NTIA Report 02-394

Wireless Link Statistical Bit Error Model

John J. Lemmon

U.S. DEPARTMENT OF COMMERCE

Donald L. Evans, Secretary

Nancy J. Victory, Assistant Secretary

for Communications and Information

June 2002
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Product Disclaimer

Certain commercial equipment, instruments, or materials are identified in this report to specify

the technical aspects of the reported results. In no case does such identification imply

recommendation or endorsement by the National Telecommunications and Information

Administration, nor does it imply that the material or equipment identified is necessarily the best

available for the purpose.
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CONTENTS

Page

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. INTRODUCTION ...........................................................1

2. THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3. PROBABILITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4. PARAMETER ESTIMATION .................................................6

5. LAND MOBILE RADIO LINK PERFORMANCE .................................8

6. WIRELESS LOCAL AREA NETWORK PERFORMANCE . . . . . . . . . . . . . . . . . . . . . . . . 22

7. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK . . . . . . . . . . . . . 25

8. ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
vi

FIGURES

Page

Figure 1. Transition diagram and bit error probabilities for the Gilbert model. ..........3

Figure 2. Land mobile radio system performance relating BER to SNR and 

parametrically to ISR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 3. Transition probability q versus SNR for the land mobile radio link 

simulations. Each curve corresponds to a different value of ISR. . . . . . . . . . . . . 11

Figure 4. Transition probability Q versus SNR for the land mobile radio link 

simulations. Each curve corresponds to a different value of ISR. . . . . . . . . . . . . 12

Figure 5. Probability h of transmitting a bit correctly (in the B state) versus SNR 

for the land mobile radio link simulations. Each curve corresponds to a 

different value of ISR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Figure 6. Comparison of error burst length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for an ìaverageî 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Figure 7. Comparison of error gap length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for an ìaverageî 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 8. Comparison of error burst length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for a FSFL 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 9. Comparison of error gap length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for a FSFL 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 10. Comparison of error burst length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for an IL 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 11. Comparison of error gap length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for an IL 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
vii

Figure 12. Comparison of error burst length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for a NL 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 13. Comparison of error gap length distributions generated by 

(a) waveform simulation and (b) the Gilbert model for a NL 

land mobile radio link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 14. Plot of the analytic representation of the error gap length distribution in the

Gilbert model for an ìaverageî land mobile radio link. . . . . . . . . . . . . . . . . . . . . 22

Figure 15. Wireless local area network performance relating BER to SNR. . . . . . . . . . . . . 23

Figure 16. Gilbert model parameters q, Q, and h versus SNR for the wireless local area

network link simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 17. Error burst length distributions generated by a waveform simulation 

and by the Gilbert model for a wireless local area network with 

SNR = 12 dB and CIR = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 18. Error gap length distributions generated by a waveform simulation

 and by the Gilbert model for a wireless local area network with 

SNR = 12 dB and CIR = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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1

The author is with the Institute for Telecommunication Sciences, National

Telecommunications and Information Administration, U.S. Department of Commerce, Boulder,

CO 80305.

WIRELESS LINK STATISTICAL BIT ERROR MODEL

John J. Lemmon1

A bit error model that enables simulations of the digital error performance of

wireless communication links has been developed. The model development has

been based on error sequences derived from waveform simulations of wireless

link performance with various modems operating under varying propagation,

noise, and interference conditions. Values of the model parameters are obtained

by analyzing the distributions of the lengths of error bursts and error gaps (errorfree intervals). Mathematical expressions have been derived for the means and

variances of the error burst and error gap distributions of the model as functions

of the model parameters. Constraining the means and variances to the values

obtained from waveform simulations uniquely determines values of the model

parameters corresponding to a given set of link conditions. Examples of error

burst and error gap distributions obtained from waveform simulations are

compared with those generated by the model for a land mobile radio system and a

wireless local area network. The simulated and model distributions are quite

similar; however, the model runs tens of thousands of times faster than the

corresponding waveform simulations, enabling rapid determination of link

performance.

