Reference

Pout'' I

NBSIR 79-1614

EVALUATION OF SIGNAL PLUS NOISE DETECTION ERROR IN

AN ENVELOPE DETECTOR WITH LOGARITHMIC COMPRESSION

A. J. Estin

W. C. Daywitt

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

-QC

100

.056

79-1G1h

1979

August 1979

Barun

JAN 3 1 1980

r NBSIR 79-1614

i

« *

EVALUATION OF SIGNAL PLUS NOISE DETECTION ERROR IN

AN ENVELOPE DETECTOR WITH LOGARITHMIC COMPRESSION

A. J. Estin

W. C. Daywitt

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

August 1979

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Luther H. Flodges, Jr., Under Secretary

Jordan J. Baruch, Assistant Secretary for Science anr< Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

1. INTRODUCTION 1 1.1 Background 1 1.2 Review of Random Processes 1 1.3 The Envelope Function 2

2. FORMULATION OF THE PROBLEM 4

2.1 The Basic PDF 4

2.2 Analysis - 6

3. EXPERIMENTAL RESULTS 12

4. USING THE CORRECTION FACTOR 12

5. ACKNOWLEDGMENT AND NOTE 14

6. REFERENCES 15

6.1 General Reference 15

iii

EVALUATION OF SIGNAL PLUS NOISE DETECTION ERROR

IN AN ENVELOPE DETECTOR WITH LOGARITHMIC COMPRESSION*

A, J. Estin and W. C. Daywitt

A correction factor is derived for the detected output of a modulated sinusoidal signal with added Gaussian noise, as processed

by an envelope detector with logarithmic compression which is

calibrated for a sinusoid. Supporting experimental data are presented which were obtained using a typical commercial system

having such a detector.

Key words: Detection; detection amplitude error; noisy signal

detection.

1. INTRODUCTION

1.1 Background

Signal detection techniques in various measurement systems serve to

convert a signal into a relatively slowly varying voltage which is

intelligible and useful. One price paid in this process is that no one signal

detection technique will so convert all different signal waveforms without

introducing some intrinsic amount of error, possibly major. For example, a root-mean-square detector will accurately represent power of the signal; but

if peak amplitude of a short rectangular pulse of low duty cycle is desired, a correction factor must be applied to remove the error, this factor being a function of both the signal characteristics and the detector properties.

One very common and useful detection system is that which envelope

detects a received signal and then logarithmically compresses the result.

Such a system is fast, stable, economical, sensitive, has wide frequency and

dynamic ranges, and provides a convenient output expressed in decibels. It is

usually calibrated for a sinusoidal waveform and is used in a large number of

commercially available spectrum analyzers and frequency-selective voltmeters.

In this report, we will analyze the detection error in this type of

detector for the case of a modulated sinusoid superimposed on Gaussian noise.

1.2 Review of Random Processes

Signals representing time-varying physical phenomena are usually

classified as "deterministic" when they can be described by an explicit

mathematical function of time, and "nondetermi ni Stic" or "random" when such an exact description is not possible. Even in the latter case, however, certain

characterizations are possible based upon the facts that a range of values

that the variable can assume may be known, and the probability of it having

each value of this range may be determined. If we are fortunate enough to

* Work partially supported by the U.S. Army Satellite Communications Agency.
have this information, many useful predictions can be made about such random

variables. A function describing this probability distribution for a continuous random variable ("continuous" in range, not in sequence of values)

is the "Probability Density Function," or PDF, which we designate as P^(x).

P^(x)dx is a real non-negative function whose value is the relative

probability of the event that ^ will lie in the interval (x,x+dx). It follows

that the probability that C falls in the interval (x]^,X 2 ) is

with the total probability being normalized to unity. The PDF can be

related to various observable quantities by means of integrals with

appropriate weighting functions over its range. If the weighting function is

chosen as x"^, the observable is the nth moment of the PDF:

where the angular bracket denotes the expected value of the argument.

In particular, the first moment represents its average value and if ^ is

a voltage across a unit resistance, the second moment represents the power

dissipated in the resistor. The variance of C is

and 0

^ is the standard deviation, which is a measure of the dispersion about

the mean.

The concept of the envelope of a voltage waveform is intuitively

understood to be that function which connects the successive positive (or

negative) peaks of, for example, a sinusoid that varies slowly in amplitude

and/or phase. For any but the simplest waveforms, precise application of this

definition encounters considerable difficulties. If the Fourier transform of

the time series representing the waveform contains significant components

far above and below the midband frequency, the envelope definition becomes

subject to the exact choice of mid-frequency. If several uncorrelated time

series were combined, the non-linear RMS process becomes difficult to

00

( 1 )

( 2 )

1.3 The Envelope Function

2
calculate, and in particular if these time series were random, the result

would rapidly become uncalcul abl e.

