectral band already occupied by

narrowband communication systems. Jamming against tactical spread-spectrum

communications is another instance of narrowband interference that may exceed the

natural resistance of a practical spread-spectrum system, which has a limited processing

gain. There are a wide variety of techniques that supplement the inherent ability of a

direct-sequence system to reject narrowband interference [24], [25]. All of the techniques

directly or indirectly exploit the spectral disparity between the narrowband interference

and the wideband direct-sequence signal. The most useful methods can be classified as

93
time-domain adaptive filtering, transform-domain processing, nonlinear filtering, or

code-aided techniques. The general form of a receiver that rejects narrowband interference

and demodulates a direct-sequence signal with binary PSK is shown in Figure 51. The

processor, which follows the chip-rate sampling of a baseband signal.

Input

pi

Output

Decision symbols

device

Sequence

generator

Despreader

Processor

Synchron.

device

∫ ∑

Tc

0

G–1

i=0

Figure 51. Direct-sequence receiver with processor for rejecting narrowband interference.

8.1 Time-Domain Adaptive Filtering

A time-domain adaptive filter [26] for interference suppression processes the baseband

sample values of a received signal to adaptively estimate the interference. This estimate is

subtracted from the sample values, thereby canceling the interference. The adaptive filter

is primarily a predictive system that exploits the inherent predictability of a narrowband

signal to form an accurate replica of it for the subtraction. Since the wideband desired

signal is largely unpredictable, it does not significantly impede the prediction of a

narrowband signal. When adaptive filtering is used, the processor in Figure 51 has the form

of Figure 52(a). The adaptive filter may be a one-sided or two-sided transversal filter.

The two-sided adaptive transversal filter multiplies each tap output by a weight except for

the central tap output, as diagrammed in Figure 52(b). This filter is an interpolator in that

it uses both past and future samples to estimate the value to be subtracted. The two-sided

filter provides a better performance than the one-sided filter, which is a predictor. The

adaptive algorithm of the weight-control mechanism is designed to adjust the weights so

that the power in the filter output is minimized. The direct-sequence components of the

tap outputs, which are delayed by integer multiples of a chip duration, are largely

uncorrelated with each other, but the narrowband interference components are strongly

correlated. As a result, the adaptive algorithm causes the interference cancelation in the

filter output, but the direct-sequence signal is largely unaffected.

An adaptive filter with 2N + 1 taps and 2N weights, as shown in Figure 52(b), has input

vector at iteration k given by

x(k)=[x1(k) x2(k)...x2N (k)]T (8-1)

94
Input

samples

(a)

To despreader

Weight

control

Adaptive

filter

Input

Output

To weight control

(b)

–

–

–

–

+

W–N W–1 W1 WN

X2N

Tc Tc Tc

d XN+1 Tc

XN X1

Σ

Figure 52. (a) Processor using adaptive filter and (b) two-sided adaptive transversal filter.

and weight vector

W(k)=[W−N (k) W−N+1(k)...W−1(k) W1(k)...WN (k)]T (8-2)

where T denotes the transpose and the central tap output, which is denoted by d, has been

excluded from x. Since coherent demodulation produces real-valued inputs to the adaptive

filter, x(k) and W(k) are assumed to have real-valued components. The symmetric

correlation matrix of x is defined as Rxx = E[xxT ]. The cross-correlation vector is defined

as Rxd = E[xd]. According to the Wiener-Hopf equation (Appendix D), the optimal weight

vector is

W0 = R−1

xx Rxd (8-3)

The least-mean-square (LMS) algorithm (Appendix D) computes the weight vector at

iteration k as

W(k) = W(k − 1) + μ2kx(k) (8-4)

where 2k = d − yk is the estimation error, yk = WT (k)x(k) is the filter output, and μ is the

adaptation constant, which controls the rate of convergence of the algorithm. The output

of the adaptive filter is 2k, which is applied to the despreader. Under certain conditions,

the mean weight vector converges to W0 after a number of iterations of the adaptive

algorithm. If it is assumed that W = W0, then a straightforward analysis indicates that

the adaptive transversal filter provides a substantial suppression of narrowband interference

[24]. Although the interference suppression increases with the number of taps, it is always

95
incomplete if the interference has a nonzero bandwidth because a finite-impulse-response

filter can only place a finite number of zeros in the frequency domain.

The adaptive transversal filter is inhibited by the presence of direct-sequence components

in the filter input vector x(k). These components can be suppressed by using

decision-directed feedback, as shown in Figure 53. Previously detected symbols remodulate

the spreading sequence delayed by G chips (long sequence) or one period of the spreading

sequence(short sequence). After an amplitude compensation by a factor η, the resulting

sequence provides estimates of the direct-sequence components of previous input samples.

A subtraction then provides estimated sample values of the interference plus noise that are

largely free of direct-sequence contamination. These samples are then applied to an

adaptive transversal filter that has the form of Figure 52 except that it has no central tap.

The transversal filter output consists of refined interference estimates that are subtracted

from the input samples to produce samples that have relatively small interference

components. An erroneous symbol from the decision device causes an enhanced

direct-sequence component in samples applied to the transversal filter, and error

propagation is possible. However, for moderate values of the signal-to-interference ratio at

the input, the performance is not degraded significantly.

Adaptive filtering is only effective after the convergence of the adaptive algorithm, which

may not be able to track time-varying interference. In contrast, transform-domain

processing suppresses interference almost instantaneously.

Input

samples

Symbols from

decision device

Adaptive

filter

Refined

interference

Bit estimate

delay

To despreader

∑

∑

+

–

–

+

ηpi–G

Figure 53. Processor with decision-directed adaptive filter.

96
8.2 Transform-Domain Processing

The input of a transform-domain processor could be a continuous-time received signal that

feeds a real-time Fourier transformer implemented as a chirp transform processor [1]. In a

more versatile implementation, which is depicted in Figure 54 and assumed henceforth, the

input consists of the output samples of a chip-matched filter. Blocks of these samples feed

a discrete-time Fourier or wavelet transformer. The transform is selected so that the

transform-domain forms of the desired signal and interference are easily distinguished.

Ideally, the transform produces interference components that are confined to a few

transform bins while the desired-signal components have nearly the same magnitude in all

the transform bins. A simple exciser can then suppress the interference with little impact

on the desired signal by setting to zero the components in bins containing the interference.

The decision as to which bins contain interference can be based on the comparison of each

component to a threshold. After the excision operation, the desired signal is largely

restored by the inverse transformer.

Input

samples

To despreader

Inverse

transformer Transformer Exciser

Figure 54. Transform-domain processor.

Much better performance against stationary narrowband interference may be obtained by

using a transform-domain adaptive filter as the exciser [27]. This filter adjusts a single

nonbinary weight at each transform-bin output. The adaptive algorithm is designed to

minimize the difference between the weighted transform and a desired signal that is the

transform of the spreading sequence used by the input block of the processor. If the

direct-sequence signal uses the same short spreading sequence for each data symbol and

each processor input block includes the chips for a single data symbol, then the

desired-signal transform may be stored in a read-only memory. However, if a long spreading

sequence is used, then the desired-signal transform must be continuously produced from

the output of the receiver’s code generator. The main disadvantage of the adaptive filter is

that its convergence rate may be insufficient to track rapidly time-varying interference.

A transform that operates on disjoint blocks of N input samples may be defined in terms

of N orthonormal, N-component basis vectors:

φi = [φi1 φi2 ... φiN ]

T , i = 1, 2,...,N (8-5)

97
which span a linear vector space of dimension N. Since the components may be complex

numbers, the orthonormality implies that

φH

i φk =

1⁄2

0, i 6= k

1, i = k (8-6)

where H denotes the complex conjugate of the transpose. The input block

x = [x1 x2 ...xN ]

T (8-7)

may be expressed in terms of the basis as

x = X

N

i=1

ciφi (8-8)

where

ci = φH

i x, i = 1, 2,...N (8-9)

If the discrete Fourier transform is used, then φik = exp(j2πik/N), where j = √−1.

The transformer extracts the vector

c = [c1 c2 ... cN ]

T (8-10)

by computing

c = BHx (8-11)

where B is the unitary matrix of basis vectors:

B = [φ1 φ2 ... φN ] (8-12)

The exciser weights each component of the transform c by computing

e = Wdc (8-13)

where Wd is the N × N diagonal weight matrix with diagonal elements W1, W2,...,WN .

The inverse transformer then produces the excised block that is applied to the despreader:

z = [z1 z2 ...zN ]

T = B e = B Wdc = B WdBHx (8-14)

If there were no weighting, then Wd = I. Since BBH = I, z = x would result, as expected

when the transformer and inverse transformer are in tandem. In general, the diagonal

elements of Wd are either set by a threshold device fed by c or they are the outputs of the

weight-control mechanism of an adaptive filter. When N equals the processing gain G and

the input comprises the unmodulated spreading sequence, the despreader correlates its

98
input block with the appropriate segment of the spreading sequence to form the decision

variable:

V = X

G

i=1

pizi (8-15)

The filtering and despreading can be simultaneously performed in the transform domain.

Let

p = [p1 p2 ... pG]

T (8-16)

denote a synchronous replica of the spreading sequence, which is generated by the receiver

code generator. Then (8-14) to (8-16) give

V = pT z = pTBWdc (8-17)

Thus, if the spreading sequence is used to produce the matrix pTBWd, then the product of

this matrix and the transform c gives V without the need for an inverse transformer and a

separate despreader.

8.3 Nonlinear Filtering

By modeling the narrowband interference as part of a dynamical linear system, one can use

the Kalman-Bucy filter [26] to extract an optimal linear estimate of the interference. A

subtraction of this estimate from the filter input then removes a large part of the

interference from the despreader input. However, a superior nonlinear filter can be

designed by approximating an extension of the Kalman-Bucy filter.

Consider the estimation of an n × 1 state vector xk of a dynamical system based on the

r × 1 observation vector zk. Let φk denote the n × n state transition matrix, Hk an r × n

observation matrix, and uk and vk disturbance vectors of dimensions n × 1 and r × 1,

respectively. According to the linear dynamical system model, the state and observation

vectors satisfy

xk+1 = φkxk + uk, 0 ≤ k < ∞ (8-18)

zk = Hkxk + vk, 0 ≤ k < ∞ (8-19)

It is assumed that the sequences {uk} , {vk} are independent sequences of independent,

zero-mean random vectors that are also independent of the initial state x0. The covariance

of uk is E £

ukuT

k

¤

= Qk. Let Zk = (z1, z2, ... zk) denote the first k observation vectors.

Let f(zk|Zk−1) and f(xk|Zk−1) denote the probability density functions of zk and xk,

respectively, conditioned on Zk−1. A fundamental result of estimation theory is that the

estimate xˆk that minimizes the mean-norm-squared error E £

kxk − xˆkk2¤is the expectation

conditioned on Zk:

xˆk = E £xk

 ̄

 ̄Zk ¤ (8-20)

99
The corresponding conditional covariance is denoted by

Pk = E

h

(xk − xˆk) (xk − xˆk)

T  ̄

 ̄Zk

i

(8-21)

From (8-18), it follows that the expectation of xk conditioned on Zk−1 is

x ̄k = E £xk

 ̄

 ̄Zk−1 ¤

= φk−1xˆk−1 (8-22)

The covariance of xk conditioned on Zk−1 is defined as

Mk = E

h

(xk − x ̄k) (xk − x ̄k)

T  ̄

 ̄Zk−1

i

(8-23)

The following theorem due to Masreliez [28] extends the Kalman-Bucy filter.

Theorem. Assume that f(xk|Zk−1) is a Gaussian density with mean x ̄k and n × n

covariance matrix Mk, and that f(zk|Zk−1) is twice differentiable with respect to the

components of zk. Then the conditional expectation xˆk and the conditional covariance Pk

satisfy

xˆk = x ̄k + MkHT

k gk (zk) (8-24)

Pk = Mk − MkHT

k Gk (zk) HkMk (8-25)

Mk+1 = φkPkφT

k + Qk (8-26)

x ̄k+1 = φkxˆk (8-27)

where gk(zk) is an r × 1 vector with components

{gk (zk)}i = − 1

f (zk |Zk−1 )

∂f

¡

zk

 ̄

 ̄Zk−1 ¢

∂zki

(8-28)

Gk(zk) is an r × r matrix with elements

{Gk (zk)}ij = ∂ {gk (zk)}i

∂zkj

(8-29)

and zkj is the jth component of zk.

Proof. When xk is given, (8-19) indicates that zk is independent of Zk−1. Therefore,

Bayes’ rule gives

f

¡

xk

 ̄

 ̄Zk ¢

= f

¡

xk

 ̄

 ̄Zk−1 ¢

f (zk |xk )

f (zk |Zk−1 ) (8-30)

With the concise notation b = [f(zk|Zk−1)]−1, (8-20) and the fact that a density is a scalar

function yield

xˆk − x ̄k = b

Z

Rn

(xk − x ̄k) f (zk |xk ) f

¡

xk

 ̄

 ̄Zk−1 ¢

dxk

= bMk

Z

Rn

f (zk |xk )M−1

k (xk − x ̄k) f

¡

xk

 ̄

 ̄Zk−1 ¢

dxk

100
Using the Gaussian density f(xk|Zk−1), (8-22), and (8-23), and then integrating by parts,

we obtain

xˆk − x ̄k = −bMk

Z

Rn

f (zk |xk ) ∂

∂xk

f

¡

xk

 ̄

 ̄Zk−1 ¢

dxk

= bMk

Z

Rn

f

¡

xk

 ̄

 ̄Zk−1 ¢ ∂

∂xk

f (zk |xk ) dxk

where the n × 1 gradient vector ∂/∂xk has ∂/∂xki as its ith component. Equation (8-19)

implies that

∂

∂xk

f (zk |xk ) = ∂

∂xk

fv (zk − Hkxk) = −HT

k

∂

∂zk

fv (zk − Hkxk)

= −HT

k

∂

∂zk

f (zk |xk )

where fv( ) is the density of vk. Substitution of this equation into the preceding one gives

xˆk − x ̄k = −bMkHT

k

Z

Rn

f

¡

xk

 ̄

 ̄Zk−1 ¢ ∂

∂zk

f (zk |xk ) dxk

= −bMkHT

k

∂

∂zk

Z

Rn

f

¡

xk

 ̄

 ̄Zk−1 ¢

f (zk |xk ) dxk

where the second equality results because f(xk|Zk−1) is not a function of zk. Substituting

(8-30) into this equation and evaluating the integral, we obtain (8-24).

To derive (8-25), we add and subtract x ̄k in (8-21) and simplify, which gives

Pk = E

h

(xk − x ̄k) (xk − x ̄k)

T  ̄

 ̄Zk

i

− (xˆk − x ̄k) (xˆk − x ̄k)

T

The second term of this equation may be evaluated by substituting (8-24). The first term

may be evaluated in a similar manner as the derivation of (8-24) except that an integration

by parts must be done twice. After a tedious calculation, we obtain (8-25). Equation

(8-26) is derived by using the definition of Mk+1 given by (8-23) and then substituting

(8-18), (8-22), and (8-21). Equation (8-27) follows from (8-22).¤

The filter defined by this theorem is the Kalman-Bucy filter if f(zk|Zk−1) is a Gaussian

density. Since (8-19) and (8-23) indicate that the covariance of zk conditioned on Zk−1 is

HkMkHT

k + Rk, where Rk = E £

vkvT

k

¤

, a Gaussian density implies that

gk (zk) = ¡

HkMkHT

k + Rk

¢−1

(zk − Hkx ̄k) (8-31)

Gk (zk) = ¡HkMkHT

k + Rk

¢−1 (8-32)

Substitution of these two equations into (8-24) and (8-25) yields the usual Kalman-Bucy

equations.

101
To apply this theorem to the interference suppression problem, the narrowband interference

sequence {ik} at the filter input is modeled as an autoregressive process that satisfies

ik = X

q

l=1

φlik−l + ek (8-33)

where ek is a white Gaussian process with variance σ2

i and the {φl} are known to the

receiver. The state-space representation of the system is

xk = φxk−1 + uk (8-34)

zk = Hxk + vk (8-35)

where

xk = [ik ik−1 ... ik−q+1]

T (8-36)

φ =















φ1 φ2 ... φq−1 φq

1 0 ... 0 0

0 1 ... 0 0 .

.

. .

.

. ... .

.

. .

.

.

0 0 ... 1 0















(8-37)

uk = [ek 0 ... 0]T (8-38)

H = [1 0 ... 0] (8-39)

The observation noise vk is the sum of the direct-sequence signal sk and the white Gaussian

noise nk:

vk = sk + nk (8-40)

Since the first component of the state vector xk is the interference ik, the state estimate

Hxˆk provides an interference estimate that can be subtracted from the received signal to

cancel the interference.

For a random spreading sequence, sk = +c or −c with equal probability. If nk is zero-mean

and Gaussian with variance σ2

n, then vk has the density

fv(v) = 1

2

Nσ2

n (v − c) + 1

2

Nσ2

n (v + c) (8-41)

where

Nσ2 (x) = 1

√2πσ

exp μ− x2

2σ2

¶

(8-42)

For this non-Gaussian density, the optimal filter that computes the exact conditional mean

given by (8-20) is nonlinear with exponentially increasing complexity and, thus, is

impractical. The density f(xk|Zk−1) is not Gaussian as required by Masreliez’s theorem.

102
However, by assuming that this density is approximately Gaussian, we can use results of

the theorem to derive the approximate conditional mean (ACM) filter [29].

Conditioned on Zk−1 and sk, the expected value of zk is Hx ̄k + sk since xk and nk are

independent of sk. From the definition of Mk and (8-35), it follows that the conditional

variance of zk is

σ2

z = H Mk HT + σ2n (8-43)

Since f(xk|Zk−1) is approximated by a Gaussian density, we obtain

f

¡

zk

 ̄

 ̄Zk−1 ¢

= 1

2

Nσ2

z (zk − H ̄xk − c) + 1

2

Nσ2

z (zk − H ̄xk + c) (8-44)

Substitution of this equation into (8-28) and (8-29) yields

gk (zk) = 1

σ2

z

∙

2k − c tanh μc2k

σ2

z

¶ ̧ (8-45)

Gk (zk) = 1

σ2

z

∙

1 − c2

σ2

z

sech2

μc2k

σ2

z

¶ ̧ (8-46)

where the innovation or prediction residual is

2k = zk − Hx ̄k = zk − z ̄k (8-47)

and

z ̄k = Hx ̄k (8-48)

is the predicted observation based on Zk−1. The update equations of the ACM filter are

given by (8-24) to (8-27) and (8-45) to (8-48). The difference between the ACM filter and

the Kalman-Bucy filter is the presence of the nonlinear tanh and sech functions in (8-45)

and (8-46).

Adaptive ACM filter

In practical applications, the elements of the matrix φ in (8-37) are unknown and may vary

with time. To cope with these problems, an adaptive algorithm that can track the

interference is desirable. The adaptive ACM filter receives zk = ik + sk + nk and produces

the interference estimate denoted by  ̄zk. The output of the filter is denoted by 2k = zk − z ̄k

and ideally is sk + nk plus a small residual of ik. An adaptive transversal filter is embedded

in the adaptive ACM filter. To use the structure of the nonlinear ACM filter, we observe

that the second term inside the brackets in (8-45) would be absent if sk were absent.

Therefore, c tanh(c2k/σ2

z ) may be interpreted as a soft decision on the direct-sequence

signal sk. The input to the adaptive transversal filter at time k is taken to be the difference

between the observation zk and the soft decision:

z ̃k = zk − c tanh μc2k

σ2

z

¶

=  ̄zk + ρ (2k) (8-49)

103
where

ρ (2k) = 2k − c tanh μc2k

σ2

z

¶

(8-50)

The input  ̃zk is a reasonable estimate of the interference that is improved by the adaptive

filter. The architecture of the one-sided adaptive ACM filter [30] is shown in Figure 55.

The output of the N-tap transversal filter provides the interference estimate

z ̄k = WT (k)z ̃k (8-51)

where W(k) is the weight vector and

z ̃k = [ ̃zk−1 z ̃k−2 ... z ̃k−N ]

T (8-52)

which is extracted from the filter taps. When  ̃zk has only a small component due to sk, the

filter can effectively track the interference, and  ̄zk is a good estimate of this interference.

A normalized version of the LMS algorithm for the adaptive ACM filter is given by the

weight-update equation:

W(k) = W(k − 1) + μ0

rk

( ̃zk − z ̄k) z ̄k (8-53)

where μ0 is the adaptation constant and rk is an estimate of the input power iteratively

determined by

rk = rk−1 + μ0

£

|z ̃k|

2 − rk−1

¤ (8-54)

The division by rk in (8-53) normalizes the algorithm by making the choice of an

appropriate μ0 for fast convergence and good performance much less dependent on the

input power level.

Interference

estimate

Tc Tc Tc

Output εk

zk-N

~ zk

W1 W2 WN c, σz

zk-2 z zk k-1

ρ( )

+ +

+

∑

∑ ∑

–

zk

^

2

Figure 55. Adaptive ACM filter.

104
The calculation of ρ(2k) requires the estimation of σ2

z . If the  ̄zk produced by the adaptive

filter approximates the prediction residual of (8-48), then (8-47), (8-35), (8-40), and (8-23)

imply that var(22

k) ≈ σ2

z + c2. Therefore, if var(22

k) is estimated by computing the sample

variance of the filter output, then the subtraction of c2 from the sample variance gives an

estimate of σ2

z .

A figure of merit for filters is the SINR improvement, which is the ratio of the output SINR

to the input SINR. Since the filters of concern do not change the signal power, the SINR

improvement is

R = E ©

|zk − sk|

2a

E ©|2k − sk|

2a (8-55)

In terms of this performance measure, the nonlinear adaptive ACM filter has been found

to provide much better suppression of narrowband interference than the linear KalmanBucy filter if the noise power in nk is less than the direct-sequence signal power in sk.

If the latter condition is not satisfied, the advantage is small or absent. Disadvantages

apparent from (8-50) are the requirements to estimate the parameters c and σ2

z and to

compute or store the tanh function. At the cost of additional complexity and delay, a

nonlinear adaptive interpolator [30] gives a slight performance gain. The preceding linear and

nonlinear methods are primarily predictive methods that exploit the inherent predictability of

narrowband interference. Further improvements in interference suppression are theoretically

possible by using code-aided methods, which exploit the predictability of the spread-spectrum

signal itself [31]. Most of these methods are based on methods that were originally developed

for multiuser detection (Section 11). Some of them can potentially be used to simultaneously

suppress both narrowband interference and multiple-access interference. However, codeaided methods require even more computation and parameter estimation than the ACM

filter, and the most powerful of the adaptive methods are practical only for short spreading

sequences.

