NASA Contractor Report 3327 

NASA 

CR 

3327 

c.1 

Preliminary Development 

Digital Signal Processing 

in Microwave Radiometers 

William D. Stanley 

CONTRACT NAS I- 15 676 

SEPTEMBER 1980 
TECH LIBRARY KAFB, NM -- 

lllilllrlllulllllsl~ll

0063953 

NASA Contractor Report 3327 

Preliminary Development of 

Digital Signal Processing 

in Microwave Radiometers 

William D. Stanley 

Old Dominion University 

Norfolk, Virginia 

Prepared for 

Langley Research Center 

under Contract NASl-15676 

NASA 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1980 

- 
INTRODUCTION 

The contract covered by this report involved a number of separate 

but closely related activities related to the field of microwave radiometry. These activities included the following specific tasks: 

(11 

(2) 

(3) 

(4) 

(5) 

(61 

(7) 

The development of several control loop and dynamic noise model 

computer programs for simulating microwave radiometer measurements 

has been essentially completed. This effort was initiated under 

an earlier task order contract, and the results of this develop- 

'ment have been published (refs. 1 and 2). 

Computer modeling of an existing stepped frequency radiometer 

has been done in an effort to determine its optimum operational 

characteristics. 

The classical second-order analog control loop has been investigated to determine its relative and possible optimum performance in 

reducing the error of the estimate in a microwave radiometer based 

on noise bandwidth and settling time criteria. Such results were 

not found anywhere in the literature. 

Several designs of digital signal-processing units for a microwave 

radiometer have been investigated extensively. One particular 

design has been identified as the most promising, and its behavior 

has been simulated extensively by means of the digital simulation 

programs mentioned in (1). 

Efforts have been initiated for developing the hardware and software 

required in the implementation of the digital signal-processing unit. 

Some of the general characteristics and peculiarities of digital 

processing of noiselike microwave radiometer signals have been 

investigated. 

Technical support has been provided in the form of computer data 

reduction of results obtained in actual ice missions. 

During the period of performance, the following specific persons were 

involved at different times (all on a part-time basis): William D. 

Stanley (principal investigator, ODU), Roland W. Lawrence (graduate research 
assistant, ODU), Sally E. Kerpelman (undergraduate research assistant, 

ODU), John M. Jeffords (graduate research assistant, ODU), and William 

H. Thornton (faculty consultant, ODU). 

Dr. Stanley was directly involved in all tasks except (7), and 

Lawrence participated in all but (3) and (7). Jeffords performed most 

of the work in (3), and the appendix of this report was authored by him. 

Prof. Thornton provided consultative assistance in tasks (4) and (5). 

Kerpelman provided virtually all the assistance required in task (7). 

This report provides a documentation of the efforts of tasks (2), 

(3), (4), (5), and (6). An attempt has been made to divide the report 

into logical units commensurate with the specific concepts investigated. 

HOWeVeT, due to a significant amount of overlap in some of these areas, 

the report does not necessarily proceed in an exact chronological format. 

Use of trade names or names of manufacturers in this report does 

not constitute an official endorsement of such products or manufacturers, 

either expressed or implied, by the National Aeronautics and Space 

Administration. 

2 
CONTROL ANALYSIS OF FEEDBACK RADIOMETER 

As a prelude to the modeling, analysis, and design of a Dicke 

noise-injection feedback radiometer, it is desirable to delineate 

the particular manner in which the control mechanism functions. A 

simplified block diagram of the important parameters of the closedloop system is shown in figure 1. The signals appearing on the figure 

represent only the "dc-like" control levels expressed in terms of 

the effective temperature values at different points in the system. 

The fluctuation components of the signals do not appear in this model. 

The model shown in figure 1 will hereafter be designated as a "control 

model." The more elaborate statistical model will be referred to as 

a "dynamic noise model." 

This simplified control model only shows one Dicke switch since 

it assumes that all frequency variations between the first Dicke switch 

and the correlation switch are sufficiently high that they can be 

neglected for control-loop analysis. The gain between the two Dicke 

switches has been momentarily lumped with the transfer function G(s). 

The frequency variation associated with G(s) is primarily that of the 

integrator and loop filter which constitute the estimation circuit 

portion of the system. 

Analysis of the loop will now be made. The input temperature TA 

undergoes a very slight attenuation due to the transmission coefficient 

1 - KDC of the directional coupler (typically 0.99). The injected noise 

is weighted by the coupling coefficient KDC of the directional coupler 

(typically 0.01) and added to the antenna temperature. The effect of 

the Dicke switch is to introduce an additional l/2 factor so that TE 

on figure 1 can be expressed as 

TE = ; (Tl - TB) (1) 

where Tl is the effective output temperature of the directional coupler 

and TB is the ambient temperature of the constant temperature enclosure. 
Effective output temperature Tl in turn can be expressed as 

- 

(2) 

where TI; is the injected excess noise appearing at the attenuator output. 

In turn, TI; can be expressed as 

TI; = KATT TEX T K, 5 (3) 

where KATT is the attenuator constant, TEX is the excess temperature 

of the noise diode, r is the pulse width of injected noise pulses, 

and Kl is a constant relating output voltage Vo to pulse frequency 

FO’ The output voltage transform Vo(s) can be expressed as the 

product of the transfer function G(s) and the error temperature transform Th(s). For convenience, the s-domain notation will be omitted in 

most of the subsequent calculations. Thus, 

V. = -G TE (4) 

Equations (l), (2), (3), and (4) may be solved simultaneously to 

yield a composite expression for the output voltage as a function of the 

input and reference temperatures. After some manipulation, the result 

is obtained as 

'0 = (1 - 'DC) [l :',' 8/2] (TB - 'A) 

where, for convenience, 6 has been defined as 

' = KDC KATT TEX -r K1 

(5) 

(61 

The output pulse frequency F. is related to the input and reference temperatures by 

F. = (1 - 'DC> [I: ",";,;I (TB - 'A) (7) 
One other possible quantity of interest is the "error" temperature TE, 

degined in terms of a hypothetical point following the addition of the 

noise-injection temperature, but after the l/2 factor of the Dicke 

switch. This function is 

T = 1 b - KDC)(TA - 'B> 

E 2 l+ G8/2 (81 

As far as the control mechanism is concerned, the significant frequency 

dependency of the loop is contained in the G(s) transfer function. 

A typical arrangement involves a near-perfect integrator and a loop 

filter such as 

G(s) = 

KO 

~(1 + ST) (9) 

The presence of this particular function results in a type 1 control 

loop. The resulting steady-state error due to a step function change in 

temperature at the input is zero on a deterministic basis. On the other 

hand, the presence of the noise fluctuations and a finite input bandwidth 

prevent the statistical error from ever reaching a zero value. 
MODEL OF ANALOG RADIOMETER SYSTEM 

The particular radiometer used as the reference for virtually all 

the analysis and design work involved in this study was the 4.5 to 7.2 

GHz stepped frequency system developed by Harrington et al. (ref. 3). A 

simplified block diagram of the system is shown in figure 2. This 

particular radiometer is a Dicke square-wave correlated type employing 

closed-loop noise-injection feedback to reduce the effects of gain 

fluctuations. Many of the salient operational features of the system 

have been adequately described (ref. 3) and will not be discussed here. 

The emphasis in this development will be directed toward the establishment and utilization of certain mathematical models for describing 

various aspects of system operation. 

While much of the emphasis in this overall study was devoted to 

the development of potential digital signal-processing schemes for 

radiometers, it was apparent at the very outset that the operation and 

limitations of the analog system had to be fully investigated first. 

Consequently, considerable effort was expended in obtaining various 

transfer functions and operational parameters for components within 

the analog system. Each of several sections will be discussed 

individually. 

RF Subsystem 

The RF subsystem will be defined as that portion of the radiometer 

from the output of the directional coupler to the input of the squarelaw detector. For the dc control model, all frequencies are sufficient 

high that the net effect of this entire portion of the system may be 

represented by a single constant gain factor. On the other hand, for 

the dynamic random model, it is necessary to utilize an appropriate 

transfer function to represent the effect of the wideband filter preceding the square-law detector. 

1Y 

An approximate gain constant for the RF subsystem is determined by 

the procedure that follows. During the time that early testing on the 

6 
radiometer was performed, R. F. Harrington of NASA/LaRC accumulated a 

significant amount of data relative to signal and noise levels used in 

the evaluation. Based on a specified noise figure of 5 dB, Harrington 

estimated that the effective receiver temperature was about TR = 627 K 

at some'nominal frequency in the operating range. The Dicke reference 

temperature was established as T8 = 308 K. With the reference temperature applied continuously to the RF subsystem, the signal level at the 

input to the square-law detector was measured at approximately -33 dBm. 

The input power density producing this output is estimated to be 

k (TB + TR) = 1.38 x 1O-23 x (308 + 627) = 1.29 x 10w20 W/Hz or 

-168.89 dBm/Hz. The corresponding gain x bandwidth factor is then 

estimated to be -33 dBm - (-168.89 dBm/Hz) = 135.89 dB-Hz. This 

corresponds to an absolute power gain x bandwidth of 3.884 x 1013 Hz. 

As it turns out, an early calculation based only on T R employed an 

input power density of -170.63 dBm/Hz, which resulted in a gain 

bandwidth factor of 137.63 dB-Hz or about 5.792 x 1013 Hz. This value was 

rounded to 5.8 x lOI Hz and used in many of the system simulations. 

This difference of about two dB is within the uncertainty range of the 

actual gain factors of the various components in the forward part of the 

loop as well as the frequency dependency range, and is believed to be 

insignificant in view of the general uncertainty concerning overall 

forward gain system parameters. This point of view is further reinforced 

by the fact that the gain involved is in the forward loop and has a 

relatively small effect on the closed-loop steady-state accuracy of the 

measurement within reasonable limits. 

Diode Detector 

The diode detector employed is an HP 5082-2350 Schottky diode which 

is biased in a square-law region of operation over the entire operating 

range of the radiometer. Within this region, the output dc voltage is 

then a linear function of the input power. Measurements by Harrington 

indicated a 200-PV output when the input was about -33 dBm (or 0.5 JJW). 

The corresponding gain constant is thus 200 x 10e6/0.5 x 10m6 = 400 V/W. 

7 
Video Amplifier 

The video amplifier is that section of the radiometer between the 

square-law detector and the output loop filter. This amplifier is 

basically a low-pass amplifier, but it need not pass dc since the mean 

temperature estimate being measured appears as a modulated component 

on the Dicke frequency. However, for square-wave correlation, it is 

necessary that a relatively large number of components of the squarewave be transmitted by the video amplifier with minimum attenuation 

and phase shift. Furthermore, noise injection with short pulses 

mandates that the amplifier have a large bandwidth compared with the 

reciprocal of the injection pulse width. (This point is pursued further 

later in this report.) 

The actual circuits used in the video amplifier will now be considered. The first two stages, designated as ICI and IC2, are identical, 

and each has the form shown in (a) of figure 3. The transfer function 

is determined to be 

sR2C2[1 + sC1 (Ro + W] 

Ga(s) = (1 + sRlC1) (1 + sR2C21 

With the element values shown on the figure, the function is 

GabI = 

66 x lo- 3s(l + 2.3892s) 

(1 + 66 x 10-3,) (1 + 79.2 x 10%) 

66 x 10-3s = [I + s/(2lr x 0.0666jJ 

[I + s/(27~ x 2.41)] [l + S/(~IT x 2.0095)J 

(10) 

(lla> 

(lib) 

The corresponding Bode plot for ICI and IC2 is shown in (b) of figure 3. 

8 
The circuit diagram for stage IC3 is illustrated in (a) of figure 4. 

The resulting transfer function Gb(s) can be determined as 

sR,ClCl + h+/R31 

Gb(4 = (12) 

1 + s(RIC1 + R2C2 + R2C1) + s2R1R2C1C2 

With the element values shown on the figure and the variable resistor 

adjusted for maximum gain, the function is 

Gb(S) = 

10.5s 

(1 + 0.53047s) (1 + 28.2767 x 10-6s) 

(13a) 

10.5s = 

[l + s/(2~ x 0.3)] p + S/(~IT x 5.63 x 103)] (13bl 

The corresponding Bode plot is shown in (b) of figure 4. 

Combining the effects of two sections of ICl and one section of IC2, 

the complete Bode plot of the original video amplifier is shown in 

figure 5. Observe that a zero of transmission occurs at dc and that a 

low-frequency rolloff of 18 dB/octave appears. Furthermore, the response 

is relatively flat from above a few hertz to about 5 or 6 KHz. However, 

it was later discovered that the bandwidth should be broadened considerably 

to accommodate the relatively narrow noise-injection pulses. 

DC Estimating Circuit 

Following the correlation switch, the signal is processed by an 

estimating circuit. The estimating network originally consisted of an 

analog integrator and a loop filter, which was a lag-lead network. The 

lag-lead network was later redesigned as a straight lag network, and this 

process will be discussed in a later section. 

9 
The circuit diagram of the integrator is shown in figure 6, and the 

circuit diagram of the original loop filter is shown in figure 7. The 

integrator has a transfer function given by 

Gi(s) = -& = 

-17.857 

S 

for the element values shown. 

The transfer function of the lag-lead network is 

(14) 

(15) 

For the particular element values shown on the figure, the transfer 

function is determined to be 

11 (1 + 2.545 x 10-3s) 

G&l = (16al 

(1 + 28 x 10-3s) 

11 F + s/(27r x 62.525)] 

= [l + s/(Z~r x 5.684)] 

The Bode plot of the lag-lead network is shown in (b) of figure 7. A 

Bode plot representing the combined effects of the integrator and laglead network is shown in figure 8. 

Noise-Injection Scheme 

The noise-injection process in the closed-loop system is achieved by 

the use of a voltage-to-frequency (V/F) converter, which produces constant 

width pulses whose frequency is directly proportional to the loop output 

voltage. The pulses are applied to a PIN diode driver circuit which is 

used as a gate for an avalanche noise diode. When the pulses are on, the 

10 
noise diode output is directly applied through an attenuator to the 

auxiliary coupling port of a directional coupler. Conversively, when 

the pulses are off, no injection noise is applied, and only the 

ambient temperature TB appears as an input from the feedback path. 

Under stable ideal closed-loop conditions, the injection noise plus the 

input antenna noise is just sufficient to balance the reference noise, 

and the effects of fluctuations in the forward path are eliminated. 

If the parameters of the feedback path are known accurately, the 

antenna temperature may be measured by determining the number of pulses 

used per unit time to establish the balance. 

Data on the noise diode indicates that the excess noise temperature 

is about 31 dB above 290 K, corresponding to an excess temperature of 

approximately 365,000 K. The V/F converter is a Date1 System VFV-lOK, 

which has a sensitivity of about 863 pulses/s/V. Each pulse has a width 

of 70 us so that the constant relating duty cycle to output loop 

voltage is about 863 x 70 x low6 = 0.0604. The injected noise encounters 

a 6-dB fixed attenuator corresponding to a power gain constant of 0.25. 

The injected noise temperature adds to the ambient temperature of 308 K 

at the attenuator output. The resulting signal temperature then adds to 

the antenna temperature through the directional coupler coupling coefficient 

of -20 dB, corresponding to a power coupling constant of 0.01. 

The manner in which the various constants contribute to the feedback 

noise is illustrated in figure 9. The possible input antenna temperature 

range is assumed to be from 0 to 300 K. The corresponding ranges of 

different temperatures are illustrated on the figure. The duty cycle 

ranges from about 0.334 to 0.00868, and the corresponding pulse rate ranges 

from 4774 pulses/s to about 124 pulses/s. Finally, the output voltage 

range is from about 5.532 v to 0.143 v. The injected noise has a 

sensitivity of about 25.55 K/pulse at the diode itself or about 0.0639 

K/pulse at the point of injection. This corresponds to a sensitivity of 

about 22,050 K/V at the diode or about 55.124 K/V at the injection point. 

