High-resolution spectral analysis using multiple-interval adaptive prediction

CIRCUITS SYSTEMS SIGNAL PROCESS VOL. 2, NO. 4, I983

HIGH-RESOLUTION SPECTRAL ANALYSIS USING MULTIPLE- INTERVAL ADAPTIVE PREDICTION* B. Egardt ''2, T. Kailath 2, and V.U. Reddy 2

Abstract. The problem of adaptively detecting two sinusoids corrupted by noise is considered, with emphasis on resolution properties. The approach is to form a spectral estimate from the coefficients of a A-step-ahead adaptive predictor. A theoretical analysis reveals that attention to the choice of the prediction horizon A gives a distinct improvement in the spectral estimate and in the resolution of the signals. The theoretical results are illustrated with numerical examples. Comparisons with previously suggested techniques are also made.

1. Introduction

In recent years , the use o f adap t ive fi l ter techniques for the de tec t ion and es t ima t ion o f s inusoida l signals in noise has a t t rac ted much a t t en t ion in the l i tera ture . W e immed ia t e ly give a typica l s t a tement o f the p r o b l e m . Assume tha t a d iscre te- t ime signal y, is received,

Y, = Z, + V, ,

where z, is a sum o f s inusoids o f u n k n o w n f requency, ampl i tude , and phase and {v,} is a sequence o f ze ro -mean r a n d o m var iables . The task is to detect the presertce o f the s inusoids and to es t imate their f requencies .

In an ear ly pape r , [1], W i d r o w e t a l . p r o p o s e d to app ly l inear p red ic t ion techniques to solve this p r o b l e m . Thei r adap t ive f i l ter c o n f i g u r a t i o n - t h e so-called adapt ive line enhancer (ALE) - was based on forming a A-step-ahead p red ic t ion P, o f y , as

f l , = a l y , _ ~ + . . . + a L Y , - a - L § ,

and then choos ing the pa rame te r s {a,} so as to min imize E{ ( y , - P,)= } . The

* Received March 30, 1983. This work was supported in part by the Joint Services Electronics Program under Contract DAAG29-79-0047, the National Science Foundation under Grant Eng78-10003, and the Air Force Office of Scientific Research under Contract AF49-620-79-C-0058. i Presently 'with ASEA, S-72183 Vasteras, Sweden. 2 Information Systems Laboratory, Stanford University, Stanford, California 94305, USA.

422 EGARDT, KAILATH, AND REDDY

positive integer A was typically chosen equal to one (at least for the case with {v,} as white noise) and an LMS algorithm was used for the adjustment of the parameters. Since the ALE was proposed, it has been studied by several authors, e.g., [2-6] , [10], especially for the case of single sinusoid.

In this paper we examine the case of two sinusoids. Different ways of forming a spectral estimate from the parameters of the predictor are compared. Furthermore, we study the effect of the prediction horizon A on the resolution, i.e., the ability to distinguish between two sinusoids that are close in frequency. The analysis is carried out for the case of [v t} being one pole low-pass noise, white noise being a special case of this model.

The main result of the paper is the demonstration that the choice of an "opt imal" prediction horizon A can substantially enhance the resolution and bias properties of this approach. A similar study was made by Zeidler et al. in [3], and we shall discuss and compare our results with theirs. The un- biased nature of the proposed method is discussed by showing the relation between the criteria for optimal A with that of an optimal notch filter whose notches are located at the frequencies of the input sinusoids.

Section 2 gives a statement of the problem and the basic idea of the method. Asymptotic expressions for the predictor characteristics are given, with the formal derivation given in the Appendix. A discussion of how to form spectral estimates and how to choose the prediction horizon is contained in Sec- tion 3. This section also contains a comparison with the technique in [3]. In Section 4, an iterative scheme for estimating the optimal value of A is developed and its effectiveness is illustrated with some examples. An example is included to illustrate the sensitivity of the spectral estimate to small variations from the optimal value. To demonstrate the variance of spectral estimate with the new method, an ensemble of 20 spectral plots that are obtained with different realizations is presented. Comparisons with other techniques are also made. Section 5 contains some concluding remarks.

2. Problem formulation and analysis

It is assumed that a discrete-time signal y~, consisting of two sinusoids corrupted by noise, is received. Thus, let

Y~ = z, + v~, (1)

where zt is the sinusoidal part,

z~ = ~/2alsin(~olt + ~bl) + x/2a2sin(~02t + ~b2) , (2)

and v, is assumed to be zero-mean, low-pass noise with covariance function

COV(V~, V~) = a2~['-s[ . (3)

It is assumed that {v~} and {z~} are uncorrelated. The problem is to estimate the sinusoidal frequencies r and w2 based on

HIGH-RESOLUTION SPECTRAL ANALYSIS 423

measurements of y, t >_ 1, and emphasis is put on good resolution between the two frequency estimates and, also, low bias in the individual estimates.