Keywords: wireless communication link, bit error model, Markov chain, waveform simulation

1. INTRODUCTION

The Institute for Telecommunication Sciences (ITS) is currently collaborating with the National

Institute of Standards and Technology (NIST) to develop national information infrastructure

research and evaluation capabilities. One area of this project, the wireless communications

performance benchmarking program, includes the development of a statistical bit error model

serving as a demonstration of technology that could be used in higher level network simulations.

Wireless network performance simulation depends on knowledge of the statistical distribution of

bit errors for each wireless link represented in the network. The distribution is a function of all

the link variables, including the channel, noise, interference, modem, coding, equalization, etc.

The bit errors encountered on a communication link can be obtained by a waveform level

simulation of the entire link. However, this kind of simulation can be computationally

prohibitive, particularly for simulations of networks comprising many links.
2

A more efficient form of simulation is discrete event simulation, whereby one generates a bit

error stream directly. Waveform simulation typically uses many samples per bit and requires

simulating the entire communication link for each sample. By contrast, discrete event simulation

of bit errors requires only one sample per bit, and, as will be seen, only requires the generation of

one or two random numbers per sample.

ITS has undertaken the development of a wireless link bit error model that enables discrete event

simulation of the bit errors encountered on wireless links. The model development has been

based on error streams derived from waveform simulations of link performance under various

conditions. Values of the model parameters have been determined by analyzing the distributions

of the lengths of error bursts and error gaps (error-free intervals). It will be shown that the

distributions generated by the waveform simulations and by the model are quite similar;

however, the calculations with the statistical model typically run tens of thousands of times

faster than the waveform simulations (the precise increase in speed depends on the type of link

being simulated).

Section 2 of this report presents the statistical model. Expressions for the error burst and error

gap length distributions are derived in Section 3 and are used to relate the model parameters to

measurable quantities, as discussed in Section 4. In Sections 5 and 6 the model is compared with

waveform simulations of the digital performance of land mobile radio systems and wireless local

area networks under various conditions. Concluding remarks and recommendations for

additional work are made in Section 7.

2. THE MODEL

Bit error models generate a sequence of noise bits (where zeros represent good bits and ones

represent bit errors) that is modulo 2 added to input bits to produce output bits. Models can be

grouped into two broad classes: memoryless models and those with memory. In memoryless

models the noise bits are produced by a sequence of independent trials. Each trial has the same

probability P(0) of producing a correct bit and probability P(1) = 1 - P(0) of producing a bit

error.

Measured data from actual communication links indicate that many links have memory, that is,

the errors occur in isolated bursts. This is because many link impairments, such as impulsive

noise, switching transients, and multipath fades, are bursty in nature. A commonly used

technique to endow a model with memory is to make the bit error probability depend on the

states of a Markov chain.

The use of Markov chains in bit error models was initiated by Gilbert [1]. The Gilbert model is

based on a Markov chain with two states: G (for good) and B (for bad or for burst). In state G,

transmission is error-free. In state B, the link has probability h of transmitting a bit correctly. A

transition diagram and bit error probabilities for the Markov chain are shown in Figure 1. For

suitably small values of the transition probabilities p = Prob(B6G) and P = Prob(G6B), the states

B and G tend to persist and the model simulates bursts of errors.
3

Figure 1. Transition diagram and bit error probabilities for the Gilbert model.

(1)

This simple model has three independent parameters (p, P, and h) and was originally used to

describe performance measurements over telephone circuits. Whether the model is adequate to

describe the error performance of wireless links has been investigated in the present work. One

can envisage more complicated models with more parameters (e.g., more than two states in the

Markov chain); however, determination of the model parameters from measured data becomes

more difficult as the number of parameters increases.

