zero-crossing analysis of fm threshold

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 5 , MAY 1982 1249

Correspondence I .

Zero-Crossing Analysis of FM Threshold

SURESH V. KIBE

Absrract-Stumpers’ idea of “zero-crossing FM detection” has been extended to analyze the phenomenon of FM threshold. The physical insight into FM threshold. In the subthreshold region, the position and numtier of zero-crossings is used to derive various formulas for calculation of noise power due to clicks in the presence of narrow-band Gaussian noise forthrightly. A two-state Markov model of the zero-crossing model for “clicks” is given. Thus, this is an alternative approach to Rice’s “clicks” analysis and offers better physical insight into FM threshold. In the subthreshold region, the experimentally determined curves are better approximated by the zero-crossing model than by Rice’s model.

INTRODUCTION

In this paper the phenomenon of FM threshold is analyzed using the zero-crossing approach. Stumpers’ [ 11 basic idea of “zero-crossing FM detection” is extended so that noise in FM is looked upon as the change in position and number of the zero-crossings of an FM wave in the presence of narrow-band Gaussian noise or any interfering signal. The physical insight gained is exploited to derive various formulas for calculation of noise power due to “clicks” or “spikes” forthrigJtly. The positive spike is likened to an ‘‘extra’’ and a negative spike to a “missing” pair of zeros. A two-state Markov process model of the zero-crossing model for a “sRike” is also pre- sented. This approach offers a clue to the time duration, frequency, and nature of “spikes” which is helpful in the design of threshold extension demodulators (TED’S). In the subthreshold region the experimentally determined curves are better approximated by the zero-crossing model than by Rice’s [2] model. Thus, this is an alternative approach and offers better physical insight into FM threshold than Rice’s “clicks” analysis. Since zero-crossing FM demodulators are basically digital devices, it is possible to conceive of an all- digital threshold extension demodulator.

I . REVIEW OF PREVIOUS WORK

Stumpers has shown that there is a direct relationship between the instantaneous frequency and the rate of zero- crossings of an FM wave.

Rice’s [ 21 conjecture of FM threshold is shown in Fig. 1. R ( t ) is the resultan’t vector.

For simplicity, the carrier is assumed to be unmodulated and is’given by

V(t) = A , cos act. (1.1)

Paper approved by the Editor for Communication Theory of the IEEE Communications Society for publication after presentation at the National Telecommunications, Conference, Houston, TX, Decem- ber 1980. Manuscript received July 18, 1979; revised October 15, 1981.

The author is with the INSAT Programme Office, Department of Space, Government.of India, Bangalore 560 009, India.

(c)

Fig. 1. Rice’s conjecture of FM threshold. (a) Locus ofR(t) and O ( t ) to cause a negative spike between t 2 and t 2 + At. (b) A plot of e ( t ) for a c a p in which the endpoint of R(r) in (a) executes a rotation around the origin. (F) .A plot of de/dt as a function of time.

Narrow-band noise n( t ) is represented by

n(t) = ~ ( t ) cos act - y ( t ) sin act

= r ( t ) cos (act + cp,)

where

and

f,: intermediate frequency. (1.3)

~ ( t ) and y ( t ) are Gaussian random processes with zero mean and deviation r/B where q is spectral density of white noise and B is the bandwidth of the rectangular IF filter. The conditions for a positive spike to occur in an interval At is given in either of the following equivalent ways [ 3 1 , [4] :

. .~

x < - A , O < y < A y , y < O

A(Pn AY At+O At

lim __ =&, lim - = y . A’t-to At

The total number of spikes [ 31 occurring per second in the

0090-6778/82/0500-1249$00.75 0 1982 IEEE

1250 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 5 , MAY 1982

presence of carrier alone (no modulation) is given by

and in the presence of modulation, the average number of spikes per second increases by a factor given by

- 2Af S N = - exp ( -- y )

n

(1.6)

where

fM: cutoff frequency of the.rectangular baseband filter

at the output

A f: maximum frequency deviation of the carrier

A S 2 NM =qfM, S. I = - : input signal power

7 = r$ 2) : input signal-to-noise ratio.

Near and below threshold, 6- % N,; therefore, the total average number of spikes is x 2 6 N . Equation (1.6) is used to account for the noise due to spike in FM receivers.