There are many different approaches to this problem (see, e.g., [7]), and here we shall investigate one particular method, based on linear prediction techniques. The idea is the following. Consider a model of the process lYe} of the form

(1 - q - ~ A ( q - 1 ) ) y , - - e , , (4)

where A ( q - ' ) is a polynomial,

A ( q -1) = a l + a 2 q -~ + . . . + a z q -~L-I~ , (5)

with q-i the delay operator defined by

q - ~ x ( t ) = x ( t - 1) .

The integer A is greater than or equal to one and e, is the residual. Note that for/x = 1, the model (4) is a conventional autoregressive (AR) model. Based on the model (4), it is straightforward to get an estimate of the spectral properties of l Y , ] by estimating the parameters { a , } . There are several estimation methods, such as gradient-based algorithms, least-squares, etc. The same technique can also be used with other parametrizations of the model (4), e.g., lattice forms, see [6-8] . Here we shall concentrate on the asymptotic (or steady-state) analysis, and for this the specific choice of the algorithm is not important.

It should finally be noted that an appropriate model for the process {y,] is n o t of the form (4) - an AR m o d e l - but rather an ARMA model [9]. The basic ~eason for looking at model (4) is its simplicity. We repeat, as stated in the introduction, that for the special case A = 1, the technique was studied by Widrow e t a l . [1] under the name adaptive line enhancer (ALE).

ASYMPTOTIC ANALYSIS. AS noted above, different algorithms to estimate the coefficients la/} can be used. Here we will analyze the steady-state characteristics, common to several different algorithms. This analysis is carried out for the problem, stated above, with two sinusoids. However, the first part of it can be extended in a straightforward way to the general case with several sinusoids.

There is one property common to all the algorithms commented on above. The steady-state solution for the estimated parameters is characterized by the fact that the covariances between the error e, and the components of the regression vector are equal to zero in the limit. Here the regression vector is

x ,_~ = [ y , _ ~ . . . y . . . . . . . ] r , (6)

and so the steady-state solution is determined by the equation

lim N-1 ~ E [ x , _ ~ e , } = 0 . (7) N--oo t=A


Defining the steady-state parameter vector

a* = [a*... a*] ~ ,

Equation (7) can also be written as

a* = R ~ r ~ ,

where N

R~ = lim 1 ]~ E{x,_~x~_,~} t=A

(8a)

N r~ = lim 1 N-~ N E E[X,-~,y,}. (8b)

t=A

The expressions (8) are further expanded and simplified in the Appendix. Two approximations are made in the calculations, namely:

A is assumed to be chosen such that o~ ~ is small (9a)

Certain couplings between positive and negative frequency components of the sinusoids are neglected (9b)

Using these approximations, explicit expressions for a* are obtained. Here we will only give the form of the final solution. The transfer function H ( z ) of the resulting (asymptotic) predictor,

H ( Z ) = z-~(a~ + a ~ z -1 + . . . + a~z -(~-~)) . (10)

can, to a good approximation, be expressed as

e-j~, H(eJ~) = ~,,h2-1(512 [c~(w)e-J~'~ + c~2(~ + 21(~ + ~2(r176 ,(11)

where h, , ~2, and (5 are constants and cr ~,.(~) are functions of 0:; see the Appendix for detailed expressions.

Equation (11) gives us a fairly explicit solution for the steady-state transfer function of the predictor. It can be noted that the dependence on A is relatively simple, and this fact gives us the possibility to derive and discuss different criteria for choosing the parameter A, as will be done in Section 3.

3. Criteria for resolution

The basic objective of the analysis in the preceding section is to facilitate a systematic study of the effect of the parameter A on the resolution. As noted above, earlier investigations of the present approach to spectral analysis have been made almost exclusively for the case A = 1. It has been pointed out that for the colored noise case, A (the "decorrelation parameter") should be chosen greater than one, however, with a few exceptions [10], a closer study of the effect of different choices of A has not been made. It should


perhaps be noted here that an assumption on A has already been made. When deriving (11), it was assumed that ot ~ is small, so that we have already departed f rom the common situation A = 1 at least if the noise is colored.

The following question now arises: what criteria should be used to pick a suitable A? Intuitively, what we want is clear - an estimate of the spectrum with two sharp peaks at the true sinusoid frequencies. Thus, we first have to decide how to form the spectrum. Even when this is fixed, it is not ob- vious how to define a quantitative measure of the accuracy of the resulting spectrum estimate.

