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IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-34, NO. 6, DECEMBER 1986 1493

A Kalman Filtering Approach to Short-Time Fourier Analysis

Abstract-The problem of estimating time-varying harmonic components of a signal measured in noise is considered. The approach used is via state estimation. Two methods are proposed, one involving pole- placement of a state observer, the other using quadratic optimization techniques. The result is the development of a new class of filters, akin to recursive frequency-sampling filters, for inclusion in a parallel bank to produce sliding harmonic estimates. Kalman filtering theory is applied to effect the good performance in noise, and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate. These filters can be seen as generalizing the DFT.

I. INTRODUCTION

S HORT-TIME Fourier analysis has been a topic of con- siderable interest in digital signal processing [ 11, [ 2 ] ,

and is concerned with the evaluation of the slowly time- varying Fourier components of a quasi-periodic signal. This problem arises in many forms in signal processing, notably speech processing and recognition, vibration analysis, power system analysis, and general time series. For example, in a power system, one is frequently inter- ested in studying the “harmonic pollution” of the fundamental mains frequency. The magnitudes and phases of the harmonics are of importance in determining the qual- ity of supply and in identifying pollution sources. Equally, the efficiency of various thyristor motor control devices can best be determined and controlled by harmonic monitoring.

The underlying signal model for situations such as these involves regularly sampled values of a continuous time signal. We presume that the period T of this signal is known or is predetermined and that there are 2N + 1 samples per period. The fundamental angular frequency is then wo = 2 r / T and we seek to determine all or part of the 2N + 1 harmonic components. The discrete-time signal model for the sampled data is

Manuscript received March 15, 1984; revised March 12, 1986. This work was supported by the Radio Research Board of Australia.

R. R. Bitmead and P. J . Parker are with the Department of Systems Engineering, Research School of Physical Sciences, Australian National University, Canberra, ACT 2601, Australia.

A. C. Tsoi was with the Department of Electrical Engineering, Univer- sity of Auckland, Auckland, New Zealand. He is now with the Department of Electronic and Electrical Engineering, Faculty of Military Studies, Uni- versity of New South Wales, Duntroon, ACT 2600, Australia.

IEEE Log Number 8610161.

where zk is the measured signal; c:, 1 = 0 , 1 , * - - ? N , andsk, E = 1 , - 1 , N are slowly time-varying functions, i.e., over time intervals of the order of one period (2N + 1) samples c: and s i are effectively constant; and u k is a stationary zero-mean white noise process independent from the remainder of zk. It is desired to estimate the functions e: and s: in such a way as to smooth the noise u k and still to track the signal variations.

The normal techniques for performing this task of short- time Fourier analysis are to block process signal values by discrete Fourier transform (DFT) together with a suitably chosen windowing function. When the short-time Fourier analysis is carried out for data compression pur- poses or signal reconstruction is otherwise envisaged, one must choose the windowing function and data block length 2N + 1 with some care to allow this synthesis [I]. The windowing methods, however, do not take into account any noise suppression properties which may be desirable in many applications. In particular, these methods to produce the harmonic analysis rely upon the DFT process which is not designed to enhance the signal to noise ratio, in spite of possibly well-known signal characteristics or redundancies.

Our approach here is to develop recursive digital filters for short-time Fourier analysis in a filter bank structure. These recursive filters are derived from an optimal state estimation specification of the short-time Fourier analysis with suitable state-variable signal models for (1.1). These are then Kalman-Bucy filters which have well-determined performance properties when used to extract signals from noise. In attempting to estimate c: and s i in ( 1 . l ) , we do not produce the exact short-time Fourier transform as de- fined in [ 2 ] , since this definition is strictly in terms of windowing followed by DFT, but rather perform the same ultimate task as that originally desired of short-time Fou- rier analysis.

The techniques that are derived do not require any block processing of signals. They generalize the DFT methods to applications where the extraction of the harmonic component signals from noise is desired. These filters are thus perhaps better regarded as a development from FIR frequency-sampling filters. Indeed, the impetus for this in- vestigation stemmed originally from the spectral observer work of Hostetter [ 3 ] and the demonstration of their link to frequency-sampling filters [ 4 ] . Section I1 of the paper is concerned with the presentation of the spectral observer approach and with an analysis of the filters generated by

0096-3518/86/1200-1493$01.00 O 1986 IEEE

1494 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-34, NO. 6, DECEMBER 1986

Kalman filtering problem. The solution to this problem is shown to be degenerate in that the Kalman filter is not exponentially asymptotically stable. This is known as “filter divergence” and several techniques are then used to avoid this, including the construction of a fictitious sig- x k = nal model which has unbounded state variance. Section IV deals with the application of these divergence allevia- tion methods to the short-time Fourier analysis problem, and Section V focuses on the evaluation of the resultant recursive filters. These filters demonstrate a very good sidelobe suppression (and hence noise rejection) and null-

this explicit methodology. An explicit design procedure related to the (2N + 1)-dimension state xk in (2 .1 ) by is developed. In Section I11 we examine in more detail the - stochastic signal model of [3] and state the associated r c1 cos ko + s1 sin ko