In this analysis some questions remain unanswered. What is the bound on the time interval At in which a spike occurs? What is the' physical equivalent of . a positive and negative spike? It ,is assumed that t o ensure a spike output it is not actually necessary to observe a complete rotation of R(r), but it is adequate that there be guaranteed a. rotation of at least n radians. Why? All of these questions are answered in the next section. Also, Rice's vectorial approach predicts a number of clicks which do 'not exist. These false clicks [ 51 further complicate Rice's model in the. threshold region. Experimental literature on the 'nature -.of the clicks exists [ 61 but the amount of' information is not adequate. The inadequacy of ' Rice's model below threshold has been mentioned in the literature I41 ~

11. ZERO-CROSSING ANALYSIS OF FM THRESHOLD

The response' of a narrow-band filter to noise looks like a sinusoid in any l/fc seconds. The output n ( r ) is represented by (1.2). The probability.density functions of r (r ) , cp,(t), and q n ' ( t ) are given by

Lo r < O

and cpn(t) is uniformly distributed with the probability density function

1

2n f,,(Vn) = - -n < cp < 71.

The probability density function of the instantaneous frequency deviation from the center frequency fc is given by [71, [81

Fig. 2 gives the heuristic zero-crossing model for FM threshold. In Fig. 2:

1) When r < A , and the noise frequency is greater or less than the signal frequency, there is no change in the number of zero-crossings of the resultant signal [Fig. 2(a) and (b)].

2 ) When r > A , for on > os the resultant signal has an extra pair of zeros, and for on < os a pair of zeros is missing [Fig. 2(c) and (d)] .

The case when, the signal and noise have an instantaneous phase difference around n has been considered, as frequency jumps are encountered only then (see also [ 101 ). It can be shown graphically tliat an extra pair or zeros is generated when the instantaneous phase difference between the noise waveform and the signal is anywhere between

l ' n * S f I n - S f n+- - and n-- -

2 fc 2 fc

a sector of width n*6f/fc radians around 77 in the phaseplane, where Sf is the instantaneous frequency difference between the noise and signal. This condition corresponds to the noise half-wave lying anywhere within the zero-crossing of the signal half-wave zero-crossings [see Fig. 2(e)l.

If the case of on < w, is to be included, a spike results if the instantaneous phase difference is in any sector of width n-16f ]/fC around n. For phase differences other than n [Fig. 2(f)] the zero-crossings of the resultant are close to those of the stronger signal but the number of zero-crossings remains same. If a spike occurs in a particular (say, ith half-cycle) interval, no spike is possible in the next half-cycle interval, as the zero-crossings of the noise would fall outside the zero- crossings of the signal in that half-cycle interval. However, a spike may occur in the (i + 2)th interval as the statistics of noise are supposed t o be constant only for l/fc seconds.

Thus, when r > A , and when the instantaneous phase difference is in any sector of width 7 7 . 1 Sf I/fc around n, a pair of zeros is either missed ( -ue spike) or an extra pair of zeros (+ue spike) is generated. This is the onset of threshold. A missing pair of zeros corresponds to a phase change of -2n. An extra pair of zeros corresponds to a +2a phase change. The area under the spike is therefore always 277.

Since the noise envelope r (r ) is Rayleigh distributed, probability P A , that it is greater than A , (r > A,) is given by

P A = exp (-As2 /2W)


AMPLiTUDE AMPLITUDE

t f

r m u R LEVEL CROSSINGS

TWO LEVEL CROSSINGS1

Fig. 2. Zero-crossing model for FM threshold.

Therefore, the joint probability p that the condition for a spike is satisfied in any 1/2fc seconds (see Appendix A) is P = P q s ' P A s o r

1 n * 1 6 f l

277 fc P = - ' - exp (-TI.

Then, the number of spikes in one second is given by

or

N = I S f I exp (-7). (2.8)

The average number of spikes 6N is given by

lii = I Sr 1 exp (-TI. (2.9)

It can be shown (see Appendix B) that for sinusoidal modulation

I d p - 2 d 277 2 f i

1 +a2 cos2 (wt)d(wt) (2.10) 1 where

Af a = 2 a -

B (2.1 1)

Substituting (2.10) in (2.9) we get

d ( u t ) exp (---y). I (2.12)

Equation (2.12) involves an elliptic integral which can be computed only by numerical methods.