3. PROBABILITIES

The parameters p, P, and h are not directly observable and must therefore be determined from

statistical measurements of the error process. Runs of G alternate with runs of B. The run lengths

have geometric distributions, with mean 1/P for the G-runs and 1/p for the B-runs. The fraction

of time spent in state B is therefore P(B) = P/(P+p). Since bit errors occur only in state B, and

with probability 1-h, the probability of error is

The sequence of states cannot be reconstructed from the sequence of bits in the error process,

because both zeros and ones (good bits and bit errors) are produced in the B state. Thus, the

distributions of run lengths of the states cannot be used to determine the model parameters from

measurements of the error process. However, the bits of the error process itself, (i.e., runs of
4

(2)

(3)

(4)

(5)

(6)

zeros and ones) are observable, and the distributions of run lengths of zeros (error gaps) and ones

(error bursts) can be used to determine the model parameters.

An expression for the error burst length distribution can be derived as follows. Error bursts can

only occur in the B state. Therefore, given the occurrence of a one, the probability that the next

K-1 bits are ones is qK-1(1-h)K-1, where q = 1-p. After a run of K ones, the error burst can

terminate in either of two ways: the random process can return to the G state, the probability for

which is p = q-1, or it can remain in the B state and produce a zero, the probability for which is

qh. Thus, the probability distribution for error bursts of length K, defined as the conditional

probability that, given a zero followed by a one, the next K-1 bits are ones followed by a zero, is

The notation 1K-1 denotes a run of K-1 ones.

Error gaps can occur in either the B state or the G state, and the derivation of an expression for

the distribution of error gap lengths is therefore less straightforward than that for the error burst

lengths. However, Gilbert [1] has shown how recurrent events theory can be used to obtain the

needed probabilities. Let fK denote the conditional probability that, given state B, the next return

to B will occur at step K:

Then f1 = q, f2 = pP, and fK = pQK-2P for K $ 2, where Q = 1-P. These probabilities can be viewed

as the coefficients of a generating function F(t) of recurrence time probabilities:

The probability fK

(m) that the mth return to B happens at step K has the generating function

Starting from a one (and hence from state B), the next one must occur at a return to B. The

probability that, given the occurrence of a one, the next one will occur on the mth return to B at

step K is
5

(7)

(8)

(9)

(10)

(11)

(12)

(13)

It follows that, given a one, the probability that the next one will occur at step K (regardless of

the number of returns to B) is

Multiplying (7) by tK, summing over K, and making use of (5) leads to the generating function

V(t) = 3<(K)tK:

Substituting (4) into (8),

where D(t) = 1 - (Q+hq)t - h(p-Q)t2

.

Now factor the quadratic D(t):

where

and L is given by the same expression with a negative square root. Substituting (11) into (9),

V(t) becomes

The coefficient of tK in the power series expansion of V(t) is

A recurrence formula for <(K) can be found by writing (9) as D(t)V(t) = (1-h)[q+(p-Q)t] and

equating coefficients of tK:
6

(14)

(15)

(16)

(17)

Initial values are

The probability distribution for error gaps of length K, defined as the conditional probability

that, given the occurrence of a one followed by a zero, the next K-1 bits are zeros followed by a

one, is

Substitution into (16) of either (13) or (14) and (15) leads to the error gap length distribution as a

function of the model parameters.

4. PARAMETER ESTIMATION

The determination of the three parameters p, P, and h from measurements of the error process

requires that the parameters be expressed as functions of three other parameters that are directly

observable. Clearly, many parameters exist that could be used for this purpose. For example,

Gilbert [1] used the probability of error P(1) and the conditional probabilities

P(11)/[P(10)+P(11)] and P(111)/[P(101)+P(111)].

The purpose of the present work has been to develop a model that can be used to simulate the

error process and correctly reproduce all of its statistical properties. To validate the model, the

error processes generated by the model must be compared to measured error processes. Since

these processes consist of alternating runs of zeros and ones, the processes can be characterized

by the probability distributions of the lengths of error bursts (runs of ones) and error gaps (runs

of zeros). From this point of view, the objective of the parameter estimation is to choose values

of the model parameters that generate error burst and error gap distributions that resemble the

corresponding measured distributions as closely as possible. For this reason the parameter

estimation has been formulated in terms of the error burst and error gap distributions.