-cl sin k% + s1 cos k%

c2 cos 2k% + s2 sin 2k%

-c2 sin 2k% + s2 cos 2k%

cN cos NkO + sN sin Nk%

-cN sin Nk% + s N cos Nk%

The magnitude of the lth harmonic component is

mr = J(c1)2 + (S1)*

= J ( c l cos lk% + s‘ sin Z I C % ) ~ + ( -c’ sin lk% + s‘ cos lk%)2

for 1 = 1 , 2 , * - , N .

ing of the harmonics, thus showing improved leakage properties over other methods. We conclude in Section VI.

11. THE SPECTRAL OBSERVER APPROACH Although our aim is to consider a general slowly time-

varying signal as modeled in (1. l), this problem is ana- lytically intractable without explicit knowledge of the time variation of c: and s:, or of a probabilistic model of their evolution. Our approach will therefore be to consider sim- pler models than (1.1) and then to tune or adjust the harmonic estimators to perform well with a large class of possible c: and s i . We begin here with the application of state observers to the deterministic spectral estimation problem, and in the next sections treat Kalman filtering solutions to the estimation problem with noise. In particular, we shall initially consider the estimation of c: and s: when these are constant functions of time c: = c‘, s i = s‘ and vk is zero.

We therefore consider the following time-invariant deterministic state-variable equation for a strictly periodic signal with samples zk, 2N + 1 samples per period.

x k + l = Fxk (2 .1)

Zk = H ’ X k (2.2) where, writing % = 2n/(2N + l) , [ [ cos 1% sin 1 0 1

-sin 1% cos le ’ F = block diag

1 = 1, . . . 7 N ] @ 1 (2.3)

H T = [ l 0 1 0 * * * 1 0 11. (2 .4) This is equivalent to the state-variable models proposed in [3] where the 2N + 1 harmonic components of zk are

The phase of the lth component is

41 = arctan ($) = [arctan ( -c l sin lk8 + s1 cos lk%

c‘ cos lk% + s‘ sin lk% )I k0 =2?rp

wherep is an integer. Equally, the value of the state vector xk at any particular time k will determine the harmonic components at any other time because F in (2.1) is invertible.

The problem of causally estimating the current state of a linear signal model from measurements of inputs and outputs is familiar from linear control systems design [5] , [6] and involves the construction of a state observer

. f k + 1 = ( F - MHT)$k + Mzk (2.5)

where M is a (2N + 1)-vector observer gain. It is well known [ 5 ] , [6] that the state estimate i k converges to x k

geometrically according to the eigenvalues of F - M H T and, further, since [F, HI is a completely observable pair, that these eigenvalues may be arbitrarily placed in the complex plane by choosing M suitably.

This observer approach to spectral estimation has been proposed in [3] where M is chosen to place all the eigenvalues of F - M H T at the origin. This leads to a “deadbeat” observer where & converges exactly to x k in 2N + 1 steps; thus yielding exactly the 2N + 1 DFT components as a simple function of the elements of the state vector. It was then shown in [4] that this was identical to the passing of the sampled signal zk through a parallel bank of FIR frequency sampling filters, which again is equivalent to computing a sliding DFT-longhand [7]. This filter bank notion and the idea of a signal model are depicted in Fig. 1. The choice of M in the deadbeat case can be made explicit, as in [4] (modulo a correction to the arith- metic in [4]), as

BITMEAD et al.: KALMAN FILTERING APPROACH TO SHORT-TIME FOURIER ANALYSIS 1495

SIGNAL ~ MEASURED ~ OBSERVER MODEL SIGNAL 1

I I I I I I I I I - 1 - state vector1

state

k

harmonic component estimate

2. ' measurement 1 noise vk I

K

I

Fig. 1. Harmonic estimation scheme

2N2a 2N27r M T = ~

2N + 1 ['Os G7 2N + 1' sin ___ .

9

cos ~

We thus have a solution to the spectral estimation problem for the signal model described by (2.1) and (2.2). For this setup, the choice of a deadbeat observer provides the fast- est convergence rate of any observer and the spectral estimation is exactly equivalent to standard short-time Fou- rier analysis using a rectangular window and DFT, except that a sliding spectrum is produced rather than by block processing as with the FFT.

In problems involving the tracking of time-varying quasi-periodic processes, this method performs well because the digital filters implementing the observer are FIR, and so have a finite memory. If, however, we consider the signal models as in (1. I) involving additive measurement noise, then fast convergence .rate of the state estimator implies poor noise smoothing properties. (This will be more fully discussed later.) Conversely, slow convergence rate implies good noise rejection but poor tracking of the time-variations of the harmonic components. It should be remarked that any block-processing spectral estimation method will have similar noise problems to the deadbeat observer because the number of data points is usually quite small when compared to the coherence length of the signal. Consequently, it is profitable to consider methods which allow observers to be designed to trade off these opposing objectives.