9. Detection of Direct-Sequence Signals

In this section, we analyze the possibility of unauthorized detection of a direct-sequence

signal, assuming that the spreading sequence is unknown to the detector, which cannot

mimic the intended receiver. The results of Section 2.3 indicate that the peak power

spectral density of a direct-sequence signal with a random spreading sequence is

A2Tc/4 = Es/2G, where Es is the symbol energy and G is the processing gain. A spectrum

analyzer usually cannot detect a signal with a power spectral density below that of the

background noise, which has spectral density N0/2. Thus, a received Es/N0 > G is an

approximate necessary, but not sufficient, condition for a spectrum analyzer to detect a

105
direct-sequence signal. If Es/N0 < G, detection may still be possible by other means. If

not, the direct-sequence signal is said to have a low probability of interception.

9.1 Ideal Detection

Detection theory provides various detection receivers depending on precisely what is

assumed to be known about the signal to be detected. We make the idealized assumptions

that the chip timing of the spreading waveform is known and that whenever the signal is

present, it is present during the entire observation interval. The spreading sequence is

modeled as a random binary sequence, which implies that a time shift of the sequence by a

chip duration corresponds to the same stochastic process. Thus, to account for uncertainty

in the chip timing, one might partition a chip duration among several parallel detectors

each of which implements a different chip timing.

Consider the detection of a direct-sequence signal with PSK modulation:

s(t) = √

2Sp(t) cos (2πfct + φ) (9-1)

where S is the average signal power, fc is the known carrier frequency, and φ is the carrier

phase assumed to be constant over the observation interval 0 ≤ t ≤ T. To determine

whether a signal s(t) is present based on the observation of the received signal , classical

detection theory requires that one choose between the hypothesis H1 that the signal is

present and the hypothesis H0 that the signal is absent. Over the observation interval

0 ≤ t ≤ T, the received signal under the two hypothesis is

r(t) = 1⁄2

s(t) + n(t), H1

n(t), H0

(9-2)

where n(t) is zero-mean, white Gaussian noise with two-sided power spectral density N0/2.

The spreading waveform p(t), which subsumes the random data modulation, is given by

(3-2) with the {pi} modeled as a random binary sequence.

The coefficients in the expansion of the observed waveform in terms of orthonormal basis

functions constitute the received vector r = [r1 r2 ... rN ]. Let θ denote the vector of

parameter values that characterize the signal to be detected. The average likelihood ratio

[32], which is compared with a threshold for a detection decision, is

Λ(r) = Eθ[f(r|H1, θ)]

f(r|H0) (9-3)

where f(r|H1, θ) is the conditional density function of r given hypothesis H1 and the value

of θ, f(r|H0) is the conditional density function of r given hypothesis H0, and Eθ is the

expectation over the random vector θ. The coefficients in the expansion of the Gaussian

106
process n(t) in terms of the orthonormal basis functions are uncorrelated and, hence,

statistically independent. Since each coefficient is Gaussian with variance N0/2,

f(r|H1, θ) = Y

N

i=1

1

√πN0

exp ∙−(ri − si)2

N0

 ̧

(9-4)

f(r|H0) = Y

N

i=1

1

√πN0

exp μ− r2

i

N0

¶

(9-5)

where the {si} are the coefficients of the signal. Substituting these equations into (9-3)

yields

Λ(r) = Eθ

(

exp "

2

N0

X

N

i=1

risi − 1

N0

X

N

i=1

s2

i

#) (9-6)

Expansions in the orthonormal basis functions indicate that if N → ∞, the average

likelihood ratio may be expressed in terms of the signal waveforms as

Λ[r(t)] = Eθ

1⁄2

exp ∙ 2

N0

Z T

0

r(t)s(t)dt − E

N0

 ̧3⁄4 (9-7)

where E is the energy in the signal waveform over the observation interval of duration T.

If N is the number of chips, each of duration Tc, received in the observation interval, then

there are 2N equally likely patterns of the spreading sequence. For coherent detection, we

set φ = 0 in (9-1), substitute it into (9-7), and then evaluate the expectation to obtain

Λ(r(t)) = exp μ− E

N0

¶X

2N

j=1

exp "

2

√

2S

N0

N

X−1

i=0

p

(j)

i r0

i

#

(coherent) (9-8)

where p

(j)

i is the ith chip of pattern j and

r0

i =

Z (i+1)Tc

iTc

r(t)ψ(t − iTc) cos (2πfct) dt (9-9)

These equations indicate how Λ(r(t)) is to be calculated by the ideal coherent detector.

The factor exp (−E/N0) is irrelevant in the sense that it can be merged with the threshold

level with which the average likelihood ratio is compared.

For the more realistic noncoherent detection of a direct-sequence signal, the received carrier

phase is assumed to be uniformly distributed over [0, 2π). Substituting (9-1) into (9-7),

using a trigonometric expansion, dropping the irrelevant factor that can be merged with

the threshold level, and then evaluating the expectation over the random spreading

sequence, we obtain

Λ(r(t)) = Eφ







X

2N

j=1

exp "

2

√

2S

N0

N

X−1

i=0

p

(j)

i (ric cos φ − ris sin φ)

#





(9-10)

107
where

ric =

Z (i+1)Tc

iTc

r(t)ψ (t − iTc) cos (2πfct) dt,

ris =

Z (i+1)Tc

iTc

r(t)ψ (t − iTc) sin (2πfct) dt (9-11)

and Eφ{ }denote the expectation with respect to φ.

The modified Bessel function of the first kind and order zero is given by

I0(x) = 1

2π

Z 2π

0

exp(x cos u)du (9-12)

Since the cosine is a periodic function and the integration is over the same period, we may

replace cos u with cos(u + φ) for any φ in (9-12). A trigonometric expansion and

x1 = x cos φ, x2 = x sin φ then yields

I0(x) = 1

2π

Z 2π

0

exp (x1 cos u − x2 sin u) du, x =

q

x2

1 + x22 (9-13)

Using this relation, the average likelihood ratio of (9-10) becomes

Λ(r(t)) = X

2N

j=1

I0

Ã

2

p2SRj

N0

!

(noncoherent) (9-14)

where

Rj =

"N

X−1

i=0

p

(j)

i ric#2

+

"N

X−1

i=0

p

(j)

i ris#2

(9-15)

Equations (9-14) , (9-15), and (9-11) define the optimum noncoherent detector for a

direct-sequence signal. The presence of the desired signal is declared if (9-14) exceeds a

threshold level.

The implementation of either the coherent or noncoherent optimum detector would be very

complicated, and the complexity would grow exponentially with N, the number of chips in

the observation interval. Calculations [33] indicate that the ideal coherent and noncoherent

detectors typically provide 3 dB and 1.5 dB advantages, respectively, over the far more

practical wideband radiometer, which is analyzed subsequently. The use of four or two

wideband radiometers, respectively, can compensate for these advantages with less

complexity than the optimum detectors. Furthermore, implementation losses and

imperfections in the optimum detectors are likely to be significant.

108
9.2 Wideband Radiometer

Among the many alternatives to the optimum detector, the wideband radiometer is notable

in that it requires virtually no detailed information about the signals to be detected other

than their rough spectral location. Not even whether the modulation is binary or

quaternary is required. Suppose that the signal to be detected is approximated by a

zero-mean, white Gaussian process. Consider two hypotheses that both assume the

presence of a zero-mean, white Gaussian process over an observation interval 0 ≤ t ≤ T.

Under H0 only noise is present, and the one-sided power spectral density is N0, while under

H1 both signal and noise are present, and the power spectral density is N1. Using N

orthonormal basis functions as in the derivation of (9-4) and (9-5), we find that the

conditional densities are

f(r|Hi) = Y

N

l=1

1

√πNi

exp μ− r2

l

Ni

¶

, i = 0, 1 (9-16)

Calculating the likelihood ratio, taking the logarithm, and merging constants with the

threshold, we find that the decision rule is to compare

V = X

N

l=1

r2

l (9-17)

to a threshold. If we let N → ∞ and use the properties of orthonormal basis functions,

then we find that the test statistic is

V =

Z T

0

r2(t)dt (9-18)

which defines an energy detector or radiometer. Although it was derived for a white

Gaussian process, the radiometer is a reasonable configuration for determining the presence

of unknown deterministic signals.

A radiometer may have one of the three equivalent forms shown in Figure 56. Consider the

system of Figure 56(a), which gives a direct realization of (918). The bandpass filter is

assumed to be an ideal rectangular filter with center frequency fc, bandwidth W, and input

r(t) = s(t) + n(t) (9-19)

where s(t) is a deterministic signal, and n(t) is bandlimited white Gaussian noise with a

two-sided power spectral density equal to N0/2. Substituting (9-19) into (9-18), taking the

expected value, and observing that n(t) is a zero-mean process, we obtain

E[V ] = Z T

0

s2(t)dt +

Z T

0

E[n2(t)]dt

= E + N0TW (9-20)

109
∫

∫

Σ

γ

i =1

| |2

| |2

( )2

Lowpass

filter

Lowpass

filter

Bandpass

filter

r(t)

r(t)

r(t)

exp(–j 2πfct)

exp(–j 2πfct)

y(t) V

 V

V

(a)

(b)

(c)

• •

Figure 56. Radiometers: (a) passband, (b) baseband with integration, and (c) baseband

with sampling at rate 1/W and summation.

which indicates that the radiometer output is an unbiased estimate of the total energy in

r(t).

According to the results of Appendix C-1, a bandlimited deterministic signal can be

represented as

s(t) = sc(t) cos 2πfct − ss(t) sin 2πfct (9-21)

Since the spectrum of s(t) is confined within the filter passband, sc(t) and ss(t) have

frequency components confined to the band |f| ≤ W/2. The Gaussian noise emerging from

the bandpass filter can be represented in terms of quadrature components as (Appendix

C-2)

n(t) = nc(t) cos 2πfct − ns(t) sin 2πfct (9-22)

where nc(t) and ns(t) have flat power spectral densities, each equal to N0 over |f| ≤ W/2.

Substituting (9-22), (9-21), and (9-19) into (9-18) and assuming that fc >> W and fc >>

1/T, we obtain

V = 1

2

Z T

0

[sc(t) + nc(t)]2 dt +

1

2

Z T

0

[ss(t) + ns(t)]2 dt (9-23)

A straightforward calculation verifies that the baseband radiometer of Figure 56(b) also

produces this test statistic.

The sampling theorems for deterministic and stochastic processes (Appendix C-3) provide

expansions of sc(t), ss(t), nc(t) and ns(t) that facilitate a statistical performance analysis.

For example,

sc(t) = X∞

i=−∞

sc

μ i

W

¶

sinc(W t − i) (9-24)

110
where sinc x = (sin πx)/πx. Since the Fourier transform of the sinc function is a

rectangular function, using Parseval’s theorem from Fourier analysis and evaluating the

resulting integral yields the approximations:

Z T

0

sinc2(W t − i)dt ≈

Z ∞

−∞

sinc2(W t − i)dt = 1

W , 0 < i ≤ TW (9-25)

Z T

0

sinc(W t − i)sinc(W t − j)dt ≈

Z ∞

−∞

sinc(W t − i)sinc(W t − j)dt = 0,

i 6= j (9-26)

The rapid decline of sinc x for |x| > 1 implies that

Z T

0

sinc2(W t − i)dt ≈ 0, i ≤ 0 or i > TW (9-27)

We define γ = bTWc, where bxc denotes the integer part of x. Substituting expansions

similar to (9-24) into (9-23) and then using the preceding approximations, we obtain

V ≈

1

2W

X

γ

i=1

∙

sc

μ i

W

¶

+ nc

μ i

W

¶ ̧2

+

1

2W

X

γ

i=1

∙

ss

μ i

W

¶

+ ns

μ i

W

¶ ̧2

(9-28)

where it is always assumed that TW ≥ 1. The error introduced by (9-27) at i = 0 and the

error introduced by (9-25) at i = TW are both nearly 1/2W. For other values of i, the

errors caused by the approximations are much less than 1/2W and decrease as TW

increases. Equation (9-28) becomes an increasingly accurate approximation of (9-23) as γ

increases. A test statistic proportional to (9-28) can be derived for the baseband

radiometer of Figure 56(c) and the sampling rate 1/W without invoking the sampling

theorems and the accompanying approximations.

Since n(t) is a zero-mean Gaussian process and has a power spectral density that is

symmetrical about fc, nc(t) and ns(t) are zero-mean, independent Gaussian processes

(Appendix C-2). Thus, nc(i/W) and ns(j/W) are zero-mean, independent Gaussian

random variables. Equation (C-40) implies that the power spectral densities of nc(t) and

ns(t) are

Sc(f) = Ss(f) = 1⁄2

N0 , |f| ≤ W/2

0 , |f| > W/2 (9-29)

The associated autocorrelation functions are

Rc(τ ) = Rs(τ ) = N0Wsinc(Wτ ) (9-30)

This expression indicates that nc(i/W) is statistically independent of nc(j/W), i 6= j, and

similarly for ns(i/W) and ns(j/W). Therefore, (9-28) becomes

V = N0

2

(X

γ

i=1

A2

i +X

γ

i=1

B2

i

)

(9-31)

111
where the Ai and the Bi are statistically independent Gaussian random variables with unit

variances and means

m1i = E[Ai] = 1

√N0W

sc

μ i

W

¶

(9-32)

m2i = E[Bi] = 1

√N0W

ss

μ i

W

¶

(9-33)

Thus, 2V/N0 has a noncentral chi-squared (χ2) distribution (Appendix B) with 2γ degrees

of freedom and a noncentral parameter

λ = X

γ

i=1

m2

1i +X

γ

i=1

m2

2i = 1

N0W

X

γ

i=1

s2

c

μ i

W

¶

+

1

N0W

X

γ

i=1

s2

s

μ i

W

¶

≈

1

N0

Z T

0

£

s2

c (t) + s2s(t)

¤

dt ≈

2

N0

Z T

0

s2(t)dt = 2E

N0

(9-34)

The probability density function of Z = 2V/N0 is

fZ(x) = 1

2

3x

λ

 ́(γ−1)/2

exp μ−x + λ

2

¶

Iγ−1

3√

xλ

 ́

u(x) (9-35)

where In( ) is the modified Bessel function of the first kind and order n (Appendix B), and

u(x) = 1, x ≥ 0, and u(x) = 0, x < 0. Using the series expansion in λ of the Bessel

function and then setting λ = 0 in (9-35), we obtain the probability density function for Z

in the absence of the signal:

fZ(x) = 1

2γΓ(γ)

xγ−1 exp 3−x

2

 ́

u(x) , λ = 0 (9-36)

where Γ( ) is the gamma function (Appendix B).

By straightforward calculations using the statistics of Gaussian variables, (9-31) and the

subsequent results yield

E[V ] = E + N0γ (9-37)

var(V )=2N0E + N2

0 γ (9-38)

Equation (9-37) approaches the exact result of (9-20) as TW increases.

Let Vt denote the threshold level to which V is compared. A false alarm occurs if V >Vt

when the signal is absent. Application of (9-36) yields the probability of a false alarm:

PF =

Z ∞

2Vt/N0

1

2γΓ(γ)

vγ−1e−v/2dv

= 1 − Γ

μ Vt

N0

, γ

¶

(9-39)

112
where the incomplete gamma function is defined as

Γ(x, a) = 1

Γ(a)

Z x

0

e−tt

a−1

dt (9-40)

and Γ(∞, a) = 1. Integrating (9-39) by parts γ − 1 times yields the series

PF = exp μ− Vt

N0

¶X

γ−1

i=0

1

i!

μ Vt

N0

¶i

(9-41)

Since correct detection occurs if V >Vt when the signal is present, (9-35) indicates that

the probability of detection is

PD =

Z ∞

2Vt/N0

1

2

3v

λ

 ́(γ−1)/2

exp μ−v + λ

2

¶

Iγ−1

3√

vλ

 ́

dv (9-42)

The generalized Marcum Q-function is defined as

Qm(α, β) = Z ∞

β

x

3x

α

 ́m−1

exp μ−x2 + α2

2

¶

Im−1(αx)dx (9-43)

where m is a nonnegative integer, and α and β are nonnegative real numbers. A change of

variables in (9-42) yields

PD = Qγ

3√

λ,

p2Vt/N0

 ́

(9-44)

The threshold Vt is usually set to a value that ensures a specified PF . To derive an easily

computed closed-form expression for Vt in terms of PF , we first approximate (9-39). When

TW >> 1, γ ≈ TW, and the central limit theorem for the sum of independent, identically

distributed random variables with finite means and variances indicates that the

distribution of V given by (9-31) is approximately Gaussian. Using (9-37) and (9-38) with

E = 0 and the Gaussian distribution, we obtain

PF ≈

1

(2πN2

0 TW)1/2

Z ∞

Vt

exp ∙−(v − N0TW)2

2N2

0 TW  ̧

dv

= Q

∙

Vt − N0TW

(N2

0TW)1/2

 ̧

, TW >> 1 (9-45)

where Q(x) is defined by (3-26). Inverting this equation, we obtain Vt in terms of PF and

N0. Accordingly, if the estimate of N0 is Nˆ0 and PF is specified, then the threshold should

be

Vt ≈ Nˆ0

√

TWQ−1(PF ) + Nˆ0TW , TW >> 1 (9-46)

where Q−1( ) denotes the inverse of the function Q( ). In the absence of a signal, (9-20)

indicates that N0 = E[V ]/TW. Thus, N0 can be estimated by averaging sampled

radiometer outputs when it is known that no signal is present.

113
In some applications, one might wish to specify the false alarm rate, which is the expected

number of false alarms per unit time, rather than PF . If successive observation intervals do

not overlap each other except possibly at end points, then the false alarms rate is

F = PF

T (9-47)

For TW > 100, the generalized Marcum Q-function is difficult to compute. Thus, we seek

an approximation that is easier to compute and to invert. If V is approximated by a

Gaussian random variable, then (9-37) and (9-38) imply that

PD ≈ Q

∙ Vt − N0TW − E

(N2

0 TW + 2N0E)1/2

 ̧

, TW >> 1 (9-48)

Figure 57 depicts PD versus E/N0 for a wideband radiometers with Nˆ0 = N0 and

PF = 10−3. Equations (9-46) and (9-48) are used to calculate Vt and PD, respectively. The

figure illustrates the increased energy required to maintain a specified PD as TW increases.

The figure also illustrates the impact of the imperfect estimation of N0 when PF = 10−3

and TW = 107. When the estimation uncertainty is enough that Nˆ0 = 1.001 N0, the

detection probability is lowered considerably.

15 20 25 30 35 40 45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Probability of Detection

Energy-to-noise-density ratio, dB

TW = 102 103 104 105 106 107 107

Figure 57. Probability of detection versus E/N0 for wideband radiometer with PF = 10−3

and various values of TW. Solid curves are the Nˆ0 = N0; dashed curve is for

Nˆ0 = 1.001N0.

114
The sensitivity of the radiometer to errors in Nˆ0 when TW is large, which has been

observed experimentally [34], is due to the fact the E[V ] contains a bias term equal to

N0TW and var(V ) contains a term equal to N2

0 TW, as indicated by (9-37) and (9-38).

Setting Nˆ0 high enough that Nˆ0 ≥ N0 is certain ensures that Vt will be large enough that

the required PF is achieved regardless of the exact value of N0. It is important that Nˆ0/N0

is as close to unity as possible to avoid degrading PD when TW is large. Consequently, the

radiometer output due to noise alone, which provides Nˆ0, should be observed often enough

that Nˆ0 closely tracks the changes in N0 that might result from small changes in the

circuitry or the environmental noise.

When Vt is specified, the value of E/N0 necessary to achieve a specified value of PD may be

obtained by inverting (9-44), which is computationally difficult but can be closely

approximated. Assuming that 2Vt > N0TW, (9-48) yields the necessary value:

E

N0

≈ £Q−1(PD)

¤2

+

Vt

N0

− TW − £Q−1(PD)

¤

r

[Q−1(PD)]2 +

2Vt

N0

− TW ,

TW >> 1 (9-49)

According to (9-46), the condition 2Vt > N0TW is satisfied if PF ≤ 1/2 and Nˆ0 ≈ N0. The

substitution of (9-46) into (9-49) and a rearrangement of terms yields

E

N0

≈ h

√

TWβ + (h − 1)TW + ψ(β, ξ, TW, h), TW >> 1 (9-50)

where

β = Q−1

(PF ), ξ = Q−1(PD), h = Nˆ0/N0 (9-51)

ψ(β, ξ, TW, h) = ξ2 − √

TWξ

∙

2h − 1 +

2βh

√

TW

+ ξ2

TW  ̧1/2

(9-52)

As TW increases, the significance of the third term in (9-50) decreases, while that of the

second term increases if h > 1. Figure 58 shows E/N0 versus TW for PD = 0.99 and

various values of PF and h.

Denoting the intercepted signal power by S and the signal duration by T1, we find from

(9-50) with E = ST1 that the intercepted power necessary to achieve specified values of PD

and either PF or F is

S

N0

≈







h

√

TW

T1

β + (h − 1)W T

T1

+ ψ

T1

, T1 < T

h

rW

T β + (h − 1)W + ψT , T1 ≥ T

(9-53)

This equation indicates that increasing the observation interval T decreases the required

power only if T ≤ T1. Although a single wideband radiometer is incapable of determining

whether one or more than one signal has been detected, narrowband interference can be

rejected by the methods of Section 8.

115
104 105 106 107

25

30

35

40

45

TW

Energy-to-noise-density ratio, dB

PF = 10–1, h = 1

PF = 10–3, h = 1

PF = 10–7, h = 1

PF = 10–7, h = 1.001

Figure 58. Energy-to-noise-density ratio versus TW for wideband radiometer with PD =

0.99 and various values of PF and h.

