I

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 40. NO. 4. AUGUST 1991

Frequency De-Aliased FFT Analysis of Step-Like Functions

G . D. Cormack and J . N. McMullin

I

Absfrucf-We present a method for decreasing aliasing error in the FFT of step-like functions. This technique substantially decreases the sole remaining significant error in the EF-FFT method [l] and is im- plemented by multiplying the EF-FFT spectral results with a simple de-aliasing function based on a piecewise linear model for the shape of the original function between data points.

I. INTRODUCTION

In the original EF-FFT paper [ 11 the periodicity assumption im- plicit in discrete Fourier transformations was utilized through time- domain pre-aliasing to obtain the spectral components of infinite duration step-like functions when they could be modeled, outside the observation window, with step and/or exponential functions. The technique was shown to be more accurate than both conven- tional windowing and the other FFT approaches described in the literature for analysis of step-like signals. In a more recent paper [ 2 ] , a time-domain padding technique designed as an augmentation to the conventional EF-FFT method and generically related to con- ventional zero-padding was developed to provide increased spec- tral resolution.

As is common with nearly all FFT methods, [1]-[2] provide spectra that contain frequency aliasing errors that can be decreased in magnitude for any desired frequency by increasing the sampling rate in the time domain. This method for decreasing frequency- domain aliasing errors cannot always be implemented, for reasons such as (i) the data acquisition equipment cannot accommodate a faster sampling rate, (ii) the data initially acquired was unique and reacquisition is not possible, and (iii) an increased sampling rate results in an increase in the size of the data set requiring FFT pro- cessing and can place an unacceptable demand on computer facil- ities.

Several authors [3]-[6] have presented methods for decreasing the frequency aliasing error in FFT results but this is the first paper that presents a simple method involving only one additional step- a multiplication of each spectral component by a precalculated de- aliasing factor. The emphasis in this paper is on de-aliasing the standard EF-FFT results but the procedure is also usable with the enhanced resolution EF-FFT [ 2 ] .

11. THEORY

We wish to derive a simple de-aliasing technique which can be applied to the spectral analysis of step-like signals. To do this we will compare the exact spectrum of a continuous piecewise linear function with its aliased spectrum. W e start with a step-like func- tion,f(t), and assume that N + 1 samples off(?) are known at equal time intervals of A t spanning from time t = 0 to t = T = N . A t , thatf(t > I N - ,) = f ( t N - I) and thatf(0) = 0. The sampled Fourier

Manuscript received April 25, 1990; revised February 13, 1991. The authors are with the Telecommunications Research Laboratories

IEEE Log Number 9100364. (TlUubs), Edmonton, Alberta, Canada T6E 5Y7.

713

transform at the frequency w,, = 2 m / N of a piecewise linear ap- proximation to this generalized step-like function is then given by

T 2 . [l - exp ( j 2 a n / N ) ] 4 a z n 2 A t F(w,,) =

N

. C [ f ~ - f@- 111 exp ( - j 2 . r m k / ~ ) . ( 1 ) h = I

Because the spectrum of any step-like function is singular at w = 0, we cannot evaluate ( 1 ) at n = 0. A general expression for the aliased version of (1) is

m

= F ( w n - n ~ . N ) ( 2 ) 01 = - m

where the - symbol denotes frequency aliased components. Thus,

N T m

%,) = 4x2 ,c, [f(Q - f(L 111 nz = - m

(1 - exp [ j 2 a ( n / N - m ) ] } exp [ - j 2 a k ( n / N - m ) ] (n - . N)*

( 3 )

which simplifies to

N T [ l - exp ( j 2 a n / N ) ] P(w, ) = C [ f ( t r ) - f(4- 111 4a2n2 k = I

m n’ . exp ( - j 2 a k n / N ) c (4) m = - w (n - m . N)”

The summation over m can be computed using contour integration

1 5 n 2 n l = - m (n - N . m)* - sinc’ ( n / ~ )

where sinc (x) = sin (ax)/(ax). Therefore

N T [ l - exp ( j 2 a n / N ) ] &U,) = c [ f ( t k ) - f(6- 111 4a2n’ sinc’ ( n / N ) k = I

1 exp ( - j Z a n k / N ) .

Comparing (1) and (6) we see that

Now we show the relation between f ( w , ) and the DFT of the sam- ple set. Simplifying ( 6 ) by rearranging subscripts and using the relation

1 - e i / * a n l N = 2 J e i/a’‘/N sin ( a n / N )

yields

P(wn) f ( t N )

T N . [l - exp ( - j 2 a n / N ) ]

+ kFo f ( t k ) exp ( - j 2 a n k / N ) . (8) 1 N - l

The last term in (8) is the DFT of the original data set and thef(tN) term is the DFT of the ramp compensation function which has the

0018-9456/91/0800-0773$01 .OO 0 1991 IEEE

~

114

-1 ’

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 40, NO. 4, AUGUST 1991

same spectral components as the step extension function [l], [8]. Thus, (8) is identical to the EF-FFT of the original data set and it follows that the de-aliased Fourier transform can be obtained from the standard EF-FFT components, EF-FFT,, with

FO = sine' (n/N) . EF-FFT, T (9)

The de-aliased spectrum obtained in this way is expected to be more accurate than the standard EF-FFT as long as the time-domain function can be reliably represented by piecewise linear modeling. Schutte [3] used a similar piecewise linear modeling method as ours in his de-aliasing method but did not obtain the preceding easy- to-use equation and did not extend the results to step-like functions.