The first and second moments (means and variances) of the measured distributions can be used

to determine the model parameters as follows. Use of the identity
7

(18)

(19)

(20)

(21)

(22)

(23)

enables one to calculate the mean error burst length :EB from (2),

and the mean error gap length :EG from (13) and (16),

Similarly, the identity

in conjunction with (2), (13), and (16) implies that the variance FEB

2

 of the error burst

distribution is

and the variance FEG

2

 of the error gap distribution is

Note that the functional relationships in (18) and (21) involve only two of the model parameters,

q and h, in the combination q(1-h). Therefore, these two relationships are not independent and

cannot be used to simultaneously determine values of q and h; each relationship determines a

value of the quantity q(1-h), and these values are likely to be different if the measured error burst

lengths do not have a geometric distribution. To avoid this difficulty, (21) has not been used in

the parameter determination. Instead, (18), (19), and (22), which provide three independent

functions of the model parameters, have been used to simultaneously determine values of the

three parameters.

Using (18) and (19) to solve for Q and h in terms of q, :EB, and :EG, one finds
8

(24)

(25)

and

Substitution of (23) and (24) into (22) results in a relationship between q and the measured

quantities FEG

2

, :EB, and :EG. This relationship is evaluated numerically and a value of q

determined. Substitution of this value of q into (23) and (24) determines values of h and P=1-Q,

respectively. In this way values of the model parameters have been determined for a variety of

link conditions.

Note that using :EB and :EG to determine the model parameters guarantees that the model will

generate the measured bit error probability, because the measured means determine the error

probability through the relationship

As a consistency check, substitution of (18) and (19) into (25) yields the expression in (1) for

P(1).

5. LAND MOBILE RADIO LINK PERFORMANCE

The viability of the statistical model to simulate the digital performance of wireless links has

been investigated by comparing the bit error processes generated by the statistical model and by

waveform simulations for a variety of land mobile radio link conditions. Error burst and error

gap length distributions have been derived from the waveform simulations and used to determine

values of the model parameters for various link conditions. The dependence of the model

parameters on link conditions has been investigated to demonstrate that well-defined

relationships exist between the link conditions and the model parameters. Finally, these

parameter values have been used in the model to generate error burst and error gap length

distributions to be compared with those generated by the waveform simulations.

5.1 Waveform Simulations

The waveform simulations were performed with a commercial communication link simulation

tool that was enhanced with additional modules developed at ITS. This tool enables the user to

configure a communication link with modular transmitter, channel, and receiver models, and has

been used to simulate the performance of a variety of wireless links, as described by Quincy and

Achatz [2].
9

Three types of channel distortion were used in the simulations: time-varying multipath, additive

noise, and interference. The simulations illustrate the performance of a binary frequency shift

keying (BFSK) mobile system operating at a frequency of 900 MHz and traveling through an

urban canyon. The source is random data at a rate of 14.4 kbps.

The propagation model of the urban canyon, based on measurements by Cox [3], uses three

independent, time-varying propagation paths separated by delays. The first path is not delayed

and is, on average, the strongest path. The second path is delayed by 5 :s and has an average

power 8 dB below that of the first path. The third path is delayed by 8 :s and has an average

power 15 dB below that of the first path. Doppler shifts as high as 80 Hz are possible in any of

the paths. The amplitude and phase of each path are controlled by a time-varying Jakes model.

After the transmitted signal is distorted by the propagation channel, noise and interference are

added to the received signal. Receiver front-end noise is modeled as additive white Gaussian

noise. Cochannel interference is modeled by a transmitter also operating with BFSK modulation

at 14.4 kbps.

Simulations were performed with signal-to-noise ratios (SNR) that varied from 0 to 50 dB in

steps of 10 dB and interference-to-signal ratios (ISR) that varied from 0 to -40 dB in steps of 10

dB. The simulations varied in length, depending on the error performance. Each simulation was

run in blocks of 5000 bits until a minimum of 1500 errors occurred.