Two approaches to the problem will be described below, and in Section 4 a comparison will be made via several examples. The first method was previously suggested and analyzed in [3].

THE METHOD OF ZEIDLER et al. [3]. The basic idea behind this approach is to compute the frequency response of the predictor transfer function H ( z ) , [H(ei~)[. 'This was in fact the method suggested in the early paper [1] on the "adaptive line enhancer". The criterion used in [3] to find an appropriate value of A is to minimize Ht(~oav) I, where

( . O a v - -

2

This criterion thus concentrates on obtaining a deep null between the two sinusoid frequencies. The minimizing value of A, say A1, is found in [3], with the same approximations we have used here, as

lr + 2n~r L - 1 A1 = - - , n integer, 2

where ~ = o~1 - o~2. Notice that this choice can only be made approximately, since we normally consider only integer values of A.

A NEW METHOD. In this method, the input spectrum is estimated by computing IG(~0)] 2, where G(~) = G ( z ) l z = e J~ with

1 1 G(z) = I1 - z-~A(z-1)l - I 1 - H ( z ) l '

and the locations of the distinct peaks are taken as the estimates of the input signal frequencies. The function [G(o~)l 2 with A = 1 was used in [2] for the measurement of digital instantaneous frequency; this is widely known as the maximum entropy method of spectrum analysis, see for example, Lang and McClellan [14] and Papoulis [15]. Here we shall show that other choices of A can be advantageous.

One measure of the accuracy of the model (4) is the variance of the residual e,. Thus, one is tempted to "opt imize" A by minimizing the residual variance at steady-state, which is given by


2 1 12" I1 - H(eJ~) t2cb , (~o) d w

+ ~2~11 - H ( e i~,) 12 + o~ll - H(ei~0l 2, (13)

where r162 is the spectral densi ty o f the noise v~. However , minimiz ing this express ion with respect to A is very cumbersome . Fo l lowing [10], we m a k e an a s sumpt ion tha t the noise var iance is small c o m p a r e d to the signal var iances , and ins tead minimize

V = p~ll - H(eJ~)l ~ + o211 - H ( e ~ ) t 2 , p~ = _ (14) 2

with respect to A. Later , in Sect ion 4, we show by numer ica l examples tha t this a s sumpt ion is not crucial . To min imize (14), we need expressions for

H ( e J ~ ) , eva lua ted at r and r Such expressions can be der ived f rom (11) with s imilar a p p r o x i m a t i o n s as before , neglecting the coupl ing between posi t ive and negat ive f requency componen t s o f the s inusoids (cf. (9b) ) . In terms o f the quant i t ies in (11) , this implies using the a p p r o x i m a t i o n s

~ , ( ~ ) = ~2(~,) = 0,

~(*02) = c~2(~o~) = 0.

F r o m (11), we then have

e-J~l~ H ( e J ~ ' ) - (k,k2 -1612) [~'(~~ + 5~(~~ (15a)

e-J~2~ H ( e ~ O - (XiX2 -161 ~) KB'(~~ + 22(~~ (15b)

where (by use o f (A.20) , (A.18) , (A.15) ) we can write the explicit expressions

~ ( ~ ) = x~x~ -1612 - 2x2(1 - c~2),

/32(o~) = 2(1 - a2)N/~2 6- p~

/32(co2) = 2(1 - a2)N/~2 6 p~

/32(~o~) = X,X2 - 1612 - 2X~(1 - o~2),

(16)

with

k~ = p , [ L - ( L - 1)2u cosco/ + a2 (L - 2)] + 2(1 - c~2), i = 1,2 .


Note that ),1 and X2 are real and 6 is complex. (See (A.12), (A.13), and (A.15)). With (15) and (16), we can write

1 - H(eJ'~O 2 ( 1 - c~ 2) [ X 2 - , j~ 161e-J'e-J~'~], (17a) klk2 -- ~2 p~

where

= arg (6) and ~ = o~, - w2.

From (14) and (17), it is clear that the magnitude of each component of V is minimized by the choice

+ ~A = 2mr.

It can be shown that arg (6) = ~ (L - 1)/2, and using this in the above relation gives the "op t ima l" delay value

A2 - 2n____~ (L - 1____~), n integer . (18) 2

(This derivation, which is somewhat simpler than our original proof, was suggested by one of the referees.) We need to know ~ and L to use formula (18). In Section 4 we shall describe an iterative technique for estimating A2 without prior knowledge of ~.

