Signal Detection and Extraction by Cepstrum Techniques

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-18, NO. 6, NOVEMBER 1972 745

R,(t,s). Definition (Al) also has meaning when x(.) has covar iance R,(t,s) + K(t,s) because the probability measures associated with zero-mean Gaussian processes with covar iances R,(t,s) and R,(t,s) + K(t,s) are equivalent ‘(mutually absolutely continuous): almost sure propert ies are preserved under absolutely cont inuous changes of measure.

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Inform. Theory, vol. IT-16, pp. 276-288, May 1970. [3] T. Kailath and R. A. Geesey, “An innovations approach to least

squares estimation-Part IV: Recursive estimation given lumped covar iance functions,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 720-727, Dec. 1971.

[4] I. Gohberg and M. Krein, “Theory and applications of Volterra operators in Hilbert space,” Transl. Math. Mono., vol. 24, Amer. Math. Sot:, Providence, R.I., 1970.

[5] D. Duttweller, “Reproducing kernel Hilbert space techniques for detection and estimation problems,” Ph.D. dissertation, Dep. Elec. Eng., Stanford Univ., Stanford, Calif., June 1970.

[6] T. Kailath, “RKHS approach to detection and estimation problems-Part 1: Deterministic signals in Gaussian noise,” IEEE Trans. Inform. Theory, vol. 17, pp. 530-549, Sept. 1971.

[7] W. A. Porter, ‘Some circuit theory concepts revisited,” Znt. J. Contr., vol. 12, pp. 433-448, 1970.

[8] -, “A basic optimization problem in linear systems,” Math. Syst. Theory, vol. 5, pp. 2044, 1971.

[9] R. Saeks, “Causality in Hilbert space,” SIAM Rev., vol. 12, pp. 357-383, 1970.

[lo] -, “State in Hilbert space,” SIAM Rev., to be publ ished; also Univ. Notre Dame. Notre Dame. Ind.. Tech. Memo. EE 6912a. , , 1969.

[l l] J. C. W illems, “Stability, instability, invertibility and causality,” SIAM J. Contr., vol. 7, pp. 645-671, 1969.

[12] J. C. W illems, “The generat ion of Lyapunov ftinctions for input- output stable systems,” SIAM J. Contr., vol. 9, pp. 105-134, 1971.

[13] G. Kall ianpur and H. Oodaira, “The equivalence and singularity of Gaussian measures,” in Proc. Symp. Time Series Analysis, M. Rosenblatt, Ed. New York: W iley, 1963.

[14] N. Aronszajn, “Theory of reproducing kernels,” Trans. Amer. Math. SOL, vol. 63, pp. 337-404, May 1950.

[15] F. Riesz and B. S. Nagy, Functional Analysis. New York: Ungar, 1955.

[16] T. Kailath and R. Geesey, “Covar iance factorization-An ex- plication via examples,” in Proc. 2nd Asilomar Co& Systems and Circuits, Monterey, Calif., Nov. 1968.

[17] T. Kailath, “Fredholm resolvents, W iener-Hopf equations, and Riccati differential equations,” IEEE Trans. Inform. Theory, vol. IT-15, pp. 665-672, Nov. 1969.

[18] W. L. Root, “Singular Gaussian measures in detection theory,” in Proc. Symp. Time Series Analysis, M. Rosenblatt, Ed. New York: W iiey,. 1963.

1191 J. L. Doob. Stochastic Processes. New York: W iley. 1953. (2Oj L. A. Shepp, “Radon-Nikodym derivatives of G&Sian mea-

sures,” Ann. Math. Statist., vol. 37, pp. 321-354, 1966. [21] I. Gohberg and M. G. Krein, “On the factorization of operators

in Hilbert space,” Acta Sci. Math. (in Hungarian), vol. 25, pp. 90-123, 1964; also translated in Amer. Math. Sot. Trans., vol. 51, 1966.

[22] I. M. Gelfand and N. Ya. Vilenkin, General ized Functions, vol. 4. New York: Academic Press, 1964.

[23] D. E. Varberg, “On equivalence of Gaussian measures,” Pac. J. Math.., vol. 11, pp. 751-762, 1961.

[24] J. HaJek, “Linear statistical problems,” Czech. Math. J., vol. 87, pp. 404-444, Dec. 1962.

[25] M. Hitsuda, “Representat ion of Gaussian processes equivalent to W iener process,” Osaka J. Math., vol. 5, pp. 299-312, Dec. 1968.

[26] C. DolBans-Dade, “Quelques applications de la formule de changement de variables pour les semimartingales,” Z. Wahr. verw. Geb., vol. 16, pp. 181-194, 1970.

[27] G. Kall ianpur and H. Oodaira, “Nonanticipative representat ions of equivalent Gaussian processes,” submitted for publication.

Signal Detection and Extraction by Cepstrum Techniques

R. C. KEMERAIT, MEMBER, IEEE, AND D. G. CHILDERS, SENIOR MEMBER, IEEE

Abstract-A technique for decomposing a composite signal of unknown multiple wavelets overlapping in time is described. The computation algorithm incorporates the power cepstrum and complex cepstrum techniques. It has been found that the power cepstrum is most efficient in recognizing wavelet arrival t imes and amplitudes while the complex cepstrum is invaluable in estimating the form of the basic wavelet and its echoes, even if the latter are distorted.

Digital data-processing problems such as the detection of multiple echoes, various methods of linear filtering the complex cepstrum, the picket-fence phenomenon, min imum-maximum phase situations, and amplitude- versus phase-smoothing for the additive-noise case are examined empirically and where possible theoretically, and are discussed.

Manuscript received November 15, 1971; revised April 26, 1972. This work was supported in part by the Office of Naval Research, Contract N00014-68-A-0173-0014. Task NR 042-278 and by the National Eye Institute under the b.S. Public Health Services Grants EY00581 and EY00077.

R. C. Kemerait is with the Department of Electrical Engineering, Florida Institute of Technology, Melbourne, Fla.

D. G. Childers is with the Department of Electrical Engineering, University of Florida, Gainesville, Fla. 32601.

A similar investigation is performed for some of the preceding problems when the echo or echoes are distorted versions of the wavelet, thereby giving some insight into the complex problem of separating a composite signal composed of several additive stochastic processes. The threshold results are still empirical and the results should be extended to multi- dimensional data.

Applications are the decomposit ion or resolution of signals (e.g., echoes) in radar and sonar, seismology, speech, brain waves, and neuro- electric spike data. Examples of results are presented for decomposit ion in the absence and presence of noise for specified signals. Results are tendered for the decomposit ion of pulse-type data appropriate to many systems and for the decomposit ion of brain waves evoked by visual stimulation.

I. INTRODUCTION

A. Motivation and Applications

w E HAVE formulated the basic problem in the following way. Given a signal that is a finite summation of

a basic wavelet and its echoes (distorted or undistorted wavelets), determine the waveshape of the wavelet and/or

146 IEEE TRANSACTIONS ON INFORMATION THEORY, NOVEMBER 1972

echoes, the number of wavelets present, their amplitudes, and their times of occurrence [l], [2]. This problem has been investigated for three types of signals. Two are felt to be representative of multiple echoes in a noisy environment for radar and sonar systems; the third is actual brain wave signals [electroencephalogram (EEG) and visual evoked responses (VER)], which are thought to be actively generated by multiple sources of spatially differentiated electrical activity [I] ,- [2].

