Direct Frequency-Domain Deconvolution when the Signals Have No ...

Direct Frequency-Domain Deconvolution when the Signals Have No Spectral Inverse Damjan Zazula and Ludvik Gyergyek Faculty of Electrical Engineering and Computer Science, Smetanova 17, 2000 MARIBOR, SLOVENIA Faculty of Electrical Engineering, Department of Systems and Control Engineering, Trzaska 25, 1000 LJUBLJANA, SLOVENIA Keywords:

Digital Signal Processing, Signal Reconstruction, Deconvolution

Abstract We describe a new method of frequency-domain deconvolution when the kernel has no spectral inverse. Discrete frequency interpolation is used to aviod zero-valued frequency samples. The algorithm does not suer from the spectral singularities of the original kernel, its complexity is proportional to the fast Fourier transform, and a comparative noise study showed improved performance relative to the direct frequency-domain method.

1 Introduction When we are interested in characteristics of a system, it is very natural to observe the system operation referring to its output signal. Then, the characteristic behaviour is generally explained either by the impulse response in an input-output model or by the internal state changes in a state-space model. Nevertheless, to establish the backward connection from the system output to its characteristics or input signals, we face the inverse problem that is to be solved by a kind of deconvolution. Many dierent approaches to deconvolution have been introduced. Some of them attempt to avoid instability of frequency-domain deconvolution, some, however, obtain the deconvolution result given the system output signal, but neither its impulse response nor the input signal. The former get the answer in time-domain algorithms using smoothing lters [1] or iterative gradient methods [2]. The latter is important especially when dealing with hardly controllable systems, e.g. in observing natural and biomedical phenomena [3]. Another approach, used mainly in speech and seismic signal processing is homomorphic deconvolution [4]. 1

All these approaches are computationally more complex than simple direct deconvolution in the frequency domain, realized by the fast Fourier transform (FFT). Unfortunately, frequency-domain deconvolution is not feasible when the kernel has no spectral inverse. Some successful, though computationally very complex algorithms (proportional to N ) have been developed in the time domain [5]. In this paper, we de ne a frequency-domain method using interpolated frequency samples that does not suer from the spectral singularities. We present a computer algorithm based on the FFT to implement the method, and we study the in uence of noise added to the signals. 4

2 Disctere Fourier transform of real sampled signals In all discrete signal processing applications where the processed signal period must be relatively long with respect to the sampling interval, the number of samples is rather high. The z transform of the sampled signal is a high-degree polynomial in z which has integer coecients, with the maximum value restricted to the range of quantization levels of the A/D converter used. Such polynomials can have roots only in certain regions of the z plane [6], so the distribution of zeros on the unit circle is not arbitrary. Consider a polynomial p(z ) in z of order N with random coecients. Let M ( ;  ) be the number of zeros of p(z ) having angles between  and  . It is shown in [7] that, with probability 1, 1

1

2

2

M ( ;  )  ?  lim = 2 ; 0      2: N ?!1 N This result indicates that, as N becomes large, the zeros of p(z) tend to become evenly distributed in angle. Thus the spacing of the zeros tends to N , which is the spacing of the frequency samples computed by an N -point DFT. If one then samples p(z) with a DFT of length 2N (interpolation), it is unlikely that adjacent even- and odd-numbered Fourier coecients will equal 0 at the same time because their angular spacing is N , while the zeros tend to be N apart. 1

2

1

2

1

2

2

2

3 An algorithm for frequency-domain deconvolution The previous discussion suggests that if a signal's N -point DFT has zeros, then interpolated spectral samples would dier from zero. Therefore, using the sequence of the interpolated samples as a deconvolution kernel, the singularities would be eliminated. Consider an arbitrary discrete system with unit-sample response h(n) and output signal y (n); n = 0; : : : ; N ? 1. Denote their DFTs H (k) and Y (k); k = 0; : : : ; N ? 1. Let the sequences H (k) and Y (k), k = 0; : : : ; 2N ? 1, be the 2N -point DFTs of the 2N -point signals h (n) and y (n); n = 0; : : : ; 2N ? 1, where 1

1

1

1

(

h(n); 0; ( y(n); y (n) = 0;

h1 (n) = 1

n=0, . . . , N-1 n=N, . . . , 2N-1 n=0, . . . , N-1 n=N, . . . , 2N-1 2

Next, note that H1 (2l + 1) = Y1 (2l + 1) =

NX ?1 n=0

NX ?1 n=0

NX ?1

h1 (n)W2nN(2l+1) = y1 (n)W2nN(2l+1)

=

n=0

NX ?1 n=0

h(n)W2nN WNnl

h

= DFT h(n)e?j

 )n (N

h

y (n)W2nN WNnl = DFT y (n)e?j ( N )n 

i

(1)

i

(2)

where the weights WN are de ned as 2 ? j N . WN = e If the kernel H (k) contained zeros, the new kernel H (2l + 1) is unlikely to have zeros and a direct frequency-domain deconvolution is feasible, leading to an input signal of length N: V (l) = Y (2l + 1)=H (2l + 1); l = 0; : : : ; N ? 1: (3) Now, recalling the time-domain convolution sum and (2), we can write: (