11 
The manner in which the pulses add to the unsymmetrical open-loop 

signal in order to balance out the net temperature is illustrated in 

figure 10. The waveform in (a) depicts the initial situation in which 

the reference temperature plus the receiver temperature is greater 

than the input brightness temperature plus the receiver temperature. 

However, the effect of added noise-injection pulses brings about a 

net balance of energy as illustrated in (b) before filtering and 

smoothing occurs. The levels in the tables are the typical values 

appearing at the input to the estimation circuit; and of course, the 

results are smoothed by that portion of the system. The fluctuations 

about the mean levels are not shown in these illustrations; i.e., 

these are the "dc-like" levels. 

Other components in the system whose effects have been considered 

as constant for the purpose at hand include a tunnel diode amplifier 

manufactured by AERTECH having a noise figure of 6 dB and a gain of 

26 dB + 1 dB, a mixer and preamplifier stage produced by RHG having a 

noise figure of 7.5 dB + 0.5 dB and a gain of 47 dB, and bandpass filters 

produced by CIR-Q-TEL having an insertion loss of less than 1 dB in the 

passband. 

Combining all the constants discussed in this section, a control 

loop model for the stepped frequency radiometer is shown in figure 11 

based on maximum gain for the variable gain stage. After several steps of 

manipulation, the closed-loop transfer function for this system is obtained 

from standard control loop analysis as 

v, (sl 

GCL(S) = TB(s) - TA(s) 

= 20.214(1 + 2.545 x 10-3s) 

s2 + 38.5496s + 1114.1 

(17) 

The poles of the transfer function are readily determined to be 

s = -19.27 2 j 27.25. This means that the damping factor is c1 = 19.27 and 

the damped radian frequency is ad = 27.25 rad/s or fd = wd/2n = 4.34 Hz. 

12 
The undamped natural radian frequency is w. = 33.38 rad/s and f. = 

wo/2~ = 5.31 Hz. The damping ratio is 5 = a/w0 = 0.577. 

It can be shown that the equivalent one-sided noise bandwidth BN 

for this transfer function is given by 

BN = 

1 + boT21 2 

85 1 w 

0 (18) 

where T2 is the numerator time constant in the lag-lead network (2.545 ms 

in this case). The value BN = 7.28 Hz was calculated for the parameters 

just considered. 

The sensitivity AT of the closed loop feedback radiometer can be 

determined from the relationship 

AT = 2(TB + TR> 2 BNO 

- 

BSI 

(19) 

where BNO is the equivalent output noise bandwidth of the loop and 

BSI is the equivalent statistical bandwidth of the input wideband filter. 

Using values obtained from the system, the values of AT for the four 

possible input bandwidths are summarized as follows: 

BSI 20 MHz 100 MHz 500 MHZ 2 GHz 

AT 1.60 K 0.71 K 0.32 K 0.16 K 

These values represent sensitivities based only on the noise-reduction 

filtering and do not include variations in coupling factor, stability of 

the noise diode temperature, or Dicke ripple, which will be discussed 

later. 

13 
REDESIGN OF THE ANALOG LOOP 

Although a primary objective of this study was the development of 

techniques for digital processing of radiometer signals, certain 

immediate operational objectives suggested that some design modifications 

of the existing stepped frequency analog radiometer would be worthwhile. 

The first area of concern was that of the loop filter following the 

integrator in the estimation circuit. Prior to this study, the circuit 

used for this purpose was the combination of circuits shown in figures 

6 and 7. This circuit is a lag-lead network whose transfer function was 

given in equation (16). 

The lag-lead network, in contrast to the lag network, has a larger 

equivalent noise bandwidth for a given damping ratio and thus does not, 

in general, smooth the fluctuations as effectively. (Although, for the 

values initially used in this system, the difference was not appreciable). 

A separate study was made by a graduate student, John M. Jeffords, to 

determine various tradeoffs and possible optimum strategies for designing 

a second-order loop in a feedback radiometer. While much standard 

information concerning the usual control design of a second-order loop 

is widely available, peculiar aspects of the radiometer design necessitated 

a separate study for this purpose. The results of Jeffords' study are 

given in the Appendix. 

Jeffords used the equivalent noise bandwidth times the settling time 

as a criterion for optimization. This quantity will hereafter be referred 

to as the BNts product. Jeffords found that the absolute minimum BNts 

product did result from the simple lag network, and this occurred for 

5 z50.95. When a numerator lead time constant (denoted as T2) is used, 

the BNts constant increases rapidly. Simultaneously, the value of 5 

which results in minimum BNts also increases. 

An immediate reaction to this result might be to set T2 = 0 corresponding 

to the lag network. However, other criteria must be considered before 

this conclusion is completely obvious, such as the sensitivities of c 

14 
and the steady-state error to changes in loop parameters. After all, 

the loop parameters, particularly forward loop parameters, are subject 

to fluctuations, so the stability of the error to changes in parameters 

should be considered. 

Jeffords found that the sensitivity of the damping factor as a 

function of gain variations is a minimum in the vicinity of 5 = 1 for 

woT2 = 1. It was further found that the sensitivity of the error to 

changes in loop gain is smaller over a wide range of time for w,T~ 

products of 1 to 1.5 or so. The result of these last two findings could 

suggest the use of a lead network for certain applications in which a 

wide variation of gain fluctuations is expected. The results of Jeffords' 

study are developed in depth in the appendix. 

As it turns out, the variation in gain parameters for a typical 

radiometer is not expected to be excessive. Furthermore, if some additional time above the minimum settling time is provided, the increase 

in error is not expected to be significant even though the sensitivity 

is high. Typical calculations suggest that this conclusion is correct. 

After considering various aspects of the loop properties and the 

results of the study, a decision was made to select a value of 5 in the 

vicinity of unity (i.e., a critically damped loop) using a lag network 

only. The critically damped loop has certain design advantages, and 

it is sufficiently close in behavior to the optimum value that most of 

the properties are essentially identical. The BNts product for the 

critically damped loop is approximately 1.5. 

time should be allowed for the loop to fully 

higher sensitivity of the error to parameter 

lag network. 

However, an additional 

settle because of the 

changes in the case of the 

The redesign of the loop based on a critically damped response was 

then formulated as follows: If the lag-lead network is replaced with a 

simple lag network, its transfer function GE(s) is 

Gg(S) = 1 + sT1 (201 

15 
The closed-loop transfer function is then determined to be of the 

form 

v. (& 

TBb) - TAbI = (21) 

where K1 is the forward gain constant and KLOOp is the loop gain 

constant. 

Following a review of the settling time specifications, it was 

decided to design around a damping constant c1 = 50, thus producing a 

transfer function of the form 

Vo(s) Kl 

TB(s) - TA(S) = cs + 5012 (221 

Comparing equations (21) and (22), it can be deduced that T1 = 0.01 s and 

KR = 8.82. The circuit form used to realize the appropriate lag network 

is shown in figure 12. This circuit replaces the original lag-lead 

network. 

In order to properly study the effects on the actual radiometer of 

the lag-lead and lag networks, the test circuit of figure 13 was connected 

in the system, and the parameters adjusted to visually determine what 

appeared to be an optimum response. The results compared favorably with 

those obtained from the mathematical analysis. 

One other design consideration for the loop will now be discussed. 

The injected noise pulses appearing at the radiometer input are relatively 

narrow, i.e., they are 70-JJS wide. In the Dicke correlation process, it 

is necessary that all noise injected into the first half-cycle actually 

appear in the first half-cycle at the output switch. Any energy that 

16 
"spills over " due to bandwidth limitations will result in some error in 

the measurement process. Because of the short pulse widths, it was 

discovered that the bandwidth of the video amplifier was insufficient 

to process the noise-injection pulses without some distortions. 

Consequently, a degradation in the calibration factor as a function of 

the noise level was observed. 

As a first step, the capacitor was removed. When the problem was 

still observed, it was noted that the particular operational amplifier 

used had an insufficient bandwidth. Consequently, it was replaced 

with an operational amplifier having a much higher bandwidth and slew 

rate, and the problem was thus resolved. 

17 
DIME RIPPLE ANALYSIS 

Along with the noise fluctuations present in the output of a 

Dicke radiometer, the ripple corresponding to the Dicke switching 

frequency and its harmonics must be considered. This ripple is present 

due to the modulation and demodulation processes of the switched signal. 

While these components may be kept to a sufficiently low level, they 

are often ignored in the design process, and this can lead to troublesome 

operation. Judging from an extensive survey of the literature, it 

appears that many investigators are not particularly concerned with 

their presence since virtually no discussion of their effects has been 

found by this investigator. 

For the purpose of the ripple analysis, consider the block diagram 

shown in (a) of figure 14. It is assumed that steady-state conditions 

exist, in which case the square-ware is perfectly balanced with a 

sufficient amount of noise injection at the input. The constant K1 

represents all gain up to the point of the Dicke correlator, and this 

gain is assumed to be frequency independent as far as the frequency range 

of the Dicke switch and its harmonics are concerned. Note that this 

constant does not include the 0.5 Dicke constant shown in figure 11, 

since that constant arises in the switching process and does not appear 

explicitly in the figure. 

The function G(s) represents the net transfer function of the 

estimation circuit, which includes the integrator and the lag-lead or 

lag network. All of the significant frequency dependency in the loop is 

contained in this block. 

The constant K2 represents the feedback constant. As in the case 

of Kl, no significant frequency-dependent effects are assumed in this 

block. 

The signal vi(t) appearing at correlator output is shown in (b) of 

figure 14 under balanced conditions. Fluctuations about the positive 

and negative mean levels are not shown. Since the function shown is a 

simple square-wave, the harmonic components may be readily determined. 

18 
The fundamental sine or cosine component of a symmetrical square-wave 

such as this is 4/II times the peak amplitude. This component is then 

weighted by the transfer function evaluated at s = jw,, where md = 

2nfd and fd is the Dicke switching frequency. Letting V1 represent the 

peak output component at the Dicke frequency, this value is 

VI = $ Kl(TB + TR> IG2(jmd) I(1 - KDcI (23) 

While the actual magnitude of the fundamental component is certainly 

important, a more meaningful measure is the relative level of the fundamental 

compared with the dc level. The dc level V. in the steady state is 

v. = 

(TB - TA) 

K2 

(1 - KDC> (24) 

Let a1 = V1/Vo represent the ratio of the fundamental ripple level to the 

dc component. This quantity is determined from equations (23) and (24) 

as 

Vl 4 (TB + TR) 

“l=v=- r~ (TB - TA> K1 kb(jWd) 1K2 

0 

(25) 

The quantity K11G2(jwd) IK2 in equation (25) represents the net 

loop gain for that particular frequency without the 0.5 Dicke switching 

factor. In the dc control loop analysis, it was convenient and proper 

to include the 0.5 factor in the definition of the loop gain. In order 

to make this analysis consistent with the control model, the 0.5 factor 

is absorbed in the loop gain definition, and an additional factor of 

2 is put in front to compensate. Thus, the ripple factor al can be 

expressed as 

8 (TB + TR) 

al = ?T (TB - TA, 1 GL(b@ 1 (261 

19 
where IGL(j 'JJd) 1 is the magnitude of the loop gain transfer function 

evaluated at the Dicke frequency with the 0.5 factor included. 

A symmetrical square-wave of the form shown in (b) of figure 14 has 

only odd harmonics. The ratio of the nth harmonic magnitude Vn to 

the dc component V. will be denoted as an. This function can be 

determined as 

(TB + TR> 

a 

n = +I (TB - TA) IGLmJ$ I (27) 

For a typical loop gain function, the fundamental component V, is 

usually much larger than any of the other components. Consequently, the 

assumption that the total harmonic level is approximately the same as a1 

is usually valid. This assumption will be made in the analysis that 

follows. In the existing radiometer, the Dicke frequency fd is 

approximately 120 Hz. 

For the original loop design, the ripple level of the fundamental 

can be determined as 

8 (TB + TRI 

“1 = rr 

31.195 (1 + 2.545 x 10S3jw) 

(TB - TA) 

x 100% 

jw(1 + 28 x 10S3j,) 

(281 

For the modified design, the corresponding function is 

8 (TB + TR> 

a1 = - II (TB - TA) 

25 

j,(l+ O.Oljo) x 100% (291 

As it turns out, the ripple levels for these two functions are almost the 

same, so only one will be shown graphically. The output peak ripple 

level expressed as a percentage of the output signal reading is shown 

in figure 15. This does not include the effect of post-loop filtering, 

which further reduces the ripple level. 

20 
From this curve, it is readily observed that the ripple level 

increases markedly with increasing input temperature and could be most 

troublesome if not properly eliminated. Computer simulation of the 

two systems did indeed verify that the ripple level was (a) dominated 

by the fundamental component, and (b) equal to the value predicted by 

the mathematical model. 

One way in which the ripple level could be reduced significantly 

in the analog system is by means of additional loop filtering. In order 

not to disturb the control mechanism of the loop, which is dominated by 

two very low frequency poles, it is necessary that the loop filter 

have a cutoff or filtering range well above a few hertz (the range of 

the dominant poles), but at the same time it must have a high attenuation at f d = 120 Hz. 

Several possible analog filters were investigated for this purpose 

and simulated with the available computer programs. Some three-pole 

Butterworth filter designs with cutoff frequencies below 100 Hz were 

used, and the simulations verified that they would be quite effective 

in reducing this ripple. 

One particular filter of interest for this purpose was an analog 

notch filter whose notch was placed at the Dicke frequency. The 

particular filter used had the transfer function GN(s) given by 

GN(S) = 

(1 + 1.7591 x 10-V) 

1 + 2.5678 x 10-3s + 5.0555 x 10-6s2 + 4.375 x 1O-qs3 

(30) 

or 

GN(sl = 

402.124(s2 + 568.49 x 103) 

s3 + 1.1557 x 103s2 + 587.01 x 103s + 228.60 x lo6 

(31) 

The amplitude response of this filter is shown in figure 16. This 

filter was found to to quite effective since it completely eliminates the 

fundamental component. However, there is a small effect due to the 

higher harmonics which are not attenuated by the notch filter. 

21 
GENERAL DIGITAL CONSIDERATIONS 

In this section several general considerations concerning digital 

processing of radiometer data will be investigated. Most of these 

considerations are general in that they apply to systems other than the 

radiometer, but it was necessary to review their significance in the 

context of the radiometer analysis. Some of these factors arise because 

of conversion between the analog signal and the digital signal and the 

reverse process. 

First, consider the analog system shown in figure 17 with an input 

x(t), an output y(t), and a transfer function G(s), where G(s) = 

Y(s)/X(s>. Assume now that the input x(t) is chosen to be a wideband 

noise source; i.e. x(t) = n,(t). Assume that the "power" spectral 

density of n,(t) is uniform and given by Gnx (0 = nx V2/Hz (or W/Hz) 

on a one-sided basis, and assume that n,(t) = 0. The variance of the 

output 0 Y 2 = E[y220j can be expressed as 

Y2(t) = E[y2(t)l =Sm( G(jd 12rlx df 

0 

This result can be expressed in countour integral form as 

1 Y2U> = 2r[j / c tf G(s)G(-s) ds 

(32) 

(33) 

where C is a contour encompassing the jw-axis and an infinite semicircle in the left-hand half-plane. It is further assumed that all 

poles of G(s) are located in the left-hand half-plane and that the 

high-frequency rolloff rate of IG(jw)I is no less than 6 dB/octave. 

Now assume that a sampled-data version of x(t) is formed, and let 

x(n) denote this quantity. Assume that all the samples of x(n) are 

statistically independent. (More will be said about this assumption in 

a later section.) Assume also that a discrete transfer function H(z) 

22 
corresponding in some sense to G(s) is formed, and let y(n) represent 

the discrete output. From the properties of z-transforms, Y(z) = H(z)X(z). 