Such a decomposition technique would contribute to our understanding of the electrogenesis of the EEG and VER as well as to overcoming, for example, multipath problems in radar systems with low look-angles. The solution to the decomposition problem is severely complicated, however, by the presence of ambient noise. Often little is known about the statistical structure of this activity and its statistics can often vary depending upon whether a signal is present or not.

Other applications have relevance as well. For example, in seismology a related problem arises, Here the data consist of the arrival times of various waves that can be processed to yield information about the location of the seismic source. The waves may be due to multiple reflections and are in- fluenced by irregularities in the transmission media. This problem may be considered more difficult than that of the radar or sonar problem in the sense that the signals very likely differ greatly in waveshape.

Several other procedures for achieving the decomposition of superimposed signals have proved successful; the more familiar methods include inverse filtering and decision theory [I], [2].

Each of these methods has proved successful under individual circumstances; e.g., in the application of inverse filtering, a signal is transformed by a linear time-invariant system, whose Fourier transform is the reciprocal of the transform of the signal components to be removed. Even so, this method has a serious limitation; the signal must be known and the signal-to-noise ratio (SNR) must be quite large. Investigations that attempted to employ a Wiener filter to improve the estimates of decomposition proved to be of little value [l]. For high-input SNR this filter is unnecessary; for low-input SNR the system acts like a matched filter that is unsuitable for decomposition unless one wishes to decompose signals where the time of occurrence between signals is approximately twice the length of the signal duration. Decision theory can be applied when noise is present to estimate the echo amplitude and arrival times. This is a straightforward approach if the signal waveshape is known. The technique to be described in this paper assumes that the wavelet waveshape and number of echoes are unknown; the only assumption made is that the echoes of this basic wavelet be “reasonable” replicas. It will be empirically demonstrated via the examples that limited echo distortion can occur and yet the technique will still produce satisfactory results.

B. History

The particular method of decomposition of a composite signal that is investigated was proposed in two seemingly

unrelated parallel studies. One approach [3]-[5] dealt with an echo-detection method called cepstrum (an anagram of the word spectrum); the other work [6] dealt with homomorphic systems. This latter work was later applied to echo detection [7], [8] d an was named the compIex cepstrum. The power cepstrum (so referred to in this paper to avoid confusion with the complex cepstrum) has been used in seismology [9] and speech [4], but homomorphic decon- volution appears to have greater potential, e.g., in speech, echo, and photographic processing [2], [7], seismology [lo], and separation of probability density functions [l I].

C. Antecedent

The power cepstrum of a function is the power spectrum of the logarithm of the power spectrum of that function. Similarly, the complex cepstrum of a function is defined as the inverse Fourier transform of the logarithm of the Fourier transform of that function. Decomposition is accomplished by utilizing the power cepstrum to detect the epochs of the echoes and their relative amplitudes; then the complex cepstrum is used to determine the signal waveform. The problems encountered in the presence of noise and echo distortion are considered; consideration is given to Hanning smoothing of the amplitude and phase information, to reducing the picket-fence effect, and to three methods of complex cepstrum filtering.

II. DECOMPOSITION OF COMPOSITE UNKNOWN SIGNALS

A. Considerations for Decomposition in the Absence of Noise

Power Cepstrum: To explain the power cepstrum analysis, an example of a single additive echo is examined; the formula for the total signal can be written as follows:

x(t) = s(t) + a,s(t - t,).

The power spectrum of x(t) can be represented as

(1)

$JJo) = 4,(0)[1 + a02 + 2a, cos o&J. (2)

This is the product of two terms and thus one is led to apply the logarithmic function to the above expression. Such an application gives

log C&(O) = log 4,(o) + log (1 + a02 + 2a, cos W,). (3)

The second term can now be expanded into an appropriate convergent infinite series; when a, is much less than unity, the infinite series can be approximated by 2a, cos ot,. Thus, the logarithm of the power spectrum is obtained with a nearly cosinusoidal ripple, the parameters of which are related to the echo parameters a, and t,. For the general case when a, is not much less than unity, higher order terms in the series expansion must be considered, which in turn adds ripples of different quefrencies.

A ripple of the kind just mentioned will usually be obscured by irregularities in the log power spectrum itself. Therefore, one can borrow from the available techniques used for the detection of periodic phenomena obscured by noise; i.e., the power spectrum can again be applied to

KEMERAIT AND CHILDERS: SIGNAL DETECTION AND EXTRACTION 747

yield the power cepstrum (see Appendix I). When this is done, delta functions (peaks) occur in the power cepstrum at quefrencies corresponding to the echo arrival time and integral multiples thereof.

Complex Cepstrum: Similarly, a simple example of a signal with one echo will be considered to explain the complex cepstrum procedure. The expression for a composite signal consisting of a wavelet convolved with a train of weighted samples for the discrete case is

with

x(n) = s(n) * P(n) (4)

P(n) = 5 a, c?(n - nk) k=O

where s(n) is the basic wavelet, x(n) is the composite signal, and N is the number of echoes present. At this point, a method of separating the impulse train from the wavelet s(n), is desired; therefore, the complex logarithm of the z-transform of x(n) is determined and then linear filtering is introduced to achieve signal separation. For the single- echo case, the impulse train is written as

P(n) = 6(n) + a,d(n - no). (5)

The z-transform X(z) (assuming that x(n) is z-transform- able) evaluated on the unit circle is

X(Z)Izzejw = X(ejw) = ,S(eiw)[l + uOe-jwno] (6)

where the contribution of the echo is a periodic function of o with a period of 2n/n,. The log spectrum, i.e., the log of X(ejw), is

log X(eiw) = log S(ejw) + log (1 + uOe-jono). (7)

Since the logarithm of the periodic waveform remains periodic with the same repetition rate, the echo is represented in the log spectrum as an additive periodic component. The necessary use of the complex logarithm function results in multivalued phase information in the calculation of the complex logarithm unless corrected for.

The complex cepstrum is found by inverse transforming (7) (see Appendix II). Delta functions (peaks) occur in the complex cepstrum at times corresponding to the echo arrival time and integer multiples thereof. If the echo amplitude is less (greater) than the basic wavelet, then the peaks occur at positive (negative) times.

Phase: In general, the phase, arg [X(e”+jw)], will be a discontinuous function of o since the phase is evaluated as the principal value, i.e., modulo 27r. This causes abrupt changes to appear in the phase, which can be corrected by “unwrapping” in order to satisfy the requirement that it be continuous and odd. The phase must be continuous and odd, and the log magnitude must be even, to ensure that the complex cepstrum (the inverse Fourier transform of the log magnitude and phase) will be a real quantity for a real input signal.

In addition to the difficulty presented by phase unwrapping, two other problems associated with the phase were

investigated. One included a linear phase component in the incoming signal and the other encompassed the minimum- maximum phase situations. If a linear phase component is present in the signal and is of sufficiently large slope, it will dominate the complex cepstrum. Should this characteristic be undesirable, it can be removed. This changes the ap- pearance of the complex cepstrum but more importantly, it displaces in time the recovered wavelet.