There is a way to process the time domain data before the FFT transformation that will provide the same spectral components as the above de-aliasing technique. Since the de-aliasing involves a multiplication of spectral functions, the time domain equivalent is a discrete convolution with the inverse discrete Fourier transform of the sampled sinc2 function. This convolution is computationally inefficient, involving on the order of N 2 multiplications and addi- tions.

An alternative to the EF-FFT de-aliasing procedure described by (9) would be to take the FFT of the original data set, to de-alias this result by multiplying with the sinc’ (n /N) de-aliasing function and then to add to this result the de-aliased version of an appropri- ate extension function spectrum. In this case the spectral compo- nents to be added to each frequency, for a step extension function, is the first term on the right of (8), multiplied with sinc’ (n /N) .

Should a decaying exponential extension function of the form f ( t N - I ) e-“(‘-T) u( t - T) be more appropriate than a step extension function, both of the preceding de-aliasing procedures are valid, with minor changes. The standard EF-FFT procedure requires an exponential compensation function, as explained in [ 11 whereas the procedure described in the preceding paragraph requires that the (-j27rn/N) exponent in (8) be changed to ( - U N - j 2 a n / N ) where a is the decay constant of the extension function. The en- hanced resolution EF-FFT [2] results also can be de-aliased with (9).

111. EXAMPLES

A. A Continuous Step-Like Function

The function shown in Fig. 1 is of the form

f ( t ) = {U, + U, exp [ -a(? - t,)] sin [27rL(t - t,)]} u(t - t,)

+ 0.5Us[1 - COS ( ~ t / t , ) ] [ ~ ( t ) - ~ ( t - t,)] (10)

where U(? - t,) is the unit step function turned on at time t = t,, t, = 200T/512, 0 I n I N - 1, N = 512, a = 12.5/(N - l ) , L = 75/(N - l ) , U, = 1 and U, = 0.125. The sampled true Fourier transform at the frequencies w, = 27rn / T is

F(wn) = us ~ X P ( - j ~ n t s ) / [ j ~ n I + u o 2 ~ L

* exp ( -jwn ts) / [(a + j w Y + ( ~ T L Y I

+ 0.5us11 - exp (-jwnts)l/Ij~nl

+ 0.5U,jwn[1 + exp (-jw,t,)]/[w~ - (7r/t,)21. (11)

A plot of this transform, divided by T, is given in Fig. 2(a). The normal EF-FFT method (subtractingf(t,,,) . n / N from the origi- nal data set prior to the FFT transformation) gives the curve of Fig. 2(b). Any difference between (a) and (b) is due to aliasing. The de- aliased version of Fig. 2(b), obtained with (9) is plotted in Fig.

n = SAMPLE NUMBER 0.0

Fig. 1. A continuous step-like function.

-1WdB.d . . . . . . . . I

1 n = HARMONIC NUMBER

Fig. 2. Magnitude of spectral components versus harmonic number for the function shown in Fig. 1 using: (a) Sampled true Fourier transform, times 1 /T, (b) Original EF-FFT method [l], (c) De-aliased version of (b).

2(c). Any difference between curves (a) and (c) is attributable to assuming a piecewise linear approximation forf (t). The maximum value for this difference is 0.9 dB (11 % relative error in “ F ( w , ) ” ) whereas the maximum difference between curves (a) and (b) is 7.6 dB (140% relative error). The de-aliasing procedure has decreased the relative error in the magnitude of the spectral components by a factor of thirteen.

B. A Discontinuous Step-Like Function

from ( lo) , To see the effect of a discontinuous step, we drop the last term

f ( t ) = {Us + U, exp [ -a(? - t,)] sin [27rL(t - t,)]} u(t - t,)

(12)

and therefore also drop the last two terms from (11). The resulting magnitude spectrum, divided by T, is given in Fig. 3(a). The EF-FFT spectrum for a sampled version of (12) is given in Fig. 3(b) and the de-aliased version, from (9), is given in Fig. 3(c). Fig. 3(c) deviates from Fig. 3(a) by a maximum amount of 3.3 dB because of the piecewise linear assumption and because Fig. 3(a) contains the spectral components of an infinitely fast rising step at n = 200. Any 512 point time-domain sample will contain no in- formation on the shape of the time-domain signal between the sam- ples taken at n = 199 and n = 200. However, the de-aliased EF- FFT results, Fig. 3(c), can be altered to include the effect of an approximation for this information by subtracting the spectral com-

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL

-10 dB.1 1

1 I

A . ~ n = HARMOMC NUMBER 255

Fig. 3. Magnitude of spectral components of (12) versus harmonic number using: (a) Sampled true Fourier transform, times 1 / T , (b) Original EF- FFT method [l], (c) De-aliased version of (b), (d) the same as (c) except that the spectral components of a ramp between t , - A t and I , has been subtracted.

ponents of a ramp that rises to value U, between n = 199 and n =

200,

U,[(t - t, + A f ) / A t ] [u(t - fA + A t ) - U ( t - t , ) ] . (13)

where A t = T / N . The components to be subtracted are the sam- pled true Fourier transform of (13) times 1 / T , or

The result is shown in Fig. 3(d) and the maximum deviation of curve (d) from (a) is 0.43 dB. This remaining error is due to the piecewise linear approximation model not being perfect.