However, before involving some of the properties of the discrete-time 

signal, consider the process of transforming the s-plane contour integral 

of equation (34) into the z-plane in a direct correspondence sense, 

assuming of course that H(z) corresponds closely to G(s). Since 

ST 

z = E is the relationship between the z and s variables, dz = 

Tc"Tds = Tzds and ds = dz/Tz. Substituting this expression in equation 

(33), and replacing G(s) by H(z) and G(-s) by H(z-I), the following 

form of the contour integral results: 

r2(n) = & WVW1)- dz 

Z 1 (34) 

where y2 (n) is the output variance of the sampled signal and Cl is 

the corresponding contour in the z-plane. 

A result from discrete transfer function theory is the relationship 

H(z>H(z-‘1 dz 

Z 1 x2(n) (35) 

for the case where all samples of x(n) are statistically independent. 

From the preceding results, it can be deduced that the quantity nx 

represents a power spectral density in V2/Hz for a continuous analog 

signal. However, nx/2T = (nx/2)fs represents a total variance in V2 for _~ 

a sampled-data version of an analog signal, and this corresponds directly 

to x2(n). Thus, the relationships correspond to each other with power 

density (or voltage-squared density) being the input for the analog case 

and total variance being the input for the discrete case. In both cases, 

the output variance is the total variance in volts-squared. Looking 

23 
at it from a different point of view, the total input in V2 is 

divided over the range fs = l/T for the discrete-time signal case. 

In systems involving A/D and D/A conversions, it is necessary to convert 

back and forth through these relationships to envision the manner in 

which the signal forms change. 

The point of this discussion is that the discrete-time and continuoustime forms produce identical results when viewed from the proper 

perspective. When continuous-time systems are considered, it is more 

convenient to work with power (or voltage-squared) spectral density 

and to consider the power divided over the frequency range of the 

sampling frequency (or half the sampling frequency). On the other hand, 

when discrete-time systems are analyzed, it is often more convenient 

to express variance in terms of the total mean-square value. The two 

points of view are equivalent provided that the sampling frequency fs = 

l/T is used for dividing up the power on a two-sided basis. Alternately, 

the folding frequency f. = fs/2 could be used on a one-sided basis. 

Consider now the process of representing a finite set of samples 

taken with an A/D converter by an ideal train of inpulses. In the 

frequency domain, the spectrum of an impulse train 6T(t) is expressed 

as 

F[s,ctjl = + f s(f- $1 

-co 

(361 

However, an actual train of nonzero width pulses would have a spectrum 

weighted by d = tl/T, which would approach zero as tl + 0. Consequently, 

the impulse spectrum has been effectively weighted by l/t1 by using 

the ideal impulse approximation. On the other hand, when the sampled 

signal is converted back to a continuous-time signal, it can be thought 

of as being held by a zero-order holding circuit whose magnitude is 

unity. The transfer function for this device can be expressed as 

24 
tl sine ftl so that the spectrum is weighted by tl, the net effect 

is tlx(l/tl) = 1, and the level of the signal is preserved. 

25 
SAMPLING EFFECTS OF NOISE SPECTRUM 

A major objective of this study was to identify possible schemes 

by which the processing and reduction of radiometer data could be 

achieved by digital signal-processing techniques. While the general 

field of digital signal processing is quite mature, there has been 

relatively little work reported in the literature on its application 

to radiometer type signals and systems. There are some peculiar 

aspects of the radiometer process that result in difficulty in 

applying standard signal-processing techniques directly to the signal. 

A significant portion of this study was devoted to investigating the 

nature of the signal and possible ways of circumventing these 

difficulties. 

The "signal" at the front end of a microwave radiometer is 

actually wideband noise having a Gaussian amplitude distribution with 

zero mean. The wideband input filter establishes the bandwidth over 

which the measurement is performed, which typically ranges from 

20 MHz to' 2GHz. The square-law detector changes the form of the signal 

to one with a chi-square distribution with one degree of freedom. The 

mean of this signal is proportional to the total power in the input 

signal. Along with the mean value, which represents the quantity to 

be estimated, is a large background noise signal which must be smoothed 

to obtain the value of the true estimate. This undesired background 

signal has a triangular power spectrum. Subsequent processing of 

the detector output reduces significantly the level of the background 

while providing an estimate of the true mean. 

In a Dicke radiometer, correlation of the signal in order to cancel 

out the effective receiver temperature noise is also achieved in the 

postdetector processing. In the case of a noise-injection feedback 

radiometer, an additional objective that must be performed is the 

development of a low-frequency control signal to provide a proper feedback 

mechanism for the loop. 

26 
One of the earliest questions of interest in investigating 

possible digital signal-processing schemes is to determine an 

appropriate point for sampling the analog signal. At first glance, 

one might conclude that there would be no hope of sampling the analog 

signal prior to the square-law detector since the bandwidths there 

could be as high as 2 GHz and hence result in a required sampling 

rate of 4 GHz using the conventional sampling rate interpretation. On 

the other hand, one might argue that since the mean value, not signal 

reproduction, is the only quantity of real concern, perhaps an undersampled signal could be usable. After all, the Dicke process itself 

as utilized in the front end of the radiometer is a form of sampling 

and its rate is far below the required Nyquist rate. 

As a result of these questions, an analytical investigation was 

made to determine the possible implications of undersampling a wideband noiselike signal. Certain simplifying assumptions were made in 

order to keep the analysis to a manageable level that would provide 

some useful practical interpretations. 

Assume a very wideband noise source ni(t) whose power spectrum 

is Gni(f) = n/2 on a two-sided basis as shown in figure 18(a). Assume 

first that the signal is filtered directly by a low-pass filter with an 

equivalent one-sided noise bandwidth BN, as shown in (b). The instantaneous output noise of this filter is nol(t>, and the total variance 

associated with this noise is NOl* From the basic definition of noise 

bandwidth, this variance is simply 

No1 = 2BN; = nB N (37) 

Now consider the system shown in (c) of figure 18. The signal is first 

sampled (or modulated) by a pulse signal p(t) whose form is shown in 

Cdl. Consequently, ns (tl can be expressed as 

n,(t) = ni(t> P(t) (38) 

27 
Since p(t) is periodic, it can be expressed as a Fourier series of 

the form 

p(t) =&n c 

jnwst 

-m 

(39) 

where f 

S 

= ws/211 = l/T is the sampling rate and T is the time between 

successive samples. For the pulse train p(t) with unity pulse amplitude as shown, the coefficient cn is given by 

C = d sine nd 

n (40) 

where d = tl/T is the duty cycle of the pulse train. Substitution of 

equation (39) in equation (38) yields 

n,(t) = ni(t) 2 cnc 

jnUst [ -co I 

=2 cn ni(tlE 

jwst 

-co 

The autocorrelation function R,,(r) corresponding to ns(t) is 

Rns(') = E [ns(t + p) n;(t + 1-I - T)] (42) 

where 1-1 is uniformly distributed over the range 0 < u < T and is a - - 

variable introduced as a more rigorous means for establishing stationarity 

of the process. R,,(r) is expanded as follows: 

28 
I$,,(4 = E 

jnols(t + P) 

c n 1 

[ 

T) 2 c* E 

-jnws(t + u - r) 

. ni(t + P - 

-m n 1) 

Interchanging the order of the averages results in 

’ [( 

m 

R,,,(T) = [niC t+p)ni(t+u-r) l Cc,, 

jnus(t + IJ) 

-co > 

x fc;c ( -jnws(t + P - r) 

-Co Y 

= Rni(r) &&2cJnwsr 

-03 . 1 

(43) 

(44) 

The power spectrum Gns(f) of the sampled signal is determined 

by computing the Fourier transform of equation (44). 

GnsW = c lcn12 -m 

= f lcn12 -Co 

co 

Gni(f - nfsl 

11. 2 

= ; c lc,l2 = ; c 

-m (45c) 

where C is the summation. In comparison with the unsampled signal, the 

power spectrum of the sampled signal is multiplied by the infinite 

summation shown. 

29 
The nature of this summation will now be investigated. It can 

be manipulated as follows: 

c = f IcJ2 = f d2 sinc2nd 

-m -m 

2 2 

sin rnrd sin nnd 

nrd md 7rd (46) 

The summation in equation (46) converges to a value of 71. Using this 

result, equation (46) may be expressed as 

d C=-rrxlT=d 

Finally, the output noise power No2 under this condition becomes 

No2 = d; (2BN) = dnB N (48) 

(47) 

Comparing equation (48) with equation. (37), it is seen that the 

output noise fluctuation power is d times the value that would be 

obtained without sampling. At first glance, this sounds appealing, 

since the total fluctuation power is apparently reduced by the sampling 

process. However, to truly evaluate this approach, it is necessary to 

consider the existence of a signal component as well. If such a signal 

component is present, it appears as a frequency domain impulse at dc. 

Let V1 represent the value of the signal component (mean value to be 

estimated). The signal power SOI of the mean value for the continuous 

nonsampled case is simply 

30 
With sampling, it is easy to show that the dc component v2 is simply 

v2 = d& (50) 

The signal power So2 with sampling is then 

So2 = d2(&)2 = d2Sol (51) 

The ratio of the signal-to-noise ratio after sampling to the signal-tonoise ratio before sampling is determined by 

S02/N02 

Sol/No1 = so1 No2 

& h = d2 . ; = d (52) 

Thus, the signal-to-noise ratio is degraded by a factor of d when the 

wideband noise signal is undersampled. 

A further explanation of this process will be made. The noise 

components are uncorrelated; therefore, the fluctuation power reduces 

in direction proportion to the time of observation. On the other hand, 

the signal (mean value) is a coherent component and its voltage level 

reduces in direction proportion to the time of observation, meaning that 

its power reduces in proportion to the square of the time of observation. 

Thus, the signal power decreases at a much faster rate than the noise 

power. 

The preceding development has been idealized somewhat in order to 

illuminate the concept fully. The noise was considered separately from 

the signal to simplify the derivation. When both the mean value (signal) 

and fluctuation noise appear together, an additional term appears in 

equation (44), which is the square of the mean value multiplied times the 

quantity in the brackets. The corresponding spectrum of equation (45~) 

then contains an infinite number of line components at integer multiples 

of the sampling frequency. Depending on the manner in which the result 

is filtered in order to enhance the signal with respect to the noise, these 

line components may or may not affect the net overall fluctuation level. 

The exact ratio given by equation (52) applies only when these line 

components are eliminated. However, the signal-to-noise ratio is degraded 

by the process of viewing the signal for a portion of the time when the 

signal is undersampled. 

31 
When the sampling rate is sufficiently high that no aliasing 

occurs, only the co term in equation (4%) contributes to the noise 

at low frequencies. In this case, the noise power, like the signal 

power, is proportional to d2 and both quantities decrease at the 

same rate. With proper filtering in this case, it is possible to avoid 

any degradation of the signal-to-noise ratio, 

An interesting inference from this development is that the basic 

Dicke radiometer fits the aliasing result as a special case. With the 

Dicke radiometer, the signal is effectively sampled for a half-cycle, 

and thus d = 0.5. Under optimum conditions of estimation, the noise 

fluctuation power-to-signal power has an additional factor of two, 

which agrees with the preceding results. 

From the preceding development, it can be deduced that undersampling 

a noise spectrum is not, in general, an advisable procedure in digital 

processing except when special objectives are sought, such as in the 

Dicke radiometer. 

32 
OPTIMUM SAMPLING RATE 

Analysis of sampled-data signal fluctuations can be approached 

from.two separate points of view: (1) a frequency domain approach based 

on filtering of the signal and (2) a purely statistical point of view 

based on a reduction of the variance. Both points of view are correct if 

properly interpreted. Furthermore, it appears that the frequency 

domain point of view is most amenable to interpretation when the signal 

is oversampled, while the statistical point of view is more convenient 

when the signal is undersampled. The approaches are about equal in 

complexity when the signal is sampled exactly at the minimum Nyquist 

rate. 

The development that follows should lead to some interesting and 

useful interpretations for these concepts. Consider a signal whose 

two-sided power spectral density Gi(f) is of a rectangular form 

as shown in (a) of figure 19. This function is then described by 

Gi(f) = $ -Bi < f < B' 1 

= 0 otherwise 

The total variance ai of the signal is 

u i 2 = 2Bi r~ = Bin 

2 

(53) 

(54) 

Now assume that the signal is sampled by an ideal impulse sampler 

with sampling rate fs. For convenience the impulse sampler will be 

assumed to have a relative weight T = l/fs so that the impulse train 

cST(t) can be expressed as 

fiT(t) = T c 6(t - nT) -Co (55) 

33 
The reason for this assumption is to cancel out the l/T factor that 

appears as a multiplier for the spectral components of the sampled 

spectrum of an ideal impulse train, which would create an additional 

"confusion factor" in interpreting some of the results that follow. 

This factor actually does not affect the final results anyway, since' 

the holding/reconstruction circuit effectively cancels the l/T factor, 

but by putting in the T factor as assumed, the levels of the 

spectral components can be better interpreted. 

After the signals are sampled, a sum-and-dump algorithm will be 

applied to the set of discrete numbers involved. This algorithm reads 

y=+$ x 

n=l n 

(56) 

where X 

n represents the set of N discrete output values of the sampler, 

and y represents the estimate obtained from application of the algorithm. 

The average is performed over an interval of f seconds, where 

'c = NT (57) 

Three possibilities will now be considered: (1) Nyquist rate sampling, 

(2) undersampling, and (3) oversampling. 

Nyquist Rate Sampling 

For this first case, the sampling rate will be assumed as 

fs = 2Bi (58) 

The form of the sampled spectrum GO (l)(f) is shown in (b) of figure 19. 

Observe that there is neither aliasing nor spectral gaps. The spectrum 

has exactly filled in the gaps without any "overcrowding." 

From a frequency domain point of view, the effect of the sum-anddump algorithm can be represented by an equivalent noise bandwidth EN, 

34 
which on a two-sided basis can be shown to be 

11 BN=-r=NT 

The variance a; of 

(591 

the estimate y can then be expressed as 

nfS =L=- 

2NT 2N (60) 

The several forms listed provide different interpretations, but they are 

essentially frequency domain in form. These results relate directly to 

the traditional continuous integrate-and-dump circuit in which the output 

variance is determined by a similar form. 

A different interpretation of equation (60) is obtained by noting 

that 

+ fs = nB. = uf 1 (611 

Substituting of.equation (61) in equation (60) results in 

cl.2 

02 = 1 

Y N (62) 

This form is the familiar statistical result concerned with the sample 

mean concept. The variance of the sample mean is the variance of the 

process sampled divided by the number of samples, provided that the 

samples are all statistically independent. It is seen that this concept 

agrees exactly with the frequency domain point of view when the signal 

is sampled at the Nyquist rate. However, results when,the signal is 

either undersampled or oversampled will be considered next. 

Undersampling (fs c 2Bi) 

Although the results that will be developed generally apply to any 

degree of undersampling, it is convenient to assume that the bandwidth 

35 
Bi is an integer multiple of the folding frequency. Thus, let 

(63) 

where M is an "undersampling" integer factor. For MU = 1, this case 

U 

reduces to the Nyquist sampling case. However, the case of interest 

involves M Ll > 1. 

Because the signal is undersampled, aliasing of the spectrum will 

occur. The resulting sampled spectrum Go c2)(f) will be of the form 

shown in figure 20. (This case corresponds to MU = 4.) The aliasing 

results in exactly MU components adding together in the low frequency 

range. Since these components add incoherently, the fluctuation power 

in the low-frequency range is exactly Mu times the value of the unsampled 

signal. 

From a frequency domain point of view, the variance c2 is now Y 

02 = 

q”U NU nMu n”Ufs 2BN=x=-=- 2NT 2N (64) Y 

Comparing equation (64) with equation (60), it initially appears that 

the variance has been increased by a factor Mu over the case of Nyquist 

sampling, and this result is correct if all the other factors in the 

expression are the same as before. However, the input spectral density 

rl is in a sense now divided over a greater bandwidth for a given input 

variance. The input variance is 

2 u. 

II+ (65) 

i 

Substitution of equation (65) in equation (64) results in 

02 = 

Y (f-56) 

where the assumption of equation (63) is utilized. 