The minimum-maximum phase situations are discussed in [S].

Linear Filtering: It is possible to linear filter the complex log voltage spectrum with a kernel designed to remove the periodic components introduced by the multiple echoes. This can be achieved by convolution in the frequency domain or by multiplying the complex cepstrum (time domain) by an appropriate function. Two filters have been found useful to remove the echo peaks from the complex cepstrum [8] ; a third filter can be used to remove the wavelet and recover the echo pulse train [S]. A brief discussion of these filters is included.

“Comb” Filtering: Achieved by multiplying the complex cepstrum by a function that is unity everywhere except at those points which are due to echoes. At these latter points the filter function is zero. The resulting filtered complex cepstrum is then interpolated at the zeroed points by averaging the preceding and subsequent points in the complex cepstrum (see Appendix II and Examples in Section III-B, C, D).

“Short-Pass” Filtering: Another way to remove the echo. However, this procedure requires that the echo with the shortest delay no, be relatively well separated from the time origin. If this is the case, then only modest distortion will be introduced in the recovered wavelet if the complex cepstrum is set to zero for the time samples greater than or equal to no (analogous to low-pass filtering). This method not only removes the peak caused by the echo with the smallest arrival time no, but also removes all harmonics as well as those peaks caused by other echoes with arrival times greater than no. Obviously, if there is considerable overlap of the basic wavelet with its echoes, then considerable distortion in the recovered wavelet can occur. The amount of overlap that can be tolerated is the most important criterion to be determined before the short-pass filter is applied.

“Long-Puss” Filtering: Determines the echo arrival times. This procedure is usually applied to the complex cepstrum. Here one requires an estimate of the arrival time of the first echo. Then the sample values in the complex cepstrum which precede this first arrival time are set to zero. The normal inverse procedure is then applied, but the train of delta functions at echo epochs is recovered rather than the wavelet. Echo detection becomes progres- sively more satisfactory as the complex cepstrum of the wavelet is more successfully removed or separated from the “impulse train.”

The reader is referred to Appendix II for a more detailed explanation of the complex cepstrum and a schematic illustration.


B. Considerations,for Decomposition in the Presence of Noise amplitude smoothing of the log magnitude and reducing

If the noise is additive, then the expression for a composite signal with a single echo can be modified to

x(t) = s(t) + a,s(t - to) + n(t), (8)

the Fourier transform of which is

X(o) = S(w)[l + aoe-jotO] + N(o). (9)

Because S(o) and N(o) are usually complex quantities, this expression can be rewritten as

IX(w)lejex = lS(o)lejes(l + aoe-jo’O) + \N(o)lej’“. (10)

the picket-fence phenomenon by adding zeroes to the sampled time series x(t). The results for the former case are reported in a later section. The picket-fence effect is described in [I21 along with a recommended means for correcting this phenomenon, namely adding zeroes to ex- tend the data record. This improves the frequency resolution of components reduced by spectral windows. The maximum improvement is a factor of 2.5 to 1, which corresponds to 4 dB. A patent was issued to Robertsen in which he basically introduces zeroes in the sampled time series x(t) to improve the detection of echoes in the power cepstrum. This patent

By making the substitutions A = IS(o)1 and B = IN(w)/ disclosure shows a possible improvement of 4 dB in the

and taking the logarithm of the above expression, we have probability of detection.’ Empirically, we have observed with some data that adding

log [lX(o)lejex] zeroes to the data record introduces a linear phase term in

= log A + 3 log [l + 2a, cos ot, + ao2 + (B/A)2 the complex cepstrum. This can be removed as previously

+ 2B/A cos (es - 6”) + yp

described. At this time, we have no theoretical explanation

cos (es - cot, - e”)] for this phenomenon.

III. COMPUTATION ALGORITHM AND THREE EXAMPLES + j tan-’ sin 8, + a, sin (0, - cut,) + B/A sin en

’ cos es + a0 cos (e, - wt,) + B/A cos 8, I Our objective was to determine the extent to which one or more echoes could be distorted in the nresence of noise

(11) The power cepstrum can be calculated from the real part of the above.

When the magnitude of the noise is small with respect to the magnitude of the wavelet (B CC A) and the amplitude of the echo is less than unity, the real part of (11) reduces to the expression for the noise-free case, as expected. That is,

Re [log IX(w)lejex]

= log A + a0 cos ot, - ‘O* -T cos2 cot, + * * . . (12)

When the magnitude of the noise is not much less than that of the wavelet (again assuming a, < 1) the other terms in (11) cannot be ignored; the resulting equation is

Re [log IX(o)lejex]

= 3 log [2&? COS (e, - e,)]

and still allow recovery of the echo arrival times (epochs), their relative amplitudes, and the shape of the basic wavelet. The basic wavelet can then be removed and echo wavelet waveform recovery can be achieved sequentially. The measurements that we used to judge our success were the rms-signal-to-rms-noise ratio (SNR) as a function of the mean square error (MSE) between the recovered and the original basic wavelets. Epoch detection was monitored via the power cepstrum and the SNR was noted for that value at which the peak due to the echo was lost in the noise. The SNR reported is actually a worst case figure. The noise was generated via a computer subroutine with a specified standard deviation, which was used to calculate the SNR. However, we almost always low-pass filtered the noise sequence before adding to the signal sequence, thereby reducing the standard deviation of the noise and thus improving the SNR. This latter SNR was reported, not the

ao2A A $++

former. 2B + 2B set (4 - 4) Some of our global results appear in the next section.

Here we describe the computation algorithm and illustrate

+ y cos a, set (e, - e,) + a0 cos (e, - 8, - ato) ’ cos (es - 0,) I

(13) When the power cepstrum is computed from (13), par-

ticular attention is given to the terms that contain the quantities a, and to since these parameters yield the desired echo information. If (13) is compared with (3) (taking into account that the former is the real part of the log voltage spectrum while the latter is the log power spectrum), we see that the cos cot, term in the absence of noise is multiplied by the term set (0, - e,) when noise is present.

It is apparent from (11) that the noise affects both the

the procedure with several examples.

A. Computation Algorithm

The block diagram of the overall method appears in Fig. 1. Initially, the power and complex cepstrums are calculated in parallel from the discrete data. The power cepstrum is scrutinized for information concerning the echo arrival times; this in turn is utilized to implement one or more of the linear filtering operations on the complex cepstrum previously described, namely, comb, short pass, or long pass. After filtering, the algorithm is merely the inverse operation of that performed to obtain the complex

magnitude and phase. We have attempted two procedures to improve wavelet extraction and echo detection. These are

1 G. H. Robertsen, Bell Telephone Laboratories, Whippany, N.J., U.S. Patent 3 466 540, Sept. 9, 1969.


THmx TYPES OF LINEAR FILTERS USED AR: I. cm 2. SagKr-PASS 5. L(HGw5.S

I LIltEM FILTER I 8

Fig. 1. Block diagram of cepstrum analysis.

cepstrum. The result is recovery of the wavelet or the impulse train, depending upon the type of filtering employed.