10. Direct-Sequence Code-Division Multiple Access

Multiple access is the ability of many users to communicate with each other while sharing a

common transmission medium. Wireless multiple-access communications are facilitated if

the transmitted signals are orthogonal or separable in some sense. Signals may be

separated in time (time-division multiple access or TDMA), frequency (frequency-division

multiple access or FDMA), or code (code-division multiple access or CDMA). CDMA is

realized by using spread-spectrum modulation while transmitting signals from multiple

users in the same frequency band at the same time. All signals use the entire allocated

spectrum, but the spreading sequences differ. Information theory indicates that in an

isolated cell, CDMA systems achieve the same spectral efficiency as TDMA or FDMA

systems only if optimal multiuser detection is used. However, even with single-user

detection, CDMA is advantageous for cellular networks because it eliminates the need for

frequency and timeslot coordination among cells and allows carrier-frequency reuse in

adjacent cells. Frequency planning is vastly simplified. A major CDMA advantage exists in

networks accommodating voice communications. A voice-activity detector activates the

transmitter only when the user is talking. Since typically fewer than 40% of the users are

talking at any given time, the number of telephone users can be increased while

maintaining a specified average interference power. Another major CDMA advantage is the

ease with which it can be combined with multibeamed antenna arrays that are either

116
adaptive or have fixed patterns covering cell sectors. There is no practical means of

reassigning time slots in TDMA systems or frequencies in FDMA systems to increase

capacity by exploiting intermittent voice signals or multibeamed arrays. Reassignments to

accommodate variable data rate are almost always impractical in FDMA or TDMA

systems. These general advantages of CDMA, combined with the resistance of

spread-spectrum signals to jamming, interception, and multipath interference, make

CDMA the most attractive choice for most mobile communications. The two principal

types of spread-spectrum CDMA are direct-sequence CDMA and frequency-hopping CDMA.

In direct-sequence CDMA, pulses with a large bandwidth relative to the symbol rate can

be generated by using a chip waveform that is modulated by a spreading sequence. In

frequency-hopping CDMA, the carrier frequency of a transmission is changed periodically.

Consider a direct-sequence CDMA network with K users in which every receiver has the

form of Figure 14. The multiple-access interference that enters a receiver synchronized to a

desired signal is modeled as

i(t) =

K

X−1

i=1

p2Iidi(t − τ )qi (t − τi) cos (2πfct + φi) (10-1)

where K − 1 is the number of interfering direct-sequence signals, and Ii is the average

power, di(t) is the code-symbol modulation, qi(t) is the spreading waveform, τi is the

relative delay, and φi is the phase shift of interference signal i including the effect of carrier

time delay. Each spreading waveform has the form

qi(t) = X∞

j=−∞

q

(i)

j ψ (t − jTc) (10-2)

where the chip waveforms are assumed to be identical throughout the network and

q

(i)

j ∈ {−1, 1}. Substituting (10-1) into (3-8) and (3-11) and then using (3-2), we obtain the

interference component of the demodulator output due to a received symbol:

V1 =

K

X−1

i=1

rIi

2

cos φi

Z Ts

0

di (t − τi) qi (t − τi) p(t)dt (10-3)

where a double-frequency term is neglected and p(t) is the spreading waveform of the

desired signal. In a direct-sequence CDMA network, the spreading sequences are often

called signature sequences.

10.1 Orthogonal Sequences

Suppose that the communication signals are synchronous so that all data symbols have

duration Ts, symbol and chip transitions are aligned at the receiver input, and short

117
spreading sequences with period N = G extend over each data symbol. Then

τi = 0, i = 1, 2,...,K − 1, and di(t) = di is constant over the integration interval [0, Ts].

The cross-correlation between qi(t) and p(t) is defined as

Cpi(τ ) = 1

Ts

Z Ts

0

p(t)qi(t − τ )dt (10-4)

Thus, for synchronous communications, (10-3) may be expressed as

V1 =

K

X−1

i=1

λiCpi(0) (10-5)

where

λi =

rIi

2

diTs cos φi (10-6)

Substituting (10-2) and (3-2) into (10-4) and then using (3-3) and (3-4), we obtain

Cpi(0) = 1

G

X

G

j=1

pjq

(i)

j (10-7)

where the right-hand side is the periodic cross-correlation between the sequences {q

(i)

j } and

{pj}. Let a and bi denote the binary sequences with components aj , b(i)

j ∈ GF(2),

respectively, that map into the binary antipodal sequences with components pj = (−1)aj+1

and q

(i)

j = (−1)b

(i)

j +1. Then a derivation similar to that in (2-28) gives

Cpi(0) = Ai − Di

G (10-8)

where Ai denotes the number of agreements in the corresponding bits of a and bi, and Di

denotes the number of disagreements. The sequences are orthogonal if Cpi(0) = 0. If the

spreading sequence a is orthogonal to all the spreading sequences bi, i = 1, 2,...,K, then

V1 = 0 and the multiple-access interference i(t) is suppressed at the receiver. A large

number of multiple-access interference signals can be suppressed in a network if each such

signal has its chip transitions aligned and the spreading sequences are mutually orthogonal.

Two binary sequences, each of length two, are orthogonal if each sequence is described by

one of the rows of the 2 × 2 matrix

H1 =

∙ 0 0

0 1  ̧

(10-9)

because A = D = 1. A set of 2n sequences, each of length 2n, is obtained by using the rows

of the matrix

Hn =

∙

Hn−1 Hn−1

Hn−1 H ̄ n−1

 ̧

, n = 2, 3,... (10-10)

118
where H ̄ n−1 is the complement of Hn−1, obtained by replacing each 1 and 0 by 0 and 1,

respectively, and H1 is defined by (10-9). Any pair of rows in Hn differ in exactly 2n−1

columns, thereby ensuring orthogonality of the corresponding sequences. The 2n × 2n

matrix Hn, which is called a Hadamard matrix, can be used to generate 2n spreading

sequences for synchronous direct-sequence communications. These spreading sequences are

often called Walsh sequences.

As an alternative to the Walsh sequences, consider the set of 2m − 1 maximal sequences

generated by a primitive polynomial of degree m and the 2m − 1 different initial states of

the shift register. Equation (2-28) implies that by appending a 0 at the end of each period

of each sequence, we obtain a set of 2m − 1 orthogonal sequences of period 2m. Without the

appending of symbols, a set of nearly orthogonal sequences for a synchronous network may

be generated from different time displacements of a single maximal sequence because its

autocorrelation, which is given by (2-29), determines the cross-correlations among the

sequences of the set. The low values of the autocorrelation for nonzero delay causes the

rejection of multipath signals. In contrast, the Walsh sequences do not have such favorable

autocorrelation functions.

10.2 Sequences with Small Cross-Correlations

The symbol transitions of asynchronous multiple-access signals at a receiver are not

simultaneous, usually because of changing path-length differences among the various

communication links. Since the spreading sequences are shifted relative to each other, sets

of periodic sequences with small cross-correlations for any relative shifts are desirable to

limit the effect of multiple-access interference. Maximal sequences, which have the longest

periods of sequences generated by a linear feedback shift register of fixed length, are often

inadequate. Let a = (...,a0, a1,...) and b = (...,b0, b1,...) denote binary sequences with

components in GF(2). The sequences a and b are mapped into antipodal sequences p and

q, respectively, with components in {−1, +1} by means of the transformation

pi = (−1)ai+1 , qi = (−1)bi+1 (10-11)

The periodic cross-correlation of periodic binary sequences a and b with the same period

N is defined as the periodic cross-correlation of the antipodal sequences p and q, which is

defined as

θpq(j) = 1

N

N

X−1

i=0

piqi+j , j = 1, 2,...,N − 1 (10-12)

A calculation similar to that in (2-28) yields the periodic cross-correlation

θpq(j) = Aj − Dj

N (10-13)

119
where Aj denotes the number of agreements in the corresponding components of a and the

shift sequence b(j)=(...,bj , bj+1,...,bj+N−1,...), and Dj denotes the number of

disagreements.

In the presence of asynchronous multiple-access interference for which τi 6= 0, the

interference component of the correlator output is given by (10-3). If we assume that the

data modulation is absent so that we may set di(t) = 1 in (10-3), then it is observed that

interference signal i produces a term in V1 that is proportional to Cpi(τi). Let

τi = NiTc + 2i, where Ni is a nonnegative integer and 0 ≤ 2i < Tc. A derivation similar to

the one leading to (2-34) gives

Cpi (NiTc + 2i) = μ

1 − 2i

Tc

¶

θpi(Ni) + 2i

Tc

θpi(Ni + 1) (10-14)

where θpi(Ni) is the periodic cross-correlation of the sequence p and qi and is given by

(10-13). Thus, ensuring that the periodic cross-correlations are always small is a critical

necessary condition for the success of asynchronous multiple-access communications.

Although the data modulation may be absent during acquisition, it will be present during

data transmission, and di(t) may change polarity during an integration interval. Thus, the

effect of asynchronous multiple-access interference (Section 10.3) will exceed that predicted

from (10-14).

For a set S of M periodic antipodal sequences of length N, let θmax denote the peak

magnitude of the cross-correlations or autocorrelations:

θmax = max {|θpq(k)| : 0 ≤ k ≤ N − 1; p, q ∈ S; p 6= q or k 6= 0} (10-15)

Theorem. A set S of M periodic antipodal sequences of length N has

θmax ≥

r M − 1

MN − 1 (10-16)

Proof. Consider an extended set Se of MN sequences p(i)

, i = 1, 2,...,MN, that

comprises the N distinct shifted sequences derived from each of the sequences in S. The

cross-correlation of sequences p(i) and p(j) in Se is

ψij = 1

N

X

N

n=1

p(i)

n p(j)n (10-17)

and

θmax = max ©

|ψij | , p(i)2 Se, p(j) 2 Se, i 6= j

a (10-18)

Define the double summation

Z = X

MN

i=1

X

MN

j=1

ψ2

ij (10-19)

120
Separating the MN terms for which ψii = 1 and then bounding the remaining

MN(MN − 1) terms yields

Z ≤ MN + MN(MN − 1)θ2

max (10-20)

Substituting (10-17) into (10-19), interchanging summations, and omitting the terms for

which m 6= n, we obtain

Z = 1

N2

X

N

n=1

X

N

m=1

X

MN

i=1

p(i)

n p(i)m

X

MN

j=1

p(j)

n p(j)m

= 1

N2

X

N

n=1

X

N

m=1 ÃX

MN

i=1

p(i)

n p(i)m

!2

≥

1

N2

X

N

n=1 "

X

MN

n=1

¡

p(i)

n

¢2

#2

= M2N (10-21)

Combining this inequality with (10-20) gives (10-16). ¤

The lower bound in (10-16) is known as the Welch bound. It approaches 1/

√

N for large

values of M and N. Only small subsets of maximal sequences can be found with θmax close

to this lower bound. The same is true for Walsh sequences.

Large sets of sequences with θmax approaching the Welch bound can be obtained by

combining maximal sequences with sampled versions of these sequences. If q is a positive

integer, the new binary sequence b formed by taking every qth bit of binary sequence a is

known as a decimation of a by q, and the components of the two sequences are related by

bi = aqi. Let gcd(x, y) denote the greatest common divisor of x and y. If the original

sequence a has a period N and the new sequence b is not identically zero, then b has

period N/gcd(N, q). If gcd(N, q) = 1, then the decimation is called a proper decimation.

Following a proper decimation, the bits of b do not repeat themselves until every bit of a

has been sampled. Therefore, b and a have the same period N, and it can be shown that if

a is maximal, then b is a maximal sequence [35]. A preferred pair of maximal sequences

with period 2m − 1 are a pair with a periodic cross-correlation that takes only the three

values −t(m)/N, −1/N, and [t(m) − 2]/N, where

t(m)=2b(m+2)/2c + 1 (10-22)

and bxc denotes the integer part of the real number x. The Gold sequences are a large set

of sequences with period N = 2m − 1 that may be generated by the modulo-2 addition of

preferred pairs when m is odd or m = 2 modulo-4 [35]. One sequence of the preferred pair

is a decimation by q of the other sequence. The positive integer q is either q = 2k + 1 or

q = 22k − 2k + 1, where k is a positive integer such that gcd(m, k) = 1 when m is odd and

gcd(m, k) = 2 when m = 2 modulo-4.

121
Since the cross-correlation between any two Gold sequences in a set can take only three

values, the peak magnitude of the periodic cross-correlation between any two Gold

sequences of period N = 2m − 1 is

θmax = t(m)

2m − 1 (10-23)

For large values of m, θmax for Gold sequences exceeds the Welch bound by a factor of √2

for m odd and a factor of 2 for m even.

One form of a Gold sequence generator is shown in Figure 59. If each maximal sequence

generator has m stages, different Gold sequences in a set are generated by selecting the

initial state of one maximal sequence generator and then shifting the initial state of the

other generator. Since any shift from 0 to 2m − 2 results in a different Gold sequence,

2m − 1 different Gold sequences can be produced by the system of Figure 59. Gold

sequences identical to maximal sequences are produced by setting the state of one of the

maximal sequence generators to zero. Altogether, there are 2m + 1 different Gold

sequences, each with a period of 2m − 1, in the set.

An example of a set of Gold sequences is the set generated by the preferred pair specified

by the primitive characteristic polynomials

f1(x)=1+ x3 + x7, f2(x)=1+ x + x2 + x3 + x7 (10-24)

Since m = 7, there are 129 Gold sequences of period 127 in this set, and (10-23) gives

θmax = 0.134. Equation (2-57) indicates that there are only 18 maximal sequences with

m = 7. For this set of 18 sequences, calculations [35] indicate that θmax = 0.323. If

θmax = 0.134 is desired for a set of maximal sequences with m = 7, then one finds that the

Maximal

sequence

generator

Initial

contents

generator

Maximal

sequence

generator

Clock

Gold

sequence

Figure 59. Gold sequence generator.

122
set has only 6 sequences. This result illustrates the much greater utility of Gold sequences

in CDMA networks with many subscribers.

Consider a Gold sequence generated by using the characteristic functions f1(x) and f2(x)

of degree m. The generating function for the Gold sequence is

G(x) = φ1(x)

f1(x) + φ2(x)f2(x)

= φ1(x)f2(x) + φ2(x)f1(x)

f1(x)f2(x) (10-25)

where φ1(x) and φ2(x) have the form specified by the numerator of (2-54). Since the

degrees of both φ1(x) and φ2(x) are less than m, the degree of the numerator of G(x) must

be less than 2m. Since the product f1(x)f2(x) has the form of a characteristic function of

degree 2m given by (2-50), this product defines the feedback coefficients of a single linear

feedback shift register with 2m stages that can generate the Gold sequences. The initial

state of the register for any particular sequence can be determined by equating each

coefficient in the numerator of (10-25) with the corresponding coefficient in (2-54) and then

solving 2m linear equations.

A small set of Kasami sequences comprises 2m/2 sequences with period 2m − 1 if m is even

[35]. To generate a set, a maximal sequence a with period N = 2m − 1 is decimated by

q = 2m/2 + 1 to form a binary sequence b with period N/gcd(N, q)=2m/2 − 1. The

modulo-2 addition of a and any cyclic shift of b from 0 to 2m/2 − 2 provides a Kasami

sequence. By including a , we obtain a set of 2m/2 Kasami sequences with period 2m − 1.

The periodic cross-correlation between any two Kasami sequences in a set can only take

the values −s(m)/N, −1/N, or [s(m) − 2]/N, where

s(m)=2m/2 + 1 (10-26)

The peak magnitude of the periodic cross-correlation between any two Kasami sequences is

θmax = s(m)

N = 2m/2 + 12m − 1 = 12m/2 − 1 (10-27)

For m ≥ 2 and M = 2m/2, the use of NM − 1 > NM − N in the Welch bound gives

θmax > 1/

√

N. Since N = 2m − 1,

Nθmax > √2m − 1 > 2m/2 − 1 (10-28)

Since N is an odd integer, Aj − Dj in (10-13) must be an odd integer. Therefore, the

definition of θmax and (10-13) indicate that Nθmax must be an odd integer. Inequality

(10-28) then implies that for M = 2m/2, N = 2m − 1, and even values of m,

Nθmax ≥ 2m/2 + 1 (10-29)

123
A comparison of this result with (10-27) indicates that the Kasami sequences are optimal

in the sense that θmax has the minimum value for any set of sequences of the same size and

period.

As an example, let m = 10. There are 60 maximal sequences, 1025 Gold sequences, and 32

Kasami sequences with period 1023. The peak cross-correlations are 0.37, 0.06, and 0.03,

respectively.

A large set of Kasami sequences comprises 2m/2(2m + 1) sequences if m = 2 modulo-4 and

2m/2(2m + 1) − 1 sequences if m = 0 modulo-4 [35]. The sequences have period 2m − 1. To

generate a set, a maximal sequence a with period N = 2m − 1 is decimated by q = 2m/2 + 1

to form a binary sequence b and then decimated by q = 2(m+2)/2 + 1 to form another binary

sequence c. The modulo-2 addition of a, a cyclic shift of b, and a cyclic shift of c provides

a Kasami sequence with period N. The periodic cross-correlations between any two Kasami

sequences in a set can only take the values −1/N, −t(m)/N, [t(m) − 2]/N, −s(m)/N, or

[s(m) − 2]/N. A large set of Kasami sequences includes both a small set of Kasami

sequences and a set of Gold sequences as subsets. Since t(m) ≥ s(m), the value of θmax for

a large set is the same as that for Gold sequences (10-23). This value is suboptimal, but the

large size of these sets makes them an attractive option for asynchronous CDMA networks.

10.3 Symbol Error Probability for Direct-Sequence Systems

Let di = (d

(i)

−1, d(i)0 ) denote the vector of the two symbols of asynchronous multiple-access

interference signal i that are received during the detection of a symbol of the desired signal.

A straightforward evaluation of (10-3) gives

V1 =

K

X−1

i=1

rIi

2

cos φi

h

d

(i)

−1Rpi (τi) + d

(i)

0 Rˆpi (τi)

i

(10-30)

where the continuous-time partial cross-correlation functions are

Rpi(τ ) = Z τ

0

p(t)qi(t − τ )dt (10-31)

Rˆpi(τ ) = Z T

τ

p(t)qi(t − τ )dt (10-32)

For rectangular chip waveforms and spreading sequences of period N, straightforward

calculations yield

Rpi(τ ) = Api(l − N)Tc + [Api(l + 1 − N) − Api(l − N)] (τ − lTc) (10-33)

Rˆpi(τ ) = Api(l)Tc + [Api(l + 1) − Api(l)] (τ − lTc) (10-34)

124
where l = bτ/Tcc and the aperiodic cross-correlation function is defined by

Api(l) =







N

P−1−l

j=0

pjq

(i)

j+l

, 0 ≤ l ≤ N − 1

N

P−1+l

j=0

pjq

(i)

j , 1 − N ≤ l < 0

(10-35)

and Api(l) = 0 for |l| ≥ N. These equations indicate that the aperiodic cross-correlations

are more important than the related periodic cross-correlations defined by (10-12) in

determining the interference level and, hence, the symbol error probability. Without

careful selection of the sequences, the aperiodic cross-correlations may be much larger than

the periodic cross-correlation. If all the spreading sequences are short with N = G, and

S = Ii, i = 1, 2, ··· , K − 1, then the symbol error probability can be approximated and

bounded [36], [37], but the process is complicated. An alternative approach is to model the

spreading sequences as random binary sequences, as is done for long sequences.

In a network with multiple-access interference, code acquisition depends on both the

periodic and aperiodic cross-correlations. In the absence of data modulations, Vc in (7-73)

has additional terms, each of which is proportional to the periodic cross-correlation

between the desired signal and an interference signal. When data modulations are present,

some or all of these terms entail aperiodic cross-correlations.

10.4 Complex-Valued Quaternary Sequences

Quaternary direct-sequence system (Section 4) may use pairs of short binary sequences,

such as Gold or Kasami sequences, to exploit the favorable periodic autocorrelation and

cross-correlation functions. However, Gold sequences do not attain the Welch bound, and

Kasami sequences that do are limited in number. To support many users and to facilitate

the unambiguous synchronization to particular signals in a CDMA network, one might

consider complex-valued quaternary sequences that are not derived from pairs of standard

binary sequences but have better periodic correlation functions.

For q-ary PSK modulation, sequence symbols are powers of the complex qth root of unity,

which is

Ω = exp μj

2π

q

¶

(10-36)

where j = √−1. The complex spreading or signature sequence p of period N has symbols

given by

pi = Ωai

ejφr , ai ∈ Zq = {0, 1, 2,...,q − 1} , i = 1, 2,...,N (10-37)

125
where φr is an arbitrary phase chosen for convenience. If pi is specified by the exponent ai

and qi is specified by the exponent bi, then the periodic cross-correlation between

sequences p and q is defined as

θpq(k) = 1

N

N

X−1

i=0

pi+kq∗

i = 1

N

N

X−1

i=0

Ωai+k−bi (10-38)

The maximum magnitude θmax defined by (10-15) must satisfy the Welch bound of (10-16).

For a positive integer m, a family A of M = N + 2 quaternary or Z4 sequences, each of

period N = 2m − 1, with θmax that asymptotically approaches the Welch bound has been

identified [38]. In contrast, a small set of binary Kasami sequences has only √N + 1

sequences

The sequences in a family A are determined by the characteristic polynomial, which is

defined as

f(x)=1+Xm

i=0

cixi (10-39)

where coefficients ci ∈ Z4 and cm = 1. The output sequence satisfies the linear recurrence

relation of (2-14). For example, the characteristic polynomial f(x)=1+2x + 3x2 + x3 has

m = 3 and generates a family with period N = 7. A feedback shift register that

implements the sequence of the family is depicted in Figure 60(a), where all operations are

modulo-4. The generation of a particular sequence is illustrated in Figure 60(b). Different

sequences may be generated by loading the shift register with any nonzero initial contents

and then cycling the shift register through its full period N = 2m − 1. Since the shift

register has 4m − 1 nonzero states, there are M = (4m − 1)/(2m − 1) = 2m + 1 cyclically

distinct members of the family. Each family member may be generated by loading the shift

register with any nonzero triple that is not a state occurring during the generation of

another family member.

A complex-valued symbol in the family A may be represented by d = d1 +jd2, where d1 and

d2 are antipodal symbols with values ±1/

√2 and φr = π/4 in (10-37) . If a complex-valued

symbol of the spreading sequence is p = p1 + jp2, then the complex multiplication of the

data and spreading sequences produces the complex-valued sequence y = y1 + jy2 = dp.

The implementation of this product is shown in Figure 61, in which real-valued inputs

d1, d2, p1 and p2 produce the two real-valued outputs y1 and y2. The equation y = dp gives

a compact complex-variable representation of the real variable equations:

y1 = d1p1 − d2p2, y2 = d2p1 + d1p2 (10-40)

Each chip y1 modulates the in-phase carrier, and each chip y2 modulates the quadrature

carrier. The transmitted signal may be represented as

s(t) = < ©Ad(t)p(t)ej2πfct

a (10-41)

126
1 2 3

Clock

Output

2 3

Contents

Initial 0 0 1

 1 1 0 0

 2 2 1 0

 3 3 2 1

 4 1 3 2

 5 1 1 3

 6 0 1 1

 7 0 0 1

Stage 1 Stage 2 Stage 3

Shift

(a)

(b)

Figure 60. (a) Feedback shift register for a quaternary sequence and (b) contents after

successive shifts.