36 
Hence this result is exactly the same as equation (62) for the 

case of Nyquist sampling because of the fact that all samples are 

statistically independent. Thus, the question of whether the output 

variance is greater or not with undersampling depends on the point of 

view. From a purely statistical point of view, the output variance is 

l/N times the input variance and the reduction effect is the same 

whether Nyquist sampling or undersampling is used for the given number of 

points. However, from a frequency domain point of view, the fluctuations 

appear to be greater; but, for a given input variance, the power density 

is smaller, representing the fact that noise power is spread over a 

wider bandwidth than before. The fact is, however, that undersampling 

is not optimum in the sense that a greater reduction of variance could 

have been achieved in the same time by employing more samples. In other 

words, undersampling results in the variance being reduced as much as can 

be expected for the number of points involved, but it could be reduced 

more in the same amount of time by using more samples. 

Oversampling (fs > 2Bi) 

As in the case of undersampling, an integer relationship between the 

bandwidth and the sampling rate will be conveniently chosen. In this 

case, the following form will be assumed: 

fs = Mo(2Bi) 

where M is an "oversampling 'I factor and is the ratio of the actual 0 

sampling rate to the Nyquist rate. The resulting sampled spectrum will 

be of the form shown in figure 21. (This case corresponds to MO = 2). 

Note in this case that there are "holes" in the resulting spectrum. 

The output variance UC in this case is 

+ 

nf 

ik= 2N5 (68) 

37 
A statistical intepretation of this result is obtained by substituting fs from equation (67) in equation (68) and utilizing the fact 

that 

u; 

n=B 

i 

The result is 

M 

02 = 2 c2 

Y Ni 

(69) 

(70) 

In this case, it appears that the output variance is increased by a 

factor of M 0 over that of the Nyquist sampled case, and this point 

of view would be correct if o: were the same as for the Nyquist 

sampled case. However, the input fluctuations are now confined to a 

narrower bandwidth for a given input power spectrum. From the frequency domain point of view, the variance has been reduced as much as 

can be expected for the given integration time. In this case, the 

frequency domain point of view seems to be easier to interpret due to 

the lack of aliasing in the spectrum. 

To summarize this case, oversampling results in the variance being 

reduced as much as can be expected for the particular time involved, but 

it could be reduced more by letting the same number of points extend 

over a longer period of time. In other words, the samples are too close 

together to be completely statistically independent, and increasing the 

time between samples will result in a greater reduction of the variance 

for the same number of points. 

From the preceding developments, it can be concluded that the 

Nyquist rate is an optimum rate for the reduction of variance in a sampled- 

'data signal with rectangular power spectrum both from the frequency domain 

point of view and from the statistical point of view. At that rate, 

both the filtering and the statistica 1 independency propert 

their best." 

.ies are "doing 

38 
The optimum rate for this purpose should not be confused with 

the signal reconstruction objective in other applications where a rate 

exceeding the Nyquist rate is always used. As a matter of fact, the 

assumption of a flat spectrum has been made thus far, so it is logical 

to ask what modification is necessary when the power spectrum rolls 

off more gradually, as it typically does. 

Consider, for example, the gradual spectral rolloff shown in (a) of 

figure 22. In order to make the aliasing error vanishingly small, the 

sampling rate f Sl is selected to be as shown in (b). While the 

resulting increase in the aliasing power contained up to the folding 

frequency is negligible, the variance reduction is suboptimal in that 

it is less than the l/N factor. Specifically, the output variance ug 

resulting from a sample mean definition can be most directly calculated 

from a frequency domain point of view and is 

(71) 

An alternate strategy with such a rolloff as this is to undersample 

somewhat so as to "conserve the sampling rate" with the philosophy that 

the variance won't increase much in the process. The results of this 

assumption are shown in (c) of figure 22. The output variance in this 

case is 

0.5fs 

0: =/, 5f G2W(yJ2 df 

S 

(72) 

where G2(f) is the modified spectrum with aliasing. 

39 
The case where aliasing is present (fig: 22~) results in an 

increase in the output fluctuations as compared to the case of figure 

22(b), but the decrease is more "efficient" for the same number of 

points. The tradeoff of interest is whether or not the reduction in 

the required sampling rate is worth the cost of the increased variance. 

Each case would have to be considered on its own merits. If the 

data rate is rather slow and the digital circuitry is capable of 

operating at a high rate anyway, the optimum choice will likely be to 

sample at a high enough rate to eliminate all aliasing error. 

40 
SECOND-ORDER DIGITAL LOOP DESIGNS 

Early in the course of this study, several second-order control 

loop designs were investigated and successfully simulated. The choice 

of a second-order loop was made at that time because the current analog 

system utilized a second-order loop. As will be shown later, a firstorder loop followed by post-loop sum-and-dump filtering was finally 

selected for the prototype implementation. Nevertheless, there are 

still some worthwhile features of the second-order loop that could make 

it a viable candidate for some applications. Consequently, some of 

the intrinsic features of a second-order digital loop will be 

discussed in this section, and some representative designs will be 

shown. 

The concept of utilizing the second-order loop was based on 

performing virtually all the data smoothing in the loop itself and considering little or no post-loop filtering. It was also decided to employ 

coefficients in this design that could be realized exactly by a rather 

limited number of bits. In this case exact coefficients were known, 

making the equivalent smoothing bandwidths more readily predicted. 

In establishing equivalence, or at least correspondence, between 

a digital loop and an analog loop, several approaches are possible. 

None of these are "exactl' in one sense since the discrete-time system 

is different from the continuous-time system, and the best that can be 

done is to establish a correspondence with respect to some particular 

criterion. For that reason, a comparison with respect to more than one 

criterion will be made in some of the developments that follow. 

One criterion is the direct z-transform correspondence z = E ST or, 

equivalently, s = (l/T) Rn z. This relationship computes the values of 

S and z that correspond to each other through the z-transform 

definition, and in this sense is exact. 

A second criterion is a correspondence through the bilinear 

transformation. A popular and convenient means for designing digital 

41 
filters is to employ a prototype continuous system variable p, obtain 

a continuous transfer function approximating the desired behavior, and 

then to replace p by 

p = KC1 - z-l) 

1 + z-1 

(73) 

where K is a mapping constant. For very low frequency correspondence 

between the prototype analog frequency and the final digital frequency, 

it can be shown that an optimum choice of K is 

K = $ = 2fs (74) 

The actual second-order functions chosen for the loop were based on 

a modified bilinear transformation criterion. Consider momentarily a 

first-order analog function of the form 

C(P) = e 

When the bilinear transformation is applied to equation (75), the 

corresponding H(z) is determined as 

Hd(z) = L (1 + z-1) 

K+cr 1 - Bi-l 

where 

BzKK+a 

(75) 

(76) 

(77) 

Consequently, if B is selected as a convenient value in the digital 

filter, the corresponding CL in the analog sense can be determined as 

cL = KC1 - B) 

l+B 

42 

(78) 
The first design of a digital estimating circuit considered is 

shown in figure 23. The particular structure shown in the figure 

represents only the digital processor portion starting from the A/D 

converter and continuing to the point at which the output estimate is 

determined. The feedback noise-injection scheme is not shown in the 

figure. The difference equations describing this system are as follows: 

u(n) = x(n) + x(n - 1) + u(n - 1) (791 

v(n) = 2-3 u(n) (801 

w(n) = v(n) + v(n - 1) + 0.875 w(n - 1) (81) 

y(n) = 2-3 w(n) (821 

The corresponding transfer functions are 

U(z) - 1 + z-1 

X(z) 1 - z-1 

V(z) - 2- 3 

U(z) 

W(z) - l+z-1 

V(z) 1 - 0.8752-l 

‘cz) - 2-3 

W(z) 

The composite transfer function is 

Y(z) - 

H(z) = X(z) (1 

2-6(1 + z-l>2 

- z-1)(1 - 0.8752-l) 

(83) 

(84) 

(85) 

(86) 

(87) 

43 
The low-frequency behavior of H(z) may be determined by substituting z = esT and letting s become very small. The function H(csT> 

then asymptotically approaches 

H(cST) z 2-6(2)2 0.5 

sT(0.125) = --% (881 

(for small s) 

For this particular design, with the sampling rate selected as fs = 750 Hz, 

the function approximates 375/s. 

Based on the constant B = 0.875 in equation (76), the corresponding 

analog c1 may be approximated by either setting E -clT = 0.875 or by 

using the bilinear transformation. In the first approach, cx = 100.15 

rad/s. In the second approach, c1 = 100 rad/s based on equation (78). 

The two values are in close agreement, and correspond very closely to 

the optimum analog value selected in an earlier redesign effort. 

The actual analog transfer function G(s) of the estimation circuit 

was earlier selected as 

G(s) : 17.857 x 8.82 

s(1 + 0.01s) (89) 

For small values of s, this function approximates 

G (~1 ~ 157.5 (901 S 

The corresponding digital function has more than twice the gain required, 

so the signal should be attenuated by a factor of 157.5/375 = 0.42 before 

processing. 

The second approach considered in second-order loop designs was one 

in which two separate modes could be employed. The first mode, designated 

mode A, is the lower resolution, faster response mode. On the other 

44 
hand, mode B provides a greater resolution at the expense of a 

longer integration time. For this particular design, an Intel 8086 

microprocessor development unit had been identified for prototype 

development work. Consequently, its clock frequency of 4.9 MHz was 

selected for timing, and the sampling rate was chosen as 

fs = 4.9 x 106/213 = 598.14 Hz 

The discrete transfer function of the lag-lead digital filter was 

selected in the form 

HR(z) = 

Kd 

1 - Bz-l 

(911 

The value of B for mode A was selected as 

B = 0.11012 = 0.812510 (92) 

where the subscripts 2 and 10 represent the binary and decimal bases, 

respectively. The corresponding value of cx in the analog representation is determined from the direct z-transform definition as 

c1 = -(l/T) Rn 0.8125 = 124.20 rad/s 

and from the bilinear transformation as 

cx = 2 x 598.14 x(1 - 0.8125)/(1 + 0.8125) = 123.75 rad/s 

To simplify further references, the value Q = 124 rad/s for mode A will 

be assumed in subsequent discussions. 

The layout of mode A is shown in figure 24. The various difference 

equations are tabulated as follows: 

45 
u(n) = x(n) + x(n - 1) + u(n - 1) 

v(n) = 2-4u(n) 

w(n) = v(n) + v(n - 1) + 0.8125 w(n - 1) 

y(n) = 2-2 w(n) 

The corresponding transfer functions are 

U(z) - l+z -1 

X(z) 1 - z-1 

V(z) 

u(z) = 2-4 

W(z) - 1 + z-1 

-- 

V(z) 1 

- 0.8125z-1 

- Y(z) = 2-2 

W(z) 

The composite transfer function is 

Y(z) - 2-6(1 + z-q2 

x(z) (1 - z-1)(1 - 0.81252-l) 

(93) 

(94) 

WI 

(96) 

(97) 

(98) 

(991 

(100) 

As observed in equation (go), the analog estimating circuit approximates 157.5/s for small s. The function of equation (101) approximates 

2-6(2)2/sT(0.1875) = 199.38/s. Thus, a small attenuation of the analog 

signal before sampling should result in nearly equal behavior of the 

digital system as compared with the analog system. 

46 
The value of B for mode B was selected as 

B = 0.1111012 = 0.95312510 (1021 

The corresponding value of a from the direct z-transform was computed 

as a = -(l/T) Rn 0.953125 = -28.716 rad/s and from the bilinear transformation as a = 2 x 598.14(1-O-953125)/(1 + 0.953125) = 28.71. The 

latter value will be selected for subsequent references. 

The layout for mode B is shown in figure 25. The various difference 

equations are tabulated as follows: 

u(n) = x(n) + x(n - 1) + u(n - 1) (103) 

v(n) = 2-5x(n) (104) 

w(n) = v(n) + v(n - 1) + 0.953125 w(n - 1) (105) 

y(n) = 2-5 w(n) (106) 

The corresponding transfer functions are 

U(z) - 1 + z-1 

X(z) 1 - z-1 

v(z) - 2-5 

X(z) 

W(z) - 1 + z-1 

V(z) 1 

- 0.9531252-l 

y(z) = 2- 5 

W(z) 

(107) 

(108) 

(1091 

(110) 

47 
The composite transfer function is 

Y(z) - 2-91 + z-l)2 

X(z) (1 

- z-1)(1 - 0.9531252-l) 

(111) 

The function of equation (111) approximates 2-10(2)2/sT(0.046875) = 

49.845/s for small s. The corresponding analog filter transfer function 

was quite close when redesigned for this higher accuracy mode. Note 

that the present analog system does not operate in a higher resolution 

mode such as this. Instead, higher resolution is achieved by postloop processing of the data obtained from the lower resolution operating 

mode. 

48 
CANCELLATION OF DICKE RIPPLE 

A troublesome problem in an analog Dicke radiometer system is 

the presence of the Dicke ripple component in the data output. 

Because of the periodic switching in such a radiometer, an undesired 

disturbance at the switching frequency appears in the signal output. 

This disturbance consists of a fundamental component at the switching 

frequency plus components at odd integer multiples of the switching 

frequency if symmetrical switching is employed. Due to the normal 

low-pass nature of the forward transfer function, the most troublesome 

component is the fundamental, and its magnitude is a close approximation to the total ripple level in most cases. 

While it is possible to set the loop parameters to adjust this 

ripple to a tolerable level by careful design, an interesting concept 

came to light in investigating possible digital-processing schemes. 

Consider the discrete transfer function H(z) of the trapezoidal 

integration approximation as given by 

H(z) = 

K(l + z-1) 

1 - z-1 

(1121 

The steady-state transfer function H(E jwT ) corresponding to this function 

is readily shown to be 

H(EjwT) = 

K cos $ 

WT = K cot nfT 

sin - 2 

(113) 

The form of the magnitude response [H(E jwT) 1 corresponding to equation 

(113) is shown in figure 26. The frequency response approximates that 

of an ideal integrator in the very low frequency range, i.e., for f << 1/2T. 

However, for f = 1/2T, the steady-state transmission is identically 

zero. 

49 
Assume now that the sampling frequency fs = l/T is chosen to be 

twice the Dicke frequency fd, i.e., fs = 2fd. This results in a 

zero or null of transmission at the Dicke frequency and at odd 

integer multiples of the Dicke frequency. Thus, as long as the 

switched signal satisfies the half-wave symmetry conditions of Fourier 

theory, all undesired components in the ripple disturbance will be 

completely eliminated. 

As it turns out, the proposed feedback schemes utilize a pulseinjection process which disturbs the half-wave symmetry of the switched 

signal. The resulting signal in this case does contain even harmonics 

which are not eliminated by this process. However, such components are 

easier .to handle since they are higher in frequency and smaller in 

magnitude. The major point of interest is that a significant reduction, 

or possibly an elimination, of the switching ripple can be achieved 

directly through a digital algorithm. 

50 

I I I I n I II I 
PROPOSED CONTROL LOOP 

After investigating in some detail the several second-order 

loop designs previously discussed, attention was directed toward 

the concept of a first-order loop followed by post-loop sum-and-dump 

filtering. Because of the presence of the post-loop filtering, the 

system was actually of higher order, but for subsequent references the 

term "first-order processor" or simply "first-order loop" will be 

used. 

Some of the advantages of the first-order loop determined from 

this investigation are (1) the form of the response is less sensitive 

to parameter deviations than the second-order loop. Small variations 

in gain parameters can sometimes adversely change the damping ratio of 

a second-order loop. (2) The loop can be made to respond faster, 

thereby allowing a greater reduction of fluctuations by the sum-and-dump 

post-loop filter. (3) It is easier to take advantage of the Dicke 

cancellation scheme previously discussed when the loop is a first-order 

design. 