The algorithm includes a procedure by which zeroes can be added to the input data as well as a feature by which Hanning smoothing of the log magnitude can be accomplished. The discrete Fourier transform (DFT) is achieved via an IBM scientific subroutine called HARM. Band- limited additive Gaussian noise was available through a separate subroutine.

The steps followed by the algorithm are 1) transform the discrete input data, 2) find the real logarithm of the absolute value of this transform, 3) perform Hanning smoothing, 4) transform again and take absolute value to get the power cepstrum, 5) compute the phase from step l), 6) use the results from 3) and 5) to calculate the inverse transform, the result of which is the complex cepstrum, 7) apply one of the three linear filters, 8) transform the result, 9) find the exponential, and finally 10) find the inverse transform to obtain an estimate of the wavelet (or impulse train).

The Appendixes provide additional mathematical background relevant to an understanding of the examples.

B. Example I: te-” Wavelet

In this example s(n) has the waveshape of a te-” pulse felt to be representative of degraded sonar or radar pulses. The composite waveform was generated by adding a delayed, attenuated (or amplified) replica to the basic pulse; noise could then be added to the composite waveform to yield a specified SNR if desired. These data provided the input to the algorithm. The power cepstrum detected the echo arrival times and determined the relative amplitudes of the echoes; the complex cepstrum extracted the basic wavelet waveform.

The results for one particular case appear in Figs. 2-4. Here, there was only one echo delayed 55 units relative to the basic wavelet; the amplitude was 0.4 relative to the basic wavelet. The damping coefficient was a = 0.06. No noise was present to facilitate interpretation of the power cepstrum (Fig. 3). The peak that is most easily seen is that at 55. The other peaks beyond this occur at integer multiples

of 55. Note the symmetry about 256. The spectrum above 256 represents the negative frequency terms or those about the sampling frequency. This is a standard representation. Further, it will be noted that little information can be obtained from an examination of the complex cepstrum (Fig. 4). This is primarily a scaling problem, since an examination of a computer listing of the points will yield the peaks introduced by the echo, which for this case occur in the right half (positive time) of the complex cepstrum since the echo is less than that of the basic wavelet (see Appendix II). If one then comb filters the complex cepstrum at these peaks and performs the inverse operation, the basic wavelet is recovered with a MSE of 0.06 for this example. All wavelets are shown in Fig. 2; the basic wavelet is dashed, the composite dotted, and the recovered solid. Note that the major error introduced in recovering the basic wavelet is a negative dc bias with some change in overall amplitude. We arbitrarily normalized each curve relative to its positive maximum value. Obviously, other manipulations could be and have been performed to reduce the MSE.

When noise is present, the wavelet can still be extracted, as will be shown later. However, one must rely upon the power cepstrum to detect the peaks introduced by the echo or echoes since smoothing more readily improved the detection of these peaks in the power cepstrum than in the complex cepstrum.

It is also interesting to demonstrate that if the echo is greater in amplitude than the basic wavelet, then it is the echo that is recovered. This is theoretically predicted in Appendix II. An example of this appears in Fig. 5 where the basic wavelet, the composite signal, and the recovered echo are shown. Here the wavelet and echo are of the form tewLlf with a = 0.04 and the echo delay is 55 units with an amplitude of 1.8 relative to the basic wavelet. Since the echo was greater than the basic wavelet, the complex cepstrum was comb filtered in the left half of the complex cepstrum (negative time) (see Appendix II). Generally, this may not be known in advance. Then comb filtering can be


0.50

0.25

-0.05

-0.X

-10 10 50 90 130 170 190

Fig. 2. Wavelets for Example l-echo less than basic wavelet teeat wavelet, a = 0.06, echo delayed 5.5 units with an amplitude 0.4 relative to the basic wavelet. Basic wavelet (dashed), composite signal (dotted), recovered wavelet (solid).

100

60

40

20

0

110

I 165

II ,I 220 I 1 1, i 0 51.2 153.6 256.0 358.4 460.8 512.0

Fig. 3. Power cepstrum for Example 1. Peaks occur at the echo delay and integer multiples thereof.

performed on both sides of the complex cepstrum with no noticeable degradation in results.

C. Example 2: Utiit Step Function Wavelets

For this example s(n) is a unit step function. However, the composite waveform has the form of a pulse. This is shown in Fig. 6. Here the composite waveform is generated by adding together three step functions of appropriate amplitudes and selected delays. The power cepstrum appears in Fig. 7. For this case, there are two echoes, one

greater and one less than the basic wavelet. A number of peaks appear in the power cepstrum, all of which can be accounted for from the theory in Appendix II and by taking into account that aliasing also occurred.

The peaks labeled in Fig. 7 are accounted for as follows.

95 Due to echo 3u(t - 95). 190 Due to echo - u(t - 190) plus a multiple of echo

at 95. 37 Aliased version of 5 x 95 = 475. Aliasing is about

512. Thus 512 - 475 = 37.

KEMERAIT AND CHILDERS: SIGNAL DETECTION AND EXTRACTION

0.65

0.25

7.51

0 51.2 153.6 256.0 358.4 460.8 512.0

Fig. 4. Complex cepstrum for Example 1. This actually represents the smoothed envelope since the discrete points alternate in sign.

2.5

0.7

0.1

-0.5

-10 10 50 90 130 170 190

Fig. 5. Wavelets for Example l--echo greater than basic wavelet temat wavelef, a = 0.04, echo delayed ,55 units with an amplitude 1.8 relative to the basic wavelet. Basic (dashed), composite (dotted), recovered (sohd).

58 Aliased version of 6 x 95 = 570 (570 - 512 = 58). 132 Aliasedversion of4 x 95 = 380(512 - 380 = 132). 153 Aliased version of 7 x 95 = 665 (665 - 512 = 153). 227 Aliased version of 3 x 95 = 285 (512 - 285 = 227).

The other points to the right of 256 are explained in the same manner as for Example 1. The complex cepstrum is comb filtered on both sides at these points. The inverse operation is then performed to yield the recovered echo shown in Fig. 6, namely the echo greater than the basic wavelet, as predicted in Appendix II. In the actual computer

plots, one notes that periodic extension occurs; this has been suppressed here for clarity. The recovered wavelet is distorted because of the severe aliasing.

D. Exemple 3: VER Wavelet and Echo in the Presence of Noise

Fig. 8 shows a typical normalized VER wavelet, the noisy composite signal, and the recovered wavelet for the case of a single echo delayed 55 units with an amplitude of 0.4 for a SNR of 20 dB. The MSE was less than 0.006. The power cepstrum appears in Fig. 9 for both cases when


3 r r --- --- 3u (t - 95) 3u (t - 95)

I I

I I

CDPCK?IlE CDPCK?IlE ~ . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . ..- _( . ..- _(

._..., ._ _ ._..., ._ _

-u (t - 190) -u (t - 190)

-10 15 40 90 140 190 240

Fig. 6. Wavelets for Example 2. Unit step functions (dashed) make up the composite signal (dotted). The recovered signal is solid.

IS4 -

71 .