127
where <{x} denotes the real part of x, A is the amplitude, and d(t) and p(t) are waveforms

modulated by the data and spreading sequences.

A representation of the receiver in terms of complex variables is illustrated in Figure 61. If

fcTc >> 1, two cross-correlation terms are negligible, and the actual implementation can

be done by the architectures of Figures 17 and 19 except that the final multiplications in

the two branches are replaced by a complex multiplication. Thus, y is extracted by

separate in-phase and quadrature demodulation. Since the complex quaternary symbols

have unity magnitude, the despreading entails the complex multiplication of y by p∗ to

produce d|p|

2 = d along with the residual interference and noise. As illustrated in Figure

62, the summation of G multiplications produces the decision variable, where G is the

number of chips per bit.

Although some complex-valued quaternary sequences have more favorable periodic

autocorrelations and cross-correlations than pairs of standard binary sequences, they do

not provide significantly smaller error probabilities in multiple-access systems [39]. The

reason is that system performance is determined by the complex aperiodic functions

analogous to those of Section 10.3. However, complex sequences have the potential to

provide better acquisition performance than the Gold or Kasami sequences because of their

superior periodic autocorrelations.

∑

∑

p1

y2

y1

d2

d1

p2

–p2

p1

Figure 61. Product of quaternary data and spreading sequences.

128
Received

signal

Decision

variable

i=0

G-1

p* exp(-j2πfct)

CMF Σ

Figure 62. Receiver for direct-sequence system with complex quaternary spreading sequences. CMF is chip-matched filter.

10.5 Direct-Sequence Systems with PSK and Random Sequences

If all the spreading sequences in a network of asynchronous CDMA systems have a

common period equal to the data-symbol duration, then by the proper selection of the

sequences and their relative phases, one can obtain a system performance better than that

theoretically attainable with random sequences. However, the number of suitable sequences

is too small for many applications, and long sequences that extend over many data symbols

provide more system security. Furthermore, long sequences ensure that successive data

symbols are covered by different sequences, thereby limiting the time duration of an

unfavorable cross-correlation due to multiple-access interference. Even if short sequences

are used, the random-sequence model gives fairly accurate performance predictions.

Consider the direct-sequence receiver of Figure 14 when multiple-access interference is

present. If the spreading sequence of the desired signal is modeled as a random binary

sequence and the chip waveform confined to [0, Tc], then (3-15) indicates that an input V

to the decision device has mean value

E[V ] = d0

rS

2

Ts (10-42)

for the direct-sequence system with coherent PSK. The interference component is

V1 =

G

X−1

ν=0

pνJν (10-43)

where

Jν =

Z (ν+1)/Tc

νTc

i(t)ψ(t − νTc) cos 2πfct dt (10-44)

and i(t) is given by (10-1). Since the data modulation di(t) in an interference signal is

modeled as a random binary sequence, it can be subsumed into qi(t) given by (10-2) with

no loss of generality. Since qi(t) is determined by an independent, random spreading

sequence, only time delays modulo-Tc are significant and, thus, we can assume that 0

≤ τi < Tc in (10-1) without loss of generality.

129
Since ψ(t) is confined to [0, Tc] and fcTc >> 1, the substitution of (10-1) and (10-2) into

(10-44) yields

Jν =

K

X−1

i=1

rIi

2

cos φi

(

q

(i)

ν−1

Z νTc+τi

νTc

ψ(t − νTc)ψ[t − (ν − 1)Tc − τi]dt

+q(i)

ν

Z (ν+1)Tc

νTc+τi

ψ(t − νTc)ψ(t − νTc − τi)dt)(10-45)

The partial autocorrelation for the chip waveform is defined as

Rψ(s) = Z s

0

ψ(t)ψ(t + Tc − s)dt, 0 ≤ s<Tc (10-46)

Substitution into (10-45) and appropriate changes of variables in the integrals yield

Jν =

K

X−1

i=1

rIi

2

cos φi

h

q

(i)

ν−1Rψ(τi) + q(i)

ν Rψ(Tc − τi)

i

(10-47)

For rectangular pulses in the spreading waveform,

ψ(t) = 1⁄2

1 , 0 ≤ t ≤ Tc

0 , otherwise. (10-48)

Consequently,

Rψ(s) = s, rectangular pulse (10-49)

For sinusoidal pulses in the spreading waveform,

ψ(t) = (√2 sin 3 π

Tc

t

 ́

, 0 ≤ t ≤ Tc

0 , otherwise. (10-50)

Substituting this equation into (10-46), using a trigonometric identity, and performing the

integrations, we obtain

Rψ(s) = Tc

π

sin μ π

Tc

s

¶

− s cos μ π

Tc

s

¶

, sinusoidal pulse (10-51)

Since both Jν and Jν+1 contain the same random variable q

(i)

ν , it does not appear at first

that the terms in (10-47) are statistically independent even when φ = (φ1, φ2,..., φK−1)

and τ = (τ1, τ2,..., τK−1) are given. The following lemma [40] resolves this issue.

Lemma. Suppose that {αi} and {βi} are statistically independent, random binary

sequences. Let x and y denote arbitrary constants. Then αiβjx and αiβky are statistically

independent random variables when j 6= k.

130
Proof. Let P(αiβjx = a, αiβky = b) denote the joint probability that αiβjx = a and

αiβky = b where |a| = |x| and |b| = |y|. From the theorem of total probability, it follows

that

P(αiβjx = a, αiβky = b)

= P(αiβjx = a, αiβky = b, αi = 1) + P(αiβjx = a, αiβky = b, αi = −1)

= P(βjx = a, βky = b, αi = 1) + P(βjx = −a, βky = −b, αi = −1)

From the independence of {αi} and {βj} and the fact that they are random binary

sequences, we obtain the further simplification for j 6= k, x 6= 0, and y 6= 0:

P(αiβjx = a, αiβky = b)

= P(βjx = a)P(βky = b)P(αi = 1) + P(βjx = −a)P(βky = −b)P(αi = −1)

= 1

2

P

μ

βj = a

x

¶

P

μ

βk = b

y

¶

+

1

2

P

μ

βj = −a

x

¶

P

μ

βk = −b

y

¶

Since βj equals +1 or −1 with equal probability, P(βj = a/x) = P(βj = −a/x) and thus

P(αiβjx = a, αiβky = b) = P

μ

βj = a

x

¶

P

μ

βk = b

y

¶

= P(βjx = a)P(βky = b)

A similar calculation gives

P(αiβjx = a)P(αiβky = b) = P(βjx = a)P(βky = b)

Therefore,

P(αiβjx = a, αiβky = b) = P(αiβjx = a)P(αiβky = b)

which satisfies the definition of statistical independence of αiβjx and αiβky. The same

relation is trivial to establish for x = 0 or y = 0. ¤

The lemma indicates that when φ and τ are given, the terms in (10-43) are statistically

independent. Since p2

ν = 1, the conditional variance is

var(V1) =

G

X−1

ν=0

var(Jν ) (10-52)

The independence of the K spreading sequences, the independence of successive terms in

each random binary sequence, and (10-47) imply that the conditional variance of Jν is

independent of ν and, therefore,

var(V1) =

K

X−1

i=1

1

2

GIi cos2 φi[R2

ψ(τi) + R2ψ(Tc − τi)] (10-53)

131
Since the terms of V1 in (10-43) are independent, zero-mean random variables that are

uniformly bounded and var(V1) → ∞ as G → ∞, the central limit theorem [6] implies that

V1/

pvar(V1) converges in distribution to a Gaussian random variable with mean 0 and

variance 1. Thus, when φ and τ are given, the conditional distribution of V1 is

approximately Gaussian when G is large. Since the noise component V2 in (3-10) has a

Gaussian distribution and is independent of V1, V has an approximate Gaussian

distribution with mean given by (10-42), and var(V ) = var(V1) + var(V2).

A straightforward derivation using the Gaussian distribution of the decision statistic V

indicates that the conditional symbol error probability given φ and τ is

Ps(φ, τ ) = Q

Ãs

2Es

N0e(φ, τ )

!

(10-54)

where Es = STs is the energy per symbol in d(t), and the equivalent noise power spectral

density is defined as

N0e(φ, τ ) = N0 +

K

X−1

i=1

2

Ii

Tc

cos2 φi[R2

ψ(τi) + R2ψ(Tc − τi)] (10-55)

For rectangular pulses, this equation simplifies to

N0e(φ, τ ) = N0 +

K

X−1

i=1

2IiTc cos2 φi

Ã

1 − 2

τi

Tc

+ 2

τ 2

i

T2

c

!

(10-56)

Numerical evaluations [40] give strong evidence that the error in (10-54) due to the

Gaussian approximation is negligible if G ≥ 50. For an asynchronous network, it is

assumed that the time delays are independent and uniformly distributed over [0,Tc) and

that the phase angles θi, i = 1, 2, ...,K − 1, are uniformly distributed over [0,2π).

Therefore, the symbol error probability is

Ps =

Ã

2

πTc

!K−1

Z π/2

0

... Z π/2

0

Z Tc

0

... Z Tc

0

Ps(φ, τ )dφ dτ (10-57)

where the fact that cos2 φi takes all its possible values over [0,π/2) has been used to shorten

the integration intervals. The absence of sequence parameters ensures that the amount of

computation required for (10-57) is much less than the amount required to compute Ps

when the spreading sequence is short. Nevertheless, the computational requirements are

large enough that it is highly desirable to find an accurate approximation that entails less

computation. The conditional symbol error probability given φ is defined as

Ps(φ) = μ 1

Tc

¶K−1 Z Tc

0

... Z Tc

0

Ps(φ, τ )dτ (10-58)

132
A closed-form approximation to Ps(φ) greatly simplifies the computation of Ps, which

reduces to

Ps =

μ 2

π

¶K−1 Z π/2

0

... Z π/2

0

Ps(φ)dφ (10-59)

To approximate Ps(φ), we first obtain upper and lower bounds on it.

For either rectangular or sinusoidal pulses, elementary calculus establishes that

R2

ψ(τi) + R2ψ(Tc − τi) ≤ T2c (10-60)

Using this upper bound successively in (10-55), (10-54), and (10-58), and performing the

trivial integrations that result, we obtain

Ps(φ) ≤ Q

Ãs

2Es

N0u(φ)

!

(10-61)

where

N0u(φ) = N0 +

K

X−1

i=1

2IiTc cos2 φi (10-62)

To apply Jensen’s inequality (4-22), the successive integrals in (10-57) are interpreted as

the evaluation of expected values. Consider the random variable

X = R2

ψ(τi) + R2ψ(Tc − τi). (10-63)

Since τi is uniformly distributed over [0, Tc), straightforward calculations using (10-49) and

(10-51) give

E[X] = 1

Tc

Z Tc

0

[R2

ψ(τi) + R2ψ(Tc − τi)]dτi = hT2c (10-64)

where

h =







2

3 , rectangular pulse

1

3 +

5

2π2 , sinusoidal pulse.

(10-65)

The function (10-54) has the form given by (4-23), Therefore, (10-55), (10-60), and

cos2 φi ≤ 1 yield a sufficient condition for convexity:

Es ≥

3

2

"

N0 +

K

X−1

i=1

2IiTc

#

(10-66)

Application of Jensen’s inequality successively to each component of τ in (10-58) yields

Ps(φ) ≥ Q

Ãs

2Es

N0l(φ)

!

(10-67)

133
where

N0l(φ) = N0 +

K

X−1

i=1

2hIiTc cos2 φi (10-68)

If N0 is negligible, then (10-68) and (10-62) give N0l/N0u = h. Thus, a good approximation

is provided by

Ps(φ) ≈ Q

Ãs

2Es

N0a(φ)

!

(10-69)

where

N0a(φ) = N0 +

K

X−1

i=1

2

√

hIiTc cos2 φi (10-70)

If N0 is negligible, then N0u/N0a = N0a/N0l = 1/

√

h. Therefore, in terms of the value of Es

needed to ensure a given Ps(φ), the error in using approximation (10-69) instead of (10-58)

is bounded by 10 log10(1/

√

h) in decibels, which equals 0.88 dB for rectangular pulses and

1.16 dB for sinusoidal pulses. In practice, the error is expected to be only a few tenths of a

decibel because N0 6= 0 and Ps coincides with neither the upper nor the lower bound.

As an example, suppose that rectangular pulses are used, Es/N0 = 15 dB, and K = 2.

Figure 63 illustrates four different evaluations of Ps as a function of GEs/ITs = GS/I, the

despread signal-to-interference ratio, which is the signal-to-interference ratio after taking

into account the beneficial results from the despreading in the receiver. The accurate

approximation is computed from (10-54) and (10-57), the upper bound from (10-61) and

(10-59), the lower bound from (10-67) and (10-59), and the simple approximation from

(10-69) and (10-59). The figure shows that the accurate approximation moves from the

lower bound toward the simple approximation as the symbol error probability decreases.

For Ps = 10−5, the simple approximation is less than 0.3 dB in error relative to the

accurate approximation. Figure 64 compares the symbol error probabilities for K = 2 to K

= 4, rectangular pulses and Es/N0 = 15 dB. The simple approximation is used for Ps, and

the abscissa shows GS/I where I is the interference power of each equal-power interfering

signal. The figure shows that Ps increases with K, but the shift in Ps is mitigated

somewhat because the interference signals tend to partially cancel each other.

The preceding bounding methods can be extended to the bounds on Ps(φ) by observing

that cos2 φi ≤ 1 and setting X = cos2 φi during the successive applications of Jensen’s

inequality, which is applicable if (10-66) is satisfied. After evaluating (10-59), we obtain

Q

Ãr 2Es

N0 + hItTc

!

≤ Ps ≤ Q

Ãr 2Es

N0 + 2ItTc

!

(10-71)

where

It =

K

X−1

i=1

Ii (10-72)

134
5 6 7 8 9 10 11 12 13 14 15

100

10–1

10–2

10–3

10–4

10–5

10–6

Despread signal to interference ratio, dB

Symbol error probability

upper bound 

___ 

- - - 

accurate approx. 

simple approx. 

lower

bound 

Figure 63. Symbol error probability of direct-sequence system with PSK in presence of

single multiple-access interference signal and Es/N0 = 15 dB.

4 6 8 10 12 14 16

100

10–1

10–2

10–3

10–4

10–5

10–6

Despread signal-to-interference ratio, dB

Symbol error probability

K = 4 

3 

2 

Figure 64. Symbol error probability of direct-sequence system with PSK in presence of

K − 1 equal-power multiple-access interference signals and Es/N0 = 15 dB.

135
A simple approximation is provided by

Ps ≈ Q

Ãs

2Es

N0 + √2h ItTc

!

(10-73)

If Ps is specified, then the error in the required Es/It caused by using (10-73) instead of

(10-57) is bounded by 10 log10p2/h in decibels. Thus, the error is bounded by 2.39 dB for

rectangular pulses and 2.66 dB for sinusoidal pulses.

The lower bound in (10-71) gives the same result as that often called the standard

Gaussian approximation, in which V1 in (10-43) is assumed to be approximately Gaussian,

each φi in (10-47) is assumed to be uniformly distributed over [0, 2π), and each τi is

assumed to be uniformly distributed over [0, Tc). This approximation, gives an optimistic

result for Ps that can be as much as 4.77 dB in error for rectangular pulses according to

(10-71). The substantial improvement in accuracy provided by (10-69) or (10-54) is due to

the application of the Gaussian approximation only after conditioning V1 on given values of

φ and τ . The accurate approximation given by (10-54) is a version of what is often called

the improved Gaussian approximation.

Figure 65 illustrates the symbol error probability for 3 interferers, each with power I,

rectangular pulses, and Es/N0 = 15 dB as a function of GS/I. The graphs show the

standard Gaussian approximation of (10-71), the simple approximation of (10-73), and the

upper and lower bounds given by (10-61), (10-67), and (10-59). The large error in the

standard Gaussian approximation is evident. The simple approximation is reasonably

accurate if 10−6 ≤ Ps ≤ 10−2.

For synchronous networks, (10-54) and (10-55) can be simplified because the {τi} are all

zero. For either rectangular or sinusoidal pulses, we obtain

Ps(φ) = Q

Ãs

2Es

N0e(φ)

!

(10-74)

where

N0e(φ) = N0 +

K

X−1

i=1

2IiTc cos2 φi (10-75)

A comparison with (10-61) and (10-62) indicates that Ps for a synchronous network equals

or exceeds Ps for a similar asynchronous network when random spreading sequences are

used. This phenomenon is due to the increased bandwidth of a despread asynchronous

interference signal, which allows increased filtering in the receiver.

The accurate approximation of (10-54) follows from the standard central limit theorem,

which is justified by the lemma. This lemma depends on the restriction of the chip

136
5 10 15 20

100

10–1

10–2

10–3

10–4

10–5

10–6

Despread signal to interference ratio, dB

Symbol error probability

upper bound 

___ 

- - - 

standard Gaussian approx. 

simple approx. 

lower

bound 

Figure 65. Symbol error probability of direct-sequence system with PSK in presence of 3

equal-power multiple-access interference signals and Es/N0 = 15 dB.

waveform to the interval [0, Tc]. If the chip waveform extends beyond this interval but is

time-limited, as is necessary for implementation with digital hardware, then an extension of

the central limit theorem for m-dependent sequences can be used to derive an improved

Gaussian approximation [41]. Alternatives to the analysis in this section and the next one

abound in the literature (e.g., [41], [42]), but they are not as amenable to comparisons

among systems.

10.6 Quadriphase Direct-Sequence Systems with Random Sequences

Consider a network of quadriphase direct-sequence systems, each of which uses dual QPSK

and random spreading sequences. Each direct-sequence signal is given by (4-1) with t0 = 0.

A suitable model for multiple-access interference is

i(t) =

K

X−1

i=1

[

√

I q1i(t − τi) cos(2πfct + φi) + √I q2i(t − τi) sin(2πfct + φi)] (10-76)

where q1i(t) and q2i(t) both have the form of (10-2) and incorporate the data modulation.

The decision variables are given by (4-2) and (4-4) with G = Ts/Tc. A straightforward

137
calculation using (10-44) indicates that

Jν =

K

X−1

i=1

pIi{cos φi[q

(1i)

ν−1Rψ(τi) + q(1i) ν Rψ(Tc − τi)]

− sin φi[q

(2i)

ν−1Rψ(τi) + q(2i) ν Rψ(Tc − τi)]} (10-77)

The statistical independence of the two sequences, the lemma of Section 10.5, and

analogous results for J0

ν yield the variances of the interference terms of the decision

variables:

var(V1) = var(U1) =

K

X−1

i=1

1

2

Ts

Tc

Ii[R2

ψ(τi) + R2ψ(Tc − τi)] (10-78)

The noise variances and the means are given by (4-8) and (4-7). Since all variances and

means are independent of φ, the Gaussian approximation yields a Ps(φ, τ ) that is

independent of φ:

Ps =

μ 1

Tc

¶K−1 Z Tc

0

... Z Tc

0

Q

Ãs

2Es

N0e(τ)

!

dτ (10-79)

where

N0e(τ) = N0 +

K

X−1

i=1

Ii

Tc

[R2

ψ(τi) + R2ψ(Tc − τi)] (10-80)

Since a similar analysis for direct-sequence systems with balanced QPSK yields (10-80)

again, both quadriphase systems perform equally well against multiple-access interference.

Application of the previous bounding and approximation methods to (10-79) yields

Q

Ãr 2Es

N0 + hItTc

!

≤ Ps ≤ Q

Ãr 2Es

N0 + ItTc

!

(10-81)

where the total interference power It is defined by (10-72). A sufficient condition for the

validity of the lower bound is

Es ≥

3

2 (N0 + ItTc) (10-82)

A simple approximation that limits the error in the required Es/It for a specified Ps to 10

log10(1/

√

h) is

Ps ≈ Q

Ãs

2Es

N0 + √hItTc

!

(10-83)

This approximation introduces errors bounded by 0.88 dB and 1.16 dB for rectangular and

sinusoidal pulses, respectively, which are much less than the maximum errors for (10-73).

In (10-81) and (10-83), only the total interference power is relevant, not how it is

distributed among the individual interference signals.

Figure 66 illustrates Ps for a quadriphase direct-sequence system in the presence of 3

interferers, each with power I, rectangular pulses, and Es/N0 = 15 dB. The graphs plot the

138
5 10 15 20

100

10–1

10–2

10–3

10–4

10–5

10–6

Despread signal to interference ratio, dB

Symbol error probability

upper bound 

___ 

- - - 

accurate approx. 

simple approx. 

lower

bound 

Figure 66. Symbol error probability of quadriphase direct-sequence system in presence of

3 equal-power multiple-access interference signals and Es/N0 = 15 dB.

accurate approximation of (10-79), the simple approximation of (10-83), and the bounds of

(10-81) as functions of GS/I. A comparison of Figures 66 and 65 indicates the advantage

of a quadriphase system.

For synchronous networks with either rectangular or sinusoidal pulses, we set the {τi) equal

to zero in (10-79) and obtain

Ps = Q

Ãr 2Es

N0 + ItTc

!

(10-84)

Since this equation coincides with the upper bound in (10-81), we conclude that

asynchronous networks accommodate more multiple-access interference than similar

synchronous networks using quadriphase direct-sequence signals with random spreading

sequences. To compare asynchronous quadriphase direct-sequence systems with

asynchronous systems using binary PSK, we find a lower bound on Ps for direct-sequence

systems with PSK. Substituting (10-54) into (10-57) and applying Jensen’s inequality

successively to the integrations over φi, i = 1, 2, ..., K − 1, we find that a lower bound on

Ps is given by the right-hand side of (10-79) if (10-82) is satisfied. This result implies that

asynchronous quadriphase direct-sequence systems are more resistant to multiple-access

interference than asynchronous direct-sequence systems with binary PSK. The equations

for Ps allow the evaluation of the information-bit error probability Pb for error-correcting

codes with hard-decision decoders. To facilitate the analysis of soft-decision decoding, two

139
assumptions are necessary. Assume that there is enough symbol interleaving that symbol

errors are independent of each other and that K is large enough that the multiple-access

interference after despreading is approximately Guassian rather than conditionally

Gaussian. Since the equivalent power spectral density N0e is derived from the variance of a

zero-mean process, it can be obtained by averaging N0e(φ, τ) over the distributions of φ

and τ . With the application of (10-64), (10-72) and (10-80) yield

N0e = N0 + hItTc (10-85)

Thus, for a binary convolutional code with rate r, constraint length K, and minimum free

distance df , Pb is upper-bounded by (5-6) with

P2(l) = Q

Ãr2Esl

N0e

!