The basic design of the first-order estimation loop filter is 

shown in figure 27. The difference equations for this system are 

w(n) = x(n) + x(n - 1) + w(n - 1) (114) 

y(n) = 0.25 w(n) (115) 

The corresponding transfer function H(z) for the filter is 

Y(z) H(z) = x(z> = 

0.25(1 + z-l) 

1 - z-1 

Cl161 

The design of the loop estimation and feedback system is shown in 

figure 28. The timing references for this system are derived from the 

4.9 MHz and the 2.45 MHz signals available in the microprocessor unit. 

51 
The sampling rate for the analog signal is fs = 2.45 MHz/212 = 598.14 Hz. 

The A/D converter is an Analog Devices type AD-572 12-bit successive 

approximation unit. The drive for the Dicke frequency fd is 

obtained by dividing fs by 2, thus giving fd = 299.07 Hz. This 

results in zeros of transmission at odd integer multiples of the Dicke 

frequency as previously discussed. 

The analog aliasing filter is a cascade of 3 simple one-pole 

filters, each having a 3-dB break frequency at 100 Hz. An investigation 

was made to determine the need for more sophisticated aliasing filter 

designs having complex poles, inasmuch as this particular filter has a 

rather pronounced amplitude "droop" in its passband. However, since the 

only component of primary interest is the dc component, and since the 

actual passband shape is relatively unimportant, this filter was found 

to be perfectly adequate, and it simplifies the overall design. There is 

a small aliasing error, but its level is insignificant in view of 

other error contributions. 

The noise-injection scheme proposed for this design differs 

considerably from the original design in that it will be injected 

continuously during a portion of the Dicke cycle. In this manner, the 

bandwidth requirements of the broadband amplifier preceding the Dicke 

correlator are eased considerably. The noise is injected once per 

cycle as illustrated by figure 29. (Actually, it is injected on both 

half-cycles, but since the Dicke switch only sees it once per cycle, 

the extra injection does not appear on the figure). The length of time 

that the noise-injection pulse is on is determined by the value of the 

digital word at the output of the loop digital filter. A word is 

loaded in the down-counter once per sample time T. The noise diode is 

gated on through the PIN switch at the beginning of this interval. When 

the down-counter reaches zero, the "status" signal turns off the switch, 

thereby reducing the noise injection to zero. 

The maximum duty cycle corresponds to minimum input temperature, 

which results in maximum feedback temperature injection. Based on a 

reference temperature of 308 K, a directional coupler coefficient of 

52 
-20 dB, an excess noise temperature of 365,000 K, an additional attenuation of 10 dB in the feedback loop, and an input temperature of 0 K, - 

the maximum duty cycle was found to be 0.835. Conversely, the minimum 

duty cycle corresponding to 300 K input was found to be 0.0217. The 

range of temperatures and duty cycle are illustrated on figure 30. 

Operation near minimum duty cycle poses the same sort of problems as 

the current analog system since the resolution is degraded seriously 

in that range. 

It was decided not to operate at a duty cycle of unity due to 

"spillover" resulting from finite bandwidth limitations. Consequently, 

the duty cycle of 0.835 was judged to be a good choice for the 

maximum value of the duty cycle. 

While odd harmonics of the Dicke frequency are eliminated by the 

digital filter, even harmonics will appear due to the asymmetrical noise 

injection. Referring to figure 29, an analysis of the levels of even 

harmonics present may be made from a combination of the energy balance 

and Fourier theory. At a condition of zero steady-state error, the 

area of the positive half-cycle must equal in magnitude the area of the 

negative half-cycle. This means that 

C TItl + C(TA + TR)T = C(TB + TR)T 

which results in 

TItl = (TB - TA)T 

or 

TId = TB - TA 

(117) 

(1181 

(1191 

where d = tl/T is a duty cycle defined as the ratio of the "on" interval 

to half of the Dicke period. (Equivalently, it is the total on time, 

53 
including the superfluous extra injection, per Dicke period to the 

total period.). Full injection corresponds to d = 1. 

The even harmonics that arise from the asymmetry may be considered 

to originate from a pulse train having a period 2T, a duty cycle d/2, 

and an amplitude CTT. The magnitudes An of these components on a 

conventional sine-cosine basis can be expressed as 

CT1 si",,;;'2 = C(TB - TA) sin s (120) 

. 

Since the fundamental and odd harmonics are eliminated by the loop 

filter, the most significant component of concern is the second harmonic 

term. This value is 

(1211 

As one would expect, A2 = 0 for d = 1. Conversely, the maximum value 

of A2 occurs for d = 0.5, meaning that noise is injected half of the 

time between successive analog samples. This maximum value A2 = CTT/?T 

occurs for an input temperature TA = 124 K. This could be serious 

except for the fact that the 3-pole aliasing filter has an attenuation of about 47 dB at the sampling frequency, plus the rolloff 

of the loop itself provides an attenuation of about 37.5 dB. The overall 

level of the even harmonics was found to be negligible in all the 

simulations. 

A simplified block diagram of the control model for the discretetime system is shown in (a) of figure 31. The gain of 4.61 x 10m3 

represents the net effect of the original gain of 2.88 x 10s3 times an 

additional gain of 1.6 that was later added to establish an optimum 

transient response. The second block is a "redefinition" gain 

established by the following procedure: The input range of the A/D 

converter is set at 52.5 V, which was determined by simulation to be 

54 
the proper level. This means that the least significant bit at the 

input corresponds to 5V/212 = 1.22 mV. In passing through the 

processor, a 4 to 1 expansion of the signal level occurs, meaning that 

two additional bits are required. However, it is desired to "redefine" 

the signal on a normalized basis so that each digital word has a 

maximum value of unity. Considering the 2.5 V maximum at the input 

coupled with the 4 to 1 expansion, the output would really have a 

maximum of 10 V on an analog basis, so a "gain" factor of 0.1 is used 

to establish it at unity maximum level. 

The forward gain function Hi(z) is then 

Hi(z) = 115.25 x 10 -6 (1 + Z-l) (1221 

1 - z-1 

For very small s, this function behaves approximately like an analog 

gain function 

Gl(s) z 

115.25 x 10+(2) = 0.138 

ST S (123) 

This equivalence is shown in (b) of figure 31, which clearly 

illustrates the nature of the corresponding first-order analog system. 

The appropriate loop gain content is 0.138 x 365 = 50.32, which will 

be rounded off to 50 for subsequent calculations. The closed loop transfer 

function G(s) will then be expressed as 

G(s) = s = 

2.74 x 1O-3 

l+s 

50 

(124) 

The overall proposed system is shown in figure 32. 

55 
POST-LOOP PROCESSOR 

The digital filter contained in the loop was designed primarily to 

provide the necessary dynamic loop behavior for optimum control of 

the noise-injection process. In contrast, the primary objective of 

the post-loop filter was to provide a smoothed estimate of the brightness temperature in as short a time as possible commensurate with the 

sensitivity specifications. 

The concept used to reduce the variance of the temperature estimate 

in the post-loop processor was the well-known sample mean algorithm. 

This process will hereafter be denoted as a "sum-and-dump" algorithm 

due to its close similarity to the integrate-and-dump circuit used in 

analog matched filter and estimation systems. Indeed, the mathematical 

behavior of the sum-and-dump algorithm on a discrete-time basis is 

virtually identical to the behavior of the integrate-and-dump filter 

on a continuous-time basis. 

The integrate-and-dump analog filter is optimum in that the 

product of the equivalent noise bandwidth times the settling time is 

minimum for all analog lumped filters. The corresponding sum-and-dump 

algorithm possesses the same optimum property among the class of discretetime filters. 

The post-loop processing is actually a set of several similar sumand-dump algorithms of different lengths as shown in figure 33. Each 

algorithm involves a scanning window that picks up all applicable loop 

samples over the duration of the loop. The number of points chosen in 

a window is always selected as an integer power of two. This simplifies 

the multiplication and division required by the microprocessor, and it 

simplifies the structure of the dependency between shorter and longer 

versions of the algorithm. 

Let y(n) represent the discrete values appearing at the output 

of the loop, and let yN(i) represent the smoothed discrete values 

appearing at the output of different length post-loop algorithms. The 

56 
integer N represents the number of points in the particular algorithm. 

Note that n is an integer changing at the loop sampling rate while 

i changes only once for every 16 values of n. Thus, 

n = 16i 

or 

i = integer part of (*) dgf I(+) 

(125) 

(126) 

The first value is simply yl(i), which is not actually a 

smoothed value but is the output of the loop as held for an interval of 

16 periods at the sampling rate, i.e.,To = 26.75 ms. Thus, 

yl(i) = y(i) for i = I* 

0 

(127) 

The remaining values at the outputs of different stages can all be 

expressed directly in terms of yl(i), or they can be expressed in 

terms of smoothed estimates of lower order. All estimates are updated 

every 16 loop samples even though the statistical dependency of 

subsequent samples increases with the order of the estimate. 

A tabulation of the various estimates follows: 

Yl(i) + yl(i - 1) 

y2(il = 2 

n (i> + yl(i - 1) + yl(i - 2) + yl(i - 3) 

y4(il = 4 

Y2(il + y21i - 21 

= 2 

(1281 

(129a) 

(129b) 

57 
7 

Ye(i) =+C Yl(i - I> 

I=0 

Y4(i) + y4(i - 4) 

= 

2 

15 

y16(i) = +& yl(i _ 1) 

I=0 

= Y8(i) +Y8(i - 8) 

2 

51 

Y32(i) = &C yl(i - I> 

I=0 

(130a) 

(130b) 

(131a) 

(131b) 

(132a) 

Ylf5 (i> + Ylfj(i - 161 

= 2 (132b) 

63 

1 

Y64(i) = 64 c Yl(i - 1) (133a) 

I=0 

Y32(i) + y32(i - 32) 

= 2 (133b) 

Additional representations of the sum-and-dump algorithms from a 

digital filter perspective and a z-transform point of view are shown 

in figures 34 and 35,respectively. 

The variance reduction of the sum-and-dump algorithm will now be 

analyzed. First, the assumption of a uniform power spectrum will be 

made, and this will be modified later to include loop frequency dependency. 

Consider then the general form of the algorithm as given by 

58 
N-l 

yNci) = + c yl(i - I) (134) 

I=0 

Let Y,(z) represent the z-transform of y,(i) and let Yl(z) represent 

the z-transform of yl(n). Transformation of both sides of equation (134) 

yields 

N-l 

‘,(‘> = ; c Z-I yl(z) (135) 

I=0 

The transfer function yIJ(z) of the post-loop sum-and-dump algorithm 

can be expressed as 

‘NC’> 1 N-1 

HN(Z) = y1cz) = E c z-1 

I=0 

(1361 

The steady-state frequency response will be denoted as s(f), and 

this quantity is determined by substituting z = E 

j wTo 

in equation (136). 

(Note that To is the post-loop sample time, not the loop sample time, 

i.e.,To = 16T.) The result initially is 

N-l . 

S(f) = + CE-'WTo 

I=0 

By means of the summation formula 

N-l 

c a1 = - 

1 - aN 

0 l-a 

(137) 

(138) 

59 
and the exponental definition of the sine function, equation (137), can 

be expressed as 

sin(NwTo/2) UT 

H;(f) = N sin(wTo/2) 

E-j(N - 1) -$ 

The equivalent one-sided noise bandwidth BN can be expressed as 

0.5fos 

BN = 

s 

\Wf) 1 2df 

0 

= /,-5fos ( ;ri;;;:f)2 df 

0 

(139) 

(140a) 

(140b) 

where f OS is the post-loop sampling rate, i.e. f OS = l/To = fo/16. 

For integration purposes, a change of variables was made by setting 

x = Tof, and the integral then becomes 

It can be shown that the value of this integral is given by 

BN=+2 

-[log2 N + 11 

0 

1 0.5 =-=- 

2NTo NT0 

(141) 

(142a) 

(142b) 

60 
Let r = NT0 = total time interval for averaging. Substituting this 

value of 'c in equation (142), the one-sided equivalent noise bandwidth is simply 

BN = 0,5 

T (143) 

This result is exactly the same as for the continuous-time integrateand-dump filter with t as the integration time. Thus, the sum-anddumb algorithm for a discrete time signal functions exactly the same 

as the integrate-and-dump filter for a continuous-time signal provided 

that the summation interval in the discrete-time case is equal to the integration interval in the continuous-time case. A further subtle assumption 

is that the power spectra for the two cases would have to be compared 

on the same basis. This would imply optimum sampling at the Nyquist 

rate for the discrete-time system as previously discussed. 

Assume momentarily that the input yl(i) has a uniform power 

spectral density Gyl(f) = n on a one-sided basis over the frequency 

range from dc to 0.5 fso. The output variance o 2 is then given by YN 

2 _ 0.5n 0.5n --=- 

'YN NT0 T (144) 

The fact is, however, that the input noise power spectrum in the 

case of interest is not flat but has the shape of the power transfer 

function of the loop. The loop relative amplitude-squared response 

function A:(f) expressed as a function of steady-state frequency is 

dominated by a single pole and is of the form 

(145) 

61 
where fl is the 3-dB loop frequency. As a close estimate of the 

anticipated design value, the damping constant of the loop is about 

c1 = 50 corresponding to fl = 50/21~ = 7.96 Hz, which was rounded to 

8 Hz, giving 

A;(f) = 1 

f2 1+ 8 

0 

(146) 

An equivalent noise bandwidth BNO based on the combination of the 

loop response given by equation (146) and the sum-and-dump algorithm can 

now be derived. The form for the equivalent noise bandwidth is 

(147) 

This function was modified for integration by substituting x = Tof as 

before. After some simplification, the result becomes 

(148) 

This integral was evaluated numerically for values of N ranging from 2 

to 64 (in integer powers of 2). It is convenient to tabulate the 

integrals in the form BNOr = BNONTo = y. These data are shown below 

plus the case for N = 1 

N 1 2 4 8 16 32 64 

Y = BNOr 0.3362 0.3459 0.4177 0.4586 0.4793 0.4896 0.4948 

62 
Observe that as the number of points increases, y. approaches 0.5 or 

BNO approaches O.~/T, which is the value without the additional loop 

filtering. In this limiting range, the post-loop algorithm reduces 

the spectrum at such a low frequency range that the effect of the 

loop filter is negligible. 

63 
SENSITIVITY OF PROPOSED DESIGN 

.In this section, an analysis of the sensitivity of the proposed 

loop design will be performed. Although the loop is a hybrid system in 

that it is partly a continuous-time system and partly a discrete-time 

system, the input-output relationships on a continuous-time basis will 

be maintained since it is from this point of view that the results must 

be interpreted and applied. 

As previously shown, the transfer function of the control loop is 

dominated by a single pole and is approximately equivalent to a 

continuous-time system of the form 

G(s)= ’ = +& 

TB - TA 

(1491 

where Y is the output loop estimate before post-loop filtering. The 

steady-state transfer function G(jw) corresponding to equation (149) 

can be expressed as 

G(jw) = oe137 = 

2.74 x 1O-3 2.74 x 1O-3 

50 + jw z 

1 + j& f (159) 

1 + js 

where the 3-dB break frequency is rounded slightly to 8 Hz for convenience. 

The digital words from the output of the loop are then processed 

through the post-loop filter. The equivalent noise bandwidth BNO for 

the whole system is determined from the results of equation (148), which 

were tabulated in the last section. 

The sensitivity AT of the closed-loop noise-injection feedback 

radiometer is determined from the equation 

J 2BN0 AT=2(TB+TR) - 

BSI 

(151) 

64 

-_. _. 
where TB is the reference temperature (308 K for this system), TR 

is the input noise temperature of the receiver, BNO is the equivalent noise bandwidth, and BSI is the input statistical bandwidth. 

For this system BNO, can be expressed as 

B =I- Y 

NO T-T (152) 

where y is the constant corresponding to the summation interval as 

given in the previous section. 

Substitution of equation (152) in equation (151) along with present 

radiometer values yields 

16.1696 x lo3 (153) 

where the value T 

0 = 26.7496 ms was used. For different values of N, 

the results of equation (153) are summarized below. 

N=2 

AT = 6.725 x lo3 

+I- BSI 

N=4 

AT = 5.225 x lo3 

II- BSI 

N=8 

AT = 3.871 x lo3 

d- B SI 

(154) 

(155) 

(156). 