IV. DISCUSSION A N D CONCLUSIONS

It has been shown that frequency aliasing errors introduced dur- ing the FFT analysis of step-like signals can be substantially de- creased by a simple de-aliasing procedure. For example, the results shown in Fig. 2(c) were obtained with a de-aliased version of the original EF-FFT method, namely by postmultiplication of the orig- inal EF-FFT results with sinc‘ ( n / N ) . Of course de-aliasing with (9) does not mean removing all alias error because to do so would mean that the function is precisely known between sample points. Rather, use of (9) implies that a piecewise linear approximation is acceptable between data points. To obtain even more accurate final spectra it would be necessary to increase the sampling rate or to use a quadratic or higher order curve-fitting procedure in the de- aliasing technique [ 5 ] , [6], [9].

The attractive features of the aliasing error reduction method introduced here, compared to increasing the sampling rate, are that data reacquisition is not required, computer requirements are small, and the spectra are of high accuracy up to the Nyquist frequency. The applicability is limited to functions that can be modeled with linear transitions between data points, as opposed to step transi- tions unless the precise timing and shape of the step transitions are known. Since most data sets are comprised of samples from slowly varying analog signals, the de-aliasing procedure provides en- hanced spectral accuracy with but minor additional mathematical complexity.

40, NO. 4. AUGUST 1991 775

REFERENCES

[ I ] G. D. Cormack and J . 0. Binder, “The extended function fast Fourier transform (EF-FFT),” IEEE Trans. Insrrum. Meas., vol. 38, pp. 730- .. 735, June, 1989. G. D. Cormack, D. A. Blair, and J . N. McMullin, “Enhanced spectral resolution FFT for step-like functions,” IEEE Trans. Insrrum. Meas., vol. 40, pp. 34-36, Feb. 1991. J . Schutte, “New fast Fourier transform algorithm for linear system analysis applied in molecular beam relaxation spectroscopy,” Rev. Sci. Insrrum. 52(3), pp. 400-404, Mar. 1981. S . Makinen, “New algorithm for the calculation of the Fourier trans- form of discrete signals,” Rev. Sci. Instrum. 53(5), pp. 627-630, May 1982. S . Sorella and S . K . Ghosh, “Improved method for the discrete Fourier transform,” Rev. Sei. Instrum. 55(12), pp. 1348-1352, August. 1984. M. Froeyen and L. Hellemans, “Improved algorithm for the discrete Fourier transform,’’ Rev. Sci. Insrrum. 56( 12), pp. 2325-2327, Dec. 1985. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953, p. 413. A. M. Nicolson, “Forming the fast Fourier transform of a step re- sponse in time-domain metrology,” Elecrron. Lef t . , vol. 9, no. 14,

G. D. Cormack and J. N. McMullin, “De-aliasing fast Fourier trans- forms,” submitted for publication to IEEE Trans. Acoust., Speech Sig- nal Processing, 1990.

pp. 317-318, July 12, 1973.

A Real-Time C-V Measurement Circuit for MOS Capacitors Under Current Stressing

Chung Len Lee, Tan Fu Lei, Jaw Huang Ho, and Wen Tong Wang

Abstract-A dynamic large signal C-V measurement circuit is pre- sented to measure real time C-V characteristics of a MOS structure under current stressing conditions. The real time information on the generation and filling of traps and surface states can thus be obtained. With this circuit it has been shown that, for an AI-gate MOS structure, charges on traps and interface surface states recover immediately after the stressing is removed.

I. INTRODUCTION

As very large scale integration (VLSI) technology advances, metal-oxide-semiconductor (MOS) devices keep shrinking their dimensions, not only on the channel length and the source and drain regions, but also on the thickness of the gate dielectric. For ex- ample, 16-Mb dynamic random access memories (DRAM) require an oxide thickness of 12.5 nm for the gate dielectric [l], and 1-Mb electrically erasable programmable read only memories (EE- PROM) use oxynitride as the tunneling dielectric [ 2 ] . As a result, one of the key requirements for making high density MOS VLSI

Manuscript received October 31, 1989; revised January 17, 1991. This work was supported by the National Science Council of the Republic of China under Contract NSC77-0404-E009- 10.

C. L. Lee, T. F. Lei, and W. T. Wang are with the Department of Electronics Engineering and the Institute of Electronics, National Chiao Tung University, Hsinchu, Taiwan, ROC.

J. H. Ho is with ERSO Industrial Technology Research Institute, Hsin- chu, Taiwan, ROC.

IEEE Log Number 9100374.

0018-9456/91/0800-0775$01 .OO 0 1991 IEEE