0 51.2 153.6 256.0 358.4 460.8 512.0

Fig. 7. Power cepstrum for Example 2. Peaks occur at the echo delays and aliased versions (see text).

Harming smoothing was and was not employed. Note that for the unsmoothed case, no conclusions can be reliably drawn. However, a peak can be discerned at 55 when smoothing is employed. Other peaks are also present. For example, one is at 220 (4 x 55) and another at 233, which is presumed to be due to noise and/or smoothing. Essen- tially no change occurred in the MSE of the recovered wavelet when smoothing was used as compared to the unsmoothed case. However, realistically the echo could not have been detected without smoothing; thus, when smoothing was not used, the wavelet was recovered only because we knew a priori where the echo occurred.

IV. SUMMARYOF OTHER RESULTS

A. The Power Cepstrum in the Absence of Noise

Single Echo: As a result of analysis and numerous computer runs we can say that when a single echo is present no difficulty is encountered in detecting echo arrival times whether the amplitude of the echo is less than or greater than that of the basic wavelet. As shown by the analysis in Section II-A and Appendix I, the peaks in the power cepstrum occur at the echo arrival time and integer multiples thereof. However, there is an ambiguity in interpreting the power cepstrum: namely, if one attempts to estimate the

KEMERAIT AND CHILDERS : SIGNAL DETECTION AND EXTRACTION 753

1.00

0.66

-10 10 50 90 130 170 190

Fig. 8. Normalized basic VER wavelet (dashed), noisy composite (dotted), recovered (solid) for Example 3. Single echo delayed 55 units with an amplitude of 0.4 relative to basic wavelet. The composite signal also contained additive noise for a

SNR of 20 dB.

45

25

15

10

0

0 51.2 353.6 256.0 358.4 460.0 511.0

Fig. 9. Power cepstrum for Example 3. The Hanning-smoothed version is shown shaded. Note that the peak can be detected at 55 units.

amplitude of the echo from the normalized power cepstrum peaks, it is not possible to tell whether the echo’s amplitude is a fraction or multiple or a negative fraction or multiple of the wavelet. This is readily explained theoretically in Appendixes I and II. The experimental results show that a plot of the normalized power cepstrum peak magnitude (for a single echo) versus the echo amplitude expressed as a percentage of the basic wavelet amplitude is a multivalued function (somewhat dome shaped, with the apex at 100 percent). Thus, for example, if an echo is detected and the value of the peak in the power cepstrum is the normalized value of 0.7, then our experimental results would predict

that for an echo arrival time of 0.4 units the echo’s amplitude would be either 0.575 or 1.65 times that of the wavelet. This ambiguity can be overcome with the complex cepstrum, since the echo is recovered with correct sign for echo amplitudes greater than the reference wavelet. When the echoes are less than the basic wavelet amplitude, then the basic wavelet is recovered first. The wavelet can then be subtracted from the composite signal and the echoes recovered successively.

Multiple Echoes: The mathematical analysis is complicated, but is outlined in Appendix I [ 131. If the echoes occur at integer multiples of one another with amplitudes less


Fig. 10. Magnitude of power cepstrum peaks as a function of the summation of the echo amplitudes for the two-echo case.

than the basic wavelet, then the complex cepstrum can still recover the basic wavelet; however, the number of echoes present cannot be reliably estimated from the power cepstrum. The main complications arise if the echo delays are not integer multiples of one another. First, consider the two-echo case. Fig. 10 shows that ambiguous peaks in the power cepstrum are encountered when the sum of the amplitudes of the two echoes is in the range 0.8 to 1.0. (A similar condition exists when the sum of echo amplitudes is in a corresponding range greater than unity.) If the echo arrival times are denoted as t, and t,(t, > to), then peaks in the power cepstrum appear at (tI - to), t,,t,,. . . (see Appendix I). Several factors contribute to the amplitudes of these peaks. One, of course, is the amplitudes of the echoes themselves as shown in Appendixes I and II. But another less obvious factor is that peaks in the power and complex cepstrum are contributed to by multiple sum and difference expressions introduced by the series expansion of the logarithmic expression. For example, it may occur that one term that contributes to the amplitude of the (tJ peak, in addition to the cos w(tJ term, is a term of the form cos w(5t, - 3t,), depending upon the specific values of t, and t,, e.g., t, = 0.4 and t, = 0.3. Further, the amplitude of this latter term may add to or subtract from the final overall amplitude of the peak at t,. Naturally, no closed- form analytical expression can be derived for such cases since the final expression is dependent upon the specific echo arrival times. Thus, each case must be handled individually.

Another important result for two echoes appears in Fig. 11 where it is shown that the peak in the power cepstrum caused by the second echo is affected by the amount of separation between the two echoes in the composite signal. This dependence can be derived mathematically by a series expansion of the logarithmic expression and investigating a truncated expansion (e.g., 5 terms) for several cases involving different echo separations.

We have worked examples when four echoes are present (arrival times not multiples of one another). The peaks appear clearly, but must be interpreted correctly for amplitude information as per the theory. The basic wavelet was successfully recovered via the complex cepstrum with a MSE of 0.002.

In summary it can be said that when two echoes are present, a threshold phenomenon was found whereby un- ambiguous echo arrival detection became impossible when the sum of the echo amplitudes exceeded 0.8. (See Fig. 10.) When there are more than two echoes, erroneous cepstrum peaks are observed at the sum and difference of the echo arrival times. If the total number of echoes is small, an algorithm might be implemented to determine if an observed peak is the sum or difference of the others and, if it is, then this peak could be omitted. The implementation would consist of having the algorithm check all the sums and differences of the detected peaks to determine if any corre- spond to one or the other of the detected peaks. In the event that a sum or difference of any two of the detected peaks corresponds to a third detected peak, this third detected peak could be eliminated as one that is not caused by a true echo. If the multiple echoes are small enough in amplitude (or, correspondingly, sufficiently greater than unity) so that the problem mentioned above does not occur, it appears that there is no limit to the number of echoes that can be detected. Epoch (arrival-time) detection has been found to be as nearly dependent upon the echo amplitudes as on their number.

B. The Complex Cepstrum in the Absence of Noise

Single Echo and Multiple Echoes: The echo arrival time is difficult to detect using this method, i.e., the peaks are easily masked. Knowledge of the location of these peaks is necessary for satisfactory wavelet extraction. Thus, the power cepstrum is recommended for this function.

Minimum-Maximum Phase Impulse Trains: Section II-B

KEMERAIT AND CHILDERS : SIGNAL DETECTION AND EXTRACTION 7.55

Fig. 11.

0 Jo * co ma 1m

-*alnl*9-Q-~

Magni tude of the power cepstrum peak of the second echo as a function of the time difference between the two echoes.

provided the derivations for the summary conclusions provided here. If the echo amplitudes are less (greater) than that of the basic wavelet (minimum (maximum) phase), peaks are mathematically predicted (see Appendix II) to be located in the complex cepstrum for positive (negative) time only. However, if the logarithm of the voltage spectrum is smoothed prior to calculating either cepstrum, then we have noted empirically that this prediction remains valid for the minimum-phase case, but that peaks appear in both the negative and positive regions for the maximum-phase case. Since most real data may be nonminimum phase, the algorithm used to calculate the complex cepstrum must avoid aliasing.