Figure 2 shows the bit-error-rate (BER) plotted versus the SNR. Each curve corresponds to a

different value of the ISR. As expected, the BER increases with increasing ISR and decreasing

SNR. From these curves, three limiting cases of performance can be identified: (1) noise limited

(NL), (2) interference limited (IL), and (3) frequency selective fading limited (FSFL). In the NL

case an increase in SNR decreases the BER. In the IL case, a change in the ISR dramatically

affects the BER for the same SNR. In the FSFL case, the interference is minimal and

performance is not improved by increasing the SNR; the frequency selective fading, caused by

multipath, is limiting the performance.
10

Figure 2. Land mobile radio system performance relating BER to SNR

and parametrically to ISR.

5.2 Dependence of Model Parameters on Link Conditions

For each waveform simulation, the error gap length and error burst length distributions were

generated and values of the model parameters determined, as described in Section 4. The results

for q, Q, and h are shown in Figures 3, 4, and 5, respectively. Like the BER curves in Figure 2,

each parameter is plotted versus the SNR, and each curve corresponds to a different value of the

ISR.

The results in Figure 3 indicate that q decreases with increasing SNR and decreasing ISR (i.e.,

decreasing BER). This is consistent with heuristic expectations, because one expects the Markov

chain to spend less time lingering in the bad state as the error probability decreases.

Conversely, the curves in Figure 4 show that Q increases with increasing SNR and decreasing

ISR (except at the lowest value of SNR). Thus, the Markov chain tends to spend more time

lingering in the good state as the error probability decreases.

The results in Figure 5 seem less amenable to heuristic interpretation, since they indicate that the

variations in h are not monotonic in either SNR or ISR. However, at the lower values of SNR, h

tends to increase with increasing SNR and decreasing ISR, as one would expect (probability of a

correct bit in the bad state increases as the error probability decreases).
11

Figure 3. Transition probability q versus SNR for the land mobile radio

link simulations. Each curve corresponds to a different value of

ISR.

Heuristic interpretations aside, what is important about these results is that the parameters vary

in a smooth, deterministic fashion with varying link conditions (SNR and ISR). A meaningful

correspondence therefore exists between the model parameters and the link conditions. The

determination of analytic expressions for these relationships via empirical curve fitting has not

been carried out. However, such expressions would provide a useful tool for evaluating the

model parameters for given link conditions, without having to carry out waveform simulations

and statistical measurements of the bit error process or interpolations between the parameter

values shown in Figures 3-5.
12

Figure 4. Transition probability Q versus SNR for the land mobile radio

link simulations. Each curve corresponds to a different value of

ISR.

Figure 5. Probability h of transmitting a bit correctly (in the B state)

versus SNR for the land mobile radio link simulations. Each

curve corresponds to a different value of ISR.
13

5.3 Comparison of Model with Waveform Simulations

Having determined values of the model parameters, it remains to show that the model correctly

reproduces the characteristics of the error process. To this end, the model was used to generate

error processes for an NL link (SNR=10 dB, ISR=-40 dB), an IL link (SNR=50 dB, ISR=-10

dB), an FSFL link (SNR=50 dB, ISR=-40 dB), and an “average” link (SNR=30 dB, ISR=-20 dB).

The lengths of the simulations using the model are the same as those of the corresponding

waveform simulations. It was found that the model runs approximately 50,000 times faster than

the waveform simulations.

Comparisons of the error burst length and error gap length distributions generated by the model

with those generated by the waveform simulations for these four cases are shown in Figures 6-13.

In each figure the probability that an error burst or error gap is of a given length is plotted versus

that length (in bits) in the form of a histogram. The vertical scale in the error gap distributions has

been made logarithmic to more clearly reveal the long tail in the distributions at large error gap

lengths.

The distributions generated by the statistical model and by the waveform simulations show

generally good agreement with the exception of one feature in the error gap distributions: the

distributions generated by the waveform simulations have a local minimum at gap lengths in the

vicinity of 25 followed by a slight bump in the distributions at gap lengths in the vicinity of 75.