65 
N = 16 

N = 32 

(157) 

(158) 

N = 64 

AT = 1’422 

II-- B SI 

The AT values for different bandwidths were computed and are listed 

in table 1. Along with the values of AT, it is also necessary to 

determine the length of time required for a measurement. The total 

measurement time TM can be represented as the sum of two components 

of the form: 

rM =T+T S (16’31 

The quantity r represents the time required to perform the post-loop 

algorithm and is simply -c = NT0 as previously discussed. On the other 

hand, fS is the settling time of the loop, and its value depends on 

the exact criteria of final value closeness chosen for the loop. 

The following arbitrary but reasonable criterion was employed for 

defining the settling time of the loop. The buildup of the output 

estimate of the loop on a continuous-time basis would be of the form 

y(t) = C1(TB - TA) (1 - $Ot) (161) 

where Cl is a constant. The error Ay based on a finite settling 

time is 

Ay = C1(TB - TA)c-'Ot 

66 

(1621 
A worst-case value of TB - TA = 308 was selected. The time TS was 

then calculated such that the error was no greater than AT/lo; i.e., 

the settling time error could be no greater than 10 percent of the 

fluctuation error. This criterion results in the following value for 

T as a function of AT. S 

These data are also tabulated in table 1 along with the calculated 

values of the total measurement time. 

67 
SIMULATION OF PROPOSED DESIGN 

The proposed system was simulated on the digital computer using 

Advanced Continuous System Language (ACSL) and some special programs 

that were developed earlier under this and a previous contract. The 

actual programs have been described in some earlier publications (refs. 

1 and 2). 

Because it would consume far too much computer time to simulate 

the radiometer with the actual bandwidths employed in the real-life 

system, a decision was made use a simulated input statistical bandwidth 

of l/(2)* = l/256 of the smallest RF bandwidth. The resulting simulation bandwidth is then 20 MHz/256 = 78.125 kHz. The reason for this 

seemingly odd choice will now be explained. Since the sensitivity of 

the measurement varies as l/ d- BSI, the AT of the simulation is 

d- 256= 16 times the corresponding value of the actual design at the 

smallest bandwidth. A factor of 16 corresponds to 4 bits in a binary 

representation. While the actual system will employ 12-bit words, it 

was desired in the simulation to have quantization effects appear in 

the same relative level as they would in the final design. This could 

then be readily achieved by employing an g-bit A/D converter and a 256- 

step down-counter for the noise-injection counter in the simulation. 

Thus, in the simulation, the relative quantization error as compared 

to the AT level would be in the same proportion as for the final 

design. 

Even with the scaling employed, the computer simulations were quite 

expensive, and only a limited number were used. Nevertherless, the 

results obtained were quite good, and they verify the predicted behavior 

quite well. 

The results of four separate runs with all post-loop outputs are 

tabulated in tables 2 through 7. In interpreting these results, bear 

in mind that the limited number of runs coupled with the dynamic nature 

68 
of the statistical parameters result in some high statistical variations 

from run to run. However, the general trends and averages here are 

most significant in establishing the validity of the concept. 

In each of the tables, the following parameters are tabulated: 

“time” = value of time at which averages are computed. 

temperature estimates 1 - 4 = the tabulated results obtained at the 

times listed. The input temperature is TA = 100 K. The output 

is scaled so as to read 0.99(TB - TA) = 205.92 K, which would 

be the ideal value. 

% = estimate of mean value of process obtained by an ensemble average 

for the four runs 

syN 

= estimate of standard deviation of process 

uyN 

= theoretical predicted value of standard deviation obtained 

using results of preceding section 

r\ 

YN = mean value of estimated means 

2 = mean value of estimated standard deviations 

YN 

Table 2 provides a tabulation of the various data for yz(n), 

i.e., the estimate based on two post-loop samples. Note that the mean 

of the standard deviations is 23.99 K, which compares favorably with 

the theoretical value of 24.06 K. 

As N increases, the statistical dependency between averages performed at different values of time increases so that estimations of the 

values down the column decrease in effectiveness. In other words, 

the greater the resolution of the estimate itself, the less one is 

able to assess the accuracy of the resolution. 

Table 3 provides a tabulation of the results for ys(n). The mean 

of the standard deviations is 17.75 K compared with a predicted value 

of 18.69 K. The values given in tables 4 through 7 provide similar 

results for ya(n), ylc(n), y32(n), and yak(n) respectively. 

The results are illustrated graphically in figure 36. 

69 

APPENDIX 

ANALYSIS OF CLOSED LOOP NETWORK 

AND NOTCH FILTER 

by 

John M. Jeffords 

Old Dominion University 

Electrical Engineering Department 

71 

- 
INTRODUCTION 

Old Dominion University and NASA/Langley Research Center are 

involved in an effort to develop design techniques for improved microwave radiometers. This report documents part of the analysis 

performed on circuits used to model a portion of the radiometer system 

in support of the continuing design effort. 

A microwave radiometer is a receiver used to estimate the mean 

value of a noise signal in the presence of background thermal noise. 

Although microwave radiometers exist in a variety of configurations with 

different complexities, sensitivities, and accuracies, the current 

interest of this effort is the noise-injection feedback system. In 

this configuration, the output of a controlled noise source is added 

to the input noise and the sum is compared to a stable reference 

noise. Feedback is used to adjust the controlled noise source so that 

a balance is achieved. The input noise is then the difference between 

the known reference noise and the output of the controlled noise source. 

Although digital signal processing is contemplated for the 

microwave radiometers to be developed, a better understanding of the 

current analog systems was desired. The current system evolved through 

a combination of both analytical and empirical techniques. An analysis 

of this system provides a baseline upon which future design can be 

established. 

Although the actual microwave radiometer involves the interaction 

of a number of random processes, the control mechanism of the feedback 

loop can be modeled by a simpler "deterministic" form which approximates 

the loop operation at very low frequencies. 

STATEMENT OF PROBLEM 

The specific objectives of this study were to: 

1. Determine the optimum value of damping factor for the loop to 

minimize the product of noise bandwidth and settling time. 

72 
2. Extend the concept of settling time as usually given in books on 

control theory to include a more general criterion for precisionbounded measurements. 

3. Determine the relative sensitivity of the loop to changes in the 

open-loop gain factor, particularly in regard to the error. 

4. Investigate the use of a lead time constant as it affects error 

and sensitivity of the loop. 

5. Evaluate the performance of a notch filter. 

GENERAL APPROACH 

Loop Analysis 

Figure A-l is the block diagram of the closed-loop, noise-injection, 

feedback deterministic radiometer model upon which this analysis is 

based (ref. 1). For analysis purposes, this block diagram was simplified 

to the block diagram of figure A-2 in which Boltzmann's constant, the 

square-law detector constant, the gains of the RF and AF amplifiers, the 

Dicke constant, and the gains of the integrator and the lead-lag network 

have all been combined into Kl, and the gains of the directional 

coupler and isolator and the conversion factors associated with the noise 

diode have been combined into K2. 

A basic measure of system performance used in the evaluation was 

the response of the system to a unit step input. If such an input is 

applied to the system of figure A-2, some steady-state output will 

eventually be reached as indicated in figure A-3. If we define settling 

time, ts, as the time required for the output to reach and remain 

within some arbitrary percentage of the final value, settling time can be 

used as a measure of system performance. For this evaluation, the 

settling time was defined as the time after which the output would not 

deviate from the final value by more than 0.01 percent. 

The noise-equivalent bandwidth, BN, of a system with transfer 

function H(f), is the bandwidth of an ideal filter with the same midband 

73 
gain, Ho, as H(f), and which passes the same average noise power 

as H(f) when white noise is applied as its input. The standard 

definition of BN is 

03 

IH(f) I2 df (A-1) 

To determine the noise-equivalent bandwidth of the system of figure A-2, 

temporarily use p as the independent variable of the transfer function. 

Then, H(p) is given by 

+ T P) 

2 

H(P) = 1 + K1K2T2 Tl 

1 + Klb > p + K1K2 p2 

where 

25 1 + K1K2T2 K1K.z 

-= 

w 0 K1K2 

and .t~$ = 7 

Making the change of variable, s = p/we, 

+ woT2sl 

H(s) = 

1 + 25s + s2 

(A-2) 

(A-3) 

74 
I 

Evaluating equation (A-l) for this transfer function, 

BN = 

1 + boT212 

82; (A-4) 

where BN is a measure of the average noise power that will be present 

in the output of the system when white noise is present at the input. 

In general, as the parameters of the system of figure A-2 are 

varied, both BN and ts will change. Although optimum performance 

involves minimizing both BN and ts, parameter values which minimize 

BN tend to make ts large, and vice versa. Therefore the product of 

BN and t S was selected as a performance criterion. 

The underdamped, critically damped, and overdamped responses to 

the system of figure A-2 to a unit step input are given by equations 

(A-5), (A-6) and (A-7) respectively. In these equations, the actual 

unit step response is yl(t) where y(t) = Kpyl(t). 

Underdamped response 

y(t) = 1 - AemSt sin [d- t + 0) 

where 

and 

0 = tan -1(tqF2) 

(A-5) 

75 
Critically damped response 

Y(t) = 1 - [(l - woT2)t + 1] e 

-t 

Overdamped response 

-St 

y(t) = 1 - Be sinh t + $ 

> 

(A-6) 

(A-7) 

where 

and 

Figure A-4 indicates how the settling time was determined for the 

underdamped case where the overshoot exceeds the target percentage of 

deviation (figure A-4a)-, the underdamped case where the overshoot is less 

than the target percentage of deviation (figure A-4b), and the critically 

damped and overdamped cases (figure A-4~). 

Since both BN and ts are functions of the damping factor, 5, 

changes in the damping factor can be expected to change the BNts 

product. If the per unit sensitivity of damping factor to KL is 

defined as 

ar; KL -- 

aKL 5 (A-81 

76 
where K L = Klktr the sensitivity of damping factor to changes in loop 

gain for the system of figure A-2 is 

which can be expressed as 

(A-91 

(A-10) 

Maximum overshoot and the time at which it occurs for the underdamped case were evaluated by setting the partial derivative of y(t) 

with respect to t equal to zero and solving the resulting equation 

for t. The result is equation (A-11). This value of t is substituted 

into the expression for l-y(t) to obtain the maximum overshoot of 

equation (A-17). 

1 

0 m,Tz L 1 

n= 

1 woT2 L 1 

(A-11) 

E = 

max i 

cos d-t,,, +; - WoT2 sin j/3 tmax )eeCtmax 

1 - 52 

(A-12) 

Defining the error as l-y(t), the per unit sensitivity of the 

error to changes in K1 can be evaluated. Define the per unit sensitivity 

of the error to changes in K, as 

77 
(A-13) 

Kl 

By the chain rule for partial derivatives, this can be expressed as 

ac a5 K1 

'El = % aK1 F (A-14) 

But 

aE -ct Cl - 5woT21 

-= e 

a6 1 - c2 

l sin 1 - c2t 

4 

(A-15) 

so that 

Notch Filter Analysis 

Consider a filter with transfer function 

s2 + d 

H(s) = 

0 

s2 + 25wos + Id2 

0 

(A-17) 

78 
After making the substitution s = j,, this transfer function can be 

put in the following form: 

H(f) = 

1 

l+j 

25 If/f,) 

l- cf/fo)2 

(A-18) 

The half-power points for this function occur when 

25 (f/f01 

= +1 (A-19) 

1 - (f/fo12 

so that for positive frequencies, the half-power points are at the 

following frequencies: 

the bandwidth of the filter, BW = f2 - fl, is then 

BW = 2cfo 

The response of this filter to a unit step input is 

y(t) = 1 - 

[ $% 2 

sin cd- Uotj]e-CUo' 

for the underdamped case. 

(A-20) 

(A-21) 

(A-22) 

79 
Let 

E =d121 52 sin(dZ Wet) emTWot (A-23) 

be the error. For the underdamped case this error will have a 

maximum value when 

t = uod&F lzan -1 (GjT) (A-24) 

as indicated in figure A-5a, after which the error will have a series 

of peaks of alternating sign and decreasing magnitude. Consider the 

first negative maximum of the error. Because of the exponential factor 

in the expression for the error, no succeeding maxima or minima will 

exceed it in magnitude, so it represents the maximum overshoot of the 

step response, the maximum amount by which the output exceeds the final 

value of the output, although it is not the largest error. In figure 

A-5a, the settling time tl is the time after which the error never 

exceeds the maximum overshoot. 

For the critically damped and overdamped cases, the response is 

as indicated in figure A-5b. 

PRESENTATION OF RESULTS 

BNts Product 

In figure A-6, the product of the noise-equivalent bandwidth BN 

and the settling time t 

S 

is plotted versus the damping factor 5 for 

five values of lead time constant. The simple lag network, in which 

the lead time constant T2 is zero, appears best from a consideration of 

just the BNts product, since it not only has the lowest values, but 

80 
the values remain relatively constant over 

The abrupt decreases in the BNts product 

curves for moT2 = 2.0 and woT2 = 1.5 occur 

5 = 

1 + (woT2> 2 

2woT2 

a wide range of damping factor. 

that are apparent in the 

when 

Sensitivity of Damping Factor to KL Changes 

The sensitivity of damping factor to changes in the loop gain is 

plotted as a function of damping factor for five values of lead time 

constant in figure A-7. 

Sensitivity of Error to K1 Changes 

Figure A-8 is a plot of the sensitivity of the error to changes in 

the forward gain plotted as a function of time for three lead time 

constants and for a damping factor of 0.95, the value at which the noiseequivalent bandwidth and settling time product is minimum when the lead 

time constant is zero. 

Damping Factor and Settling Time versus Maximum Overshoot 

Figures A-9, A-10, A-11, A-12, and A-13 are plots of damping factor 

and wt 0 s versus maximum overshoot for the underdamped loop, each 

figure for a different lead time constant. A comparison of the effect 

of lead time constant on overshoot and the resultant settling time can 

be obtained from these curves. 

Notch Filter Damping Factor and Settling Time 

Figure A-14 presents damping factor and settling time as a function 

of maximum overshoot for the notch filter. 

81 
SUMMARY 

Loop Analysis 

Figures A-6, A-7, and A-8 indicate that as long as loop gain variations are not encountered, the simple lag network provides superior 

performance as long as damping factors above approximately 0.8 are 

used. For damping factors in the vicinity of critical damping, which 

appears to be the logical area of damping factor to operate, uOT2 = 1.0 

minimizes the sensitivity of the damping factor to changes in loop gain. 

Although figure A-8 indicates that the error is still very sensitive to 

changes in K1 at most times, this is probably not as serious as it 

appears at first. 

For example, considering the curve of figure A-8 for w,T~ = 1.0, 

it can be seen that the error is insensitive to changes in K1 in the 

vicinity of wet = 10.5. From figure A-11 it can be seen that the 

maximum overshoot corresponding to a settling time of this value is 

less than 10s6. Since w. can be expected to be on the order of 100 

rad/s, this operation would require allowing times on the order of 

0.1 s for settling, which is not unreasonable. 

Although figure A-6, which is plotted for an error of 10m4, indicates 

that the simple lag network produces the minimum bandwidth-time product 

for any damping factor, the sensitivity of the error to changes of 

loop gain will likely preclude operating the system at minimum settling 

times. Allowing additional time for settling before taking a measurement will reduce the magnitude of the error. However, so much time may 

be required that lead compensation may be preferable, even though its 

minimum bandwidth-time product is higher than that of the simple lag 

network. 

82 
GAIN 

FLUCT. 

TBFigure A-l. Block diagram of closed-loop, noise-injection, feedback deterministic 

radiometer models. 
+ 

--q 

K1 1 + T2s 

* - . S 1 + Tls 

K2 4 

Figure A-2. Block diagram of system analyzed. 
Y(t) 

<= 1 

--cm 5’1 

--P 5< 1 

Figure A-3. System step responses. 
L 

t 5 

(a) Underdamped: over sh 

y(t) I 

E 

0 

is target 

percentage of error 

t 

loot exceeds target error 

tS 

(b) Underdamped: overshoot less than target error 

t 

tS 

(c) Critically damped and overdamped 

Figure A-4. Examples of settling time determination. 