= Q

Ãr 2rEbl

N0 + hItTc

!

(10-86)

The system capacity is the number of equal-power users that can be accommodated while

achieving a specified Pb. For equal-power users, It = (K − 1)Es/Ts. Let γ1 denote the value

of Es/N0e necessary for a specific error correcting code to achieve the specified Pb. Equation

(10-85) implies that the system capacity is

K =

1

1 + G

h

μ 1

γ1

− 1

γ0

¶o , γ0 ≥ γ1 (10-87)

where bxc is the integer part of x, γ0 = Es/N0, G = Ts/Tc is the processing gain, and the

requirement γ0 ≥ γ1 is necessary to ensure that the specified Pb can be achieved for some

value of K. Since h < 1 in general, the factor G/h reflects the increased gain due to the

random distributions of interference phases and delays. If they are not random but

φ = τ = 0, then h = 1 and the number of users accommodated is reduced.

As an example, consider a system that resembles the downlink of the IS-95 CDMA

network. The data modulation is balanced QPSK and G = 64. The error-correcting code is

a rate-1/2 binary convolutional code with constraint length 9. The chip waveform is

approximately sinusoidal so that h = 1/3+5/2π2 = 0.587. If Pb = 10−5 or better is

desired, the performance curves for convolutional codes indicate that γ1 ∼= 3 dB is required.

Equation (10-87) then indicates that the system capacity is K = 44 if γ0 = 10 dB and

K = 54 if γ0 = 20 dB.

10.7 Wideband Direct-Sequence CDMA

A direct-sequence CDMA system is called wideband if it uses a spectral band with a

bandwidth that exceeds the coherence bandwidth of a frequency-selective fading channel.

The two most commonly proposed types of wideband systems are single-carrier and

140
multicarrier systems. A single-carrier direct-sequence CDMA system uses a single carrier

frequency to transmit signals. A multicarrier direct-sequence CDMA system partitions the

available spectral band among multiple direct-sequence signals, each of which has a distinct

carrier frequency. The main attractions of the multicarrier system are its potential ability

to operate over disjoint, noncontiguous spectral regions and its ability to avoid

transmissions in spectral regions with strong interference or where the multicarrier signal

might interfere with other signals. These features have counterparts in frequency-hopping

CDMA systems.

A single-carrier system provides diversity by using a rake receiver that combines several

multipath signals. A multicarrier system provides diversity by the maximal-ratio

combining of the parallel correlator outputs, each of which is associated with a different

carrier. Bit error probabilities are determined subsequently for ideal multicarrier and

single-carrier systems with lossless diversity combining in the presence of white Gaussian

noise and Rayleigh fading.

Multicarrier Direct-Sequence CDMA

A typical multicarrier system divides a spectral band of bandwidth W into L regions, each

of bandwidth W/L approximately equal to the coherence bandwidth [43]. In the

transmitter, the product d(t)p(t) of the data modulation d(t) and the spreading waveform

p(t) simultaneously modulates L carriers, each of which has its frequency in the center of

one of the L spectral regions, as illustrated in Figure 67(a). The receiver comprises L

parallel demodulators, one for each carrier, the outputs of which are suitably combined, as

indicated in Figure 67(b). The total signal power is divided equally among the L carriers.

The chip rate and, hence, the processing gain for each carrier of a multicarrier

direct-sequence system is reduced by the factor L. However, if strong interference exists in

a spectral region, the associated carrier can be omitted and the saved power redistributed

among the remaining carriers. Error correcting codes and interleaving can be used to

provide both time diversity and coding gain.

141
•

•

•

•

•

•

• •

d(t) p(t)

Transmitted

signal

cos (2πfLt +θL)

cos (2πf2t + θ2)

cos (2πf1t + θ1)

Σ

•

•

•

•

•

•

•

•

Received

signal

cos (2πfLt + θL)

cos (2πf2t + θ2)

cos (2πf1t+ θ1)

Demodulator

Demodulator

Demodulator

p(t)

p(t)

p(t)

Combiner Output

(a)

(b)

Figure 67. Multicarrier direct-sequence CDMA system: (a) transmitter and (b) receiver.

Since the spectral regions are defined so that the fading in each of them is independent and

frequency nonselective, rake combining is not possible, but the frequency diversity provided

by the regions can be exploited in a diversity combiner. Whether or not the diversity gain

exceeds that of a single-carrier system using the entire spectral band and rake combining

depends on the multipath intensity profile of the single-carrier system.

Consider a multicarrier system that uses binary PSK to modulate each carrier. Each

received signal copy with a different carrier frequency experiences independent Rayleigh

fading that is constant during a symbol interval. The received signal for a symbol in

branch i is

ri(t) = Re £

αiejθixψ(t)ej2πfit

¤

+ ni(t), 0 ≤ t ≤ Ts, i = 1, 2,...,L (10-88)

where x = +1 or −1 depending on the transmitted symbol, each αi is a fading amplitude,

each θi is a phase shift, fi is the carrier frequency, Ts is the symbol duration, and ni(t) is

the noise. Assume that either the interference is absent or, more generally, that the

received interference plus noise in each diversity branch can be modeled as independent,

zero-mean, white Gaussian noise with the same two-sided power spectral density N0/2.

143
Ideal lossless power splitting among the L carriers is assumed. Let E = Es/L denote the

received symbol energy per carrier in the absence of fading, where Es is the total received

energy per symbol. Assume that the spectral division among the carriers prevents

significant interference among them in the receiver. For coherent detection, a classical

analysis [44] indicates that the conditional symbol error probability given the {αi} is

Ps|α (γs) = Q

3p2γs ́

(10-89)

where

γs = E

N0

X

L

i=1

α2

i (10-90)

The symbol error probability is determined by averaging Ps|α(γs) over the distribution of

γs, which depends on the {αi} and embodies the statistics of the fading channel.

Suppose that independent Rayleigh fading occurs so that each of the {αi} is independent

with the identical Rayleigh distribution and E {α2

i } = α2. Since α2i is exponentially

distributed, γs is the sum of L independent, identically and exponentially distributed

random variables. It follows that (Appendix B-5) the probability density function of γs is

fγ(x) = 1

(L − 1)! ̄γL xL−1 exp μ

−x

γ ̄

¶

u(x) (10-91)

where u(x) = 1 if x ≥ 0 and u(x) = 0 if x < 0, and the average signal-to-noise ratio (SNR)

per branch is

γ ̄ = E

N0

α2 = Esα ̄2

LN0

(10-92)

The symbol error probability is determined by averaging (10-89) over the density given by

(10-91). Thus,

Ps(L) = Z ∞

0

Q

3√

2x

 ́ 1

(L − 1)! ̄γL xL−1 exp μ

−x

γ ̄

¶

dx (10-93)

Direct calculations verify that since L is an integer,

d

dxQ

3√

2x

 ́

= − 1

2

√π

exp(−x)

√x (10-94)

d

dx "

e−x/γ ̄X

L−1

i=0

(x/γ ̄)

i

i!

#

= − 1

(L − 1)! ̄γL xL−1 exp μ

−x

γ ̄

¶

(10-95)

Applying integration by parts to (10-93), and using (10-94), (10-95), and Q(0) = 1/2, we

obtain

Ps(L) = 1

2 −X

L−1

i=0

1

i! ̄γi2

√π

Z ∞

0

exp £−x

¡

1+ ̄γ−1¢¤ xi−1/2dx (10-96)

143
This integral can be evaluated in terms of the gamma function Γ(). A change of variable in

(10-96) yields

Ps(L) = 1

2 − 12

r γ ̄

1+ ̄γ

X

L−1

i=0

Γ(i + 1/2)

√πi!(1 +  ̄γ)i (10-97)

Since Γ(1/2) = √π, the symbol error probability for no diversity or a single branch is

p = Ps(1) = 1

2

μ

1 −

r γ ̄

1+ ̄γ

¶

(PSK, QPSK) (10-98)

Since Γ(x)=(x − 1)Γ(x − 1), it follows that

Γ(k + 1/2) =

√πΓ(2k)

22k−1Γ(k) =

√πk!

22k−1

μ

2k − 1

k

¶

, k ≥ 1 (10-99)

Solving (10-98) to determine  ̄γ as a function of p and then using this result and (10-99) in

(10-97) gives

Ps(L) = p − (1 − 2p)

X

L−1

i=1

μ

2i − 1

i

¶

[p(1 − p)]i (10-100)

This expression explicitly shows the change in the symbol error probability as the number

of diversity branches increases. Equations (10-98) and (10-99) are valid for

quadriphase-shift keying (QPSK) because the latter can be transmitted as two independent

binary PSK waveforms in phase quadrature.

An alternative expression for Ps(L), which may be obtained by a far more complicated

calculation entailing the use of the properties of the Gauss hypergeometric function, is [44],

[45]

Ps(L) = pLX

L−1

i=0

μ

L + i − 1

i

¶

(1 − p)

i (10-101)

By using mathematical induction, this equation can be derived from (10-100) without

invoking the hypergeometric function.

Figure 68 plots Ps(L) for multicarrier systems as a function of Esα2/N0, the average symbol

signal-to-noise ratio (SNR). The diminishing returns as the diversity level L increases is

apparent. If the required bit error probability is 10−6 or more, than increasing L beyond

L = 32 is not likely to be useful because of the hardware requirements and the losses

entailed in the power division in the transmitter.

144
0 2 4 6 8 10 12 14 16 18 20

Average symbol SNR, dB

Symbol error probability

100

10–1

10–2

10–3

10–4

10–5

10–6

L = 64

4

8

16

32

Figure 68. TSymbol error probability for multicarrier systems with L carriers.

Rake Receiver for Single-Carrier Direct-Sequence CDMA

Consider a direct-sequence signal that has a random spreading sequence and is

accompanied by multipath components in addition to the direct-path signal. If the

multipath components are delayed by more than one chip, then the independence of the

chips ensures that the multipath interference is suppressed by at least the processing gain.

However, since multipath signals carry information, they are a potential resource to be

exploited rather than merely rejected. A rake receiver [44] provides path diversity by

coherently combining the resolvable multipath components present during

frequency-selective fading, which occurs when the chip rate of the spreading sequence

exceeds the coherence bandwidth. Consider a multipath channel with frequency-selective

fading slow enough that its time variations are negligible over a signaling interval. To

harness the energy in all the multipath components, a receiver should decide which signal

was transmitted among M candidates, s1(t), s2(t),...,sM(t) only after processing all the

received multipath components of the signal. Thus, the receiver selects among the M

baseband signals or complex envelopes

νk(t) = X

L

i=1

cisk (t − τi) , k = 1, 2,...,M, 0 ≤ t ≤ T + Td (10-102)

where T is the duration of the transmitted signal, Td is the multipath delay spread, L is

the number of multipath components, τi is the delay of component i, and the channel

145
parameter ci is a complex number representing the attenuation and phase shift of

component i.

When the data modulation is binary PSK, only a single symbol waveform s1(t) and its

associated decision variable are needed. Assume the presence of zero-mean, white Gaussian

noise with two-sided power spectral density N0/2. If αi = |ci|, then it can be shown [44]

that for a rake receiver with perfect tap weights, the conditional bit error probability given

the {αi} is provided by (10-89). However, for a rake receiver, each of the {αi} is associated

with a different multipath component, and hence each E[α2

i ] has a different value in

general. Since there is only a single carrier, we may set E = Es in (10-90), which may be

expressed as

γs = X

L

i=1

γi, γi = Es

N0

α2

i (10-103)

If each αi has a Rayleigh distribution then each γi has the exponential probability density

function

fγi = 1

γ ̄i

exp μ− x

γ ̄i

¶

u(x), i = 1, 2,...,L (10-104)

where the average SNR for a symbol in branch i is

γ ̄i = Es

N0

E £α2

i

¤

, i = 1, 2,...,L (10-105)

If each multipath component fades independently so that each of the {γi} is statistically

independent, then γs is the sum of independent, exponentially distributed random

variables. Therefore, if  ̄γi 6=  ̄γj , j 6= k, then the probability density function of γs is

(Appendix B-5)

fγ(x) = X

L

i=1

Ai

γ ̄i

exp μ− x

γ ̄i

¶

u(x) (10-106)

where

Ai =







Q

L

k=1

k6=i

γ ̄i

γ ̄i−γ ̄k , L ≥ 2

1 , L = 1

(10-107)

The symbol error probability is determined by averaging the conditional symbol error

probability of (10-89) over the density given by (10-106). A derivation similar to that

leading to (10-97) yields

Ps(L) = 1

2

X

L

i=1

Ai

μ

1 −

r γ ̄i

1+ ̄γi

¶

(10-108)

146
Since only white Gaussian noise is present, the processing gain of the system is irrelevant.

To assess the potential performance of the rake receiver, it is assumed that the largest

multipath component has  ̄γ1 =  ̄γ and that L ≤ 4 components are received. The other three

or fewer minor multipath components have relative average symbol SNRs specified by the

multipath intensity vector

Γ =

μγ ̄2

γ ̄ ,

γ ̄3

γ ̄ ,

γ ̄4

γ ̄

¶

(10-109)

Figure 69 plots the symbol error probability as a function of  ̄γ = Esa2

1/N0, the average

symbol SNR of the main component, for representative multipath intensity vectors. This

figure and other numerical data establish two basic features of single-carrier systems with

rake receivers.

0 2 4 6 8 10 12 14 16 18 20

100

10-1

10-2

10-3

10-4

10-5

10-6

Average symbol SNR, dB

Symbol error probability

(.505, .495, 0)

(.340, .335, .325)

(.995, .990, 0)

(1, 0, 0)

no rake

. . . . .

. .

Figure 69. Symbol error probability for single-carrier systems and L ≤ 4 multipath components with different multipath intensity vectors.

1. System performance improves as the total energy in the minor multipath components

increases.

2. When the total energy in the minor multipath components is fixed, the system

performance improves as the number of resolved multipath components L increases and as

the energy becomes uniformly distributed among these components.

147
The improvement as L increases depends on the assumption that ideal channel estimation

is available to the rake receiver. However, when a practical channel estimator is used,

measurements indicate that only three or fewer components are likely to have a sufficient

signal-to-interference ratio to be useful in the rake combining [46]. An increase in L due to

an increase in the bandwidth W is not always beneficial for a practical system. Although

new components provide additional diversity and may exhibit the more favorable Ricean

fading rather than Rayleigh fading, the average power per multipath component decreases

because some composite components fragment into more numerous but weaker components.

Hence, the estimation of the channel parameters becomes more difficult, and the fading of

some multipath components may be highly correlated rather than independent.

In mobile CDMA networks, typically three significant multipath components for which

γ ̄2 +  ̄γ3 has a value between  ̄γ and 2 ̄γ are potentially available. Thus, the graphs with

Γ = (.505, .495, 0) and Γ = (.995, .990, 0) represent reasonable approximations of relatively

unfavorable and favorable multipath intensity vectors, respectively.

A comparison of Figures 68 and 69 indicates that a multicarrier system with diversity

L = 32 outperforms single-carrier systems with diversityL = 4 if  ̄γ is sufficiently large.

However, this value  ̄γ is much larger than is required in practical systems. To make a more

realistic comparison, we assume that an error-correcting code with ideal channel-symbol

interleaving is used. For a loosely packed, binary block code and hard-decision decoding

with a bounded-distance decoder, the information-bit error probability is [47]

Pb ≈ Xn

i=t+1

μ

n − 1

i − 1

¶

Pi

s (1 − Ps)

n−1 (10-110)

where n is the code length, t is the number of symbol errors that the decoder can correct,

and Ps is the channel-symbol error probability. The signal energy per channel symbol is

Es = rEb, where r = k/n is the code rate, k is the number of information bits per codeword,

and Eb is the energy per information bit. We may evaluate Ps by using the expressions for

Ps(L) with  ̄γ = rγ ̄b, where  ̄γb is the average bit SNR.

As an example, we assume that a BCH(63, 36) code with n = 63, k = 36, and t = 5 is used.

Figure 70 plots Pb for a multicarrier system with L = 32 and single-carrier systems with

Γ1 = (.505, .495, 0) and Γ2 = (.995, .990, 0). If the required Pb exceeds 10−5, then the rake

combining provides more than a 0.5 dB advantage when the multipath intensity vector is

Γ1 and more than a 2.1 dB advantage when it is Γ2.

The preceding results imply that in a benign environment, devoid of partial-band

interference, a multicarrier system suffers a potential performance loss relative to the less

costly single-carrier system. The underlying reason is that the rake receiver of the

signel-carrier system harnesses energy that would otherwise be unavailable. In contrast, the

multicarrier receiver recovers energy that has been redistributed among the L carriers but

148
0 1 2 3 4 5 6 7 8

100

10-1

10-2

10-3

10-4

10-5

10-6

Average bit SNR, dB

Bit error probability

multicarrier

Γ2

Γ1

Figure 70. Information-bit error probability for multicarrier system with L = 32 and for

single-carrier systems with Γ1 = (.505, .495, 0) and Γ2 = (.995, .990, 0). Errorcorrecting code is BCH(63, 36).

is available to the single-carrier system even without rake combining. Despite its potential

disadvantage in a benign environment, a multicarrier direct-sequence CDMA system will

often be preferable to single-carrier systems because of its substantially superior

performance against partial-band interference [43].

10.8 Cellular Networks and Power Control

In a cellular network, a geographic region is partitioned into cells, as illustrated in Figure

71. A base station that includes a transmitter and receiver is located at the center of each

cell. Ideally, the cells have equal hexagonal areas. Each mobile (user or subscriber) in the

network transmits omnidirectionally and communicates with the base station from which it

receives the largest average power. Typically, most of the mobiles in a cell communicate

with the base station at the center of the cell, and only a few communicate with more

distant ones. The base stations act as switching centers for the mobiles and communicate

among themselves by wirelines in most applications. Cellular networks with

direct-sequence CDMA allow universal frequency reuse in that the same carrier frequency

and spectral band is shared by all the cells. Distinctions among the direct-sequence signals

are possible because each signal is assigned a unique spreading sequence.

149
0

Sector coverage

Figure 71. Geometry of cellular network with base station at center of each hexagon. Two

concentric tiers of cells surrounding a central cell are shown.

Cells may be divided into sectors by using several directional sector antennas or arrays at

the base stations. Only mobiles in the directions covered by a sector antenna can cause

multiple-access interference on the reverse link or uplink from a mobile to its associated

sector antenna. Only a sector antenna serving a cell sector oriented toward a mobile can

cause multiple-access interference on the forward link or downlink from the mobile’s

associated sector antenna to the mobile. Thus, the numbers of interfering signals on both

the uplink and the downlink are reduced approximately by a factor equal to the number of

sectors.

To facilitate the identification of a base station controlling communications with a mobile,

each spreading sequence for a downlink is formed as the product or concatenation of two

sequences often called the scrambling and channelization codes. A scrambling code is a

sequence that identifies a particular base station when the code is acquired by mobiles

associated with the base station and its cell or sector. A long sequence is preferable to

minimize the possibility of a prolonged outage. If the set of base stations use the Global

Positioning System or some other common timing source, then each scrambling code may

be a known phase shift of a common long pseudonoise sequence. If a common timing

source is not used, then at the cost of increased acquisition time or complexity, the

scrambling codes may comprise a set of long Gold sequences that approximate random

binary sequences. Walsh or other orthogonal sequences are suitable as channelization codes

that allow each mobile to extract its messages while blocking messages to other mobiles

within the same cell or sector. For the uplinks, channelization codes are not strictly

necessary, and the scrambling codes that identify the mobiles may be drawn from a set of

long Gold sequences.

150
The principal difficulty of direct-sequence CDMA is called the near-far problem. If all

mobiles transmit at the same power level, then the received power at a base station is

higher for transmitters near the receiving antenna. There is a near-far problem because

transmitters that are far from the receiving antenna may be at a substantial power

disadvantage, and the spread-spectrum processing gain may not be enough to allow

satisfactory reception of their signals. A similar problem may also result from large

differences in received power levels due to differences in the shadowing experienced by

signals traversing different paths or due to independent fading. In cellular communication

networks, the near-far problem is critical only on the uplink because on the downlink, the

base station transmits orthogonal signals synchronously to each mobile associated with it.

For cellular networks, the usual solution to the near-far problem of uplinks is power control,

whereby all mobiles regulate their power levels. By this means, power control potentially

ensures that the power arriving at a common receiving antenna is almost the same for all

transmitters. Since solving the near-far problem is essential to the viability of a

direct-sequence CDMA network, the accuracy of the power control is a crucial issue.

In networks with peer-to-peer communications, there is no cellular or hierarchical structure.

Communications between two mobiles are either direct or are relayed by other mobiles.

Since there is no feasible method of power control to prevent the near-far problem,

direct-sequence CDMA systems are not as attractive an option as frequency-hopping

CDMA systems in these networks.

An open-loop method of power control in a cellular network causes a mobile to adjust its

transmitted power to be inversely proportional to the received power of a pilot signal

transmitted by the base station. An open-loop method is used to initiate power control,

but its subsequent effectiveness requires that the propagation losses on the forward and

reverse links be nearly the same. Whether they are or not depends on the duplexing

method used to allow simultaneous or nearly simultaneous transmissions on both links.

Frequency-division duplexing assigns different frequencies to an uplink and its

corresponding downlink. Time-division duplexing assigns closely spaced but distinct time

slots to the two links. When frequency-division duplexing is used, as in the IS-95 and

Global System for Mobile (GSM) standards, the frequency separation is generally wide

enough that the channel transfer functions of the uplink and downlink are different. This

lack of link reciprocity implies that power measurements over the downlink do not provide

reliable information for subsequent uplink transmissions. When time-division duplexing is

used, the received local-mean power levels for the uplink and the downlink will usually be

nearly equal when the transmitted powers are the same, but the Rayleigh fading may

subvert link reciprocity. For these reasons, a closed-loop method of power control, which is

more flexible than an open-loop method, is desirable. A closed-loop method requires the

base station to transmit power-control information to each mobile based on the power level

received from the mobile or the signal-to-interference power ratio.

151
When closed-loop power control is used, each base station attempts to either directly or

indirectly track the received power of a desired signal from a mobile and dynamically transmit

a power-control signal. The effect of increasing the carrier frequency or the mobile speeds

is to increase the fading rate. As the fading rate increases, the tracking ability and, hence,

the power-control accuracy decline. This problem and an analysis of intercell and intracell

interference are presented in references [48] and [49].