The characteristic equation of the system matrix F of (2 .1) i.s X 2 N f 1 - 1 = 0, i.e., all the eigenvalues of F are equally spaced around the unit circle. The characteristic equation of F - M H T of [3] is X Z N f ' = 0, since all the observer poles have been shifted to X = 0. We may at- tempt to compromise between these two extremes of mar- ginally stable and deadbeat observers by designing observers with poles equally spaced around a circle of radius K C 1. That is, we desire to choose M such that

det (XI - F + M H T ) = X Z N f l - K ~ ~ + ~ . (2.7)

The convergence rate of the observer will then be K ~ .

Given that we have computed the M-matrix and we choose to implement the resultant spectral observer using

(2 .5) , then one may compute the transfer function of the observer as the (2N + 1) X 1 rational matrix

W(z) = ( z l + F + MHT)-'M

which takes the input Z ( z ) , the z-transform of { z k } , to f ( z ) . The components of this vector of rational functions are the transfer functions of the individual filters which take as input z k and produce as output the short-time Fou- rier harmonic components. Each of these outputs, or rather sine/cosine pairs of outputs, may then be processed to yield magnitude and phase information if desired.

The filters proposed in this section are recursive and were designed to provide an alternative approach to the DFT for the estimation of harmonic signal components. By choosing an IIR filter, we have incorporated a capability to smooth additive signal noise at the cost of a slower convergence rate. For the short-time Fourier analysis problem, one would choose the design parameter K

to balance noise rejection versus tracking speed. Thus, the information regarding the noise variance and signal time variation can be used to select an appropriate filter design. However, there is an arbitrariness about the pole positions, and consequently, there may be a better approach. Indeed, there is an enormous body of literature concerned with optimal state estimation in the presence of measurement and state noise, and we shall explore the application of these techniques in the next section to derive Kalman filtering solutions to the short-time Fourier analysis problem.

111. THE KALMAN FILTER APPLIED TO SPECTRAL ANALYSIS

Consider the following problem. We have available measurements of the signal zk which we presume to be generated by the following signal model:

x k + 1 = F X k + GWk (3.1)

zk = H T X k + u k (3.2)

where wk and u k are zero-mean Gaussian white noise processes satisfying

,r 1 -l

I

where Q 2 0, R > 0, and the initial state x. is an Zo-mean Gaussian random variable with covariance Po 2 0. We ask the question: how can we causally estimate the state xk using measurements zk so that E 1 xk - &k- l 2 is min- imum, wherizk/k- I represents the estimate of the x k given (zi: i = 0, * * , k - l}? This is a particular variant of the optimal filtering problem as stated in [ 101 and [ 111 and the solution is the Kalman filter.

& + I l k = (F - K k H T ) . f k / k - l + K k Z k , 2 0 l - l $0 (3.4)

K k = F C k / k - l H ( H T C k , k - l H + R)-' (3 .5 )


e k + Ilk = F[ck/k- 1 - c k / k - 1

- H ( H T & _ l H + R)-’

- HTEk,k- l ] F T + GQGT (3.6)

EOi-1 = Po. (3.7)

If we wish to generate a best estimate &k of xk given information up to and including zk, this may be generated as

K k = Ek/k-lH(HTEk/k-lM + R)-’

i& = (Z - E k H T ) . t k , k - l + Ekzk. (3.8)

The immediately noticeable features of the Kalman filter are, first, that the filter is a time-varying system, and, second, that the implementation via (3.4) is identical to the observer form (2.5) except that the time-invariant gain vector M has been replaced by a time-varying Kk computed according to (3.5)-(3.7). This time variation is an artifact of the initial conditions, and the requirement of optimality at each time instant - C k + is the covariance of ik+ I/k - and a time-invariant filter may be derived for this time-invariant signal model either by letting time k tend - to infinity or by selecting X, = 0 and Eo, - equal to E, the solution of steady-state Riccati equation

E = F [ C - C H ( H T C H + R ) - ’ H T C ] F T + GQGT. -

(3.9) In this case, Kk becomes fixed and the Kalman filter is a time-invariant observer [lo] whose gain matrix is chosen to satisfy an optimality condition of state estimation.

In applying the Kalman filtering formalism to the short- time Fourier analysis problem, it is natural to consider modifying the signal model (2. I), (2.2) by adding a zero- mean white measurement noise, i.e.,

x k + l = Fxk (3.10)

zk = H’Xk + uk (3.11)

where R is the covariance of u k and F, H are given by (2.3) and (2.4). Thus, the signal model corresponds to a strictly periodic process obscured by white measurement noise. Given initial conditions io/ - and Eo, - , (3.4)-(3.6) allow us to implement the optimal state estimator for this problem-optimal in terms of minimizing the covariance of the state estimation error. Thus, for this particular signal model, all other methods of spectral analysis including DFT procedures cannot outperform the Kalman filter according to the optimality measure above.