The apparent absence of this feature in the error gap distributions generated by the model cannot

be attributed to statistical noise in the distributions due to the finite length of the simulations,

because the analytical expression (13) for the error gap distribution in the statistical model does

not have this feature. As an illustration of this, Figure 14 shows the error gap distribution for the

“average” link computed with the recurrence formula given by (14)-(16) and clearly indicates that

this feature is absent. This relatively minor feature could perhaps be simulated by generalizing the

model; for example, the number of states could be increased. However, this has not been

investigated due to the increased complexity it would add to the model.

Quantitative comparisons of the distributions generated by the model and by the waveform

simulations are difficult, because analytical expressions for the distributions generated by the

waveform simulations, analogous to those for the model, do not exist. Thus, any measure of

disagreement would largely reflect the statistical noise in the distributions generated by the

waveform simulations, which could only be eliminated by running simulations with excessively

long computation times.
14

Figure 6. Comparison of error burst length distributions generated by (a)

waveform simulation and (b) the Gilbert model for an ìaverageî land

mobile radio link.
15

Figure 7. Comparison of error gap length distributions generated by (a)

waveform simulation and (b) the Gilbert model for an ìaverageî land

mobile radio link.
16

Figure 8. Comparison of error burst length distributions generated by (a)

waveform simulation and (b) the Gilbert model for a FSFL land mobile

radio link.
17

Figure 9. Comparison of error gap length distributions generated by (a)

waveform simulation and (b) the Gilbert model for a FSFL land mobile

radio link.
18

Figure 10. Comparison of error burst length distributions generated by (a)

waveform simulation and (b) the Gilbert model for an IL land mobile

radio link.
19

Figure 11. Comparison of error gap length distributions generated by (a)

waveform simulation and (b) the Gilbert model for an IL land mobile

radio link
20

Figure 12. Comparison of error burst length distributions generated by (a)

waveform simulation and (b) the Gilbert model for a NL land mobile

radio link.
21

Figure 13. Comparison of error gap length distributions generated by (a)

waveform simulation and (b) the Gilbert model for a NL land mobile

radio link.
22

Figure 14. Plot of the analytic representation of the error gap length distribution

in the Gilbert model for an ìaverageî land mobile radio link.

6. WIRELESS LOCAL AREA NETWORK PERFORMANCE

To further investigate whether the Gilbert model can describe the error performance of wireless

links, error processes generated by waveform simulations and by the model have been compared

for a wireless local area network (WLAN). The WLAN uses an IEEE 802.11 proposed standard

modulation, namely, frequency-hopped Gaussian minimum shift keying (GMSK). Multipath

distortion was derived from impulse response measurements performed by ITS in an office

environment at a center frequency of 1500 MHz [4]. Since the bandwidth of the channel

measurements was considerably narrower than the 83.5 MHz hopping bandwidth of the proposed

IEEE 802.11 standard, no hopping sequence was used in the simulations.

Direct sequence and frequency hopping modems will coexist in the same ISM band and are

likely to generate cochannel interference to one another. Interference caused by a direct sequence

modem was modeled as wideband noise and white Gaussian noise was also added to the

distorted signal.

The signal source was an image derived from a standard bit map file obtained by scanning a U.S.

Geological Survey topographic map. Pixel resolution was limited to 16 shades of gray

represented by 4 bits.
23

Figure 15. Wireless local area network performance relating BER to SNR.

Simulations were performed with a fixed carrier-to-interference ratio (CIR) of 10 dB and an

SNR that varied from 0 to 28 dB in steps of 4 dB. Each simulation used 300,000 bits at a

transmission rate of 1 Mbps. Details of the simulations, including the channel measurements,

processing of the measured data, the GMSK modulation, and the modem architecture, have been

discussed by Achatz and Qiuncy [5].