11. Multiuser Detectors

The conventional single-user direct-sequence receiver of Figure 14 is optimal against

multiple-access interference only if the spreading sequences of all the interfering signals and

the desired signal are orthogonal (Section 10.1). Orthogonality is possible in a synchronous

communication network, but in an asynchronous network, it is not possible to find

sequences that remain orthogonal for all relative delays. Thus, the conventional single-user

receiver, which only requires knowledge of the spreading sequence of the desired signal, is

suboptimal against asynchronous multiple-access interference. The price of the

suboptimality might be minor if the spreading sequences are carefully chosen and the noise

is relatively high, especially if an error-correcting code and a sector antenna or adaptive

array are used. If a potential near-far problem exists (Section 10.8), power control may be

used to limit its impact. However, power control is imperfect, entails a substantial

overhead cost, and is not feasible for peer-to-peer communication networks. Even if the

power control is perfect, the remaining interference causes a nonzero error floor, which is a

minimum bit error probability that exists when the thermal noise is zero. Thus, an

alternative to the conventional receiver is desirable.

A multiuser detector is a receiver that exploits the deterministic structure of

multiple-access interference or uses joint processing of a set of multiple-access signals. An

optimum multiuser detector almost completely eliminates the multiple-access interference

and, hence, the near-far problem, thereby rendering power control unnecessary. A more

practical multiuser detector alleviates but does not eliminate the power-control

requirements of a cellular network on its uplinks. Even if a multiuser detector rejects

intracell interference from mobiles within a cell, it cannot reject intercell interference,

which arrives from mobiles associated with different base stations than the one receiving a

desired signal. Since intercell interference is typically more than one-third of the total

interference on an uplink, even ideal multiuser detection will increase network capacity by

a factor less than three. Multipath components can be accommodated as separate

interference signals or rake combining may precede the multiuser detection.

152
11.1 Optimum Detectors

Consider a direct-sequence CDMA network with K users, each of which uses PSK to

transmit a block of N binary symbols. A jointly optimum detector makes collective symbol

decisions for K received signals based on the maximum a posteriori (MAP) criterion. The

individually optimum detector selects the most probable set of symbols of a single desired

signal from one user based on the MAP criterion, thereby providing the minimum symbol

error probability. In nearly all applications, jointly optimum decisions would be preferable

because of their lower complexity and because both types of decisions will agree with very

high probability unless the symbol error probability is very high. Assuming equally likely

symbols are transmitted, the jointly optimum MAP detector is the same as the jointly

optimum maximum-likelihood detector, which is henceforth referred to as the optimum

detector.

For synchronous communications in the presence of white Gaussian noise, the symbols are

aligned in time, and the detection of each symbol of the desired signal is independent of the

other symbols. Thus, the optimum detector can be determined by considering a single

symbol interval 0 ≤ t ≤ Ts. Let dk denote the symbol transmitted by user k. The

customary (highly idealized) assumption is that a perfect carrier synchronization enables

the receiver to remove a common carrier frequency and phase. Thus, the composite

baseband received signal is

r(t) = X

K

k=1

Akdkpk(t) + n(t) 0 ≤ t ≤ Ts (11-1)

where Ak is the received symbol amplitude from user k, pk(t) is the unit-energy spreading

waveform of user k, dk = ±1, and n(t) is the baseband Gaussian noise. If it is assumed that

each of the K signals has a common carrier frequency but a distinct phase relative to the

phase of the receiver-generated synchronization signal, then each Ak in (11-1) is replaced

by Ak cos φk, where φk is the relative phase of the signal from mobile k.

Assuming that all possible values of the symbol vector d = [d1 ...dK]

T are equally likely,

the optimum detector is the maximum-likelihood detector [44], [50], which selects the value

of d that minimizes the log-likelihood function

Λ(d) = Z Ts

0

"

r(t) −X

K

k=1

Akdkpk(t)

#2

dt (11-2)

subject to the constraint that dk = +1 or −1. The vector of the cross correlations between

r(t) and the spreading sequences is denoted by θ = [r1 r2 ...rK]

T , where

rk =

Z Ts

0

r(t)pk(t)dt , k = 1, 2,...,K (11-3)

153
Let A denote the K × K diagonal matrix with diagonal components A1, A2,...,AK. Let R

denote the K × K correlation matrix with elements

Rik =

Z Ts

0

pi(t)pk(t)dt , i, k = 1, 2,...,K (11-4)

where Rii = 1 and |Rik| ≤ 1 because the spreading waveforms are normalized to unit

energy. Expanding (11-2), dropping an integral that is irrelevant to the selection of d, and

then substituting (11-3) and (11-4), we find that the maximum-likelihood detector selects

the value of d that maximizes the correlation metric

C = 2dTAθ − dTARAd (11-5)

subject to the constraint that dk = +1 or −1, k = 1, 2,...,K.

This equation implies that the optimum detector uses a filter bank of K parallel

correlators. Correlator k computes rk given by (11-3) and can be implemented as the

single-user detector of Figure 14. Equation (11-5) also indicates that the K spreading

sequences must be known so that R can be calculated, and the K signal amplitudes must

be estimated. Short spreading sequences are necessary or R must change with each

symbol. The optimum detector is capable of making joint symbol decisions for all K

signals or merely the symbol decisions for a single signal.

As an example, consider synchronous communications with K = 2 and R12 = ρ. After the

elimination of terms irrelevant to the selection, (11-5) implies that the optimum detector

evaluates C = 2A1d1θ1 + 2A2d2θ2 − 2ρA1A2d1d2 for the four pairs with d1 = ±1 and d2 ± 1.

The pair that maximizes C provides the decisions for d1 and d2.

For asynchronous communications over the AWGN channel, the derivation of the

maximum-likelihood detector is analogous but more complicated [44], [50]. A major

difference is that a desired symbol overlaps two consecutive symbols from each interference

signal. The optimum detector uses a filter bank of K parallel correlators, but N symbols

from each correlator must be processed to make decisions about NK binary symbols. The

vector d is NK × 1 with the first N elements representing the symbols of signal 1, the

second N elements representing the symbols of signal 2, etc. The detector must estimate

the transmission delays of all K multiple-access signals, and the NK × NK correlation

matrix R has components that are partial cross correlations among the signals. In

principle, the detector must compute 2NK correlation metrics and then select K symbol

sequences, each of length N, corresponding to the largest correlation metric. The Viterbi

algorithm simplifies computations by exploiting the fact that each received symbol overlaps

at most 2(K − 1) other symbols. Nevertheless, the computational complexity increases

exponentially with K.

In view of both the computational requirements and the parameters that must be

estimated, it is highly unlikely that the optimum multiuser detector will have practical

154
applications. Subsequently, alternative suboptimal multiuser detectors are considered. All

of them follow carrier removal with a filter bank of correlators.

11.2 Decorrelating detector

The decorrelating detector may be derived by maximizing the correlation metric of (11-5)

without any constraint on d. For this purpose, the gradient of f with respect to the

n-dimensional, real-valued vector x is defined as the column vector ∇xf with components

∂f/∂xi, i = 1, 2,...,n. From this definition, it follows that for column vectors x and y

∇x

¡

xT y

¢

= ∇x

¡

yT x

¢

= y (11-6)

If A is an n × n symmetric matrix, then expressing xTAx in component form and using

the chain rule yields

∇x

¡

xTAx¢= 2Ax (11-7)

Applying (11-6) and (11-7) to the correlation metric, we find that ∇xC = 0 implies that C

is maximized by d = d0, where

A d0 = R−1θ (11-8)

provided that R is invertible. Since each component of the vector A d0 is a positive

multiple of the corresponding component of d0, there is no need to solve for d0. A suitable

estimate of the transmitted bits is

dˆ = sgn ¡R−1θ

¢ (11-9)

where each component of the vector sgn(x) is the signum function of the corresponding

component of the vector x. The signum function is defined as sgn(x)=1, x ≥ 0, and

sgn(x) = −1,x< 0. The decorrelating detector, which implements (11-9), has the form

diagrammed in Figure 72. For asynchronous communications, each of the K correlators in

the filter bank produces N bits. The accumulator constructs the NK-dimensional vector θ

and the linear transformer computes R−1θ. The decision devices evaluate (11-9) to

produce dˆ.

A second derivation of the decorrelating detector assumes that the detector has the filter

bank as its first stage. If (11-1) gives the input, then the output of this stage is

θ = RAd + n (11-10)

where n is the NK-dimensional noise vector. This equation indicates that the coupling

among components of d, which causes the correlation among components of θ, is due solely

to the presence of the matrix R. The effect of this matrix is removed by computing

R−1

θ = A d + R−1n (11-11)

155
•

•

•

•

•

•

Received

signal

DD-K

DD-2

DD-1

Decision

devices

Accumulator

and linear

transformer

Correlator K

Correlator 2

Correlator 1

Filter bank

Figure 72. Architecture of decorrelating detector and MMSE detecter. Filter bank comprises parallel correlators.

If (11-9) is used to determine the NK transmitted bits, the multiple-access interference is

completely decorrelated from dˆ.

A third derivation assumes the presence of the filter bank. If zero-mean, white Gaussian

noise with two-sided power spectral density N0/2 enters each correlator, then a

straightforward calculation indicates that the NK × NK covariance matrix of θ is

E £nnT ¤

= N0

2

R (11-12)

The probability density function of θ given A d is

f(θ|A d)=(N0π det R)

−K/2

exp h−N −1

0 (θ − RAd)

T R−1 (θ − RAd)

i

(11-13)

The maximum-likelihood estimate of A d is the vector that maximizes

(11-13) or, equivalently, minimizes the log-likelihood function

Λ(A d)=(θ − RAd)

T R−1

(θ − RAd) (11-14)

Using (11-6) and (11-7), we again obtain the estimate given by (11-8), which leads to dˆ

given by (11-9).

Although the decorrelating detector eliminates the multiple-access interference, it increases

the noise by changing n to R−1n. From (11-12), and the symmetric character of R, it

follows that the covariance matrix of the noise vector R−1n entering the decision devices is

E £R−1nnTR−1

¤

= N0

2

R−1 (11-15)

The variance of the noise that accompanies one of the symbols of user k is (N0/2)(R−1)kk.

Therefore, the symbol error probability is

Ps(k) = Q

Ãs

2Ek

N0 (R−1)kk!

, k = 1, 2,...,K (11-16)

156
where Ek = A2

k is the symbol energy. The symbol error probability for single-user detection

by user k in the absence of multiple access interference is

Ps(k) = Q

Ãr2Ek

N0

!

, k = 1, 2,...,K (11-17)

Thus, the presence of multiple-access interference requires an increase of energy by the

factor (R−1)kk when the decorrelating detector is used if a specified Ps is to be maintained.

As an example, consider synchronous communications with K = 2 and R12 = R21 = ρ. The

correlation matrix and its inverse are

R =

∙

1 ρ

ρ 1

 ̧

, R−1 = 1

1 − ρ2

∙ 1 −ρ

−ρ 1

 ̧

(11-18)

Equation (11-9) indicates that the symbol estimates are ˆd1 = sgn(r1 − ρr2) and

ˆ

d2 = sgn(r2 − ρr1). Since (R−1)11 = (R−1)22 = (1 − ρ2)−1,

Ps(k) = Q





s

2Ek (1 − ρ2)

N0



 , k = 1, 2 (11-19)

If ρ ≤ 1/2, the required energy increase or shift in each Ps curve is less than 1.25 dB.

To demonstrate analytically the advantage of the decorrelating detector, consider

synchronous communications and a receiver with a filter bank of K conventional detectors.

Each conventional detector is a single-user matched filter. If perfect carrier synchronization

removes a common phase shift of all the signals and produces the baseband received signal

of (11-1), then (11-3) implies, that the output of detector k is

rk = dkAk +X

K

i=1

i6=k

diAiRik +

Z Ts

0

n(t)pk(t)dt (11-20)

The set of K symbols is estimated by

ˆ

d = sgn(θ) (11-21)

By symmetry, we can assume that dk = 1 in the evaluation of the symbol error probability.

Let Dk denote the (K − 1)-dimensional vector of all the di, i 6= k. Conditioning on Dk and

calculating that var(rk) = N0/2, we find that the conditional symbol error probability for

user k is

Ps (k|Dk) = Q

Ãr2Ek

N0

Bk

!

(11-22)

157
where

Bk =1+X

K

i=1

i6=k

diRik

Ai

Ak

(11-23)

If all symbol sets are equally likely, then the symbol error probability for user k is

Ps(k)=2−(K−1)

2

XK−1

j=1

Ps (k| Dkj ) (11-24)

where Dkj is the jth choice of the vector Dk, which can take 2K−1 values.

For K = 2 with R12 = R21 = ρ and A2/A1 = √E2/

√E1, (11-22) to (11-24) yield the symbol

error probability for user 1:

Ps(1) = 1

2

Q

Ãr2E1

N0

Ã

1 − ρ

rE2

E1

!! +

1

2

Q

Ãr2E1

N0

Ã

1 + ρ

rE2

E1

!!

= 1

2

Q

Ãr2E1

N0

Ã

1 − |ρ|

rE2

E1

!! +

1

2

Q

Ãr2E1

N0

Ã

1 + |ρ|

rE2

E1

!!

(11-25)

The symbol error probability for user 2 is given by (11-25) with the roles of E1 and E2

interchanged. The second term in (11-25) is usually negligible compared with the first one

if ρ 6= 0. Thus, if

ρ2E2/E1 >

3

1 − p1 − ρ2

 ́2

(11-26)

then a comparison of (11-25) with (11-19) indicates that decorrelating detector usually

outperforms the conventional detector. However, if E2 is sufficiently small, then the

conventional detector gives a lower Ps than the decorrelating detector.

In a more realistic model of the decorrelating and conventional detectors, the received

signal in Figure 72 is passband. Correlator k uses a synchronized carrier to remove carrier

k at the common frequency fc. Since each carrier has a distinct phase φk, the elements of

the correlation matrix are

Rik = cos (φk − φi)

Z Tc

0

pi(t)pk(t)dt , i, k = 1, 2,...,k (11-27)

if fcTc >> 1. For synchronous communications with K = 2, (11-19) and (11-25) with

ρ = R12 = R21 then represent the conditional symbol error probability given the value of

φ = φk − φi. Averaging over φ is necessary to obtain Ps(1) and Ps(2).

Compared with the optimum detector, the decorrelating detector offers greatly reduced,

but still formidable, computational requirements. There is no need to estimate the signal

158
amplitudes, but the transmission delays of asynchronous signals must still be estimated.

The inversion of the correlation matrix R in real time is not possible for asynchronous

signals with practical values of NK. Suboptimal partitioning and short spreading

sequences are generally necessary and degrade the theoretical performance given by (11-16).

11.3 MMSE Detector

The minimum mean-square-error (MMSE) detector is the receiver that results from a

linear transformation of θ by the K × K matrix L such that the metric

M = E £kd − Lθk2¤ (11-28)

is minimized. Let L0 denote the solution of the equation

E £(d − L0θ) θT ¤

= 0 (11-29)

Let tr(B) denote the trace of the matrix B. Since kxk2 = tr(xxT ) for a vector x,

kd − Lθk2 = tr n[d − L0θ + (L0 − L) θ] [d − L0θ + (L0 − L) θ]

T

o

= kd − L0θk2 + k(L0 − L) θk2 + 2tr h(d − L0θ) θT (L0 − L)

T

i

(11-30)

Substitution of this equation into (11-28) and the application of (11-29) yields

M = E £kd − L0θk2¤+ E £k(L0 − L) θk2¤≥ E £kd − L0θk2¤ (11-31)

which proves that L0 minimizes M. If the data symbols are independent and equally likely

to be +1 or −1, then E[ddT] = I, where I is the identity matrix. Using this result, (11-10),

(11-12), E[n] = 0, and the independence of d and n, we obtain

E £dθT ¤

= A R , E £θθT ¤= R A2R +

N0

2

R (11-32)

Substitution of these equations into (11-29) yields

L0 =

μ

RA +

N0

2

A−1

¶−1

= A−1

μ

R +

N0

2

A−2

¶−1

(11-33)

provided that the inverses exist. Since A and, hence, A−1 are diagonal matrices with

positive diagonal components if all signals are active, (11-33) may be simplified to the

linear transformation matrix

L0 =

μ

R +

N0

2

A−2

¶−1

(11-34)

159
without any change in the MMSE estimate of the transmitted symbols:

ˆ

d = sgn(L0θ) (11-35)

The MMSE detector has the structure of Figure 72.

The MMSE and decorrelating detectors have almost the same computational requirements,

and they both have equalizer counterparts, but they differ in several ways. The MMSE

detector is near-far resistant, but does not obliterate the multiple-access interference and,

hence, does not completely eliminate the near-far problem. However, it does not accentuate

the noise to the degree that the decorrelating detector does. As N0 → 0, L0 → R−1 and

the MMSE estimate approaches the decorrelating detector estimate. As N0 increases, the

MMSE estimate approaches that of the conventional detector given by (11-21), and the

symbol error probability generally tends to be lower than that provided by the

decorrelating detector. A disadvantage of the MMSE detector is that the signal amplitudes

must be estimated so that A in (11-34) can be computed.

For either the MMSE or decorrelating linear detectors to be practical, it is necessary for

the spreading sequences to be short. Short sequences ensure that the correlation matrix R

is approximately constant for significant time durations if the communication channel and

the amplitudes of the interference signals are slowly varying. The price of short sequences

is a security loss and the occasional but sometimes persistent performance loss due to a

particular set of relative signal delays. Even with short spreading sequences, adaptive

versions of the MMSE detector are much more practical than the nonadaptive versions of

either linear detector.

An adaptive multiuser detector [50] is one that does not require explicit knowledge of either

the spreading sequences or the timing of the multiple-access interference signals. The

receiver samples the output of a wideband filter at the chip rate. The use of short

spreading sequences affords the opportunity for the adaptive detector to essentially learn

the sequence cross-correlations and thereby to suppress the interference. The learning is

accomplished by processing a known training sequence of symbols for the desired signal

during a training phase. This operational phase is followed by a decision-directed phase

that continues the adaptation by feeding back symbol decisions. Adaptive detectors

potentially can achieve much better performance than conventional ones at least if the

transmission channel is time-invariant, but coping with fast fading and interference changes

requires elaborate modifications. A blind adaptive detector [51] is one that does not require

training sequences. These detectors are desirable for applications such as system recovery

but entail some performance loss and complexity increase relative to other adaptive

detectors. Long sequences do not possess the cyclostationarity that makes possible many of

the advanced signal processing techniques used for blind multiuser detection and adaptive

channel estimation.

160
11.4 Interference Cancellers

An interference canceller is a multiuser detector that explicitly estimates the interference

signals and then subtracts them from the received signal to produce the desired signal.

Interference cancellers may be classified as successive interference cancellers in which the

subtractions are done sequentially, parallel interference cancellers in which the subtractions

are done simultaneously, or hybrids of these types. Only the basic structures and features

of the successive and parallel cancellers are presented subsequently. A large number of

alternative versions, some of them hybrids, adaptive, or blind, have been proposed in the

literature [50]. Some type of interference canceller is by far the most practical multiuser

detector for an asynchronous direct-sequence CDMA network, especially if long spreading

sequences are planned [52].

11.5 Successive Interference Canceller

Figure 73 is a functional block diagram of a successive interference canceller, which uses

nonlinear replica generations and subtractions to produce estimates of the symbol streams

dˆ1, dˆ2,..., dˆK transmitted by the K users. The input may be the sampled outputs of a

chip-matched filter for PSK modulation or the complex-valued samples derived from a

quadrature demodulator for quaternary modulation. Detector-generator i produces a

replica of the signal transmitted by user i. Its basic structure is depicted in Figure 74. The

correlator, which comprises a multiplier and summer as in Figure 14, despreads signal i.

The channel estimate is a stream of complex numbers that are applied to the correlator

output to remove the effects of the propagation channel. The decision device produces the

estimated symbols transmitted by user i. These symbols are remodulated and modified to

account for the effects of the channel. After a respreading, the replica of signal i is

generated and sent to the corresponding subtracter. The input or output of the decision

device may produce the estimated symbol stream, depending on whether the decoder uses

soft or hard decisions. The channel estimator uses known pilot or training symbols to

determine the effect of the channel. Hard decisions are used in the replica generation, but

may not be appropriate if the channel estimate is inaccurate because a symbol error doubles

the amplitude of the interference that enters the next stage of the canceller of Figure 73.

The enhanced interference adversely affects subsequent symbol estimates and replicas.

Input

samples residual

d1 d2 dK

Replica K Detectorgenerator K Replica 2 Detectorgenerator 2 Replica 1 Detectorgenerator 1

^ ^ ^

∑ ∑ ∑

• • •

+ + +

– – –

Delay Delay Delay

Figure 73. Successive interference canceller with K detector-generators to produce signal

estimates for subtraction.

161
Input

samples

Replica i

Channel estimate

Channel

estimator

Remodulator Respreader Decision

device

Correlator i

Figure 74. Structure of detector-generator for signal i.

The outputs of a set of K correlators and level detectors are applied to a device that orders

the K received signals according to their power levels. This ordering determines the

placement of the detector-generators in Figure 73. Detector-generator i, i = 1, 2,...,K,

corresponds to the ith strongest signal.

The first canceller stage eliminates the strongest signal, thereby immediately alleviating

the near-far problem while exploiting the superior detectability of the strongest signal. The

first difference signal is applied to the detector-generator for the second strongest received

signal, etc. The amount of interference removal from a signal tends to increase from the

strongest received signal to the weakest one. The delay introduced, the impact of

cancellation errors, and the implementation complexity may limit the number of canceller

stages to fewer than K, and a set of conventional detectors may be needed to estimate

some of the symbol streams. At a low signal-to-noise ratio, inaccurate cancellations may

cause the canceller to lose its advantage over the conventional detector. The successive

interference canceller of Figure 73 requires known spreading sequences and timing of all

signals.

A multistage canceller comprising more than one successive interference canceller

potentially improves performance by repeated cancellations if the delay and complexity can

be accommodated. The second canceller or stage of a multistage canceller is illustrated in

Figure 75. The input is the residual of canceller 1, which is shown in Figure 73. Replica 1

of canceller 1 is added to the input and then an improved replica of signal 1 is subtracted.

Subsequently, other replicas from canceller 1 are added and corresponding improved

replicas are subtracted. The symbol streams are produced by the final canceller. Rake

combining of multipath components may be incorporated into a multistage or single-stage

canceller to improve performance in a fading environment [53].