If we now consider utilizing the steady-state time-invariant version of the filter, we see that we need to solve (3.9) for E when Q = 0. We have the following lemma.

Lemma I [ I 71: If [F, HI is a completely detectable pair, then there exists a unique maximally nonnegative definite solution E to the steady-state Riccati equation (3.9).

It then is apparent that the limiting value of for the problem specified by (3.10) and (3.11) is zero because Q

= 0. This can be interpreted from two viewpoints. First, if our knowledge of the initial state is exact, i.e., Eo, - 1

= 0, then, since we presume xk to be truly periodic, we know the state exactly at all subsequent times since it evolves as (3.10). Equivalently, if we do not know no precisely, it is clear that, as we have a periodically repeating state obscured by independent noise, we may recover the state-to-arbitrary accuracy by averaging over a suffi- ciently long interval. Thus, in the limit our state error covariance’is zero. We have the following proposition. Proposition I : The optimal (Kalman) filter for gener-

ating ik,k-l from zk, which evolves according to (3.4)- (3.6), is degenerate in the sense that Kk -+ 0 as k -+ 00. This filter is thus stable but not exponentially asymptotically stable.

This property of filter degeneracy, and asymptotically ignoring the incoming measurements, is well known in the optimal filtering circles as “filter divergence” [lo], [l 11 , and arises from the uncontrollability of the state- variable signal model (3. lo), (3.11). The stability of the filter follows by Lyapunov methods [IO] , [ 111. The Kal- man filtering solution to the problem of extraction of harmonic components from noise is optimal for this formu- lation, however, it is clearly inappropriate for use in short- time Fourier analysis since it cannot maintain a tracking capability for signals with slowly varying spectra.

Several methods have been proposed to help deal with the problem of filter divergence [IO] and, in spite of their seemingly disparate origins, these methods may be interpreted as acting in similar manners by effectively intro- ducing an extra, fictitious state noise term into (3.10) as in (3.1). By thus altering the signal model slightly, it becomes possible to utilize the Kalman filtering formalism to derive an optimal filter for the new problem which has a useful effect on the performance when applied to the original signal model.

Our aim will be to derive a suitable problem formula- tion whose stationary Kalman filter solution will have the desired frequency-sampling properties to allow its use in Fourier analysis, but which is exponentially asymptotically stable with a tunable rate, to allow compromise between noise smoothing and tracking properties. The particular methods which we shall investigate will be: adding state noise, exponential data weighting, and covariance setting. These methods and their effectiveness will be considered in the next sections.

Iv. KALMAN FILTER SOLUTIONS TO THE SHORT-TIME FOURIER ANALYSIS PROBLEM

We shall consider three different approaches to the filter divergence problem and illustrate their effectiveness in producing desirable filter properties for the retrieval of the slowly time-varying harmonic components of a quasi-periodic signal in white measurement noise. As remarked earlier, each of the methods is similar in effect, and this will be shown in spite of their seemingly different raisons d’etre. The required filter properties which we shall con-

BITMEAD et ai.: KALMAN FILTERING APPROACH TO SHORT-TIME FOURIER ANALYSIS 1497

centrate on wjll be: tunable exponential degree of stability to facilitate filter design for differing signal-to-noise ratios and rates of change; goad noise suppression and harmonic isolation for adequate performance with stationary periodic signals in noise; and quantifiable performance characteristics. The degree of satisfaction of these aims will be considered after each of the methods of modifying the signal model is presented.

I ) State Noise Znsertion: Here we consider the signal model

X k + l = FXk + GWk (4.1)

zk = HTXk + uk (4.2)

with Fand Has in ( 2 . 3 ) and (2 .4) , (wk, uk) satisfying (3 .3 ) with Q and A nonsingular, and G chosen so that [F, GI is completely controllable. (G = I will dg.) The aim of adding state noise is to disturb the modeled state xk to prevent its being deterministically predictable, thus forcing the state estimator not to disconnect from the measurements asymptotically.

The steady-state Riccati equation (3 .9) now no longer admits the solution E = 0, because of the addition of the nonzero GQGT term. Indeed, since [F, G] is completely controllable, must be positive definite and the resulting F - KHT must be exponentially stable [ 1 2 ] , [ 1 3 ] . We note two important features of this approach. First, it is difficult a priori to quantify the precise convergence rate gained by selecting a particular G and Q , although computation should be straightforward once F - KHT is derived. Second, since F has all its eigenvalues on the unit circle and wk is a white process, (4.1) represents an equation forxk which implies that it has unbounded moments- effectively, xk will be a collection of modulated random walk processes whose variance increases linearly with time. It is precisely this aspect of state behavior in ( 4 . 1 ) and ( 4 . 2 ) which guarantees the exponential forgetting of the steady-state Kalman filter (see [ 10, ch. 61, and [ll]).