A rigorous generalization of the envelope function has been obtained in

terms of Hilbert transforms [1]. If h(t) is a real time series, its

corresponding pre-envelope (or pre-detection) function z(t) is given by

z(t) = h(t) + jh(t) (3)

where h(t) is the Hilbert transform of h(t). The envelope is defined as the

absolute value of z(t):

z(t) I = Vz(t)z*(t) (4)

where the asterisk denotes the complex conjugate. It will be useful to

show, at this time, that the power in a time series which is

Fourier-expandable is related to the envelope function by:

P(t) (5)

If s(t) is a voltage time series, which can be expressed as a Fourier

series, then

s(t) =y a cos (jj t + b sin oo t . (6)

n n n n

n

In order to obtain a power that is "instantaneous" with respect to the

envelope variations, we shall integrate over a time AT that is long compared

with any rf period ZTi/wp, but short compared with any modulation period

27T/((jL)p - (%). (This constrains us to the case when the signal bandwidth

is small relative to its rf components.)

3
Hence,

or

P(T) . ^

s'(t)dt

P(T) = ~Y1H ua a + b b ) cos (o) - w ,)T + 2nrn [nm nm n m

(a b - a b ) sin (oo - co )T m n n m n m

( 7 )

The Hilbert transforms of the sine and cosine functions are: [2]

^ ( 8 )

g^(t) = sin (jot, g^(t) = cos wt

g (t) = cos (jot, g (t) = -sin wt

2 c J

From eqs (3), (6), and (8),

z(t) = X! + Jb^)e

n

-j03 t

n

It follows that

I z(t)2 |/2 is equal to the right side of eq (7)

2. FORMULATION OF THE PROBLEM

2.1 The Basic PDF

The PDF for the envelope of Gaussian noise added to a signal composed of

a sum of sinusoids is given by the Rice-Nakagami relation [3],

P^ (x)

A

x>0 (9)

= 0 , x<0

4
where: z(t) is the signal function,

jpQ is the mean-squared value of the Gaussian noise power,

and Iq is the modified Bessel function of the first kind and

order zero.

Note that | z(t) | is the envelope function of the signal in the absence of

noise. (For convenience, we are henceforth dropping the subscripts used in

eq (1) and replacing z(t) by z.) It is helpful to examine the limiting cases

of eq (9) for high and low si gnal -to-noi se (S/N) power ratios.

Since Io(0) = the low S/N limit is:

1 i m P ( X

)

z->0

This is the familiar Rayleigh distribution for band-limited noise.

Using the asymptotic form of Io(u) for very large argument.

and with the definition q z 1/(2\Pq), the limit for high S/N ratio is:

lim P(x) = lim

/ |^'i exp/-q(x -

[ z |)‘ qx» qx3o v I I

This is recognizable as the Dirac delta function [4]. Hence,

1 im P(x) = /r^ 6(x -

I z |) ip^O V I ^ I 0

and as might be anticipated, in the

distribution is concentrated at the

absence of noise, the probability

signal envelope.

5
2.2 Analysis

We will next develop a rigorous expression for the si gnal -to-noi se power

ratio. The white noise power delivered into the detector through a filter of

bandpass characteristic H(f) is

( 10 )

where Wq is the noise power density (watts/Hertz ) and B is the equivalent

noise bandwidth.

If the modulated carrier is assumed to be of the form of eq (6), in which

the sidebands lie within a constant gain portion of H(f), the power in the

carrier is given by eq (5). Thus, the si gnal -to-noise power ratio is

( 11 )

Letting x represent the random variable of the detected signal, i.e., the

envelope variable, then the logarithmically compressed output is given by

y

= In X . (12)

The average value of this output is its first moment, which from eq (1) is

<y> L

yP(y)dy (13)

Since y has a one-to-one correspondence with x, as shown in eq (12), its

PDF can be obtained [5] from the PDF of x as

P(y)dy = P(x)dx . ( 14 )

Substituting eqs (12) and (14) into eq (13), and reducing the limits to

correspond to the range of y.

6
dx . (15)

no

Combining eqs (9) and (11) into eq (15), and defining a new variable of

integration w = x2/(2i^o)s we have an expression for the average value of

the output of the detected, logarithmically compressed voltage;

<y> I^(2/W)e '^dw

0

In order to eliminate 'Pq from this expression, we shall evaluate the

power before logarithmic compression. Again, using eq (1),

0 0 \ 0 / ' o'

This can be integrated directly in terms of the, confl uent hypergeometric

function [6] as

<x > = Z'P 1^1 - 721' ).

0

which reduces to a finite two-term series:

<x^> = 2(ip (18)

Alternatively, eq (18) could have been obtained directly by appealing to the

following theorem [7]: "The ensemble average of the square of the envelope

function is equal to twice the ensemble average of the square of the original

time series."