11.6 Parallel Interference Canceller

A parallel interference canceller detects, generates, and then subtracts all multiple-access

interference signals simultaneously, thereby avoiding the delay inherent in successive

interference cancellers. A parallel interference canceller for two signals is diagrammed in

162
+ ∑ ∑ ∑ ∑

+

+ +

+

+

– – • • • residual of

canceller 1

d2 d1

Replica 2

canceller 1

Replica 1

canceller 1

Detectorgenerator 2

Detectorgenerator 1

^ ^

Delay Delay

Figure 75. Second canceller of multistage canceller using successive interference cancellers.

Figure 76. Each detector-generator pair may be implemented as shown in Figure 74. Each

of the final detectors includes a digital matched filter and a decision device that produce

soft or hard decisions, which are applied to the decoder. Since all signals are processed in

the same manner and the initial detections influence the final ones, the parallel interference

canceller is not as effective in suppressing the near-far problem as the successive interference

canceller unless the CDMA network uses power control. Power control also relieves the

timing synchronization requirements. A better suppression of the near-far problem is

provided by the multistage parallel interference canceller, in which each stage is similar but

has an improved input that results in an improved output. Figure 77 shows the multistage

canceller for two signals. Each stage has the form of Figure 76 without the final detectors.

Detector

1

To

decoder

Final

detector 2

Final

detector 1

Input 

samples

Generator

2

Detector

2

Generator

1 ∑

∑

+

–

–

+

Delay

Delay

Figure 76. Parallel interference canceller for two signals.

• • •

Input

samples To

decoder

Final

detector 2

Final

Stage detector 1

N

Stage

2

Stage

1

D

D

D

D D

D

Figure 77. Multistage parallel interference canceller for two signals. D=delay.

163
A. Inequalities

A.1 Jensen’s Inequality

A function g(x) defined on an open interval I is convex if

g(px + (1 − p)y) ≤ pg(x) + (1 − p)g(y) (A.1)

for x, y in I and 0 ≤ p ≤ 1. Suppose that g(x) has a continuous, nondecreasing derivative

g0

(x) on I. The inequality is valid if p = 0 or 1. If x ≥ y and 0 ≤ p < 1,

g(px + (1 − p)y) − g(y) = Z px+(1−p)y

y

g0(z)dz ≤ p(x − y)g0(px + (1 − p)y)

≤ p

1 − p

Z x

px+(1−p)y

g0(z)dz

= p

1 − p

[g(x) − g(px + (1 − p)y)]

Simplifying this result, we obtain (A-1). If y ≥ x, a similar analysis again yields (A-1).

Thus, if g(x) has a continuous, nondecreasing derivative on I, it is convex.

Lemma. If g(x) is a convex function on the open interval I, then

g(y) ≥ g(x) + g−(x)(y − x) (A.2)

for all y, x in I, where g−(x) is the left derivative of g(x).

Proof. If y − x ≥ z > 0, then substituting p = 1 −z/(y − x) into (A-1) gives

g(x + z) ≤

μ

1 − z

y − x

¶

g(x) + z

y − x

g(y)

which yields

g(x + z) − g(x)

z

≤ g(y) − g(x)

y − x

, y − x ≥ z > 0 (A.3)

If v > 0 and z > 0, then (A-1) implies that

g(x) ≤

z

v + z

g(x − v) + v

v + z

g(x + z)

which yields

g(x) − g(x − v)

v

≤ g(x + z) − g(x)

z

, v, z > 0 (A.4)

Inequality (A-3) indicates that the ratio [g(y) − g(x)]/(y − x) decreases monotonically as

y → x from above and (A-4) implies that this ratio has a lower bound. Therefore, the right

165
derivative g+(x) exists on I. If x − y ≥ v > 0, then (A-1) with p = 1 − v/(x − y) implies

that

g(x − v) ≤

μ

1 − v

x − y

¶

g(x) + v

x − y

g(y)

which yields

g(x) − g(y)

x − y

≤ g(x) − g(x − v)

v

, x − y ≥ v > 0 (A.5)

This inequality indicates that the ratio [g(x) − g(y)]/(x − y) increases monotonically as

y → x from below and (A-4) implies that this ratio has an upper bound. Therefore, the left

derivative g−(x) exists on I, and (A-4) yields

g−(x) ≤ g+(x) (A.6)

Taking the limits as z → 0 and v → 0 in (A-3) and (A-5), respectively, and then using

(A-6), we find that (A-2) is valid for all y, x in I.¤

Jensen’s inequality. If X is a random variable with a finite expected value E[X], and

g( ) is a convex function on an open interval containing the range of X, then

E[g(X)] ≥ g(E[X])

Proof. Set y = X and x = E[X] in (A-2), which gives

g(X) ≥ g(E[X]) + g−(E[X])(X − E[X]). Taking the expected values of the random

variables on both sides of this inequality gives Jensen’s inequality.¤

A.2 Chebyshev’s Inequality

Consider a random variable X with distribution F(x). Let E[X] = m denote the expected

value of X and P[A] denote the probability of event A. From elementary probability it

follows that

E[|X − m|

k] = Z ∞

−∞

|x − m|

kdF(x) ≥

Z ∞

|x−m|≥α

|x − m|

kdF(x)

≥ αk

Z ∞

|x−m|≥α

dF(x) = αkP[|X − m| ≥ α]

Therefore,

P[|X − m| ≥ α] ≤

1

αk E[|X − m|

k] (A.7)

Let σ2 = E[(X − m)2] denote the variance of X. If k = 2, then (D.1.1) becomes

Chebyshev’s inequality:

P[|X − m| ≥ α] ≤

σ2

α2 (A.8)

166
As an application, let α = 3σ. Then

P[|X − m| < 3σ] ≥

8

9

This inequality indicates that the probability that a random variable is within three standard

deviations of its mean value is at least 8/9.

167
B. Probability Distributions

B.1 Chi-Square Distribution

Consider the random variable

Z = X

N

i=1

A2

i (B.1)

where the {Ai} are independent Gaussian random variables with means {mi} and common

variance σ2. The random variable Z is said to have a noncentral chi-square (χ2)

distribution with N degrees of freedom and a noncentral parameter

λ = X

N

i=1

m2

i (B.2)

To derive the probability density function of Z, we first note that each Ai has the density

function

fAi (x) = 1

√2πσ

exp ∙−(x − mi)2

2σ2

 ̧

(B.3)

From elementary probability, the density of Yi = A2

i is

fYi (x) = 1

2

√x

[fAi (

√x) + fAi (−√x)] u(x) (B.4)

where u(x)=1, x ≥ 0, and u(x)=0,x< 0. Substituting (B-3) into (B-4), expanding the

exponentials, and simplifying, we obtain the density

fYi (x) = 1

√2πxσ

exp μ−x + m2

i

2σ2

¶

cosh μmi

√x

σ2

¶

u(x) (B.5)

The characteristic function of a random variable X is defined as

CX(jν) = E[ejvX] = Z ∞

−∞

fX(x) exp(jνx) dx (B.6)

where j = √−1, and fX(x) is the density of X. Since CX(jν) is the conjugate Fourier

transform of fX(x),

fX(x) = 1

2π

Z ∞

−∞

CX(jν) exp(−jνx) dν (B.7)

From Laplace or Fourier transform tables, it is found that the characteristic function of

fYi (x) is

CYi (jν) = exp[jm2

i ν/(1 − j2σ2ν)]

(1 − j2σ2ν)1/2 (B.8)

169
The characteristic function of a sum of independent random variables is equal to the

product of the individual characteristic functions. Because Z is the sum of the Yi, the

characteristic function of Z is

CZ(jν) = exp[jλν/(1 − j2σ2ν)]

(1 − j2σ2ν)N/2 (B.9)

where we have used (B-2). From (B-9), (B-7), and Laplace or Fourier transform tables, we

obtain the probability density function of noncentral χ2 random variable with N degrees of

freedom and a noncentral parameter λ:

fZ(x) = 1

2σ2

3x

λ

 ́(N−2)/4

exp ∙−x + λ

2σ2

 ̧

IN/2−1

Ã√

xλ

σ2

!

u(x) (B.10)

where In( ) is the modified Bessel function of the first kind and order n. This function may

be represented by

In(x) = X∞

i=0

(x/2)n+2i

i! Γ(n + i + 1) (B.11)

where the gamma function is defined as

Γ(x) = Z ∞

0

yx−1 exp(−y)dy , x > 0 (B.12)

The probability distribution function of a noncentral χ2 random variable is

FZ(x) = Z x

0

1

2σ2

3y

λ

 ́(N−2)/4

exp μ−y + λ

2σ2

¶

IN/2−1

μ√yλ

σ2

¶

dy , x ≥ 0 (B.13)

When N is even so that N/2 is an integer, then using FZ(∞) = 1 and a change of variables

in (B-13) yields

FZ(x)=1 − QN/2

Ã√

λ

σ ,

√x

σ

!

, x ≥ 0 (B.14)

where the generalized Q-function is defined as

Qm(α, β) = Z ∞

β

x

3x

α

 ́m−1

exp μ−x2 + α2

2

¶

Im−1(αx) dx (B.15)

and m is an integer. Since Qm(α, 0) = 1, it follows that 1 − Qm(α, β) is an integral with

finite limits that can be numerically integrated. However, the numerical computation of

the generalized Q-function is simplified if it is expressed in alternative forms [2].

170
The mean, variance, and moments of Z can be easily obtained by using (B-1) and the

properties of independent Gaussian random variables. The mean and variance of Z are

E[Z] = Nσ2 + λ (B.16)

σ2

z = 2Nσ4 + 4λσ2 (B.17)

where σ2 is the common variance of the {Ai}.

From (B-9), it follows that the sum of two independent noncentral χ2 random variables

with N1 and N2 degrees of freedom, noncentral parameters λ1 and λ2, respectively, and the

same parameter σ2 is a noncentral χ2 random variable with N1 + N2 degrees of freedom

and noncentral parameter λ1 + λ2.

B.2 Central Chi-Square Distribution

To determine the probability density function of Z when the {Ai} have zero means, we

substitute (B-11) into (B-10) and then take the limit as λ → 0. We obtain

fZ(x) = 1

(2σ2)N/2Γ(N/2)xN/2−1 exp 3

− x

2σ2

 ́

u(x) (B.18)

Alternatively, this equation results if we substitute λ = 0 into the characteristic function

(B-9) and then use (B-7). Equation (B-18) is the probability density function of a central

χ2 random variable with N degrees of freedom. The probability distribution function is

FZ(x) = Z x

0

1

(2σ2)N/2Γ(N/2)yN/2−1 exp 3

− y

2σ2

 ́

dy, x ≥ 0 (B.19)

If N is even so that N/2 is an integer, then integrating this equation by parts N/2 − 1

times yields

FZ(x)=1 − exp 3

− x

2σ2

 ́

N/

X

2−1

i=0

1

i!

3 x

2σ2

 ́i

, x ≥ 0 (B.20)

By direct integration using (B-18) and (B-12) or from (B-16) and (B-17), the mean and

variance of Z are

E[Z] = Nσ2 (B.21)

σ2

z = 2Nσ4 (B.22)

B.3 Rice Distribution

Consider the random variable

R =

q

A2

1 + A22 (B.23)

171
where A1 and A2 are independent Gaussian random variables with means m1 and m2,

respectively, and a common variance σ2. The probability distribution function of R must

satisfy FR(r) = FZ(r2), where Z = A2

1 + A22 is a χ2 random variable with two degrees of

freedom. Therefore, (??) with N = 2 implies that

FR(r)=1 − Q1

Ã√

λ

σ , rσ

!

, r ≥ 0 (B.24)

where λ = m2

1 + m22. This function is called the Rice probability distribution function. The

Rice probability density function, which may be obtained by differentiation of (B-24), is

fR(r) = r

σ2 exp μ

−r2 + λ

2σ2

¶

I0

Ã

r

√

λ

σ2

!

u(r) (B.25)

The moments of even order can be derived from (B-23) and the moments of the

independent Gaussian random variables. The second moment is

E[R2]=2σ2 + λ (B.26)

In general, moments of the Rice distribution are given by an integration over the density in

(B-25). Substituting (B-11) into the integrand, interchanging the summation and

integration, changing the integration variable, and using (B-12), we obtain a series that is

recognized as a special case of the confluent hypergeometric function. Thus,

E[Rn] = (2σ2)

n/2 exp μ

− λ

2σ2

¶

Γ

3

1 +

n

2

 ́

1F1

μ

1 +

n

2

, 1; λ

2σ2

¶

, n ≥ 0 (B.27)

where the confluent hypergeometric function is defined as

1F1(α, β; x) = X∞

i=0

Γ(α + i)Γ(β)xi

Γ(α)Γ(β + i)i!

, β 6= 0, −1, −2,... (B.28)

The Rice density function often arises in the context of a transformation of variables. Let

A1 and A2 represent independent Gaussian random variables with common variance σ2 and

means m and zero, respectively. Let R and Θ be implicitly defined by A1 = R cos Θ and

A2 = R sin Θ. Then (B-23) and Θ = tan−1(A2/A2) describes a transformation of variables.

A straightforward calculation yields the joint density function of R and Θ:

fR,Θ(r, θ) = r

2πσ2 exp μ

−r2 − 2rm cos θ + m2

2σ2

¶

, r ≥ 0, |θ| ≤ π (B.29)

The density function of the envelope R is obtained by integration over θ. Since the

modified Bessel function of the first kind and order zero satisfies

I0(x) = 1

2π

Z 2π

0

exp(x cos u) du (B.30)

172
this density function reduces to the Rice density function of (B-25). The density function

of the angle Θ is obtained by integrating (B-29) over r. Completing the square in the

argument of (B-29), changing variables, and defining

Q(x) = 1

√2π

Z ∞

−x

exp μ−y2

2

¶

dy = 1

2

erfc μ x

√2

¶

(B.31)

where erfc( ) is the complementary error function, we obtain

fΘ(θ) = 1

2π

exp μ− m2

2σ2

¶

+

m cos θ

√2πσ

exp μ−m2 sin2 θ

2σ2

¶ ∙

1 − Q

μm cos θ

σ

¶ ̧ ,

|θ| ≤ π (B.32)

Since (B-29) cannot be written as the product of (B-25) and (B-32), the random variables

R and Θ are not independent.

Since the density function of (B-25) must integrate to unity, we find that

Z ∞

0

r exp μ− r2

2b

¶

I0

μr

√

λ

b

¶

dr = b exp μ λ

2b

¶

(B.33)

where λ and b are positive constants. This equation is useful in calculations involving the

Rice density function.

B.4 Rayleigh Distribution

A Rayleigh-distributed random variable is defined by (B-23) when A1 and A2 are

independent Gaussian random variables with zero means and a common variance σ2. Since

FR(r) = FZ(r2), where Z is a central χ2 random variable with two degrees of freedom,

(B-20) with N = 2 implies that the Rayleigh probability distribution function is

FR(r)=1 − exp μ− r2

2σ2

¶

, r ≥ 0 (B.34)

The Rayleigh probability density function, which may be obtained by differentiation of

(B-34), is

fR(r) = r

σ2 exp μ

− r2

2σ2

¶

u(r) (B.35)

By a change of variables in the defining integral, any moment of R can be expressed in

terms of the gamma function defined in (B-12). Therefore,

E[Rn] = (2σ2

)

n/2

Γ

3

1 + n

2

 ́

(B.36)

Certain properties of the gamma function are needed to simplify (B-36). An integration by

parts of (B-12) indicates that Γ(1 + x) = xΓ(x). A direct integration yields Γ(1) = 1.

173
Therefore, when n is an integer, Γ(n)=(n − 1)!. Changing the integration variable by

substituting y = z2 in (??), it is found that Γ(1/2) = √π.

Using these properties of the gamma function, we obtain the mean and the variance of a

Rayleigh-distributed random variable:

E[R] = rπ

2

σ (B.37)

σ2

R =

3

2 − π

2

 ́

σ2 (B.38)

Since A1 and A2 have zero means, the joint probability density function of the random

variables R = pA2

1 + A22 and Θ = tan−1(A2/A1) is given by (B-29) with m = 0. Therefore,

fR,Θ(r, θ) = r

2πσ2 exp μ

− r2

2σ2

¶

, r ≥ 0, |θ| ≤ π (B.39)

Integration over θ yields (A-35), and integration over r yields the uniform probability

density function:

fΘ(θ) = 1

2π , |θ| ≤ π (B.40)

Since (B-39) equals the product of (B-35) and (B-40), the random variables R and Θ are

independent. In terms of these random variables, A1 = R cos Θ and A2 = R sin Θ. A

straightforward calculation using the independence and densities of R and Θ verifies that

A1 and A2 are independent, zero-mean, Gaussian random variables with common variance

σ2. Since the square of a Rayleigh-distributed random variable may be expressed as

R2 = A2

1 + A22, where A1 and A2 are zero-mean independent Gaussian random variables

with common variance σ2, R2 has the distribution of a central chi-square random variable

with 2 degrees of freedom. Therefore, (B-18) with N = 2 indicates that the square of a

Rayleigh-distributed random variable has an exponential probability density function with

mean 2σ2.

B.5 Sum of Independent, Exponentially Distributed Random Variables

Consider the random variable

Z = X

N

i=1

Yi (B.41)

where the {Yi} are independent, exponentially distributed random variables with unequal

positive means {mi}. The exponential probability density function of Yi is

fYi (x) = 1

mi

exp μ− x

mi

¶

u(x) (B.42)

174
A straightforward calculation yields the characteristic function

CYi (jν) = 1

1 − jνmi

(B.43)

Since Z is the sum of independent random variables, (B-43) implies that its characteristic

function is

CZ(jν) = Y

N

i=1

1

1 − jνmi

(B.44)

To derive the probability density function of Z, (B-7) is applied after first expanding the

right-hand side of (B-44) in a partial-fraction expansion. The result is

fZ(x) = X

N

i=1

Bi

mi

exp μ− x

mi

¶

u(x) (B.45)

where

Bi =







Q

N

k=1

k6=i

mi

mi − mk

, N ≥ 2

1 , N = 1

(B.46)

and mi 6= mk, i 6= k. A direct integration and algebra yields the probability distribution

function

FZ(x)=1 −X

N

i=1

Bi exp μ− x

mi

¶

, x ≥ 0 (B.47)

Equations (B-45) and (B-12) give

E[Zn] = Γ(1 + n)

X

N

i=1

Bimn

i , n ≥ 0 (B.48)

When the {mi} are equal so that mi = m, 1 ≤ i ≤ N, then CZ(jν) = (1 − jνm)−N .

Therefore, the probability density function of Z is

fZ(x) = 1

(N − 1)!mN xN−1 exp 3

− x

m

 ́

u(x) (B.49)

which is a special case of the gamma density function. Successive integration by parts yields

FZ(x)=1 − exp μ− x

m

¶ N

X−1

i=0

1

i!

μ x

m

¶i

(B.50)

From (B-49) and (B-12), the mean and variance of Z are found to be

E[Z] = Nm (B.51)

σ2

Z = Nm2 (B.52)

175
C. Signal Representations

C.1 Bandpass Signals

The Hilbert transform of a real-valued function g(t) defined on the time interval

−∞ <t< ∞ is

H[g(t)] = ˆg(t) = 1

π

Z ∞

−∞

g(u)

t − u

du (C.1)

Since its integrand has a singularity, the integral is defined as its Cauchy principal value:

Z ∞

−∞

g(u)

t − u

du = lim2→0

" Z t−2

−∞

g(u)

t − u

du +

Z ∞

t+2

g(u)

t − u

du#(C.2)

provided that the limit exists. Since (C-1) has the form of the convolution of g(t) with

1/πt, gˆ(t) results from passing g(t) through a linear filter with an impulse response equal to

1/πt. The transfer function of the filter is given by the Fourier transform

F

1⁄2 1

πt

3⁄4

=

Z ∞

−∞

exp(−j2πf t)

πt

dt (C.3)

where j = √−1. This integral can be rigorously evaluated by using contour integration.

Alternatively, we observe that since 1/t is an odd function,

F

1⁄2 1

πt

3⁄4

= −2j

Z ∞

0

sin 2πf t

πt

dt

= −j sgn(f) (C.4)

where sgn(f) is the signum function defined by

sgn(f) =







1, f> 0

0, f = 0

−1, f< 0

(C.5)

Let G(f) = F{g(t)}, and let Gˆ(f) = F{gˆ(t)}. Equations (??) and (??) and the

convolution theorem imply that

Gˆ(f) = −j sgn(f)G(f) (C.6)

177
Because H[ˆg(t)] results from passing g(t) through two successive filters, each with transfer

function −j sgn(f),

H[ˆg(t)] = −g(t) (C.7)

provided that G(0) = 0.

Equation (C-6) indicates that taking the Hilbert transform corresponds to introducing a

phase sift of −90 degrees for all positive frequencies and +90 degrees for all negative

frequencies. Consequently,

H[cos 2πf0t] = sin 2πf0t (C.8)

H[sin 2πf0t] = − cos 2πf0t (C.9)

These relations can be formally verified by taking the Fourier transform of the left-hand

side of (C-8) or (C-9), applying (C-6), and then taking the inverse Fourier transform of the

result. If G(f) = 0 for |f| > W and fc > W, the same method yields

H[g(t) cos 2πf0t] = g(t) sin 2πfct (C.10)

H[g(t) sin 2πf0t] = −g(t) cos 2πfct (C.11)

A bandpass signal is one with a Fourier transform that is negligible except for

fc − W/2 ≤ |f| ≤ fc + W/2, where 0 ≤ W < 2fc and fc is the center frequency. If

W << fc, the bandpass signal is often called a narrowband signal. A complex-valued signal

with a Fourier transform that is nonzero only for f > 0 is called an analytic signal.

Consider a bandpass signal g(t) with Fourier transform G(f). The analytic signal ga(t)

associated with g(t) is defined to be the signal with Fourier transform

Ga(f) = [1 + sgn(f)]G(f) (C.12)

which is zero for f ≤ 0 and is confined to the band |f − fc| ≤ W/2 when f > 0. The

inverse Fourier transform of (C-12) and (C-6) imply that

ga(t) = g(t) + jgˆ(t) (C.13)

The complex envelope of g(t) is defined by

gl(t) = ga(t) exp[−j2πfct] (C.14)

where fc is the center frequency if g(t) is a bandpass signal. Since the Fourier transform of

gl(t) is Ga(f + fc), which occupies the band |f| ≤ W/2, the complex envelope is a baseband

178
signal that may be regarded as an equivalent lowpass representation of g(t). Equations

(C-13) and (C-14) imply that g(t) may be expressed in terms of its complex envelope as

g(t) = Re[gl(t) exp(j2πfct)] (C.15)

The complex envelope can be decomposed as

gl(t) = gc(t) + jgs(t) (C.16)

where gc(t) and gs(t) are real-valued functions. Therefore, (C-15) yields

g(t) = gc(t) cos(2πfct) − gs(t) sin(2πfct) (C.17)

Since the two sinusoidal carriers are in phase quadrature, gc(t) and gs(t) are called the

in-phase and quadrature components of g(t), respectively. These components are lowpass

signals confined to |f| ≤ W/2.