The error processes derived from the simulations were analyzed analogously to those for the land

mobile radio links discussed in Section 5. The BER is plotted versus SNR in Figure 15. Error

burst length and error gap length distributions were computed, and the means and variances of

the distributions were used to determine values of q, Q, and h versus SNR. The results are shown

in Figure 16. The dependencies of the parameters on SNR are qualitatively similar to those for

the land mobile radio links at the corresponding power levels of signal, noise, and interference.

The values of the model parameters determined from the waveform simulations were then used

in the model to generate error burst and error gap length distributions. The distributions closely

resemble those generated by the waveform simulations. Example comparisons are shown in

Figures 17 and 18 for an SNR of 12 dB. Note that these plots differ from those in Figures 6-13

for the land mobile radio links in that semi-log scales (rather than linear scales) and the

analytical expressions for the model distributions (rather than simulated distributions) are used in

the former. For these WLAN scenarios, the model runs approximately 15,000 times faster than

the corresponding waveform simulations.
24

Figure 16. Gilbert model parameters q, Q, and h versus SNR for the wireless

local area network link simulations.

Figure 17. Error burst length distributions generated by a waveform simulation

and by the Gilbert model for a wireless local area network with SNR

= 12 dB and CIR = 10 dB.
25

Figure 18. Error gap length distributions generated by a waveform

simulation and by the Gilbert model for a wireless local area

network with SNR = 12 dB and CIR = 10 dB.

7. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK

The simple three-parameter model (Gilbert model) with parameter determination as described in

this paper enables accurate simulation of the error processes for the wireless links that were

investigated (land mobile radio and wireless local area networks). The error distributions derived

from the model generally agree with those derived from waveform simulations.

The model is a powerful tool that can enhance network simulations and provide rapid evaluation

of link quality. For the land mobile radio and local area network scenarios discussed in this

work, the model runs faster than the corresponding waveform simulations by factors of

approximately 50,000 and 15,000, respectively. As more elements are added to the link

simulations (antennas, coding, etc.) these factors may increase considerably, due to the increased

signal processing load in the waveform simulations. It is therefore recommended that this work

be extended to other link conditions (propagation conditions, modulations, coding, antennas,

etc.).

The model parameters appear to be well-defined, deterministic functions of the link conditions

(signal-to-noise and signal-to-interference ratios). The dependence of the model parameters on

link conditions could therefore be represented as functional relationships determined by

empirical curve fitting. Such relationships would obviate the need to carry out additional

waveform simulations or to carry out interpolations between previously determined values of the

model parameters every time link conditions are varied.
26

8. ACKNOWLEDGMENTS

This work has been supported by the National Institute of Standards and Technology as part of

the wireless communications performance benchmarking program. The author wishes to

acknowledge E.A. Quincy for suggesting this problem and for helpful advice, R.J. Achatz for

helpful discussions on discrete channel models and for assistance in carrying out the waveform

simulations, and J.F. Mastrangelo for implementing the model and generating the numerical

results presented in this report.

9. REFERENCES

[1] E.N. Gilbert, ìCapacity of a burst-noise channel,î Bell Syst. Tech. J. 39, pp. 1253-1256,

1960.

[2] E.A. Quincy and R.J. Achatz, ìWireless performance prediction via software

simulation,î in Proc. of the IEEE Pac. Rim Conf. on Comm., Computers, and Signal

Processing (PACRIM), Victoria, British Columbia, Canada, July 11-12, 1994.

[3] D.C. Cox, ìDistributions of multipath delay spread and average excess delay for 910

MHz urban mobile radio paths,î IEEE Trans. Ant. Prop. AP-23, pp. 206-213, 1975.

[4] P.B. Papazian, Y. Lo, E.E. Pol, M.P. Roadifer, T.G. Hoople, and R.J. Achatz, ìWideband

propagation measurements for wireless indoor communications,î NTIA Report 93-292,

Jan. 1993.

[5] R.J. Achatz and E.A. Quincy, ìPerformance prediction of WLAN speech and image

transmission,î in Proceedings of the 1995 IEEE 45th Vehicular Technology Conference,

Vol. 2, Chicago, IL, Jul. 1995, pp. 559-563.