184 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-28, NO. 2, APRIL 1980

Recursive Discrete Fourier Transformation

Abstract-This paper presents new discrete Fourier transform methods which are recursive, expressible in state variable form, and which in- volve real number computations. The algorithms are especially useful for running Fourier transformation and for general and multirate sampling.

Numerical examples are given which illustrate the ability of these spectral observers to operate at sampling rates other than the Nyquist rate, to perform one-step-per-sample updating, and to converge to the spectrum in the presence of severe numerical truncation error.

M I. INTRODUCTION

ODERN signal processing relies heavily upon spectral computation. There exists, in practice, a number of

situations for which conventional discrete Fourier transform (DFT) methodsare complicated or awkward. This paper pre- sents new recursive DFT algorithms which are especially useful for running Fourier transformation and in general multirate sampling situations.

Although these methods do not rival the speed of the fast Fourier transform [ l ] - [4] for large numbers of sample points, they offer the following advantages:

1) All operations are with real numbers, simplifying pro- gramming on many minicomputers and microprocessors.

2) A state variable format may be used, if desired. 3) Sampling rates other than the Nyquist rate are easily

accommodated, as are various multirate sampling situations. 4) Progressive incorporation of new samples, discarding the

old as in running Fourier transformation, requires only a single iteration. If desired, a fading memory may instead be used for discarded samples.

5 ) Numerical errors in most calculations, such as roundoff or truncation (if they are not so severe to cause instability), may be corrected by recirculating the signal samples.

These results are a natural extension of the work of many re- searchers. The approach of modeling unknown system input dynamic dates from the work of Johnson [ 5 ] , [ 6 ] , Dav- ison [7], [8] . Bryson and Luenberger [9], Belanger [lo], a d Young and Willems [I I] considered similar problems. Hostetter and Meditch related Davisons work to observer theory [12] and investigated system algebraic structure and properties in detail [ 131 , [ 141 .

Manuscript received June 24, 1979; revised November 8,1979. This work was supported in part by a grant from the Long Beach Founda- tion, California State University, Long Beach, CA.

The author is with the Department of Electrical Engineering, Cali- fornia State University, Long Beach, CA 90840.

A. Discrete Fourier Transform Consider a band-limited periodic function

y ( t ) = d o + 5 (an COST+& n2nt n = i T

where T is the period of the function, N is the harmonic number of the highest frequency present in y ( t ) , and do, the as, and the 0s are the functions Fourier coefficients. In quadrature (sine and cosine) form, the discrete Fourier trans- form of y ( t ) is the determination of its 2N + 1 Fourier coef- ficients from a set of 2N + 1 evenly spaced time samples ~ 5 1 - ~ 7 1

Of course, the complex exponential form of the series offers notational advantages in most situations. This development, however, will involve real number calculations exclusively.

The Fourier transform of the periodic function y ( t ) is

Y ( j o ) = 2nd06 (a) + n [ (an -ion) 6 ( - - ?) n =I

where 6 (e) is the Dirac delta function.

B. Band-Limited Periodic Signal Representation

A linear, time-invariant homogeneous differential equation, with characteristic polynomial

s [P. + ( 3 2 ] [? + ( 3 7 . - [.2 t ( y 2 ] , has a general solution of the form (l), where the Fourier coefficients do, a1, * , a ~ , P I , * * * , P N are arbitrary constants dependent upon the system initial conditions. The general solution of such an equation is an arbitrary periodic function with period T and band limit beyond the Nth harmonic.

A convenient state variable representation for this homo- geneous system is the following:

0096-3518/80/0400-0184$00.75 0 1980 IEEE

HOSTETTER: RECURSIVE DISCRETE FOURIER TRANSFORMATION 185

0 1 0 0 . .- 0

0 0 0 1 . * . 0

0 0 0 0 * . * 0

0 0 0 0 ... 0

=Ax,

y = [ l 0 1 0 * * . 1 0 1 ] x

= c x. T

In this form of the state equations, the constant and each har- monic term in the response are decoupled from one another.