We remark here that the analysis used to arrive at (18) is an extension of the analysis of [10] to the case of two sinusoids, but with the following difference. In the case of a single sinusoid [10], no assumption was made on the coupling between the positive and negative frequency components of the sinusoid. In the case of two sinusoids, which is considered in the present paper, we have two positive and two negative frequency components. When the two sinusoids are closely spaced and their frequencies are not close to zero or to the Nyquist frequency, the separation between the two positive frequency components (or the two negative frequency components) is significantly smaller than the separation between the positive frequency component of one sinusoid and the negative frequency component of the other sinusoid. Consequently, the coupling between the two positive frequency components (or the two negative frequency components) is significantly higher than the coupling between the positive frequency component of one sinusoid and the negative frequency of the other sinusoid. To simplify the analysis, we neglected the latter coupling in comparison to the former. The numerical examples of Section 4 show that this approximation is adequate.

The comment about the approximate character of (12) (because A has to be an integer) also holds in this case. It should be emphasized that the


A2 given by (18) differs from the A1 of (12) given by the method of Zeidler, et al.

To see how the optimal A, given by (18), affects the bias in the estimates, consider the transfer function of the prediction-error filter 1 - H(eJ~). Ideally, we would like to have this as a perfect notch filter with the notches at the frequencies of the input sinusoids. However, this is impossible to achieve in practice. We therefore wish to make it as close to the ideal notch filter as possible. Thus, we want [ 1 - H ( e J ~ ) ] to be minimum at ~o = ~ol, i = 1, 2. Note that this was the condition used in deriving the relation (18) for A2. It is then clear that 1/[(1 - H(eJ~)] will have at least local maxima at ~o = o~1 and ~0~, and the level of these peaks depends on how good the notches are.

We are thus motivated to compare the above two methods with some numerical examples. This will be done in Section 4.

4. Numerical examples

In this section we shall investigate the significance of choosing A according to the rule (18) and also compare our method with that of Zeidler et al. We shall suggest an iterative scheme for estimating the optimal A, and demonstrate its effectiveness with some examples. We shall also illustrate the sensitivity to variations f rom the optimal A, and also the variance of the spectral estimate with the new method.

We used a lattice-form A-step-ahead predictor and the corresponding recur- sive least-squares algorithm [6] in the simulations. The transfer H ( Z ) can easily be calculated f rom the parameters in the lattice realization, see, e.g., [6].

Figures 1 through 9 show the results of the various simulation examples to be described below. Let us first note that the true sinusoid frequencies are marked with vertical lines in the figures. In the figure captions, N denotes the number of data samples used in the simulation. In all examples, except the one corresponding to Fig. 7, the noise v, is white. Finally, note that wherever a comparison is made, the same noise realization has been used in all the simulations involved in the comparison.

COMPARISON OF THE NEW METHOD WITH THAT OF ZEIDLER et al. Two dif-

ferent methods of computing spectral estimates, together with the corresponding " o p t i m a l " values of A, were described in Section 3. The two techniques are compared in Figs. 1 and 2. Figure 1 shows a plot of IH(eJ~)l according to the method of [3]with the choice of A given by (12). The corresponding plot of 1/I 1 - H(eJo) l for our new method with A from (18) is shown in Fig. 2. Two equal-power sinusoids of normalized frequencies 0.15 and 0.2 are used in this example. A comparison clearly reveals that the first method suffers from considerable bias in the frequency estimates as well as wide peaks


-!0

~q

,d <

�9 Z

-20

-30 - i i

.i .2

? /

I i I i !

.3 .4 .5

N O R M A L I Z E D F R E Q U E N C Y

Figure 1. Spectral est imate with the me thod of [3] (pl = o5 = 0 dB, L = 15, N = 500, A = 3) .

m

r ~

<

O 2;

- I 0

- 2 0

- 3 0 ' '

0 .1

V J

! ,~ / 1 / 1 '

i ~ i l , I

2 .3 .4 .5


Figure 2. Spectral estimate with the method of this paper (01 = 05 = 0 dB, L = 15,

N = 500, A = 13).


in the spectrum, whereas no such effects occur in our method. It is true that the criterion chosen for the method of [3] gives a deep null between the peaks, but this fact is o f less significance than the high side-lobe level. Also note that a relatively large predictor order (15) was needed to get any information at all f rom this method.

Experiences similar to the above in several examples motivate us to drop further comparison with the method of [3], and the examples below all use the new method.

COMPARISON WITH A NOISE COMPENSATION TECHIQUE OF KAY [11]. In [11], Kay describes a fairly elaborate noise compensation techique and applies it to an example with two sinusoids in noise. The sinusoid frequencies are 0.14 and 0.20. Our method was applied to his example to provide some comparison, and the result is shown in Fig. 3. Note that due to the high signal- to-noise ratio, only 100 data points were sufficient to get a good spectral estimate. The result for the near optimal A (optimal value ~ 13.17) is fairly close to the result in [11]. From Fig. 3 it is also seen that A = 13 gives con- siderably sharper peaks and much lower bias of the frequency estimates compared to the choice A = 1.