Solving eq (18) for 24>q, and with eq (11),

1 + r

* ( 19 )

7
Substituting eq (19) into eq (16), and solving for the power in the detected

envelope.

2 <x > = ex

r

2<y> - e'*"!. PV

“'"t e I. V

•N

+ ln(l + r)V (20)

where Ij and I 2 are the following integrals:

00

The first integral can be easily evaluated in closed form. First, we expand

the Bessel function in its pov/er series (Dwight 813.1)* as

oo = L

n=0

(23)

Substituting this into eq (21), interchanging the order of summation and

integration, and removing the quantities independent of the integration, we find that the integral reduces to the gamma function, r(n+l), which, for

integral n becomes nl. The remaining series is identified as the

exponential e>^, giving

e I. = 1 . (24)

The second integral, eq (22), is somewhat more cumbersome. Again, we substitute eq (23) for the Bessel function and interchange the order of

* Dwight, H. B., Tables of Integrals and Other Mathematical Data (MacMillan,

N.Y., 1957).

8
integration and summation, obtaining

(n!) •'o

We designate this integral as l 2 (n); using u = e~^ and

dv = 1n w dw, this can be integrated by parts (Dwight 610.9) to produce a recursion relation in n.

I 2

(n + 1) = (n + l)l

2

(n) + n^, n>:0 .

Since l2(0) = -y (Dwight 852.1), where y is Euler's Constant = 0.57721 56649..., I 2 (n) can be obtained explicitly as

= n'.[-y + H(n)] (26)

where H(n) is the harmonic function

H(n) = 1 + 1/2 + ... + 1/n, and H(0) = 0 .

Substituting eq (26) into eq (25), the factorial cancels one of those in the

denominator, and we have for I 2

:

00 n

I 2 = -ye'' + Ijj ^H(n), H(0) = 0 . (27)

Substituting eqs (24) and (27) into eq (20), and taking the natural

logarithm of both sides, we finally arrive at the following expression:

(28)

(29a)

where

1 n<x > = 2<y> + [y + ln(l + r) - c(r)]

w I I

Each term of eq (28) is in nepers. The left-hand side represents the true

power in the detected waveform. The first term on the right-hand side is the

power indicated by the detector, and the term in square brackets is the

9
desired correction term which depends only upon the si gnal -to-noi se power

ratio.

Unfortunately, the series in eq (29a) is slow and cumbersome to evaluate.

We can, however, convert it to a much more tractable form in the following

way. Let us differentiate s(r) with respect to r (interchanging the order of

summation and differentiation), and collect terms. Again, identifying the

power series for the exponential, this has a simple form:

dc(r) _1 - e dr r

To regain ?(r), we integrate from a to r, with a to be determined as a constant of integration:

dr . (29b)

In order that ?(0) = 0, as can be seen from eq (29a), it follows that a ^-0. By first expanding e"*^ in a power series, then integrating term-by-term,

and finally letting a ^0, we arrive at a much more convenient form for <;(r):

C

n+1 n (- 1 ) nn| (29c)

This series is absolutely and uniformly convergent for finite r, and

furthermore may be truncated with any desired maximum error because it is

alternating. Computation for very large r, however, may still prove

difficult. The magnitude of the terms of the series will increase from r (for

the first term) to the order of e'^r’^/^/y^ (for the (r-2)"^*^ or

(r-l)"*^!^ term) and only thereafter converge. Inasmuch as the computation

process requires subtraction of successive terms, this could demand retention

of an enormous number of significant figures for only modest accuracy in the

result. To circumvent this difficulty, we return to eqs (29b) and (28). The

correction term in eq (28), which we designate

F(r) = y + 1 n(l + r) - ^ (r) (30)

can also be written as

10
F(r) = - Ej(r) (31)

where E]^(r) is the real limit of the exponential integral (MS 5.1.11)*.

An asymptotic expansion of eq (31) for large r is easily obtained (A&S 5.1.51,

Dwight 601) as

...]

or

r>>l. (32a)

For small r, we use the series expansion of ln(l+r). Combining with the

terms of eq (29c) into eq (30):

F(r) = y • • • > r<<l. (32b)

Values of F(r) over the midrange of r are readily evaluated from the

exact solution

F(r) = Y

+ 1 n(l + r)

(-1) r nnl (32c)

For convenience, these expressions for F(r) are collected following:

* Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions (NBS

Applied Math Series No. 55, 1964).