86 
la; .i 

I 

I t 

% 

(a) Settling time determination for maximum error curveunderdamped notch filter response. 

y(t) 

(b) Notch filter response: critically damped and overdamped. 

Figure A-5. Notch filter responses. 

87 

-- 
BNts 

10 

9 

5 

4 

\ 

\ 

uoT2 = 0 

I\ \ 

-mm- 

\ 

woT2 = 0.5 \ 

I 

I i\ \ \ - -- moT2 = 1.0 

\, \ ' 

\ ---- UT 

\ 

o 2 = 1.5 

\ ' \ -/- 

----- uoT2 = 2.0 //--- 

-1 

\ / / 

I I I I I 1 I I I I I I I I I I 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 

5 

Figure A-6. Product of noise-equivalent bandwidth and settling time plotted versus damping 

factor and lead time constant. 
woT2 = 0 

--- woT2 = 0.5 

--- w,T1 = 1.0 

---- woT2 = 1.5 

----- woT2 = 2.0 

-S I I I I I I I I 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 

5 

Figure A-7. Sensitivity of damping factor to changes in loop gain as 

function of damping factor and lead time constant. 

89 
uoT2 = 0 

----- woT2 = 1.0 

--- uoT2 = 1.5 

Figure A-8. Sensitivity of error to changes in forward gain as function of lead time constant 

and time. 
12.0 

11.0 

10.0 

9.0 

wt 0 s 

8.0 

7.0 

6.0 

5.0 

- L \ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

5 \ 

\ 

\ 

---- wt 0 s \ 

\ 

\ 

\ 

\ 

\ 

\ 

I I I I I1111 I I I I1111 I I I111111 I I III1 IL 

1.0 

0.9 

0.8 

0.7 

5 

0.6 

0.5 

0.4 

0.3 

lo-6 10-5 10-4 10-3 10-Z 

Emax 

Figure A-9. Damping factor and settling time versus maximum overshoot for woT2 = 0. 
11.0 

10.0 

9.0 

8.0 

wt 0 s 

7.0 

6.0 

5.0 

4.0 

- 

\ 

\ 

- \ 

\ 

\ 

\ 

- \ 

\ ‘\ \ 

- \ \ \ \ 

\ 

\ 

\ 

\ 

\ 

I I lllrll I I I I1111 

‘1 I I I lllll I I I I 

10-5 10-4 10-3 

Emax 

lo-6 

-1.0 

. 

0.8 

0.7 

r 

0.6 

Figure A-10. Damping factor and settling time versus maximum overshoot for woT2 = 0.5. 
14.0 

12.0 

10.0 

8.C 

wt 0 s 

6.t 

4.c 

2.t 

l- 

)- 

I- 

)- 

IO 

-e- wt 0 s 

-1 

-C 

-( 

-( 

-( 

I 

10-6 10-j 

Figure A-11. Damping factor and settling time versus maximum overshoot 

for wOT2 = 1.0. 

I.9 

1.8 

I.7 

5 

3.6 

0.5 

0.4 

0.3 
6.0 

5.0 

4.0 

wt 0 s 

3.0 

2.0 

1.0 

0 I I I Ill111 I I I lllll I I I111111 I I IIII 

10-4 10-j 10-2 10-l loo 

1.0 

0.9 

0.8 

0.7 

5 

0.6 

0.5 

0.4 

0.3 

Figure A- 12. Damping factor and settling time versus maximum 

overshoot for uoT2 = 1.5. 
7.0 

6.0 

5.0 

4.0 

3.0 

lot 0 s 

2.0 

1.0 

0 

3 

--- wt 0 s 

\ 

‘I. 

I I I111111 I I I lllll I I I lllll I I III1 

- 

JL 

1.0 

0.9 

0.8 

0.7 

0.6 ' 

0.5 

0.4 

0.3 

10-4 10-3 10-2 10-l 100 

E max 

Figure A-13. Damping factor and settling time versus maximum overshoot for 

moT2 = 2.0. 
10 

9 

8 

7 

wt OS 

6 

\ 

\ 

I I I lllll I I I111111 I I I lllll I 

\ 0.3 

I IIIII 

10-s 10-4 10-3 10-2 10-l 

E max 

Figure A-14. Damping factor and settling time versus maximum 

overshoot for the notch filter. 

-0.9 

-0.8 

3 

--- wt 0 s 

- 

\ 

\ 

\ 

\ - 0.4 

1. 
REFERENCES 

1. Stanley, W.D.: Digital Simulation of Dynamic Processes in 

Radiometer Systems. Final Report on NASl-14193, Task Authorization No. 46, NASA CR-159266, 1980. 

2. Stanley, W.D.; Harrington, R.F.; and Lawrence, R.W.: Dynamic 

Simulation of Random Processes in Radiometers Using CSMP and 

ACSL. Presented at IEEE SOUTHEASCON Conference (Roanoke, VA), 

April 1979; also published in Conference Proceedings. 

3. Harrington, R.F.; Couch, R.H.; and Fedors, J.C.: An Airborne 

Remote Sensing 4.5 to 7.2 Gigahertz Stepped Frequency Microwave 

Radiometer. Presented at the 1979 International Microwave 

Symposium (Orlando, FL), April-May, 1979; also published in 

Conference Proceedings. 

4. Stanley, W.D.; and Peterson, S.J.: Equivalent Statistical 

Bandwidths of Conventional Filters. IEEE Trans. Comm., Vol. 

COM-27, Oct. 1979. 

5. Bendat, J.S.; and Piersol, A.G.: Random Data: Analysis and 

Measurement Procedures, John Wiley and Sons, Inc., 1971. 

6. Tiuri, M.E.: Radio Astronomy Receivers. IEEE Trans. Antennas 

Prop., Vol. AP-12, Dec. 1964, pp. 930-938. 

7. Evans, G.; and McLeish, C.W.: RF Radiometer Handbook. Artech 

House (Dedham, MA), 1977. 

8. Dicke, R.H.: The Measurement of Thermal Radiation at Microwave 

Frequencies. Rev. Sci. Instr., Vol. 17, July 1946, pp. 268-275. 

I‘6 

97 
9. 

10. 

11. 

Hardy, W.N.; Gray, K.W.; and Love, A.W.: An S-Band Radiometer 

Design with High Absolute Prevision. IEEE Trans. Microwave Theory 

and Technique, Vol. MTT-22, Apr. 1974, pp. 382-390. 

Tretter, S.A.: Introduction to Discrete-Time Signal Processing. 

John Wiley & Sons, Inc. (NY), 1976. 

Stanley, W.D.: Digital Signa .l Processing. Reston Pub lishing Co. 

(Reston, VA), 1975. 

98 
TABLE l.- PREDICTED SENSITIVITIES BASED ON NOISE FLUCTUATIONS AND MEASUREMENT TIMES 

(INCLUDING BOTH PROCESSING AND SETTLING TIMES) FOR PROPOSED DIGITAL SYSTEM. 

BsI 

= 20 1MHz 

BsI 

= 100 MHZ 

BsI 

= 500 MHZ 

BsI 

=2 GHz 

N=l 

T 26.75 ms 

AT 2.096 K 

'S 146 ms 

\ 'M 173 ms 

I 

AT 0.938 K 

' Ts 162 ms 

'M 189 ms 

AT 0.419 K 

' 's 178 ms 

'M 205 ms 

I 

AT 0.210 K 

' 's 192 ms 

%I 219 ms 

2 

53.5 ms 

1.504 K 

153 ms 

206.5 ms 

0.672 K 

168 ms 

221.5 ms 

0.301 K 

185 ms 

238.5 ms 

0.150 K 

199 ms 

252.5 ms 

4 

107 ms 

1.168 K 

158 ms 

265 ms 

0.523 K 

174 ms 

281 ms 

0.234 K 

190 ms 

297 ms 

0.117 K 

204 ms 

311 ms 

8 

214 ms 

0.866 K 

164 ms 

378 ms 

0.387 K 

180 ms 

394 ms 

0.173 K 

196 ms 

410 ms 

0.0866 K 

210 ms 

424 ms 

16 

428 ms 

0.626 K 

170.ms 

598 ms 

0.280 K 

186 ms 

614 ms 

0.125 K 

202 ms 

630 ms 

0.0626 K 

216 ms 

644 ms 

32 

856 ms 

0.447 K 

177 ms 

1.033 s 

0.200 K 

193 ms 

1.049 s 

0.0894 K 

209 nis 

1.065 s 

0.0447 K 

223 ms 

1.079 s 

64 

1.712 s 

0.318 K 

184 ms 

1.896 s 

0.142 K 

200 ms 

1.912 s 

0.0636 K 

216 ms 

1.928 s 

0.0318 K 

230 ms 

1.942 s 
TABLE 2.- COMPUTER SIMULATION OF PROPOSED DESIGN WITH TWO POST-LOOP SAMPLES 

IN AVERAGE. (ALL UNITS EXCEPT TIME ARE EXPRESSED IN KELVIN.) 

I 

y2 SIMULATION 

Temperature Estimates 

TIME(s) Run 1 Run 2 Run 3 Run 4 

0.213 216.54 188.11 188.29 248.71 

0.320 193.82 180.90 243.18 190.79 

0.427 203.17 221.00 196.04 172.97 

0.533 166.72 218.59 213.69 177.51 

0.640 162.72 201.21 210.12 200.32 

0.747 212.08 177.42 207.01 185.71 

0.853 196.40 225.45 236.77 147.30 

0.960 257.53 222.15 212.35 223.05 

1.067 245.77 222.87 186.42 229.46 

1.173 158.80 197.65 238.19 190.61 

1.280 202.10 213.07 188.74 201.84 

1.387 202.28 195.78 219.39 224.02 

1.493 196.76 269.65 216.54 197.65 

1.600 212.08 178.22 216.90 213.33 

1.707 212.80 169.85 170.65 209.95 

1.813 198.18 212.00 180.45 161.74 

1.920 184.01 239.98 177.06 214.31 

Y2 

210.41 

202.17 

198.30 

194.13 

193.59 

195.56 

201.48 

228.77 

221.13 

196.31 

201.44 

210.37 

220.15 

205.13 

190.81 

188.09 

203.84 

203.63 

!? 

y2 

SY 2 OY 2 

28.82 24.06 

27.89 24.06 

19.88 24.06 

25.87 24.06 

21.05 24.06 

16.63 24.06 

39.92 24.06 

19.78 24.06 

25.06 24.06 

32.64 24.06 

9.95 24.06 

13.49 24.06 

34.24 24.06 

18.06 24.06 

23.77 24.06 

21.80 24.06 

29.02 24.06 

23.99 

A 

sY2 

- 

100 
TABLE 3.- COMPUTER SIMULATION OF PROPOSED DESIGN WITH FOUR POST-LOOP SAMPLES 

IN AVERAGE. (ALL UNITS EXCEPT TIME ARE EXPRESSED IN KELVIN.) 

Y4 SIMULATION 

Temperature Estimates 

TIME(s) Run 1 Run 2 Run 3 Run 4 74 S 

Y4 OY4 

0.213 202.59 211.73 208.70 228.44 212.87 11.06 18.69 

0.320 196.85 179.48 240.51 171.67 197.13 30.78 18.69 

0.427 198.09 241.05 194.89 184.59 204.66 24.94 18.69 

0.533 171.00 222.24 206.20 174.57 194.00 25.62 18.69 

0.640 184.59 195.02 188.20 206.83 193.50 24.85 18.69 

0.747 211.10 186.15 205.45 180.54 195.81 14.76 18.69 

0.853 200.37 205.67 217.48 169.31 198.21 20.55 18.69 

0.960 229.19 216.50 205.67 204.15 213.88 11.60 18.69 

1.067 231.69 224.20 192.26 224.74 218.22 17.64 18.69 

1.173 153.94 211.19 234.36 194.80 198.57 33.89 18.69 

1.280 199.70 208.21 208.34 186.33 200.65 10.36 18.69 

1.387 203.80 199.74 216.81 201.93 205.57 7.67 18.69 

1.493 214.71 217.48 204.91 207.01 211.03 6.02 18.69 

1.600 191.23 217.83 217.43 217.83 211.08 13.23 18.69 

1.707 190.83 174.44 188.07 211.64 191.25 15.37 18.69 

1.813 195.24 211.42 203.93 192.84 200.86 8.50 18.69 

1.920 188.96 235.52 178.62 207.90 202.75 24.99 18.69 

202.94 17.75 

4 

Y4 

A 

sy4 

101 

lk - 
TABLE 4.- COMPUTER SIMUUTION OF PROPOSED DESIGN WITH EIGHT POST-LOOP SAMPLES 

IN AVERAGE. (ALL UNITS EXCEPT TIME ABE EXPRESSED IN KELVIN.) 

y8 SIMULATION 

Temperature Estimates 

TIME(s) Run 1 Run 2 Run 3 Run 4 

0.320 199.72 195.60 224.61 200.05 

0.427 197.47 210.26 217.70 178.13 

0.533 184.55 231.64 200.54 179.58 

0.640 177.80 208.63 197.20 190.70 

0.747 197.85 190.59 196.82 193.68 

0.853 205.74 195.91 211.46 174.93 

0.960 214.78 211.08 211.57 186.73 

1.067 230.44 220.35 198.96 214.45 

1.173 192.81 217.70 213.31 209.77 

1.280 176.82 209.70 221.35 190.56 

1.387 201.75 203.98 212.58 194.13 

1.493 209.26 208.61 210.86 204.47 

1.600 202.97 217.65 211.17 212.42 

1.707 191.03 196.13 202.75 214.74 

1.813 193.04 192.93 196.00 202.24 

1.920 192.10 223.47 191.28 200.37 

78 

205.00 

200.89 

199.08 

193.58 

194.74 

197.01 

206.04 

216.05 

208.40 

199.64 

203.11 

208.30 

211.05 

201.16 

196.05 

201.81 

202.62 

A 

78 

sY8 OY8 

13.23 13.86 

17.32 13.86 

23.48 13.86 

12.87 13.86 

3.28 13.86 

16.06 13.86 

12.98 13.86 

13.17 13.86 

10.89 13.86 

19.80 13.86 

7.59 13.86 

2.72 13.86 

6.08 13.86 

10.24 13.86 

4.36 13.86 

15.02 13.86 

11.86 

Q _. 

I 

-I 

y8 

102 
TABLE 5.- COMPUTER SbfULATIO~ OF PROPOSED DESIGN WITH 16 POST-LOOP SAMk'LES IN 

AVERAGE. (ALL UNITS EXCEPT TIME ARE EXPRESSED IN KELVIN.) 

y16 SIMULATION 

Temperature Estimates 

TIME(s) Run 1 Run 2 Run 3 Run 4 ?16 sy16 516 

0.533 192.14 213.62 212.58 189.82 202.04 12.81 10.02 

0.640 187.64 209.44 207.45 184.42 197.24 13.03 10.02 

0.747 191.20 211.12 198.68 186.63 196.91 10.70 10.02 

0.853 191.77 202.27 204.33 182.81 195.30 9.98 10.02 

0.960 206.32 200.83 204.20 190.21 200.39 7.15 10.02 

1.067 218.09 208.13 205.21 194.69 206.53 9.63 10.02 

1.173 203.80 214.39 212.44 198.25 207.22 7.55 10.02 

1.280 203.63 215.03 210.16 202.51 207.83 5.87 10.02 

1.387 197.28 210.84 212.94 201.95 205.75 7.39 10.02 

1.493 193.04 209.16 216.11 197.52 203.96 10.57 10.02 

1.600 202.36 210.82 211.87 203.27 207.08 4.96 10.02 

1.707 200.14 202.37 206.81 209.60 204.73 4.27 10.02 

1.813 198.01 205.29 203.59 207.33 203.56 4.00 10.02 

1.920 191.57 209.80 197.01 207.55 201.48 8.65 10.02 

- - 

202.86 8.33 

A 

‘16 

Q 

y16 

103 
TABLE 6.- COMPUTER SIMULATION OF PROPOSED DESIGN WITH 32 POST-LOOP SAMPLES 

IN AT:ERAGE . (ALL UNITS EXCEPT TIME ARE EXPRESSED IN KELVIN.) 