From Parseval’s relation in Fourier analysis and (C-6), we obtain

Z ∞

−∞

gˆ2(t) dt =

Z ∞

−∞

|Gˆ(f)|

2 df =

Z ∞

−∞

|G(f)|

2 df =

Z ∞

−∞

g2(t) dt (C.18)

Therefore,

Z ∞

−∞

|gl(t)|

2 dt =

Z ∞

−∞

|ga(t)|

2 dt =

Z ∞

−∞

g2(t) dt +

Z ∞

−∞

gˆ2(t) dt

= 2 Z ∞

−∞

g2(t) dt = 2E (C.19)

where E denotes the energy of the bandpass signal g(t).

C.2 Stationary Stochastic Processes

Consider a stochastic process n(t) that is a zero-mean, wide-sense stationary process with

autocorrelation

Rn(τ ) = E[n(t)n(t + τ )] (C.20)

where E[x] denotes the expected value of x. The Hilbert transform of this process is the

stochastic process defined by

nˆ(t) = 1

π

Z ∞

−∞

n(u)

t − u

du (C.21)

where it is assumed that the Cauchy principal value of the integral exists for almost every

sample function of n(t). This equation indicates that ˆn(t) is a zero-mean stochastic

process. The zero-mean processes n(t) and ˆn(t) are jointly wide-sense stationary if their

179
correlation and cross-correlation functions are not functions of t. A straightforward

calculation using (C-21) and (C-20) gives the cross correlation

Rnnˆ(τ ) = E[n(t)ˆn(t + τ )] = 1

π

Z ∞

−∞

Rn(u)

τ − u

du = Rˆn(τ ) (C.22)

A similar derivation using (C-7) yields the autocorrelation

Rnˆ(τ ) = E[ˆn(t)ˆn(t + τ )] = Rn(τ ) (C.23)

Equations (C-20), (C-22), and (C-23) indicate that n(t) and ˆn(t) are jointly wide-sense

stationary.

The analytic signal associated with n(t) is the zero-mean process defined by

na(t) = n(t) + jnˆ(t) (C.24)

The autocorrelation of the analytic signal is defined as

Ra(τ ) = E[n∗

a(t)na(t + τ )] (C.25)

where thee asterisk denotes the complex conjugate. Using (C-20) and (C-22) to (C-25), we

obtain

Ra(τ )=2Rn(τ )+2jRˆn(τ ) (C.26)

which establishes the wide-sense stationarity of the analytic signal.

Since (C-20) indicates that Rn(τ ) is an even function, (C-22) yields

Rnnˆ(0) = Rˆn(0) = 0 (C.27)

which indicates that n(t) and ˆn(t) are uncorrelated. Equations (C-23), (C-26), and (C-27)

yield

Rnˆ(0) = Rn(0) = 1/2Ra(0) (C.28)

The complex envelope of n(t) or the equivalent lowpass representation of n(t) is the

zero-mean stochastic process defined by

nl(t) = na(t) exp(−j2πfct) (C.29)

where fc is an arbitrary frequency usually chosen as the center or carrier frequency of n(t).

The complex envelope can be decomposed as

nl(t) = nc(t) + jns(t) (C.30)

where nc(t) and ns(t) are real-valued, zero-mean stochastic processes.

180
Equations (C-29) and (C-30) imply that

n(t) = Re[nl(t) exp(j2πfct)]

= nc(t) cos(2πfct) − ns(t) sin(2πfct) (C.31)

Substituting (C-24) and (C-30) into (C-29) we find that

nc(t) = n(t) cos(2πfct)+ˆn(t) sin(2πfct) (C.32)

ns(t)=ˆn(t) cos(2πfct) − n(t) sin(2πfct) (C.33)

The autocorrelations of nc(t) and ns(t) are defined by

Rc(τ ) = E[nc(t)nc(t + τ )] (C.34)

and

Rs(τ ) = E[ns(t)ns(t + τ )] (C.35)

Using (C-32) and (C-33) and then (C-20), (C-23), and (C-24) and trigonometric identities,

we obtain

Rc(τ ) = Rs(τ ) = Rn(τ ) cos(2πfcτ ) + Rˆn(τ ) sin(2πfcτ ) (C.36)

which shows explicitly that if n(t) is wide-sense stationary, then nc(t) and ns(t) are

wide-sense stationary with the same autocorrelation function. The variances of n(t), nc(t),

and ns(t) are all equal because

Rc(0) = Rs(0) = Rn(0) (C.37)

A derivation similar to that of (C-36) gives the cross correlation

Rcs(τ ) = E[nc(t)ns(t + τ )] = Rˆn(τ ) cos(2πfcτ ) − Rn(τ ) sin(2πfcτ ) (C.38)

Equations (C-36) and (C-38) indicate that nc(t) and ns(t) are jointly wide-sense stationary.

Equations (C-28) and (C-38) give

Rcs(0) = 0 (C.39)

which implies that nc(t) and ns(t) are uncorrelated.

Equation (C-21) indicates that ˆn(t) is generated by a linear operation on n(t). Therefore, if

n(t) is a zero-mean Gaussian process, ˆn(t) and n(t) are zero-mean jointly Gaussian

processes. Equations (C-32) and (C-33) then imply that nc(t) and ns(t) are zero-mean

jointly Gaussian processes. Since they are uncorrelated, nc(t) and ns(t) are statistically

independent, zero-mean Gaussian processes.

The power spectral density of a signal is the Fourier transform of its autocorrelation. Let

S(f), Sc(f), and Ss(f) denote the power spectral densities of n(t), nc(t), and ns(t),

respectively. We assume that Sn(f) occupies the band fc − W/2 ≤ |f| ≤ fc + W/2 and that

181
fc > W/2 ≥ 0. Taking the Fourier transform of (C-36), using (C-6), and simplifying, we

obtain

Sc(f) = Ss(f) = 1⁄2 Sn (f − fc) + Sn (f + fc), |f| ≤ W/2

0, |f| > W/2 (C.40)

Thus, if n(t) is a passband process with one-sided bandwidth W, then nc(t) and ns(t) are

baseband processes with one-sided bandwidths W/2. This property and the statistical

independence of nc(t) and ns(t) when n(t) is Gaussian make (C-31) a very useful

representation of n(t).

Similarly, the cross-spectral density of nc(t) and ns(t) can be derived by taking the Fourier

transform of (C-38) and using (C-6). After simplification, the result is

Scs(f) = 1⁄2 j[Sn (f − fc) − Sn (f + fc)], |f| ≤ W/2

0, |f| > W/2 (C.41)

If Sn(f) is locally symmetric about fc, then

Sn(fc + f) = Sn(fc − f), |f| ≤ W/2 (C.42)

Since a power spectral density is a real-valued, even function, Sn(fc − f) = Sn(f − fc).

Equation (C-42) then yields Sn(f + fc) = Sn(f − fc) for |f| ≤ W/2. Therefore, (C-41) gives

Scs(f) = 0, which implies that

Rcs(τ ) = 0 (C.43)

for all τ . Thus, nc(t) and ns(t + τ ) are uncorrelated for all τ , and if n(t) is a zero-mean

Gaussian process, then nc(t) and ns(t + τ ) are statistically independent for all τ .

The autocorrelation of the complex envelope is defined by

Rl(τ ) = 1

2

E[n∗

l(t)nl(t + τ )] (C.44)

where the 1/2 is inserted so that

Rl(0) = Rn(0) (C.45)

which follows from (C-28) and (C-29). Substituting (C-30) into (C-44) and using (C-36)

and (C-38), we obtain

Rl(τ ) = Rc(τ ) + jRcs(τ ) (C.46)

The power spectral density of nl(t), which we denote by Sl(f), can be derived from (C-46),

(C-41), and (C-40). If Sn(f) occupies the band fc − W/2 ≤ |f| ≤ fc + W/2 and

fc > W/2 ≥ 0, then

Sl(f) = 1⁄22Sn (f + fc), |f| ≤ W/2

0, |f| > W/2 (C.47)

182
Equations (C-36) and (C-38) yield

Rn(τ ) = Rc(τ ) cos(2πfcτ ) − Rcs(τ ) sin(2πfcτ ) (C.48)

Equations (C-48) and (C-46) imply that

Rn(τ ) = Re [Rl(τ ) exp (j2πfcτ )] (C.49)

We expand the right-hand side of this equation by using the fact that Re[z]=(z + z∗)/2.

Taking the Fourier transform and observing that Sl(f) is a real-valued function, we obtain

Sn(f) = 1

2

Sl(f − fc) + 1

2

Sl(−f − fc) (C.50)

If Sn(f) is locally symmetric about fc, then (C-47) and (C-42) imply that Sl(−f) = Sl(f),

and (C-50) becomes

Sn(f) = 1

2

Sl(f − fc) + 1

2

Sl(f + fc) (C.51)

Many useful communication signals are modeled as having the form

s(t) = Ad1(t) cos(2πfct + θ) + Ad2(t) sin(2πfct + θ) (C.52)

where θ is an independent random variable that is uniformly distributed over 0 ≤ θ < 2π.

The modulations have the form

di(t) = X∞

k=−∞

aikψ(t − kT − T0 − ti), i = 1, 2 (C.53)

where {aik} is a sequence of independent, identically distributed random variables,

aik = +1 with probability 1/2 and aik = −1 with probability 1/2, ψ(t) is a pulse waveform,

T is the pulse duration, ti is the relative pulse offset, and T0 is an independent random

variable that is uniformly distributed over the interval (0, T) and reflects the arbitrariness

of the origin of the coordinate system. Since aik is independent of ain when n 6= k, it

follows that E[aikain] = 0, n 6= k. It follows that the autocorrelation of di(t) is

Rdi(τ ) = E[di(t)di(t + τ )]

= X∞

k=−∞

E[ψ(t − kT − T0 − ti)ψ(t − kT − T0 − ti + τ )] (C.54)

Expressing the expected value as an integral over the range of T0 and changing variables,

we obtain

Rdi(τ ) = X∞

k=−∞

1

T

Z t−kT −ti

t−kT −T −ti

ψ(x)ψ(x + τ )dx

= 1

T

Z ∞

−∞

ψ(x)ψ(x + τ )dx, i = 1, 2 (C.55)

183
This equation indicates that d1(t) and d2(t) are wide-sense stationary processes with the

same autocorrelation.

If the sequences {a1k} and {a2k} are statistically independent, then the autocorrelation of

s(t) is

Rs(τ ) = A2

2

Rd1(τ ) cos(2πfcτ ) + A2

2

Rd2(τ ) sin(2πfcτ ) (C.56)

where Rd1(τ ) and Rd2(τ ) are the autocorrelations of d1(t) and d2(t), respectively. This

equation indicates that s(t) is wide-sense stationary. If the sample functions of d1(t) and

d2(t) have Fourier transforms that vanish for |f| ≥ fc, then (C-10), (C-11), (C-24), and

(C-29) indicate that the complex envelope of s(t) is

sl(t) = Ad1(t) − jAd2(t) (C.57)

Equation (C-44) and the independence of d1(t) and d2(t) imply that the autocorrelation of

sl(t) is

Rl(τ ) = A2

2

Rd1(τ ) + A2

2

Rd2(τ ) (C.58)

The power spectral density of sl(t) is the Fourier transform of Rl(τ ). From (C-58) and

(C-55), we obtain the density

Sl(f) = A2 |G(f)|

2

T (C.59)

where G(f) is the Fourier transform of ψ(t).

In a quadriphase-shift-keying (QPSK) signal, d1(t) and d2(t) are usually modeled as

independent random binary sequences with pulse duration T = 2Tb, where Tb is a bit

duration. The component amplitude is A = pEb/Tb, where Eb is the energy per bit. If ψ(t)

is rectangular with unit amplitude over [0, 2Tb], then (C-59) yields the power spectral

density for QPSK:

Sl(f)=2Eb sinc22Tbf (C.60)

which is the same as the density for PSK. For a binary minimum-shift-keying (MSK) signal

with the same component amplitude,

ψ(t) = √2 sin μ πt

2Tb

¶

, 0 ≤ t < 2Tb (C.61)

Therefore, the power spectral density for MSK is

Sl(f) = 16Eb

π2

∙

cos(2πTbf)

16T2

b f 2 − 1

 ̧2

(C.62)

184
C.3 Sampling Theorems

Consider the Fourier transform G(f) of an absolutely integrable function g(t). The

periodic extension of G(f) is defined as

G ̄(f) = X∞

i=−∞

G(f + iW) (C.63)

where W is the period of G ̄(f) and it is assumed that the series converges uniformly.

Suppose that G ̄(f) has a piecewise continuous derivative so that it can be represented as a

uniformly convergent complex Fourier series:

G ̄(f) = X∞

k=−∞

ck exp μ−j2πk f

W

¶

(C.64)

where the Fourier coefficient ck is given by

ck = 1

W

Z W/2

−W/2

G ̄(f) exp μj2πk f

W

¶

df (C.65)

Substituting (C-63) into (C-65) and interchanging the order of the summation and the

integration, which is justified because of the uniform convergence, we obtain

ck = 1

W

X∞

i=−∞

Z W/2

−W/2

G(f + iW) exp μj2πk f

W

¶

df (C.66)

We change variables and observe the exp(j2πki) = 1 to obtain

ck = 1

W

X∞

i=−∞

Z W/2+iW

−W/2+iW

G(f) exp μj2πk f

W − j2πki¶

df

= 1

W

Z ∞

−∞

G(f) exp μj2πk f

W

¶

df (C.67)

Since g(t) is absolutely integrable, the last integral is the inverse Fourier transform of G(f)

evaluated at t = k/W, and

ck = 1

W g

μ k

W

¶

(C.68)

Substituting (C-68) into (C-64) yields the Poisson sum formula:

G ̄(f) = 1

W

X∞

k=−∞

g

μ k

W

¶

exp μ−j2πkf

W

¶

(C.69)

where the series converges uniformly.

185
Suppose that the Fourier transform vanishes outside a frequency band:

G(f)=0 , |f| > W/2 (C.70)

It follows that

g(t) = Z W/2

−W/2

G(f) exp(j2πf t)df (C.71)

Since G(f) = G ̄(f) for |f| < W/2, (C-71) and (C-69) and the interchange of a summation

and integration yield

g(t) = X∞

k=−∞

g

μ k

W

¶ 1

W

Z W/2

−W/2

exp ∙j2πf

μ

t − k

W

¶ ̧ df (C.72)

Evaluating this integral and defining sinc x = sin(πx)/πx, we obtain the sampling theorem

for deterministic signals:

g(t) = X∞

k=−∞

g

μ k

W

¶

sinc(W t − k) (C.73)

Consider a wide-sense stationary stochastic process n(t) with autocorrelation Rn(τ ) and

power spectral density Sn(f), which is the Fourier transform of Rn(τ ). If

Sn(f)=0 , |f| > W/2 (C.74)

then it follows from the sampling theorem that

Rn(τ ) = X∞

k=−∞

Rn

μ k

W

¶

sinc(Wτ − k) (C.75)

For an arbitrary constant α, the Fourier transform of R(τ − α) is Sn(f) exp(−j2πfα),

which is zero for |f| > W/2. Therefore, (eqC-75) can be applied to R0

n(τ ) = Rn(τ − α),

which gives

Rn(τ − α) = X∞

k=−∞

Rn

μ k

W − α

¶

sinc(Wτ − k) (C.76)

We define the stochastic process

nν (t) = Xν

k=−ν

n

μ k

W

¶

sinc(W t − k) (C.77)

An expansion indicates that the mean square difference between n(t) and nν(t) is

E{[n(t) − nν(t)]2

} = Rn(0) − 2

Xν

k=−ν

Rn

μ

t − k

W

¶

sinc(W t − k)

+ Xν

i=−ν

sinc(W t − i)

Xν

k=−ν

Rn

μi − k

W

¶

sinc(W t − k) (C.78)

186
Since Rn(τ ) = Rn(−τ ), the repeated use of (C-76) yields

limν→∞ E{[n(t) − nν (t)]2

} = 0 (C.79)

which states that the mean square difference between n(t) and nν(t) approaches zero.

Thus, the sampling theorem for stationary stochastic process is

n(t) = X∞

k=−∞

n

μ k

W

¶

sinc(W t − k) (C.80)

where the equality holds in the sense of (C-79).

187
D. Adaptive Filters

The input and weight vectors of an adaptive filter are

x = [x1 x2 ...xN ]

T , W = [W1 W2 ...WN ]T (D.1)

where T denotes the transpose and the components of the vectors may be real or complex.

The filter output is the scalar

y = WT x (D.2)

The derivation of the optimal filter weights depends on the specification of a performance

criterion or estimation procedure. A number of different estimators of the desired signal

can be implemented by linear filters that produce (D-2). Unconstrained estimators that

depend only on the second-order moments of X can be derived by using performance

criteria based on the mean square error or the signal-to-noise ratio of the filter output.

Similar estimators result from using the maximum-a-posteriori or the maximum-likelihood

criteria, but the standard application of these criteria entails the restrictive assumption

that any interference in X has a Gaussian distribution.

The difference between the desired response d and the filter output is the error signal:

2 = d − WT x (D.3)

The most widely used method of estimating the desired signal is based on the minimization

of the expected value of the squared error magnitude, which is proportional to the mean

power in the error signal. Let H denote the conjugate transpose and an asterisk denote the

conjugate. We obtain

E[|2|

2

] = E [2

∗

2] = E £¡d∗WHx∗¢ ¡d − xTW¢¤

= E[|d|

2

] − WHRxd − RH

xdW + WHRxxW (D.4)

where

Rxx = E £

x∗xT ¤ (D.5)

is the N × N Hermitian correlation matrix of x and

Rxd = E [x∗

d] (D.6)

is the N × 1 cross-correlation vector. If we assume that y 6= 0 when W 6= 0, then Rxx must

be positive definite.

In terms of its real part WR, and its imaginary part WI , a complex weight vector is

defined as

W = WR − jWI (D.7)

189
The gradient of f with respect to the n-dimensional, real-valued vector x is defined as the

column vector ∇xf with components ∂f/∂xi, i = 1, 2,...,n. Let ∇wr and ∇wi denote the

N × 1 gradient vectors with respect to WR and WI respectively. The complex gradient

with respect to W is defined as

∇ ̄ w = ∇wr − j∇wi (D.8)

Let Wi, WRi and WIi, i = 1, 2,...,N, denote the components of W,WR, and WI ,

respectively. Letr g(W,W∗) denote a real-valued function of W and W∗. Regarding W

and W∗ as independent variables, we assume that g is an analytic function of each Wi

when W∗ is held constant and an analytic function of each W∗

i when W is held constant.

We define ∇w∗ as the gradient with respect to W∗. Since Wi = WRi − jWIi,

∂Wi

∂WRi

= 1, ∂Wi

∂WIi

= −j, ∂W∗

i

∂WRi

= 1, ∂W∗

i

∂WIi

= j (D.9)

The chain rule of calculus then implies that

∇ ̄ wg (W,W∗

) = ∇wrg − j∇wig

= ∇wg + ∇w∗ g − j (−j∇wg + j∇w∗ g) (D.10)

Thus,

∇ ̄ wg (W,W∗)=2∇w∗ g (W,W∗) (D.11)

This result allows a major simplification in calculations. Since ∇x

¡

xTy

¢

= ∇x

¡

yTx

¢

= y,

(D-11) and (D-4) yield

∇ ̄ wE £|2|

2¤

= 2RxxW − 2Rxd (D.12)

Since ∇wrg = 0 and ∇wig = 0 imply that ∇ ̄ wg = 0, a necessary condition for the optimal

weight is obtained from setting ∇ ̄ wE[|2|

2] = 0. Thus, if Rxx is positive definite and hence

nonsingular, the Wiener-Hopf equation for the optimal weight vector is

W0 = R−1

xx Rxd (D.13)

Substituting W = W0 into (D-4), we obtain the mean square error

2

2

m = E[|d|

2

] − RH

xdR−1xx Rxd (D.14)

Equations (D-4), (D-13), and (D-14) imply that

E[|2|

2

] = 2

2

m + (W − W0)

H Rxx (W − W0) (D.15)

Since Rxx is positive definite, this equation shows that the Wiener-Hopf equation provides

a unique optimal weight vector and that (D-14) gives the minimum mean square error.

Since the computational difficulty of inverting the correlation matrix is considerable when

the number of weights is large, and insofar as time-varying signal statistics may require

190
frequent computations, adaptive algorithms not entailing matrix inversion have been

developed. Suppose that a performance measure, P(W), is defined so that it has a

minimum value when the weight vector has its optimal value. In the method of steepest

descent, the weight vector is changed along the direction of the negative gradient of the

performance measure. This direction gives the largest decrease in P(W). If the signals and

weights are complex, separate steepest-descent equations can be written for the real and

imaginary parts of the weight vector. Combining these equations, we obtain

W(k + 1) = W(k) − μ∇ ̄ wP(W(k)) (D.16)

For complex signals and weights, a suitable performance measure is P(W) = E[|2|

2]. Using

(D-12) and (D-16), we obtain the steepest-descent algorithm:

W(k + 1) = W(k) − 2μ [RxxW(k) − Rxd] (D.17)

This ideal algorithm produces a deterministic sequence of weights and does not require a

matrix inversion, but it requires the knowledge of Rxx and Rxd. However, the possible

presence of interference means that Rxx is unknown. In the absence of information about

the direction of the desired signal, Rxd is also unknown.

The least-mean-square (LMS) algorithm is obtained when Rxx is estimated by

x∗(k)xT (k), Rxd is estimated by x∗(k)d(k), and (D-3) is applied in (D-17). The LMS

algorithm is

W(k + 1) = W(k)+2μ2(k)x∗

(k) (D.18)

For a fixed value of W(k), the product 2(k)x∗(k) is an unbiased estimate of ∇ ̄ wE[22]. According to this algorithm, the next weight vector is obtained by adding to the present weight

vector the input vector scaled by the amount of error. It can be shown [26], [1] that, for an

appropriate value of μ, the mean of the weight vector converges to the optimal value given

by the Wiener-Hopf equation.

191
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197

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