11
F (r ) ~ r>>l

2 c; 2

F ( r ) = y

- — + ^ ^ ^ 4 18

r«l (32a ,b,c)

00 ^ F(r) = y + ln(l + r) - Y' ^

r< °° n=l

where y= Euler's Constant = 0.57721 56649..., and F(r) is in nepers

express this correction in decibels,

To

F^g(r) = (lOlog^Qe)F(r) (33)

and as a mil ti pi i cati ve ratio correction.

e(r) = e F(r)

(34)

Alternative computational forms are available using polynomial expansions

for the exponential integral form of eq (31). (A&S 5.1.53-5.1.56)

3. EXPERIMENTAL RESULTS

The commercial automatic spectrum analyzers are typical instruments using

the detection system analyzed in this paper. Response data were obtained with

one of the most advanced such instruments, using as input a variable noise source (solid state diode followed by a calibrated attenuator) and a nominally

calibrated signal generator. These data are shown in figure 1. The solid

line calculated curve was obtained from eq (32a) for r>10, eq (32b) for r<0.1,

and eq (32c) for the intermediate range 0.1<r<10.

4. USING THE CORRECTION FACTOR

It is apparent that making the correction described in this paper

presupposes a knowledge of the signal and noise levels being measured. Most

frequently this is not the case; and, in fact, finding the si gnal -to-noi se ratio is usually the desired result.

This can be accomplished by a rapidly convergent iterative calculation if an additional measurement can be made of the noise background in the absence

12
CORRECTION

FACTOR

[£(r)]

SIGNAL/NOISE POWER RATIO (r)

Fig. 1. Signal/noise power ratio (r)

13
of signal. Consider the expression for the observed level of a signal

consisting of signal power S and noise power N. From eqs (28) and (34), we

have

S + N = e(S/N)

N = P^e(O)

(35a, b)

where the P's are the relative observed levels. Subtracting and dividing, a transcendental relation in S/N (or r) results.

(S+N) - N

N

P c(r) - P e(0)

S+N

^ ^

Pm£(0)

(36)

Because e(r) is a fairly slowly varying function of r, a simple iterative

calculation will normally converge to r within two or three iterations.

Other reference combinations of signal and noise can be used, but the one described above is probably the simplest and most widely useful. It is

apparent that using the system noise level as a reference serves two

additional purposes: First, it enables an absolute level calibration to be

obtained if a standard noise source is available. Second, transference to the

condition when S and N are of comparable level places the least stringent

requirements upon system linearity.

5. ACKNOWLEDGMENT AND NOTE

The authors wish to acknowledge the helpful suggestion of P. L. Bender of

the Joint Institute of Laboratory Astrophysics that enabled proceeding from

eq (29a) to eq (29c). We are indebted to R. J. Matheson of the National

Telecommunication and Information Agency's Institute for Telecommunication

Sciences for the use of his Automatic Spectrum Analyzer and for his help in

setting up and conducting the experiment.

14
6. REFERENCES

[1] Dugundji, J., Envelopes and Pre-Envelopes of Real Waveforms, IRE Trans.

PGIT IT-4 , No. 1, 53 (March 1958).

[2] Erdelye, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Tables of

Integral Transforms, Vol . II, 252-253 (McGraw-Hill Book Co., Inc.,

New York , N.Y. , 1954)

.

[3] Deutsch, R., Nonlinear Transformations of Random Processes, 10

(Prentice-Hall, Inc., N.J., 1962).

[4] Lighthill, M. J., Introduction to Fourier Analysis and Generalised

Functions, 17 (Cambridge Press, England, 1962).

[5] Panter, P. F., Modulation, Noise, and Spectral Analysis, 114 (McGraw-Hill

Book Co., Inc., New York, N.Y., 1965).

[6] Watson, G. N., Theory of Bessel Functions, 394 (Cambridge Press, England,

1962).

[7] Deutsch (1962) 7.

6.1 General Reference

Magrab, E. B. and Blomquist, D. S., The Measurement of Time-Varying

Phenomena (Wi 1 ey-Interscience, N.Y., 1971).

15
NBS-114A (REV. 9-7SI

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 79-1614

3. R^clptent's Accession No,

4. TITLE AND SUBTITLE

Evaluation of Signal Plus Noise Detection Error in an Envelope Detector with Logarithmic Compression

5. Publication Date August 1979

Peiformln® Organizalion Cotie

‘\

7. AUTHOR(S)

A.J. Estin and W.C. Daywitt

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS ('Srreer, City, St^te, ZIP) Sponsored in part by: U.S. Satellite Communications Agency

Fort Monmouth, New Jersey

13. Type of Report & Period Covered

15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual aumtnary of most significant information. If document includes a significant bibliography or

literature survey, mention it here.)

A correction factor is derived for the detected output of a modulated

sinusoidal signal with added Gaussian noise, as processed by an envelope detector

with logarithmic compression which is calibrated for a sinusoid. Supporting ex- perimental data are presented which were obtained using a typical commercial

system having such a detector.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name

separated by semicolons)

Detection; detection amplitude error; noisy signal detection

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