The average time between spikes is then

1

N Ts = =- (2.13)

Schilling and Billig (cited in [4, p. 1531) used (1.6) to account for noise due to spikes in FM receivers, and (So/No) in terms of the system parameters is given by (for sinusoidal modulation)

s 0 - (3/21P2 Y (B/fM)

N, 1 + ( 1 2 ~ / 7 7 ) ~ ( ~ / m e - 7 - -

where

fM = maximum modulating frequency or the bandwidth of

the rectangular baseband filter

B = rectangular I F bandwidth.

A sketch of So/No versus y is shown in Fig. 3. The experimentally determined points included are for

B - = 5 and p = 4 . (2.15) fM

With the parameters given in (2.15), we have

Af a = 2 & - - S 1 . 1

B

and

i2n d l + a2 sin2 ( u m t ) d ( u m t ) = 9.58131 = 5.636 a.

(2.16)

Substituting (2.16) and (2.1 5) in (2.1 2), we get

n

and

(2.17)


38 I

10 0

HPVT CARRIER TO NOlSE RATIO V d b - Fig. 3. Output So/No for FM discriminator.

Equation (2.18) plotted in Fig. 3 shows that the curve corre- sponding to the zero-crossing model is closer to the experimentally determined curve. The multiplication factor in the denominator of (2.1 8) will change if the system parameters are changed, as (2.16) will have to be computed afresh.

Equation (2.8) for the number of spikes per second is the same as that obtained by Schilling.

For Af = 0 (no modulation), (2.12) reduces to

and the output signal-to-noise ratio is given by

_ - SO (3 / 2 T(B/fM - No 1 -k 6 ’y(B/fM)2e-7

(2.19)

(2.20)

A comparison of (2.1 2) and (2.19) shows that the zero- crossing model also predicts enhanced increase in the number of spikes under modulation. However, this model predicts a much steeper fall in the output signal-to-noise ratio than that predicted by Rice’s model under no modulation [compare (2.19) with (1.6)].

111. DURATION OF A SPIKE

This analysis gives a definite clue to the time duration and nature of spike. The duration of the spike is roughly 1/2fn where f, is the instantaneous frequency of noise. Since E[f,] = f, and f, is expected to be within (f, 2 B/2) with 95 percent probability [ 71

95 percent of the time. The amount of frequency jump depends upon the instanta-

neous frequency difference and the amplitude ratio of signal and noise. This information is helpful in the design of threshold extension devices (TED’S) [3, pp. 348-35 11.

A zero-crossing FM demodulator with threshold extension

capability would therefore be an accurate zero-crossing counter which is capable of ignoring sudden changes in the zero-crossing count in the interval given by (3.1) (see Fig. 4).

The detection of a “missing” or “extra” pair of zeros is possible by a level detector used in conjunction with the detector as definite patterns of the resultant are observed during spike formation (Fig. 2).

APPENDIX A

TWO-STATE MARKOV MODEL FOR FM THRESHOLD

The probability of occurrence of a spike in any half-cycle interval (1 /2fc seconds) can be shown to approach

1 n 1 6 f l - .- exp (-TI = P f, using a two-state Markov process model.

Let the time axis be divided into n slots of length 2/f, seconds (half-cycle intervals). The following assumptions are made.

1) If a spike occurs in the ith time slot, the probability of occurrence of a spike in the (i + 1)th slot is zero, i.e., the Markov process goes from an ACTIVE state t o a REST state after the occurrence of a spike (Fig. 5). If no spike occurs in the ith slot the system stays in the ACTIVE state and a spike may or may not occur in the ( i + 1)th slot. (The fact that a spike does not occur in the (i + 1)th time slot if it has occurred in the ith slot has been explained earlier.)

2) The statistics of narrow-band noise are constant in any two consecutive time slots (one cycle interval of l/f, seconds), i.e., if a spike occurs in the ith time slot, no spike shall occur in the (i -k 1)th time slot but the spike may or may not occur in the (i + 2)th interval.