Although the band-limited periodic signal y( t ) may not actually be produced by the system (2), it may be considered to be the output of this system. Were the system (2) pro- ducing y(t) , the individual harmonic components of y would be the signals

2nt 2nt T T

2nt

x1(t)=a1 cos-++I sin-

x 2 ( t ) = - J - = - d l cos-- al sin- & "( T dt T T

4nt 4nt T T

xg(t) = a2 cos - + 0 2 sin -

Discrete Fourier transformation is thus equivalent to the determination of the system (2) initial conditions

x1 (0) = a1

(I

0

0

0

1

) 2 0

0

x1

X2

x3

x4

X2N-1

X 2 N

x2N+1.

X2N+l(O) = do

from samples of y (t).

11. SPECTRAL OBSERVATION A. Discrete-Time Signal Model

At is A discrete-time system model of (2), with sampling interval

e(k + 1) = exp (AAt)e(k) = &(k) w(k) = cTe(k). (3)

This system produces samples of the corresponding continuous- time system signals. In particular,

w(k) =y(kAt)

El(k)=X1(kAt)=Q1 COS- 2nkAt t p1 sin - 2nkA t T T

- a1 sin - T

3 (k) = x3 (kA t) = 02 COS - 4nkA t 4nkA t + pz sin - T T

- CY2 sin- 4nkAt) T

HOSTETTER: RECURSIVE DISCRETE FOURIER TRANSFORMATION 187

111. NYQUIST RATE COMPUTATION A. Convergence to the Fourier Coefficients

ability of 2 N t 1 evenly spaced samples for which The most common situation in practice is that of the avail-

T A t = -

2 N t 1 .

With this choice of sampling interval, after 2N t 1 steps, the spectral observer state will be

t I ( 2 N + 1) = 1 (2N + 1) = (0) = ( ~ 1

t 2 ( 2 N + 1 ) = ~ 2 ( 2 N + l)=eZ(O)=- 27$1 T

t 3 ( 2 N + 1 ) = 3 ( 2 N t ' 1 ) = 3 ( 0 ) = ( Y 2

g2N+1(2Nt 1 ) = E w + 1 ( 2 N t 1 ) = E w + I ( o ) = d J * (5 1 B. Simplified Gain Vector Solution

The Nyquist-rate deadbeat observer has the property that, for zero initial conditions, the first 2N output samples are zero. This result follows from the interpolation properties of the filter. Initially, the filter generates interpolations where the leading samples are zero. In terms of the discrete-time system matrices, this property requires that

c'g = 0

c%g = 0

cT@g = 0

Additionally, the gain coupling to the (2N t 1)th state (repre- senting the constant term in the response) is

1 - 82N+1 -

Hence, the 2N t 1 linear algebraic equations (6) and (7) may be solved, off line, for the gain coefficients g .

Gain coefficients calculated in this manner were used for the following example.

C. Example Performance of an eight-harmonic Nyquist-rate spectral

observer is shown in Fig. 1 . The discrete-time filter samples are seen to converge to samples of the band-limited function after 2N t 1 steps. The individual harmonic samples are also shown. The plots shown were generated on an Hp 2100 system.

Fig. 1. Response of the eight-harmonic Nyquist-rate spectral observer.

IV. OTHER SAMPLING RATES

A. Convergence After 2N + 1 Samples The spectral observer may operate with any sampling inter-

val A t . In general, 2N t 1 samples are required for con- vergence to determine the harmonic content of the band- limited signal y ( t ) .

B. Gain Vector Solution Via Transformation

A general method of gain coefficient calculation is to tem- porarily transform the system to the dual of the phase variable canonical form [19] - [ 2 0 ] , [31] - [35]

where

and

188 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-28, NO. 2 , APRIL 1980

1 4 s 1 i 1 l [ r l ! l l l l ! r ! l !

i I Fig. 2. Response of a two-harmonic spectral observer operating above

the Nyquist rate.

The characteristic equation of a' is easily shown to be z2N+1 - p l z 2 N - p2z2N-1 - . . . - P2Nz - PuV+l = 0.