Linear Filtering: As described earlier, there are two types of linear filters that can be used to remove the peaks in the complex cepstrum caused by the echoes, namely, comb and short pass. However, these filters can only be applied if one has estimated the echo arrival times from either the power or complex cepstrum.

Comb Filtering: For the special case where the echo arrival times are integer multiples of one another, the comb filter is identical to that used for the single echo case. If the arrival times are random the filter is complex because of the necessity of removing the additional peaks situated at the sum and difference times of the epochs of the multiple echoes.

Short-Pass Filtering: Once the arrival time of the echo has been determined by the power cepstrum, short-pass filtering can be used, which sets all values in the complex cepstrum to zero beyond the echo epoch. However, this form of filtering introduces more distortion in the recovered wavelet than comb filtering. This becomes increasingly so when multiple echoes are present since the first peak to appear is that due to the difference in arrival times and it is this point beyond which all values in the complex cepstrum are zeroed. The amount of distortion depends upon the echo arrival time, the smaller the arrival time, the larger

the distortion. Similarly, if the complex cepstrum of the basic wavelet dies out rapidly, then less distortion is introduced by short-pass filtering. It is usually impractical to use this form of filtering in the multiple echo case if distor- tionless wavelet recovery is desired.

Impulse Train Extraction: Long-Pass Filtering: This filter zeros all complex cep-

strum points preceding the first echo peak. The impulse train is then recovered. Our experiments showed this method to work well, but that the power cepstrum procedure was superior.

Wavelet Extraction: Single Echo: The results for this case are excellent for

an echo with an amplitude less than or greater than that of the basic wavelet. However, considerable distortion (as measured by MSE) is introduced as the echo amplitude approaches that of the wavelet. (See Fig. 12.) If the echo amplitude exceeds that of the basic wavelet, then the echo is recovered. (See Appendix II.)

Multiple Echoes: Generally, if the echo amplitudes are less than the wavelet, the wavelet can be recovered. As one or more echo amplitudes approach that of the wavelet, wavelet recovery becomes more difficult. If the echo amplitudes are greater than that of the basic wavelet, but not equal, then the largest echo is recovered. (See Appendix II.) If two or more echo amplitudes are equal and greater than that of the basic wavelet, then recovery has not been possible.

C. The Power Cepstrum in the Presence of Noise

Single Echo: The SNR was decreased until the cepstrum peak was not detectable. This occurred at a SNR of 20 dB and was not affected by whether the echo amplitude was less than or greater than that of the basic wavelet.

Smoothing: Hanning smoothing can greatly improve

Fig. 12. Mean square error (MSE) as a function of the echo magnitude. Epoch time of a single echo was 0.55 s.

the detection of echoes from the power cepstrum as shown in one of the previous examples.

Multiple Echoes: The same threshold was found here. The threshold can be lowered to a SNR of 2 dB by smoothing, which is discussed in the following.

D. The Complex Cepstrum in the Presence of Noise

Single Echo and Multiple Echoes: The noise only increases the difficulty in interpreting the complex cepstrum; however, the noise has less overall effect on the complex cepstrum than on the power cepstrum in that wavelet recovery is still possible below SNR’s where the power cepstrum peaks are masked, provided, of course, that one knows the echo arrival time so that the comb filter can be properly designed.

Linear Filtering: For SNR’s above 20 dB the results previously discussed apply. Below 20 dB linear filtering was of little value since the echo arrival times could not be detected, unless smoothing was applied.

Smoothing: Hanning smoothing has allowed us to reduce the threshold to a SNR of 2 dB with the MSE near 8.1 and we believe this can be lowered even further. Smooth- ing was applied to the log magnitude, the phase, and the complex cepstrum. For wavelet recovery the MSE was reduced by smoothing the log magnitude, but no noticeable improvement was obtained from smoothing the phase and/or the complex cepstrum.

Wavelet Extraction: The log magnitude and the phase both deteriorated with the decrease in SNR. For some of our experiments, we have noted that when the data record

i I \ \ \ \ \ \

‘L\ -L-M

IEEE TRANSACTIONS ON INFORMATION THEORY, NOVEMBER 1972

length of a noisy composite signal is doubled by the addition of zeroes, a considerable improvement is effected in the recovered wavelet, e.g., the MSE is halved. No sound theoretical explanation of this is available other than that already advanced for reducing the picket-fence effect. But how this applies to the complex cepstrum and wavelet recovery is as yet unknown.

We nonetheless recommend both increasing the record length by the addition of zeroes and amplitude smoothing of the log magnitude to reduce the MSE in wavelet extraction.

E. Distorted Echo Removal

To date we have distorted the echo in two ways, namely, by adding noise to the echo only and by truncating the echo (as might occur in data recording). Other forms of echo distortion are under investigation. When noise is added to the echo only the power cepstrum peak becomes undetectable at a SNR of 18 dB; however, if the echo arrival time is known the wavelet can still be satisfactorily recovered with a MSE of less than 0.1 down to a SNR of nearly 10 dB.

For the truncated echo experiments, a threshold occurs at approximately 20 percent truncation (the last 20 percent of the echo is discarded), i.e., for truncations that exceed this amount the power cepstrum peaks become immersed in the ambient noise. No matter what the truncation, the peaks in the power cepstrum shift point for point with the amount of truncation. When the detected peaks are comb filtered and smoothed in the complex cepstrum, the recovered wavelet is only slightly distorted. The displacement of the power cepstrum peaks as a result of echo truncation does not affect the wavelet recovery capability of the complex cepstrum since the comb filter can be shifted accordingly. Thus, the comb filter effectively removes the echo from the complex cepstrum as long as the echo arrival time can be estimated in some way.

V. SUMMARY AND CONCLUSIONS A. The Power Cepstrum

The power cepstrum yields the best indication of echo arrival times even in the presence of noise. However, all phase information is lost, and thus, wavelet recovery cannot be achieved. In the presence of noise the threshold occurs at a high SNR (about 20 dB) unless some form of smoothing is used, e.g., Hanning smoothing reduces the threshold easily to a SNR of 2 dB. When multiple echoes are present erroneous peaks appear if the summation of the normalized echo amplitudes falls in the range 0.8 to 1 .O. The magnitude of the peaks is dependent on the magnitude of the initial epoch and the separation between epochs.

B. The Complex Cepstrum

This procedure can recover the basic wavelet or the echo waveform. The algorithm is complicated, however. Recovery can be effected for a wide range of echo amplitudes and in the presence of a large number of echoes. Only a few limita-


tions have been found: wavelet recovery cannot take place if one or more echoes have normalized amplitudes of unity or if two or more echoes are equal and greater than or equal to that of the basic wavelet. The MSE increases in the single-echo case as the amplitude of the echo approaches that of the reference wavelet. Wavelet recovery is effected with less distortion by comb filtering as compared to short- pass filtering.

The complex cepstrum does not appear to exhibit threshold phenomenon in the presence of noise. The log magnitude is more susceptible to noise than the phase. The MSE of the extracted wavelet can be lowered (8 dB in SNR) by simultaneously adding zeroes to the record length and amplitude smoothing.