The probability of occurrence of a spike when the process is in the ACTIVE state is p . Here,

Pr (spike) = P , (ACTIVE + REST/ACTIVE)

X P , (ACTIVE)

P , (spike) = p X P , (ACTIVE).

Let the probability that the system was in the ACTIVE state when the process was started (0th time slot) be 4 .

Pro (ACTIVE) = 4

P,, (ACTIVE) = Pro (ACTIVE) X Pr (ACTIVE + ACTIVE)

+ Pro (REST) x 1

= 4(1 - P I + (1 - 4 )

= 1 - pP,o (ACTIVE).

In general

P,i (ACTIVE) = 1 - p 4- p 2 - * - a pi*

m

P,, (ACTIVE) = (--p)‘ = - i= 0 l + P

. I

1 P

l + p 1 + p .‘. P , (spike) = p X - - - - = p

for p < 1.


EXTRA/MISSING COUNT OETECTOR

I I

FULSES AT ZERO-CROSSINGS

“OWLAlED OUTPUT

V

FM HWT d ZERO C R O S W COUNTER * ACCURATE

DETEClOR

Fig. 4. Model of FM detector with threshold extension.

Fig. 5 . Two-state Markov model for zero-crossing model for a spike.

APPENDIX B

CALCULATION OF TIME AVERAGE OF 18 f I

The probability deqsity function of the instantaneous frequency deviation pn ( t ) of narrow-band Gaussian noise is given by

where B is the bandwidth of the rectangular IF filter. Let the frequency modulated carrier be represented by

where

A , = amplitude of the FM wave Af = maximum frequency deviation of the carrier f, = frequency of the modulating signal.

Then, the instantaneous frequency deviation fd of the FM wave away from the carrier is given by

Now, the instantaneous frequency separation between the FM wave and noise is given by

= pn‘(t) - Af COS O, t . 03.4)

Hereafter, Sf = y for ease of notation.

by The probability density function of the process y ( t ) is given

For the process l y ( t ) 1, it can be shown that the mean of the time average can be computed by first computing the ensemble average and then taking the time average of the ensemble average.

The ensemble average of y ( t ) is given by

After suitable trigonometric substitution and simplification we get

where

( A n 2 B2

u2 ,= 12 - . (B. 10)

The time average of the ensemble average is given by

(B. 11)

This

[ I1

P I

r31

r41

r51

r61

[71

[81

is an elliptic integral and can be computed numerically.

REFERENCES

F. L. H. M. Stumpers, “Theory of frequency modulation noise,” Proc. IRE, vol. 36, pp. 1081-1092, Sept. 1948. S. 0. Rice, “Noise in FM receivers,” in Time Series Analysis, M. Rosenblatt, Ed. New York: Wiley, 1963, ch. 25, pp. 375-424. D. L. Schilling and H. Taub, Principles of Communications Sysrems. New York: McGraw-Hill, 1971, ch. 10, pp. 320-361. M. Schwartz, W . R. Bennett, and S . Stein, Communicufion Sysrems and Techniques. New York: McGraw-Hill, 1966, ch. 3. D. Yavuz and D. T. Hess, “False clicks in FM detection,” IEEE Truns. Commun., vol. COM-18, Dec. 1970. D. Yavuz, “FM click shapes,” IEEE Truns. Commun., vol. COM- 19, Dec. 1971. E. D. Sunde, Communications Systems Engineering Theory. New York: Wiley, 1969, ch. 3. S . 0. Rice, “Statistical properties of a sine-wave plus noise,” Bell Syst. Tech. J . , vol. 27, pp. 109-157, Jan. 1948.


[9] A. Papoulis, Probubility, Random Variables and Stochastic Processes. New York: McGraw-Hill, ,1965, ch. 10.

IO] K. Leentvaar and J . H. Flint, “The capture effect in FM receivers,” IEEE Truns. Commun., vol. COM-24, pp. 531-539, May 1976.