N <

�9 Z

-i0

-20

-30

0 .i

::113 /

.3 .4 .5

NORMALIZED FREQUENCY

Figure 3. Spectral estimate for an example in [11] (ol = o2 = 5 dB, L = 8, N = I00).

EFFECT OF A. The effect of the choice of & is illustrated in Fig. 4 for a more realistic example than the previous one. The SNR is now 0 dB and only

HIGH-REsOLUTION SPECTRAL ANALYsis 431

a 9th-orde:r predictor is used. The improvement a correct choice of A can bring is even more pronounced here.

m

s <

0 Z

- 1 0

- - 2 0

- 3 0

_/

1 ! 1

0 .1

! ~ - ~ j ~ = 116

l

. 2

I �9 1 1 | - - �9

. 3 .4 . 5


Figure 4. Spectral estimates at 0 dB SNR; A = 16 obtained by iterative technique (ol = p2 = 0dB, L = 9, N = 250).

AN ITERATIVE SCHEME FOR ESTIMATING A 2. Computation of A2 using rule (18) requires knowledge of ~ and L. For practical applications, we need a tec, hnique for estimating this value. The following method is proposed for finding an estimate 3~2.

Choosing an initial value of A as unity, obtain the estimate of the predictor coeffic:ients and compute the power spectrum I G(r 2. Estimate the values of co,and cos by detecting the two largest peaks in the spectrum and evaluate

/~2 = integer part [ (2_~+ 1 -2 L ) + 0.5].

Repeat the above procedure with ,5 = As. Terminate the process when the values of As obtained in two successive iterations are the same.

Three examples are considered to demonstrate the effectiveness of the above method. Figure 4 shows a case where two iterations gave As = As := 16. The second example, shown in Fig. 5(a), considered two sinusoids ,at 0.20 and 0.25 with 0 dB SNR. It took three iterations, and A2


-I0

Z 0~ -20

-30

A = I ~ J

,5 = 20

L"

| ; I J

,2 .3 .4 .5


Figure 5 (a). Use of the i terative scheme to est imate A2. Example with two sinusoids or normal ized frequencies 0.20 and 0.25 (p~ = p2 = 0 dB, L = 9, N = 250).

.d <

~:~ -20 �9 Z

-30 A J !

0 .i

0 .i

V !

.2

J I, ! I I

.3 .4 .5


Figure 5 (b). Example with two sinusoids of normal ized frequencies 0.14 and 0.20 (p2 = p2 = 0 d B , L = 8, N = 250) .


, . . . . . . ,

r.t]

<

0 Z

(a)

- 1 0

- 2 0

- 30

0

A= 1 7

/ : 9 A= 1 5 -

.1

<] . 2

~.j~ A = 17

, 3 . 4 . 5


- lo

s <

0 -20 z

A = 18

A = 1 4 - -

I 1

A = 14

- 3 0 l 1 1 I ;

C . ! .2 . 3 . 4

(b) N O R M A L I Z E D F R E Q U E N C Y

F i g u r e 6. Sens i t iv i ty to v a r i a t i o n s f r o m t he o p t i m a l A ( p l = 02 = 0 dB, L = 9 ,

N = 250.

~1 A = 1 8

.5


was obtained as 17. Note that the exact value obtained from (18) is A2 = 16. Figure 5 (b) shows the third example with two sinusoids at 0.14 and 0.20 with 0 dB SNR. In this case, the method took three iterations and gave 2i2 as 13, compared to the optimal value A2 = 13.17. We remark here that the method does not require new data in each iteration, since the same data can be pro- cessed in all the iterations.

We do not claim that the above scheme converges to the optimal A in all cases. This scheme has to be tried extensively before it can be established as a method for estimating the optimal A. This, however, is not the purpose of the present paper.

SENSITIVITY TO VARIATIONS FROM THE OPTIMAL A. The example of Fig. 4 is considered here, and spectral estimates with A = A2 _-+ t and A = A2 + 2 are computed. The results are shown in Fig. 6. The plots show that for values of A that differ f rom A2 by 1, the performance degradation is small. This can be seen by comparing the plots for A = 17 and 15 with that of the optimal case given in Fig. 4. It appears f rom the plots o f Fig. 6(a) that the side- lobe level has gone down with A = 15. But note that a corresponding decrease in the level of the first peak has taken place. The results also show that even when the actual A differs f rom A2 by 2, the peaks around the true frequencies are fairly sharp. We remark here that the same noise realization has been used in all simulations.