Wavelet recovery is possible even in the presence of echoes that have been distorted by the addition of separate noise or which have been truncated (such as might occur in data recording). Echo truncation of any amount causes shifts in the power cepstrum peaks; further if the truncation exceeds 20 percent of the echo wavelet length, the power cepstrum peaks become immersed in the ambient noise. However, wavelet recovery can still be achieved, by comb filtering the shifted peaks.

We believe that future research along these lines will eventually lead to the derivation of a nonlinear method that will successfully achieve the decomposit ion of composite stochastic processes. Our efforts are directed toward a better understanding of and an improvement in wavelet recovery in the presence of distorted echoes.

ACKNOWLEDGMENT

The authors wish to express their sincere appreciation to Carol Halpeny for the preparation of the figures.

APPENDIX I

POWER CEPSTRUM

The main points of the power cepstrum appear in [3], some of which are also given in Section II-A. Many conclusions reached are based upon the log series expansion

log(1 + x) = x - ; + 5 - ..‘) -1 < X 5 1 (I-l)

applied to (3), which appears to limit the range of the echo amplitude (a& This constraint also depends on the value of the epoch (to) of the echo. Under the most stringent conditions a, is bounded by -0.41 4 a0 _( 0.41. In practice these bounds are relaxed since the cos cot0 term seldom takes on the value of unity, which sets the limitation on a, given above.

Having expanded the second part of (3) in a series, we then take the power spectrum of this expression. The result is called the power cepstrum. From spectral theory we see that there will be delta functions located at quefrency to and its multiples in the power cepstrum.

Single Echo W ith Magni tude Greater than Unity

When the amplitude (ao) of the echo has a value greater than unity, the series expansion (I-l) cannot be applied per se. How- ever, the following mathematical manipulation can be applied beginning with (3), which is restated here for a, > I

log [&(o)] = log [h(w)] + log (1 + uo2 + 2 . a0 cos ox,)

log [&(w)] = log [h(w)] + log [a071 + l/ao2 + 2/a, cos cot,)]

= log [a02 . qqw)] + log(l + l/ao2 + 2/a,coswt,).

U-2)

Now we can use the same series expansion to expand the second part of this expression, which has as the most severe case limitation (2.4 5 u. 5 co). Using reasoning similar to that used in the case with a, less than unity, the actual values that a, can take are much closer to unity than the value of 2.4 shown here.

At this point we again apply the power spectrum to the log power spectrum to obtain the power cepstrum. The results will be delta functions at the quefrencies of the echo epoch and its multiples in the power cepstrum.

Two Echoes W ith Amplitudes Less than Unity The same procedure can be used to obtain the log of the power

spectrum for two echoes with each amplitude less than unity, namely

log d,(w) = log f&(w) + log (1 + 2a, cos cot, + 2a, cos wt,

+ 2a,a, cosw(t, - to) + ao2 + a12). (I-3)

The first term of the series expansion used previously is

x = (2a, cos cot, + 2a, cos wt,

+ 2aoaI cos w(t, - to) + ao2 + al’).

For the multiple-echo case, we encounter an infinite number of cosinusoidal components by expanding the second part of (I-3) in a series. The number of these components that will show up as noticeable peaks in the power cepstrum depends on the amplitudes (a0 and al) of the echoes. By calculating the first two and three terms of the series respectively, one can show the effect that the echo amplitudes have on the higher order terms. The terms with coefficients of sufficient magnitude will yield additional peaks in the computed power cepstrum.

For the two-echo case the amplitude of the second cepstrum peak depends upon the time difference between the two epochs of the echoes as well as upon the amplitude of the first echo [ 131.

APPENDIX II COMPLEX CEPSTRUM

Homomorphic filtering of convolved signals was developed by Oppenheim [6]; a specific application of this filtering, called the complex cepstrum, was introduced by Schafer [8]; the theory and applications also appear in [7]. The latter reference provides most of the necessary theoretical background. However, it is convenient to demonstrate here that echoes greater than the basic wavelet can be recovered.

Mathematical Analysis jbr the Single Echo Case For convenience the analysis will be carried out for the

continuous case; thus, the appropriate transformations will be accomplished by means of the Fourier transform.

When the composite signal consists of a wavelet and one echo, the mathematical expressions can he written as

x(t) = s(t) + a0 .s(t - to) (II-l)


and the Fourier transform is

X(w) = S(w)(l + a0 * e-jury (H-2)

where X(w) and S(w) are usually complex quantities. If the complex logarithm is applied to this expression, one obtains

log [X(w)] = log [S(w)] + log (1 + a, . e-joro). (H-3)

The only convergent log series that can be applied to expand the second term is

log (1 + x) = x - x2/2 + x3/3 - . . 1x1 < 1 (11-4)

and the substitution of ac, . e -jwto for X with a, < I gives

(11-5)

At this point this expression is inverse Fourier transformed in order to obtain the complex cepstrum to give

F-‘{log X(w)} = F-‘{log S(o)} + uO s(t - to)

- $ s(t - 2&J + . . . . (11-6)

Therefore the complex cepstrum for a single echo with an amplitude less than that of the wavelet consists of the inverse Fourier transform of the complex log of the Fourier transform Fig. 13. Schematic representation of composite signal x(n), complex

cepstrum i(n), and comb filtering to recover the echo when a > 1. of the original wavelet signal, plus a train of impulse functions with decreasing amplitudes and alternating signs in positive time.

When the amplitude of the echo is greater than that of the wavelet (a9 > l), the following mathematical manipulation can

x(t) = s(t) + a,s(t - to) + a,s(t - t,) (II-IO)

be made and the Fourier transform is

log [X(w)] = log [S(w)] + log aa . e-jWfO(l + l/a, . ej”‘O) X(w) = S(w)(l + a,e-joto + ule-jwtl). (11-l 1)

= log [S(o)a, . e-jwto] + log (1 + l/a, . ejw’O).

(11-7) Applying the complex logarithm gives

If we apply the same series expansion used previously, we obtain log [X(w)] = log [S(o)] + log (1 + nOe-jwfo + ule-jwzl).

log [X(w)] = log [S(o)a, . e-jut01 (11-12)

+ l/aoejwfo - +u02ej20to + . . . (11-g) If we expand the second part with the limitation

the inverse Fourier transform of which is iv -jut0 + ule-.iwtll < 1

F-‘{log [X(w)]) = F-‘(log [S(o)a,e-j”“‘]} + I/U, s(t + lo) then

- +uo2 6(t + 21,) + . . . . (11-9) i0g [Ox] = i0g [S(W)] + uoe-jwrO + u,e-jwtl

Thus, for a single echo with an amplitude greater than that of the wavelet, the complex cepstrum consists of the inverse Fourier transform of the complex log of the Fourier transform of the echo, plus a train of impulse functions with alternating signs and decreasing amphtudes in negative time only.