~32 SIMULATION 

Temperature Estimates 

TIME(s) Run 1 Run 2 Run 3 Run 4 is2 sY3* u 

y32 

0.960 199.23 207.23 208.39 190.01 201.22 8.51 7.15 

1.067 202.86 208.79 206.33 189.55 201.88 8.57 7.15 

1.173 197.50 212.75 205.56 192.44 202.06 8.94 7.15 

1.280 197.70 208.65 207.25 192.66 201.57 7.68 7.15 

1.387 201.80 205.83 208.57 196.08 203.07 5.43 7.15 

1.493 205.56 208.64 210.66 196.10 205.24 6.44 7.15 

1.600 203.68 212.60 212.16 200.76 207.30 5.99 7.15 

1.707 201.89 208.70 208.48 206.05 206.28 3.16 7.15 

1.813 197.64 208.06 208.26 204.64 204.65 4.96 7.15 

1.920 192.30 209.48 206.56 202.53 202.72 7.51 7.15 

203.60 6.72 

r\ A 

y32 sY 

104 
TABLE 7.- COMPUTER SIMULATION OF PROPOSED DESIGN WITH 64 POST-LOOP SAMPLES IN 

AVERAGE. (ALL UNITS EXCEPT TIME ARE EXPRESSED IN KELVIN.) 

y,, SIMULATION 

Temperature Estimates 

TIME(s) Run 1 Run 2 Run 3 Run 4 , %4 %Jt 

1.813 198.43 207.65 208.33 196.04 202.61 6.29 

1.920 197.58 209.13 206.45 197.32 202.62 6.07 

202.62 6.18 

A 

Gt 

A 

%A 

/ 

105 

L 
rA 

DICKII 1 

TI SWITCll 7 

l - KDC .a TE 

+ - G(s) 

- “0 

P 

T = pulse width (s/pulse) t 

Kl 

I I 

DUTY 

*I; - 

TN CYCLE 

KATT - *Ex +-- T - 

1 5 kl 

K1 = voltage-to-frequency constant p”‘ses’s V 

T = pulse width (s/pulse) 

T1 = (1 - KoC)TA + K&TB + T;) 

TE = $1.1 - Tgl, V. = -GTE 

6” 

‘I; = KAmTEX~K,Vo’= fl 

DC 

“Cl (l-K&W 

- = 

TB-TA l+GB/2 

FO (1-KDC)GK112 

-c 

*B-*A 1 + G6/2 

6 = KDCKATTTEX% 2.L (1-K,,C)/2 

*A-*B 1 + m/2 

Figure 1. Block diagram of control mechanism in feedback noise-injection radiometer. 
1 CHz 

L POLARIZER 

LATCHING TUNNEL DIODE HOT CARRIER DIODE 

CIRCULATOR AMPLIFIER MIXER 26 dB GAIN PREAMPLIFIER - 

DIRECTJONAL MHZ 

L 

LOOP 

* FILTER 

“IF J-L DATA 

- CONVERTER OUT b 

(70 !Js 

PULSES) 

PIN 

ATTENUATOR 4 NOISE DIODE 

DIODE c SWITCH * 

Figure 2. Block diagram of radiometer system used as a reference in this study. 
(4 

(b) 

6.6 J.IF 12 KR 350 Kn 

6.6 VF 

R2 = 10 KR 

AMPLITUDE RESPONSE 

0.0666 2.41 2.01 

Hz Hz Hz 

Figure 3. Circuit used for stages ICI and IC2 and its Bode plot. 

108 
(a) 

R1 = 30 KQ 

Cl = 1 r.lF 

R2= 500 KR 

C2 = 0.001 PF 

(b) AMPLITUDE RESPONSE 25.93 dB (19.8) 

0.3 Hz 5628 Hz 

Figure 4. Circuit used for stage IC3 and its Bode plot. 

109 
120 

80 

dB 

-80 

-160 

AMPLITUDE RESPONSE 

85.1 dB (18,000) 

(a> 

18 dB/octave 

I I I I1111 I I lllllll I I I lllll 

10-3 10-2 10-l 100 101 lo2 

Figure 5. Bode plot of video amplifier (continued on p. 111). 
120 

r 

AMPLITUDE RESPONSE 

85.1 dB(18,OOO) 

-80 

-160 1 I I I111111 I I Il,llll I I I IIl,ll I I I IIld I I I111111 

102 103 104 105 106 107 

Figure 5. (Concluded). 
C = 0.56 UF 

Figure 6. Integrator circuit used in estimation circuit. 

112 
(a) 

R. = 50 KR 

Rl = 5 KS1 

c = 0.56 pF 

(b) AMPLITUDE RESPONSE 

L 

5.68 Hz 62.5 Hz 

Figure 7. Original loop filter (lag-lead network) and its 

Bode plot. 

113 
80 

40 

dB 

0 

-80 

AMPLITUDE RESPONSE 

Figure 8. Bode plot of estimation circuit. 
O-300 K FORWARD 5.532-0.143 V 

* 0.99 SYSTEM + 

STAGES 

I 308-1lK 

0.01 

A 

30,800-1,100 K 

308 K 

T 30,492-792 K 

121,968-3,168 K 

. 70 us 4 1 

TY CYCLE RANGE 4774-124 

0.334-0.00868 PULSES/s 

Figure 9. Illustration of noise addition in loop and range 

of values. 

115 
v+= C(TA + TR) 

(b) 

ii- = -C(TB + TR) 

TA ' 100 K 200 K 300 K 

-+ v 3.58 V 4.15 V 4.72 V 5.29 V 

-- V -5.33 v -5.33 v -5.33 v -5.33 v 

c = 5.705 x 10-3 

TR = 627 K 

TA 0 100 K 200 K 

-+ V 5.33 v 5.33 v 5.33 v 

-_ V -5.33 v -5.33 v -5;33 v 

Figure 10. Manner in which noise pulses add to signal to force net 

error temperature to zero. 

116 
DIRECTIONAL RF DIODE 

COUPLER BOLTZMANN’S 

CONSTANT CONSTANT 

AMPLIFIER DETECTOR DICKE ““,“:;;“” 

GAIN CONSTANT CONSTANT ’ INTEGRATOR LAG-LEAD NETWORK 

ll(1 + 2.545 x lo-%) L 

1 + 28 x lo-% 

DIRECTIONAL FIXED EXCESS VOLTAGE TO 

COUPLER ATTENUATOR NOISE PULSE DUTY CYCLE 

CONSTANT TEMPERATURE CONSTANT 

Figure 11. Control loop model for stepped frequency radiometer. 
Ro 

I - 

Figure 12. Form of lag network used to replace lag-lead network. 

118 
O-100 KR 

10 KR 

5K.Q _ 

2 

- 

O-10 KR 

+ 0 3 

- 1 uF 

I 

0 

Figure 13. Test circuit used to optimize the loop response. 

119 
DOES NOT INCLUDE 

0.5 DICKE CONSTANT 

(al 

TB - TA Kl 

V 

- G(s) 

K2 

@I 

v, (tl 

I KI (TB + TR) 

-KI(TB + TR1 

Figure 14. System model and waveform used in Dicke ripple analysis. 

120 
100 

90 

80 

70 

60 

50 

40 

30 

20 

% 

10 

9 

8 

7 

6 

5 

PEAK RIPPLE LEVEL PERCENTAGE 

I I 

0 50 100 150 200 250 300 350 

INPUT TEMPERATURE (KELVIN) 

Figure 15. Ripple level output percentage versus input temperature. 

121 
10 

0 

dB 

-10 

-20 

-30 

-40 

-50 

AMPLITUDE RESPONSE 

10 100 

I I I I I III 

.20 1,000 

FREQUENCY (Hz) 

Figure 16. Amplitude response of possible analog loop notch filter. 
x (t> Y(t) 

X(s) Y(s) 

H(z) 

Figure 17. System forms used in digital discussions. 

b. 

123 
(a) 

CJ) 

t 

Gll. Co 1 

I n 

2 

f- 

(cl LOW-PASS 

FILTER 

BN 

“02 (t) 

* 

1 

p(t) 

Cd) 

Figure 18. Noise and system forms used in developing the concept of 

noise sampling. 

124 
(a> Gi (f) 

I! 

2 

I 

-Bi Bi fS 

f 

GO(l)(f) 

-B i B. 

fS 

f 1 

Figure 19. Sampling of bandwidth-limited signal exactly at Nyquist 

rate (fs = 2Bi). 

125 
Gi (f) 

11 

2 

-Bi f S 

B i f 

Figure 20. Sampling of bandwidth-limited signal at one-fourth the 

Nyquist rate. 

126 
-Bi Bi f S 

Cb) 

Figure 21. Sampling of bandwidth-limited signal at twice the Nyquist 

rate. 

127 
(a> G(f) 

I 

f 

S 

f 

Sl 

Figure 22. Sampling at different rates of signal whose spectrum has 

gradual rolloff. 

128 
x(n) w(n) L 

, H 2-3 r(n) 

H(z) = 

291 + z-y- 

(1 - z -I)(1 - Bz-I) 

B = 0.1112 = 0.87510 

I 1 

, 

I K+\- 

Figure 23. The first second-order digital estimating circuit investigated. 
w(n) 

> 2-2 . 

r(n) 

+ 

1’ 

2-l 

z-1 

A 

2-2 

H(z) = 

2-Q + z-92 

(1 - z-l)(l - Bz-1) 

B = 0.11012 = 0.812Slo 

Figure 24. Mode A of the second second-order digital estimating circuit investigated. 
II(z) = 

2T’O(l l z-l)2 

(1 - z -‘)(l - lw’) 

B = 0.11111112 = 0.95312510 

Figure 25. Mode B of the second second-order digital estimating circuit investigated. 
MAGNITUDE 

Figure 26. Magnitude response of integration algorithm with Dicke 

frequency equal to the folding frequency. 

132 
DIGITAL LOOP FILTER 

w(n) I y(n) 

= 2-2 4 

Figure 27. Proposed first-order control loop filter. 

133 
FROM 

DICKE 

SWITCH 

OUTPUT 

VARIABLE 

GAJN 

AMPLIFIER 

ALIASING h/D DIGITAL FILTER POST- LOOP 

=- FILTER - CONVERTER 0.25(1 + z-11 w SUM AND DUMP 

Ii(z) = PROCESSING 1 _ z-1 

/ 

598.14 t A f L CLOCK 

Hz 

2.45 MHz 

4.9 MHZ 

299.0’ HZ ,)I,r*“E 

I- . OUTrUT OUTPUT 

TO DIVIDE 

UICKE - BY 4 BY v< 

SWITCH 2 

CONTROL 2’2 

NOISE 

IN.JECTION - 

ATTEN -’ 

t , 

NOISE PIN 

DIODE - D1oDE 

STATUS wwN 

- 

I 

SWITCH COUNTER M 

Figure 28. Proposed loop estimation and feedback system. 
C(TI + TA * TR> 

I C (TA + TR) 

- t1 -4 

-TT 

,-----2T- :, -C(TB + TR) 4 

1 t 

5 

Figure 29. Noise-injection cycle in proposed design. 

135 
308 

30.492 to 792 

0.1 

i, 

304,920 to 7,YN 

OU'I'Y CYCLE 

0.835 to 0.0217 

l'Fx = 365,000 K 

Figure 30. Ranges of temperatures and duty cycle of proposed design. 
I 

(a) 

TB - TA 

~ 4.61 ) 0.1 _ 0.25(1 + z-l) 

x 10-S (1 - z-l) 

TB - TA 

Figure 31. Control model for discrete-time loop and its approximate 

continuous-time equivalent model. 

137 
RADOME 

P 

ANTENNA 

i ,, 

1 GHz # 

LATCHING 

CIRCULATOR AMPLIFIER MIXER AMPLIFIER HOT CARRIER DIODE 

DIRECTIONAL 

COUPLER 

POLARIZER LO 

VIDEO AMPLIFIER FILTERS 

AMPLIFIER DATA 

ALIASINC 

FILTER 

CONTROL 

Figure 32. Block diagram of proposed system. 
TliE ACTUAL LOOP OUTPUT y(n) CHANCES EVERY 1.6718 ms. 

HOWEVER, ALL yN(i) ESTIMTES CHANGE EVERY 26.75 111s. I 

-I 'rO L- 

= 26.75 ms 

-I 

I- 2To = 53.5 ,ns -4 

4To = 107 ms --- 

8To = 214 ms 

I I I I 

16To = 428 ms * 

I I 

L-- 

I I I I I 

32”b = 856 ms -----A 

-III 

64To = 1.712 s -- --l 

Figure 33. Sum-and-dump algorithms proposed for post-loop processor. 

ESTIMATE OF POINTS 

YI(~) 

h(i) 

Y4 (i) 4 

ye(i) 8 

Y16(i) 16 

Y32Ci) 32 

Y64(i) 64 
POST-LOOP FILTER 

+&cl + = 0.5 

Y2 Y4 Y8 Y16 Y32 1 

ouwisr 

* DATA 

‘64 

Figure 34. Digital filter format for sum-and-dump algorithms. 
37.38 Hz 18.69 Hz 9.35 Hz 

l+z -1 

- 2 

4.67 Hz 

26.75 ms 53.5 ms 107 ms 

Y16 Y32 Y64 

214 ms 428 ms 856 ms 1.71 s 

1 + z- 1 L 1+2-l 1 + z-1 

2 2 2 

L 

2.336 Hz 1.168 Hz 0.584 Hz 

Figure 35. Sum-and-dump algorithms from a z-transform point of view. 
0 MEASURED VALUES 

El PREDICTED VALUES 

NUMBER OF SAMPLES IN POST-LOOP ALGORITHM 

Figure 36. Sensitivities of proposed design resulting from dynamic 

simulation (78.125-kHz bandwidth). 

142 
1. Report No. 2. Government Accession No. 

NASA CR-3327 I 

I 

4. Title and Subtitle 

PRELIMINARY DEVELOPMENT OF DIGITAL SIGNAL 

PROCESSING IN MICROWAVE RADIOMETERS 

7. Author(s) 

William D. Stanley 

9. Performing Organization Name and Address 

3. Recipient’s Catalog No. 

I I 

5. Report Oate 

September 1980 

6. Performing Organization Code 

I 8. Performing Orgamzation Report No. 

10. Work Unit No. 

Old Dominion University Research Foundation 

P. 0. Box 6369 11. Contract or Grant No. 

Norfolk, Virginia 23508 

13. Type of Report and Period Covered 

1 2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546 

-11 

1 

1 5. Supplementary Notes 

Langley Technical Monitor: William F. Croswell 

Appendix by John M. Jeffords 

1 6. Abstract . 

The contract covered by the report involved a number of closely related 

tasks including: the development of several control-loop and dynamic noise 

model computer programs for simulating microwave radiometer measurements; 

computer modeling of an existing stepped frequency radiometer in an effort to 

determine its optimum operational characteristics; investigation of the 

classical second-order analog control loop to determine its ability to 

reduce the estimation error in a microwave radiometer; investigation of 

several digital signal-processing unit designs; initiation of efforts to 

develop required hardware and software for implementation of the digital 

signal-processing unit; investigation of the general characteristics and 

peculiarities of digital processing noiselike microwave radiometer signals; 

and computer data reduction of results obtained from actual ice missions. 

17 ‘. Key Words [Suggested by Author(s)) 

Radiometer 

Signal processing 

Digital signal processing 

1: I. Security Classif. (of this report] 

Unclassified 

I 

18. Distribution Statement 

Unclassified - Unlimited 

Subject Category 33 

I 

20. Security Classif. (of this page) 

Unclassified 

21. No. of Pages 22. Price 

142 A07 

For sale by the National Technical Information Service, Springfield. Virginia 22161 

. ..^. . _ ____ 

NAbA-Langley, IYW