Adaptivity Versus Tree Searching in DPCM

+I

t

- 2.58 a* =-1.222

aa = ,301

PARTIAL CODE TREE

(a) H. C. CHAN AND J. 3. ANDERSON

Input X,-,

Abstract-The interaction is studied between the effects of tree searching and predictor and quantizer adaptation in a differential pulse-code modulation (DPCM) speech digitzer. A 2 s male sentance is digitized at 1 and 2 bits/sample. The maximum entropy method computes a new source model every 8 ms, and a variety of strategies are used to adapt the quantizer. Results are displayed in plots of local SNR versus time. It is found that the predictor, the quantizer, and the tree search each contribute roughly independently to the total SNR. Adaptivity contributes in total 6-7 dB and tree searching 2-3 dB. Local gains of both types can be higher.

I . INTRODUCTION

In recent years the technique of code tree searching has attracted wide interest as a method of digitizing pictures and speech. Consider a code book of words { k N } in which each iN = 11, 2 2 , - a , G N is a sequence of reproducer outputs by which we wish to approximate the waveform samples xl, x2, - * a , X N . The reproducer letters are generated in such a way that their interrelation satisfies a tree structure. Virtually all practical waveform digitizers produce code words related in this way; one example, differential pulse-code modulation (DPCM), appears in Fig. 1. A tree searching encoder traces several code word “paths” in parallel, releasing after a delay the best path in terms of a fidelity measure like mean square error. An alternate term for the tree encoder is the delayed encoder.

The original studies [ I ] of tree coding made use of DPCM- like codes generated by a least mean square (LMS) predictor structure like that of Fig. 1. This structure was nonadaptive. A moderate degree of tree searching yielded about 3 dB SNR improvement over ordinary DPCM, which executes only a “single path” search along one code tree path. In more recent work, Uddenfeldt [2 ] placed the design of these nonadaptive codes on a more analytical casis. Jayant and Christensen [ 3 ] studied the case of a fixed predictor and a “Jayant” adaptive quantizer, finding 2-3 dB gain from the addition of tree searching. Wilson and Husain [ 4 ] published an early study of

Paper approved by the Editor for Data Communication Systems of the IEEE Communications Society for publication after presentation at the National Telecommunications Conference, New Orleans, LA, December 1981. Manuscript received January 22, 1981; revised July 1, 1981. This work was supported in part by the Natural Sciences and En- gineering Research Council of Canada under Strategic Grant G0359.

H. C. Chan is with the Communications Research Centre, Govern- ment of Canada, Ottawa, Ont., Canada K2H 8S2.

J. B. Anderson is with the Department of Electrical and Systems

Troy, NY 12181. Engineering, School of Engineering, Rensselaer Polytechnic Institute,

X” I 0”

I I

D P C M E N C O D E R

(b) Fig. 1. Adaptive DPCM encoder block diagram, with nonadaptive 1-bit

code tree; the 2 bit code tree has four branches out of each node. For forward adaptation the “adaptation input” is x n ; for backward adaptation it is 2,.

the adaptive predictor/adaptive quantizer case; their adaptivity was based on solution of the Yule-Walker equation and tree searching yielded an additional 2-3 dB gain. All of these papers dealt with actual speech.

Throughout all of this work, it has not been clear whether code tree searching is displacing model adaptation as a means of increasing the SNR, or whether it is an independent source of gain. One can ask as well whether searching provides an alternative to amplitude adaptation. Were tree searching merely an alternative, the decision to use it would depend on whether it were cheaper than adaptation at the same SNR. Otherwise, the decision would be more complex. In this corre- pondence we show that tree searching and the various forms of adaptivity contribute roughly independently to the SNR.

We perform a set of experiments on the speech sentence “speed and efficiency were stressed,” digitized at the rates 1 and 2 bits per speech sample. The SNR is in terms of mean square error without frequency weighting; such a measure of fidelity is not ideal for speech, but it gives a needed simplification to the interpretation of the experiments and it allows for comparison to some existing theoretical results. To sepa- rate out the various origins of the SNR gain, we pose the following questions, with reference t o Fig. 1. How much gain is obtainable from a predictor and what must its order be? How much gain originates from the quantizer alone, that is, from adaptation to amplitude variations? How do both of these vary on a local basis? Do either of these sources of gain tend t o displace the tree search gain, or is this gain independent of adaptation?

0090-6778/82/0500-1254$00.75 0 1982 IEEE

zero-crossing analysis of fm threshold

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