The characteristic equation of the transformed feedback system is

r (P1 - g I ) ( P 2 - g i ) 0 1 * * - 0 0 I : (P2N-g;N) 0 0 * ' 0 1 (P2N+l - g h + 1 ) . . '

z2N+1 - (P1 - g 9 z m - (P2 - g;)ZuV-l - ...- (P2N - giN)z - (P2N+l - g h + 1 ) = O ,

each coefficient of which may be chosen at will by appropriate choice of the elements of g'. For a deadbeat error system,

g ; = p i , i = l , 2 , - . * , 2 N t l .

Then

g = Q-'g'.

C Example Response of a two-harmonic spectral observer operating

above the Nyquist rate is shown in Fig. 2. The Fourier coeffi- cients are determined well in advance of the end of one period of y( t ) . Additional samples will not affect the spectral esti- mate if it is correct; if slight numerical errors are present, additional samples will tend to correct them.

v. UPDATING THE RECURSIVE TRANSFORM A. Next Sample Updating

In 2N + 1 steps a deadbeat Nyquist-rate spectral observer produces the Fourier coefficients (5) of the corresponding band-limited signal. The next iteration, with an additional sample, will produce the Fourier coefficients of the function with the previous 2N + 1 samples; all effects of prior samples will have been removed.

For zero initial conditions, then, this spectral observer generates initial spectral estimates corresponding to those samples not yet provided being zero.

There is also the possibility of updating intermediate samples by cycling the spectral observer, with g = 0, to that sample or by employing multiple-step system matrices.

By choosing observer eigenvalues other than all zero, a "fading memory" of earlier samples may be provided. Hence, various windows of the data may be provided without addi- tional computational complexity.

B. Example Fig. 3 shows a sequence of updatings of the Nyquist-rate

spectral observer. For each new sample, a single iteration of

HOSTETTER: RECURSIVE DISCRETE FOURIER TRANSFORMATION 189

Fig. 3. Next sample updating of the eight-harmonic Nyquist-rate spectral observer. The dotted curves show the previous estimate of the band-limited function, while the solid curves show the new esti- mate at each step.

the observer generates the new DFT corresponding to the most recent 2N + 1 samples.

VI. IMPRECISE CALCULATIONS A. Model and Observer Eigenvalue Errors

Imprecise arithmetic processing, whether by roundoff, truncation, or design, has two effects upon the spectral ob- server. Imprecision which results in placement of the observer eigenvalues at positions other than the origin may be reduced asymptotically to zero by recirculation of the data provided that the error is not so great as to result in system instability.

r t r r ~ ! r r ~ ~ ~ r ~ r ~ r r ! r ~ I

Fig. 4. Response of the eight-harmonic Nyquist-rate spectral observer with gain coefficients truncated at one significant digit.

Imprecision which is equivalent to an error in the signal model (3) will, of course, produce Fourier coefficient errors. Error in knowledge of the sampling interval is of this category. Simula- tion studies to date, with models to the 25th harmonic, have demonstrated little susceptibility of spectral observers to minor modeling errors.

B. Example In Fig. 4 is shown representative effects of imprecise arith-

metic operations. For this example, the observer gains were each truncated to a single significant digit. Convergence of the spectral observer is not precise after 2N + 1 steps. However, recirculating the samples rapidly improves the spectral esti- mate , as shown.

VII. CONCLUSIONS New, recursive discrete Fourier transform methods have

been given which are especially useful in situations involving multirate-sampling running Fourier transformation, and im- precise computational arithmetic. Numerical examples were given which illustrate important aspects of the theory.

Harmonic smoothing, filtering, and prediction are easily im- plemented with these methods, as is the tracking of slowly varying harmonic content of a signal.

The extension of spectral observation to two dimensions is straightforward, although cursed with high dimensionality as are all such methods.

By varying the observer gains, it is also possible to generate signal estimates with successively higher order harmonic con- tent, if desired.

190 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-28, NO. 2, APRIL 1980

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