Note: We should stress that the unstable signal model zk does not imply that our real signal in (1.1) has this unbounded moment property, rather it is a mechanism for adjusting the optimal filtering framework to provide for uncertainty in the evolution of the c:, s: harmonic coefficients. A stability proof of the resultant steady-state Kal- man filter will be derived shortly.

We shall next consider a different technique for guar- anteeing filter stability, but with a prescribed degree of stability.

2) Exponential Data Weighting [IO], [Ill: If the cost criterion of the optimal estimation problem is modified by weighting the cost of estimate errors at time k by a’ for constant a > 1 , then one achieves new filter design equa, tions yielding new design covariances x&- 1, and Kalman gain KE (see [lo, p. 1361). These same matrices can be derived from modifications to the signal model by either a) adding extra state noise (a - 1)1’2Fwk where wk has (a priori unknown) variance x&; or b) by replac- ing I; with aF. Since the effect of exponential data

weighting is to increase input noise, all the benefits of this latter procedure accrue here. Also, it can be demonstrated that equivalence b) guarantees that the resulting Kalman filter has degree of stability a greater than that of the Kal- man filter for the original problem ( 3 . lo), (3.11). That is to say here that all the poles of the exponentially data weighted filter are inside a circle of radius 01 - I .

This method attracts the advantages of increasing input noise and also produces a prescribed degree of stability of the steady-state filter. The implementation of such a procedure still requires the solution of the steady-state Ric- cati equation to determine K and for filter design and performance estimations, and we will develop now a method which avoids this explicit computation. 3) Covariance Setting: The approach of explicitly

adding state noise to the signal model leads to positive definite E’s and hence stabilizing K ’ s . The alternative technique to be used here and alluded to in 1191 is summarily to constrain the matrix &/k - to be suitably positive definite. In effect, we shall set = P for a given positive definite matrix P and generate K from ( 3 . 5 )

K = F C H ( H ~ C H + ~ 1 - l

= FPH(HTPH + R)-’. (4 .3)

Note that by using this design method we obviate the need to solve any Riccati equation, and so, this involves re- duced computational effort.

Equation (3.9) can be used to solve for GQGT to show that this covariance setting corresponds to state noise in- sertion equivalent to considering the signal model with state equation

X k f l = Fxk + t k

and noise variance

E[tkt:] = P - F [ P - PH(HTPH f R ) - ’ H T P ] F T ,

( 4 . 4 ) provided P is chosen to make the right-hand side of ( 4 . 4 ) nonnegative definite. We thus have an equivalence with method 1) . Equally, since F is invertible, we may factor t k = FTk and derive an equivalence with 2). Thus, covariance setting attracts the benefits of the previously proposed schemes without the need to solve a Riccati equation for the limiting solution although one must ensure that ( 4 . 4 ) is nonnegative definite. Further, if P is chosen as €1, the resultant observers will be a single-parameter family of filters and thus amenable to simple selection rules by quantifying performance in terms of E .

Having demonstrated the broad equivalence of these methods, we may now present general results concerning the Kalman filters derived by covariance setting which will be particularly applicable to the achievement of the short- time Fourier analysis goals stated at the beginning of this section.

Theorem I : Consider the matrices F and H given by (2 .3 ) and ( 2 . 4 ) . Let the 1 X 1 matrix R = r > 0 and let P = EZ for some scalar E > 0. Then defining


K = F P H ( H ~ P H + ~ 1 - l (4.5)

the matrix [F - K H T ] has all its eigenvalues strictly inside the unit circle.

Proof: Writing the right-hand side of (4.4) and noting that a) P commutes with all other matrices, b) F is orthogonal so that FFT = I , and c) ( H T P H + R ) is a scalar ( N + 1 ) ~ + r , we have

P - F [ P - PH(HTPH + R)- 'HTP]FT = a2FHHTFT

for a 2 = E'((N + 1 ) ~ + r ) - ' . Thus, P is the solution of an algebraic Riccati equation (3.9) with (F , G, H , Q , R ) replaced by ( F , FH, H , I , R ) . That is, these P and K are the steady-state solutions of a Kalman filtering problem which has these latter matrices. Since the pair [F, HI is completely detectable, and consequently, [F , FH] is completely stabilizable, F - KHT is exponentially asymptotically stable.

tiw The thrust of this result is that while proposition 1 shows

that the steady-state Kalman filter for (3.10) and (3.11) is degenerate, we can summarily construct a stable filter

i k + l l k = [F - K H T ] $ - l + H z ~ (4.6)

with K given by (4.5) simply by assigning P = EI for some E. This is, in fact, the steady-state Kalman filter for a different problem, but this generates a class of digital filters, parameterized by E , which possess favorable properties, as we shall see. The filter is only optimal in the case that the signal truly is generated by (4.1) and (4.2) with state noise variance (4.4). In this case the steady- state covariance is equal to P.