If the time series x(n), and a(n) are used to denote the composite signal and its corresponding complex cepstrum, respectively, and similarly s(n) and i(n) denote the basic wavelet time series and its complex cepstrum, then Fig. 13 illustrates schematically a composite signal with one echo, the complex cepstrum for the cases when the echo is less than (a < 1) or greater than (a > 1) the basic wavelet and comb filtering to recover the echo when the echo is greater than the basic wavelet.

Mathematical Anqlysis jbr the Two-Echo Case The same procedure is followed as that for the single echo

case; the expression for the composite waveform can be written

- u02/2e -j2wt0 - uoule-j(b+tl)w _ u12/2e-j2u~t, + . . (H-13)

and the complex cepstrum is

F-‘{log [X(w)]} = F-‘{log [S(w)]) + a0 d(t - to)

+ a, d(t - tJ - u,2/2 s(t - 2t,) - UOUl s[t - (to + t,)]

- ur2/2 s(t - 2t,) + . . . . (H-14)

Therefore, the complex cepstrum for multiple echoes with amplitudes less than that of the wavelet will consist of the inverse transform of the complex logarithm of the transform of the wavelet plus delta functions only in positive time.

When the amplitudes of the multiple echoes are greater than that of the wavelet (i.e., la, 1 > 1) a similar rearrangement of the expression can be accomplished; we start with (11-12) and make

IEEE TRANSACTIONS INFORMATION THEORY, VOL. IT-18, NO. 6, NOVEMBER 1972 759

the assumptions ai > a, and a,,, ai > 1.0; then (11-12) becomes

log [X(w)] = log [S(w)] + log [ule-jatl

(1 + uo/~~ej~(r~-‘~) + l/a,ejw’l) (11-15) or

log [X(w)] = log [S(co)ule-jot11

+ log (1 + uO/ulej”(tl-‘o) + l/ulejw’l)

and application of the same log series will give

log [X(w)] = log [S(w)ule-jwrl] + uO/ulejw(tl-to)

+ l/ulejwtl _ uo2/2u12ej2~(~~-W

_ +u12ej2wfl _ uo,ul 2ejd2tl -to) + . . . . (H-16)

The complex cepstrum is

F-‘{log [X(41) = F-‘{log [S(o)u,e-j”tl I> + aola1 art + (tl - r,,l

+ l/u, s(t + t1) - ao2/2q2 s[t + 2(t, - r,,]

- +ar2 3(t + 2t,) - ao/ulZ cS[t + (2t, - lo)] + . . . . (11-17)

Therefore, the complex cepstrum for multiple echoes with amplitudes greater than that of the wavelet consists of the inverse transform of the complex logarithm of the transform of the echo with the greatest amplitude, plus delta functions all located in either positive or negat ive time, depending upon whether to > f, or t, > to, respectively. It is also apparent that the cases concerned with amnli tudes greater than unitv are maximum

phase while those with ampli tudes less than unity are minimum phase situations.

REFERENCES

111

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UOI

[ill

H21

[I31

D. G. Childers, R. S. Varga, and N. W. Perry, Jr., “Composi te signal decomposit ion.” IEEE Trans. Audio Electroacoust.. vol. AI-J-18, pp. 471-477, bet. 1970. S. Senmoto and D. G. Childers, “Adaptive decomposit ion of a composite signal of identical unknown wavelets in noise,” IEEE Trans. Syst., Man, Cybern., vol. SMC-2, pp. 59-66, Jan. 1972. B. P. Bogert, M. J. Healy, and J. W. Tukey, “The quefrency analysis of time series for echoes: cepstrum, pseudo-autoco- variance. cross-ceostrum and saohe crackine.” in Proc. Svm~. Time &vies Anal&is, M. Rosenblatt, Ed. ‘New York: W iley, 1963, ch. 15, pp. 209-243. A. M. Noll, “Short-time spectrum and cepstrum techniques for Fepli;itth detection,” J. Acoust. Sot. Amer., vol. 36, pp. 296-302,

. . B. P. Bogert and J. F. Ossanna, “The heuristics of cepstrum analysis of a stationary complex echoed Gaussian signal in stationary Gaussian noise,” IEEE Trans. Inform. Theory, vol. IT-12, pp. 373-380, July 1966. A. V. Oppenheim, “Superposit ion in a class of nonl inear systems,” Res. Lab. Electron., M.I.T., Cambridge, Mass., Tech. Rep. 432, Mar. 31, 1965. A. V. Oooenheim. R. W. Schafer. and T. G. Stockham. Jr.. “Nonline’a; filtering of multiplied and convolved signals,” hoc: IEEE, vol. 56, pp. 12641291, Aug. 1968. R. W. Schafer, “Echo removal by discrete general ized linear filtering,” Ph.D. dissertation, Mass. Inst. Technol., Cambridge, 1968. T. J. Cohen. “Source-death determinations using soectral. vseudo- autocorrelation and cepstral analysis,” Geophyi . .J. Roy. ‘Astron. Sot., vol. 20, pp. 223-231, 1970. T. J. Ulrych, “Application of homomorphic deconvolut ion to seismology,” Geophysics, vol. 36, pp. 650-660, Aug. 1971. J. C. Prabhakar and S. C. Gupta, “Separat ion of Rayleigh and Poisson density functions through homomorphic filtering,” Nat. Electronics Conf., pp. 605-610, Dec. 1970. G. D. Bergland, “A guided tour of the fast Fourier transform,” IEEE Spectrum, vol. 6, pp. 41-52, July 1969. R. C. Kemerait, “Signal detection and extraction by cepstrum techniques,” Ph.D. dissertation, Univ. Florida, Gainesville, 1971.

Optimum Quantizers and Permutation Codes TOBY BERGER, MEMBER, IEEE

Abstract-Amplitude quantization and permutation encoding are two of the many approaches to efficient digitization of analog data. It is shown in this paper that these seemingly different approaches actually are equivalent in the sense that their opt imum rate versus distortion performances are identical. Although this equivalence becomes exact only when the quantizer output is perfectly entropy coded and the permutation code block length is infinite, it nonetheless has practical consequences both for quantization and for permutation encoding. In particular, this equivalence permits us to deduce that permutation codes provide a readily implementable block-coding alternative to buffer- instrumented variable-length codes. Moreover, the abundance of methods in the literature for optimizing quantizers with respect to various criteria can be translated directly into algorithms for generating source permutation codes that are opt imum for the same purposes.

Manuscript received January 3, 1972; revised February 25, 1972. The author is with the School of Electrical Engineering, Cornell

University, Ithaca, N.Y. 14850.

The opt imum performance attainable with quantizers (hence, permutation codes) of a fixed entropy rate is explored too. The investigation reveals that quantizers with uniformly spaced thresholds are quasi- opt imum with considerable generality, and are truly opt imum in the mean-squared sense for data having either an exponential or a Laplacian distribution. An attempt is made to provide some analytical insight into why simple uniform quantization is so good so generally.

I. INTRODUCTION AND SYNOPSIS

LTHOUGH communicat ion and information theorists A have suggested many novel digitization techniques, simple quantization continues to be used almost universally in practice. The widespread preference for quantization has a sound basis. Quantizers are relatively easy to implement and, moreover, their encoding performance usually is nearly optimum. For example, in the case of minimum-mean-

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