CORRELATED NOISE. The usefulness of the derived method for choosing A even in the colored noise case is illustrated in Fig. 7 with an example for o~ = 0.5. Here the choice A = 1 gives a poor picture of the spectrum, whereas the choice (18) clearly gives good estimates of the frequencies. The considerable difference between the two peaks probably can be explained to be due to the low-pass character of the noise spectrum. Finally note that ot ~ is small for the "op t ima l " A = 16, as assumed in the theoretical analysis of Section 2.

Low FREQUENCIES. In the analysis of Section 2, it was assumed that the coupling between the positive and negative frequency components of the sinusoids could be neglected. I f the frequencies are very low or very high, this approximation may not be very accurate. In Fig. 8, the results for an example with low frequencies (0.05 and 0.10) are shown. Clearly a remarkable improvement is obtained by going from A = 1 to A = 16. However, the two signal peaks are no longer well separated f rom the other peaks. Also, a peak at r = 0 is obtained, but this should not cause problems in a practical situation.


0

: A = I

- I 0

~ ) < -30 . . . .

o .1 .2 . 3 .4 .5


Figure 7. Spectral est imates for a case with corre la ted noise (p~ = p2 = 0 dB, L = 9,

N = 250, c~ = 0 . 5 ) .

s <

�9 Z

- 1 0

- 2 0

- 3 0

0 .1

/ ~ J . , A = 16

.4 .5 .2 . 3


Figure 8. Spectral est imates for a case with sinusoids o f low frequencies (p~ = p2 = 0

dB, L = 9, N = 250) .

436 EC;ARDT, KAILATH, AND REDDY

VARIANCE OF THE SPECTRAL ESTIMATE. To get some feeling for the variance of the spectral estimate obtained with Method 2, we computed 20 spectra with different noise realizations. Two sinusoids of normalized frequencies 0.15 and 0.20 at 0 dB SNR were used in the experiment, and A was kept fixed at A2 = 16 for the whole ensemble. We found that the variance of the estimate around the true frequencies was very small, and that the largest peaks of each spectra are essentially located at the true frequencies. The variance of the spectral estimate, however, seemed to be significant away from the true frequencies.

COMPARISON WITH BI2RG'S SPECTRAL ESTIMATE FOR SHORT RECORD LENGTHS. To compare the average performance of the new method with that of Burg's [12] for short data lengths, we simulated the sample of Fig. 3 with 50 samples of data and computed the average spectrum from 10 different spectra obtained with 10 different noise realizations. Similarly, the average spectrum was computed using Burg's technique. From the plots of average spectra, shown in Fig. 9, it is seen that on the average the new method gives better spectral estimates, both in accuracy and bias, than Burg's method. We remark here that the same noise realizations have been used in both cases.

o

-lO

-20

-30 ...... �9

o .1

f I~Z

i

.2 .3 .4

Figure 9. Average of 10 spectra for short data lengths (pl = p2 -- 5 dB, L = 8, N = 50. I -Burg ' s technique; I I -method of this paper with A = 13).

HIGH-REsOLUTION SPECTRAL ANALYSIS 437

5. Concluding remarks

In this paper, we have considered the problem of resolving two sinusoidal signals in noise using an adaptive A-step-ahead predictor. A theoretical analysis of ,the steady-state predictor performance revealed that a particular value of the, prediction horizon A is optimal in a certain sense. Computation of this value requires the knowledge of the predictor length and the difference frequency. For practical applications, therefore, an estimate of the near- optimal A !is needed, and an iterative scheme to obtain such an estimate is described.

Certain approximations and assumptions were necessary to yield simple procedures. Explicit analytical evaluation of these simplifications is not possible, but we feel that the numerical results presented here show that the new procedures have some value. Considerable improvements have been obtained with the proposed method, both in accuracy and in bias.

Appendix

The derivation of the expression (11) of Section 2 will be given here. The analysis follows the lines of those in [5], [10], and [13]. For easy reference, we first give the formula for the steady-state parameter vector a*:

a* = R;~ rxy, (A.1)

where Rxx and rxy were defined by (8). Since the noise is uncorrelated with the sinusoids,

R ~ = R . + R ~ ,

where

2 Rvv = av

R~z =

]0~ O{ , o~L-I 1

Z~-A

t=A Zt-A-L+I

Z~-~

Z t - & - L + I

(A.2)

Using complex notation, we can write

Zt-~

Z t.-~-L + I

a [ej,~,o_,~,§162 ) _ e_j~c,_,~,§162

a2 + 2-~ [eiC~'-~**~)7(~~ - e-JC~c'-~>+*'~7(-~~ (A.3)


where

7(~) = [ l e -j . . . . e-Jt~-lJ~] r . (A.4)

To s impl i fy the fo l lowing expressions, we use the def ini t ions

"r~ = 3,(~o~), 3'2 = 7(c":), and 3' = 3,(~o) �9

We then see f rom (A .4 ) tha t

~~ = 3,(-w~) , 72 = 7(-~o2) , and ~ = 3,(-~o) .