It is worth noting that these methods for handling short- time Fourier analysis using the Kalman filter structure produce time-invariant recursive digital filters described by (4.6) with input the quasi-periodic signal zk and output the vector Zk of harmonic component estimates. Since zk is a scalar measurement signal and ikIk- is a (2N + 1)- process, the Kalman filter is a single input (2N + 1)-output filter, which is best interpreted as a parallel bank of recursive digital filters as depicted in Fig. 1. This concept is quite familiar to short-time Fourier analysis [14], [15] and, indeed, the DFT-based approach can be considered as a bank of frequency sampling filters so that, in effect, the novelty of the approach here is that a different and new class of bandpass filters has been used in a standard stmc- ture. If one rewrites the state-variable form of (4.6) as a vector transfer function

H(z ) = [zI - F + KH']-'K, (4.7)

one recovers the individual transfer functions from the signal input to each harmonic component, thus allowing all the flexibility of frequency sampling design but with filters having different performance characteristics. We shall move on in the next section to consider the particular properties of these newly derived filters as they relate to the short-time Fourier analysis problem.

v. FILTER PERFORMANCE IN SHORT-TIME FOURIER ANALYSIS

Recall that the previously stated desired properties of the filters are tunable tradeoff between exponential stability and noise smoothing, harmonic isolation to prevent leakage, and quantifiable noise performance. We shall demonstrate how each of these aims is achieved beginning with an analysis of the noise performance.

Noise Performance The root of the difficulty in applying strict design meth-

ods to the short-time Fourier analysis problem is that one usually wants to avoid making hard assumptions concerning the evolution of the time-varying harmonic components. Here we have effectively modeled this variation as arising from a random state noise process. In attempting to quantify noise performance without presuming a model for harmonic evolution, we consider evaluating the steady- state estimate covariance for the situation where the filter is applied to a stationary periodic process with additive white noise. (We already know that the optimal state estimate here is asymptotically exact.) We have the following.

f ieorem 2: Consider the signal z k generated by

x k f 1 = Fxk (5.1)

z k HTXk + u k (5.2)

where F is given by (2.3), H by (2.4), and u k is zero-mean white noise with variance R. If we choose a positive definite symmetric matrix P = EZ and implement the harmonic component estimator

i $ + l , k = [F - K H T ] & - l + KZk (5.3)

where K = FPH[HTPH + R] -', then the steady-state estimation error covariance satisfies

P L lim E[xk - ikIk- ' 1 [xk - ikIk- 2 0. (5.4)

Proof: The right-hand inequality of (5.4) follows since the optimal state estimate covariance is zero, while the left-hand inequality is established by standard argu- ments of [lo, ch. 61. vvv Exponential Stability

We know from theorem 1 that if we choose any positive definite matrix P = e l , then the resultant filter (5.3) is exponentially asymptotically stable. The desired, conjec- tured, but as yet unproven stability result is that the degree of stability of the filter improves with increasingly more positive definite P matrices. We do have the following two partial results.

Lemma 2: If we choose any two positive definite matrices P1, P, satisfying P I > P, > where is the solution of the algebraic Riccati equation (3.9) and derive Ki = FPjH[HTPiH + R I P ' , i = 1, 2, we have

ldet [F - K,HT] 1 < ldet [F - K2HT] 1 .

T

k - m

BITMEAD et al.: KALMAN FILTERING APPROACH TO SHORT-TIME FOURIER ANALYSIS 1499

Lemma 3: If we choose P = EZ and form K = FPH[HTPH + RI- ' , then F - KHT has rank H eigenvalues approaching zero as E -+ 03.

The first of these results shows that the geometric mean of the eigenvalues of the filter is a monotonic function of choice of P, while the latter describes the high P limit. It should be noted that, since [ F , HI is an observable pair, an M exists which causes F - MHT to have all its eigenvalues at zero. This M does not have the form of a Kalman gain, however. Thus, the DFT is not a Kalman filter form.

In order to reinforce some of these ideas of improved degree of stability for large P, consider Fig. 2 which de- picts the root locus diagram of det[XZ - F + KHT] for a third-order system (dc term plus fundamental) for P = EZ as E varies from 0 to 03. Notice that the root loci begin at the periodic system poles equispaced around the unit circle, and that as E is increased, the filter poles move to- wards the origin. Since rank H = 1 here, lemma 4 pre- dicts that one pole will tend to the origin as E diverges to infinity.

Response of the Harmonic Filters Figs. 3 and 4 present frequency response magnitude

data of the dc filter for a 15th-order system. The filter of Fig. 3 has been designed with P = 0.1 Z and R = 1, while that of Fig. 4 with P = 0.021 and R = 1. The broken lines on these graphs show the frequency response of the 15th-order frequency sampling filter which gives a comparison between the magnitude frequency responses of the discrete Fourier transform (broken lines) and the Kalman filter derived filters (solid lines).