It is easy to check tha t all t - dependen t terms d i sappea r in evaluat ing (A .2 ) with (A .3 ) inser ted. Thus ,

2 2 (9-1 H --H 0- 2 H --H = 3,~] + [3,~3,~ + u ] R~ y [ 3 , , 3'~ + u -5- 7 ~

= FF" , (A.5)

where

E l ( A . 6 )

and H denotes H e r m i t i a n t ranspose . Not ice tha t the phases ~b, and ~2 o f the sinusoids do not enter in the result (A .5 ) even though no assumpt ions on the

~ s have been made . The vector rx~ can also be expressed in terms o f F . W e have

rxy ,im { N - o o ~ E / = A

Z ~-,a

Zt-~-L+I

Z~ +

V,-z

U r - - A - L + I

, ( A . 2 )

and assuming tha t A is chosen so tha t o~ ~ is small , we get the a p p r o x i m a t i o n

rv ,

z~

(A.7)

HIGH-REsOLUTION SPECTRAL ANALYSIS 439

where the last step fo l lows in the same way as (A .5 ) and

e -~ . . . . e . . . . e ~ : ( A . 8 ) v = ~ e - J ~ , ~ x/2

Using (A .1 ) , (A .5 ) , and (A .7 ) , a we l l -known ma t r ix f o rmu la can be appl ied to give

a* = R ~ r~ = (R~ + Frn) -~I 'v

[R:~ -1 = - R ~ r ( : + r - R : : r ) - l r - R~lrv-I

- 1 - 1 = - - R ~ r ) - v) R ~ F v R ~ F ( v ( I + r H -1

- 1 1 = R , , F R - v , (A.9)

where

R = I + I ' 'yRS.1".

In compu t ing R, we now make the fo l lowing a p p r o x i m a t i o n :

R = I + 2 1 I 0 01 , (A.10)) 2or(1 - c~2) L0 J

1 --et where

-o~ 1 + c~ 2

1 + c~2 -c~

- (x 1

and

[ 2 . . . . . . . 1 al 71 R ~ 71 a le2 3'1 1~ ~ 72

0 = n 2 n - - I (A.11) ~1o2 "Y2 /~ :~'Yl o272 R .~72


To replace the o f f -b lock -d i agona l mat r ices by zero , the a p p r o x i m a t i o n in (ADO) can be interpreted as neglecting coupling between positive and negative frequencies . This a p p r o x i m a t i o n is val id for the frequencies tha t are nei ther too low nor too high, as can be seen f rom the results in [10] . Here we jus t note tha t the a p p r o x i m a t i o n is jus t i f ied in the end by the results .

W e now def ine

~(~,,o:2) A ,, - - , = 3'~ R ~ 3'2

Then

(1 - e s(o,-~=l) [ (1 - e ~ ' ~ ' ' ~ , - ~ )

- ~(eS~, + e-S~)( l - eSC~-,~,~,-o~)

+ a*e :~ , -~ ' ( 1 _ eSC~-~,~,-~,)] . ( A . 1 2 )

and

~(~,, ~2) = ~ (w,, ~ ,) (A.13a)

~(w, w) = L - (L - 1)2~ cosw + a 2 ( L - 2 ) . (A.13b)

Inser t ing (A .11) into (A . IO) and eva lua t ing the entries of 0 gives

R = 1

2(1 - ~2)

X, 6 X2

X2

( A . 1 4 )

w h e r e

X, = o , ~ ( ~ , , ~ , ) + 2(1 - ~ ) ,

X2 = p25 (~2 ,~2 ) + 2(1 - ~2) . (A.15)

H I G H - R E s o L U T I O N SPECTRAL ANALYSIS 4 4 1

and pl , p2 are signal-to-noise ratios

Pl = ---~, P2 = --7 "

The quantities needed to calculate the steady-state parameter vector a* f rom (A.9) are now available. We will use the expression for a* to study the properties of the transfer function H ( z ) of the resulting predictor,

* 1 H ( z ) = z - ~ ( a ; + a~z- + . . . + a~z - ( L - ' , ) .