There are four main points worth noting: the sidelobe suppression, null locations, sharpness of peaks, and the effect of changing P and R. The first three of these are artifacts of the design criterion of separating the harmonic components of a periodic signal in noise. Thus, the response at the center frequency is 0 dB and the out-of-band noise is suppressed by rapidly falling sidelobes. In addi: tion, harmonic isolation is preserved by locating the filter nulls at the center frequencies of neighboring filters. This is an important consequence of choosing an unstable signal model (4.1)-that the filter zeros remain equispaced around the unit circle. Finally, the degree of stability of the filters is improved as P is increased and/or R is de- creased. This, again, is intuitively reasonable given the design philosophy,

For a periodic signal with added measurement noise, Fig. 5 shows how the estimate of one particular harmonic component of that signal evolves with time. The input signal was a square wave and the displayed plots are the estimates of the 3rd harmonic component. The abscissa values plotted are integral numbers of the fundamental period of the signal and the filters'had order 32.

The plots demonstrate clearly several aspects of the DFT and Kalman filters. The DFT converges in one period, but subsequently exhibits poor noise rejection as ex- pected. The Kalman filters, on the other hand, converge

Fig. 2. Root locus diagram for third-order filters.

-1.0 -06 -0.6 -0k -02 (D 0.2 U 0.6 QB 1.0

Fig. 3 . Magnitude response of order 15 dc filter. Curve number: 1) Kal- man filter, E = O , l , R = 1; 2) DFT (frequency sampling filters).

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 FREWENCY (x r -11

Fig. 4 . Magnitude response of order '15 dc filter. Curve number: 1) Kal- man filter, E = 0.02, R = 1; 2 ) DFT (frequency sampling filters).

noise smoothing. Convergence rate and noise rejection are traded off against each other by choosing different design parameter E . Compare the curves for E = 0.01 and E =

more sIowIy to the correct resuIt, but then show good 0.1. Fig. 5 iIIustrates the advantage of the Kalman fiIters

i500 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH. AND SIGNAL PROCESSING, VOL. ASSP-34. NO. 6, DECEMBER 1986

CURVE NO. 1- 2- 3 . . , . . . . . . . . . 4 _ _ ~ _ _ _ _

Fig. 5 . Time evolution of harmonic components. Curve number: 1) ref- erence value, DFT of signal without noise; 2) estimate from signal with noise using DFT; 3) estimate from signal with noise using Kalman filter, E = 0.01, R = 1; 4) estimate from signal with noise using Kalman filter, E = 0.1, R = 1.

over the DFT, namely, a tunable degree of noise rejection by selecting E in the design stage. The filters are linear, therefore, their response to a signal with slowly time- varying harmonic components is exactly the same as the response with zero initial conditions shown in Fig. 5.

VI. CONCLUSION The aim of short-time Fourier analysis is to estimate the

slowly time-varying harmonic components of a quasi-periodic signal. We have considered the short-time Fourier analysis problem when the signal is measured in the presence of noise. Our approach has been via state estimation theory.

Without presuming any particular form of time variation for the harmonic components of the quasi-periodic signal, we have arrived at two design procedures which produce filters with desirable properties. Therefore, we propose that the methods given here will be suitable for a large class of quasi-periodic signals. Both solutions we have proposed yield a parallel bank of recursive filters. A sliding (synchronous) estimate of the signal’s Fourier coefficients appears at the filter bank outputs.

The first approach uses pole placement of a state observer. The d,esign assures a predetermined degree of stability. The user is free to trade off the speed of response of the filter to time variations against the ability of the filter to smooth noise.

The second method is based on quadratic optimization theory. For this we use Kalman filtering theory utilizing a signal model which is different to the actual signal in that unbounded covariances of the states occur in the model. The new class of filters which arises from this method has many desirable properties. They are exponentially stable. They exhibit complete harmonic isolation by nulling other harmonic frequencies. There is a tunable noise smoothing which is balanced against variable sidelobe suppression and speed of response time variations in the signal. A comparison to the DFT has been given and

the advantages of this new class of filters over the DFT are shown. These advantages include noise smoothing and sidelobe suppression while giving a sliding spectrum. The process of windowing followed by DFT requires batch processing and has a computational advantage over the current methods. Its noise rejection properties, however, are not tunable and may be poor. In particular, it is difficult to use the known, slowly time-varying harmonic structure of the signal to advantage.

It should be stressed that the class of filters which we have derived from quadratic optimization techniques is an artifact of the assumed signal model. They are intended for use in applications where the fundamental frequency of oscillation of the actual signal is known and this is used to overcome the effects of noise. Their advantage over other methods for handling filter divergence in quadratic optimization problems is that the solution of a matrix Ric- catti equation is not needed. Therefore, the aforemen- tioned desirable properties become available without con- siderable computational effort at the design stage. Good- win et al. [18] propose a similar method to eliminate sinusoidal disturbances due to rotor motion from helicopter flight data. Recently, Kitagawa [16] has presented related Kalman filtering-based methods for the estimation of time-varying spectra which fit autoregressive models to the signal data yielding estimates of nonharmonically related frequencies. His resultant filters are time varying in both coefficients and order, and require the monitoring of an Akaike information criterion measure to determine the suitable model order at particular times. His method of using more slowly varying parameter variation models than the random walk model used here can, of course, be applied to generate different stationary filters using the methods of this paper, although this could be of question- able benefit given that an exponentially stable filter will be produced by either model structure.

ACKNOWLEDGMENT The authors wish to thank Prof. -B. Anderson, Prof. M.

Gevers, and M.-A. Poubelle for numerous helpful com- ments and suggestions.

REFERENCES [I] J. B. Allen and L. R. Rabiner, “A unified approach to short-time

Fourieranalysis and synthesis,” Proc. ZEEE, vol. 65, pp. 1558-1564, Nov. 1977.

[2] R. W . Schafer and L. R. Rabiner, “Design and simulation of a speech analysis-synthesis system based on short-time Fourier analysis,” ZEEE Trans. Audio Electroacoust., vol. AU-21, pp. 165-174, June 1973.

[3] G. H. Hostetter, “Recursive discrete Fourier transformation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASP-28, pp. 184- 190, Apr. 1980.

[4] R. R. Bitmead, “On recursive discret.e Fourier transformation,” ZEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 319- 322, Apr. 1982.

[5] T. E. Fortmann and K. L. Hitz, An Introduction to Linear Control Systems. New York: Marcel Dekker, 1977.

[6] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980.

[7] E. Gold and C. M. Rader, Digital Processing of Signals. New York: McGraw-Hill, 1969.

BITMEAD et 01.: KALMAN FILTERING APPROACH TO SHORT-TIME FOURIER ANALYSIS 1501

[SI S. Barnett, Matrices in Control Theory. London, England: Van Nostrand Reinhold, 1972.

[9] C. C. MacDuffee, The Theory of Matrices. New York: Chelsea, 1950.

[lo] B. D. 0. Anderson and J . B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979.

[ l l ] A. H. Jazwinski, Stochastic Processes and Filtering Theory. New York: Academic, 1970.

[12] V. Kucera, “The discrete Riccati equation of optimal control,” Ky- bernetika, vol. 8, no. 5, pp. 430-447, 1972.

[13] S. W. Chan, G. C. Goodwin, and K. S. Sin, “Convergence properties of the Riccati difference equation in optimal filtering of nonsta- bilizable systems,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 110-118, Feb. 1984.

[14] R. W. Schafer, L. R. Rabiner, and 0. Hemnann, “FIR digital filter banks for speech analysis,” Bell Syst. Tech. J . , vol. 54, no. 3, pp. 531-544, Mar. 1975.

[15] R. W. Schafer and L. Rabiner, “Design of digital filter banks for speech analysis,” Bell Syst. Tech. J . , vol. 50, no. 10, pp. 3097- 3115, Dec. 1971.

[16] G. Kitagawa, “Changing spectrum estimation,” J . Sound and Vi- bration, vol. 89, no. 3, pp. 433-445, 1983.

[17] M.-A. Poubelle, I. R. Petersen, M. R. Gevers, and R. R. Bitmead, “A miscellany of results on an equation of Count J . F. Riccati,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 651-654, July 1986.

[18] G. C. Goodwin, R. J . Evans, R. Lozano Leal, and R. A. Feik, “Sin- usoidal disturbance rejection with application .to helicopter flight data estimation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 479-484, Juris 1986.

[19] A. P. Sage and J . L. Melsa, Estimation Theory with Applications to Communications and Control. New York: McGraw-Hill, 1971, p. 415.

Robert R. Bitmead (S’76-”79-SM’86), for a photograph and biog- raphy, see p. 879 of the August 1986 issue of this TRANSACTIONS.

Ah Chung Tsoi (S’70-M’72-M’84) was born in Hong Kong in 1947. He received the Diploma of Technology in electrical engineering in 1969, the M.Sc. and the Ph.D. degrees in control engineering from the university of Salford, Salford, En- gland, in 1970 and 1972, respectively, and the B.D. (extra-mural) degree in theology from Otago University, Dunedin, New Zealand, in 1980.

He is currently a Senior Lecturer in Electrical Engineering at the Australian Defence Force Academy, Canberra, Australia. From 1972 to

1974 he was a Senior Research Fellow in Distributed Parameter Systems at the Inter-University Institute of Engineering Control, University College of North Wales, Bangor. From 1974 to 1977 he was a Lecturer at Paisley College of Technology, Scotland, and from 1977 to 1982 he was a Senior Lecturer in Electrical Engineering at the University of Auckland, New Zealand. He has held visiting positions at the Department of Systems En- gineering, Australian National University, Canberra. His research interests are in control and digital signal processing.

Dr. Tsoi is an associate member of the IEE, England.

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