Using the definition of % (A.4) and (A.9) ,

H(e j~) = e-J~c~*r 7 = e-Jo~vr(R-X)rFrRS~y

From (A.6),

0-1 T~-I

if2 ~r - -1

r ~ F 2 y = ( 7 1 l-l - - - I

0 2 I t -- - 1 3'~R~y_

( A . 1 6 )

(A.17)

where the quantities ~,(~) are given by

~,(~) = ~( - ,~ , , ~),

~2(w) = ~ ( - ~o2, w), (A.18) ~ 3 ( ~ ) = ~(~o , , ~o),

~,(~o) = ~(~o2, ~).

From (A.8), (A.14), and (A.17), (A.16) can be simplified to give the final expression for H(~o) :

H(eJQ - e - J ~

[o~,(~)e -j~'~ + c~2(~)e -i~=a (A.19)

+ /3,(oJ)eJ~, ~ + /32(w)e j~=a] ,

442 EGARDT, KAILATFI, AND REDDY

where

ot2(cO) -- p2X ,~ j2 (W) - - q:P,P2 a ~,(CO) ,

/S~(.) = p , X l , ~ , ( ~ ) - jp,p= E ~ , ( , , , ) .

(A.20)

The expressions (A.12), (A.13), (A.15), and (A.18)-(A.20) provide us with the desired solution of the steady-state transfer function of the predictor.

References

[1] B. Widrow et al., "Adaptive Noise Canceling Principles and Applications," Proc. IEEE 63 1975, 1692-1716.

[2] L.J. Griffiths, "Rapid Measurement of Digital Instantaneous Frequency," IEEE Trans. on Acoustics, Speech and Signal Processing ASSP-23 (1975), 209-222.

[3] J.R. Zeidler et al., "Adaptive Enhancement of Multiple Sinusoids in Uncor- related Noise," IEEE Trans. on Acoustics, Speech and Signal Processing ASSP-26 (1978), 240-254.

[4] J.R. Treichler, "Response of the Adaptive Line Enhancer to Chirped and Doppler-Shifted Sinusoids," 1EEE Trans. on Acoustics, Speech and Signal Proc- essing ASSP-28 (1980), 343-348.

[5] F.W. Symons, Jr., "Narrowband Interference Rejection Using the Complex Linear Prediction Filter," 1EEE Trans. on Acoustics, Speech and Signal Proc- essing ASSP-26 (1978), 94-98.

[6] V.U. Reddy, et al., "Optimized Lattice-Form Adaptive Line Enhancer for a Sinusoidal Signal in Broadband Noise," IEEE Trans. on Acoustics, Speech and Signal Processing ASSP-29 (1981), 702-710.

[7] D.W. Tufts and R. Kumaresan, "Improved Spectral Resolution II," Proceedings of the IEEE Conference on Acoustics, Speech and Signal Processing, Denver, CO, April 1980.

[8] L.J. Griffiths, "A Continuously-Adaptive Filter Implemented as a Lattice Struc- ture," Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 683-686, Hartford, CT, May 1977.

[9] S.L. Marple, Jr., "Resolution of Conventional Fourier, Autoregressive, and Special ARMA Methods of Spectral Analysis," Proceedings of the IEEE In- ternational Conference on Acoustics, Speech and Signal Processing, pp. 74-77, Hartford, CT, May 1977.

[10] V.U. Reddy and A. Nehorai, "Response of the Adaptive Line Enhancer to a Sinusoid in Lowpass Noise," Proc. 1EEE, 128, Pt. F (1981) 161-166.

[ 11 ] S.M. Kay, "Noise Compensation for Autoregressive Spectral Estimates," IEEE Trans. on Acoustics, Speech and Signal Processing ASSP-28 (1980), 292-303.


[12] J.P. Burg, "A New Analysis Technique for Time Series Data," NATO Adv. Study Inst. Signal Processing with Emphasis on Underwater Acoustics, The Netherlands, August 1968.

[13] A. Nehorai and M. Morf, "Enhancement of Sinusoids in Colored Noise and Whitening Performance of Exact Least-Squares Predictors," submitted to 1EEE Trans. on Acoustics, Speech and Signal Processing.

[14] S.W. Lang and J.H. McClellan, "Frequency Estimation with Maximum En- tropy Spectral Estimates," IEEE Trans. on Acoustics, Speech and Signal Proc- essing, ASSP-28 (1980), 716-724.

[15] A. Papoulis, "Maximum Entropy and Spectral Estimation: A Review," IEEE Trans. on Acoustics, Speech and Signal Processing, ASSP-29 (1981), 1176-1186.

High-resolution spectral analysis using multiple-interval adaptive prediction

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