)

1

1

y (n)W2nN

=

1

NX ?1 r=0

x(r)h(n ? r)W2nN ; n = 0; : : : ; N ? 1:

The matrix presentation of (4), using the fact that W ?Nr = ?W NN?r , is: 2 3 y (0) 6 7 6 y (1)W N 7 6 7 6 y (2)W 7 6 7= N 6 . 7 6 .. 7 4 5 N ? y (N ? 1)W N 2 h(0) ?h(N ? 1)W NN? ?h(N ? 2)W NN? : : : 6 6 h(1)W N h(0) ?h(N ? 1)W NN? : : : 6 6 h(2)W h(1)W N h(0) ::: 6 N 6 . 6 .. 4 ::: h(N ? 3)W NN? h(N ? 1)W NN? h(N ? 2)W NN? 2

(4)

2

2 2 2

2

1

1

2

2 2 2

2

2

1

2

2

2

1

2

2 6 6 6 6 6 6 6 4

2

2

3

?h(1)W N ?h(2)W N ?h(3)W N 2 2 2 3 2

h(0)

3 7 7 7 7 7 7 7 5

(5)

3

x(0) x(1)W2N x(2)W22N


. ..

x(N ? 1)W2NN?1

7 7 7 7 7 7 7 5

(6)

Denote the modi ed signals y(n)W nN and x(n)W nN in (6) as vectors ~y and ~x , respectively. The modi ed system matrix formed from h(n)W nN can be replaced by a dierence of matrices H and Hd, where 2 3 h(0) h(N ? 1)W NN? h(N ? 2)W NN? : : : h(1)W N 6 7 6 h(1)W N h(0) h(N ? 1)W NN? : : : h(2)W N 7 6 7 h(1)W N h(0) : : : h(3)W N 7 H = 666 h. (2)W N 7 7 6 .. 7 4 5 N? N? N? h(N ? 1)W N h(N ? 2)W N h(N ? 3)W N : : : h(0) 2

0

2

0

2

0

2

0

2 2 2

1

2

2

2

1

2

2

1

2

3

2

2

3

2 2 2 3 2

2 6 6 6 6 6 6 6 4

Hd =

0 0 0 ... 0

h(N ? 1)W2NN?1 h(N ? 2)W2NN?2 : : : h(1)W2N 0 h(N ? 1)W2NN?1 : : : h(2)W22N 0 0 : : : h(3)W23N

0

0

Hence, (6) can be written

... 0

3 7 7 7 7 7: 7 7 5

~y0 = H0~x0 ? 2Hd~x0

and

H? ~y = (I ? 2H? Hd)~x :

(8)

C = I ? 2H? Hd :

(9)

0

De ne the correction matrix:

(7)

1

0

0

0

1

0

1

The matrix (9) depends entirely on the system response h(n). Then 1 ~x0 = C?1 H? y0 0 ~

(10)

and the desired x(n) is extracted from x(n) = x0 (n)W2?Nn ; n = 0; : : : ; N ? 1;

(11)

which results from the de nition of ~x . Because of (1), (2), and Hd~x , a frequency-domain equivalent of (8) equals 0

0

hP

i

(r)h(l ? r)W NN?r = X (2l + 1) ? ; l = 0; : : : ; N ? 1 (12) H (2l + 1) where X stands for the 2N -point DFT of the original input signal. When the second term on the right side of (12) equals 0 for all l; V (l) = X (2l + 1); l = 0; : : : ; N ? 1 and (11) gives the desired input signal. This condition is ful lled when Y1 (2l + 1) H1 (2l + 1)

2DFT

1

N ?1 r=l+1 x

2

1

1

1

(8n 2 [1; N ? 1])(x(n)h(N ? n) = 0):

(13)

The condition (13) is equivalent to requiring that y(n) be the linear convolution of x(n) and h(n).

4 Realization of a computational algorithm We now state the deconvolution algorithm explicitly. We are given y (n); n = 0; : : : ; N1

and h(n); n = 0; : : : ; N ; 2

where N  N . Let N  N and let N be a power of 2. Then the algorithm is as follows: a. Form y (n) and h (n); n = 0; : : : ; 2N ? 1, by padding y(n) and h(n) with zeros. b. Compute the length 2N FFT's Y (k) and H (k), k = 0; : : : ; 2N ? 1. 1

2

1

1

1

1

1

4

c. ]Compute x(n) according to x(n) = IDFT

or x(n) = IDFT

"

"

Y1 (2l) H1 (2l)

#

; if H (2l) 6= 0 for l = 0; : : : ; N ? 1 1

#

Y1 (2l + 1) j ( N )n e ; H1 (2l + 1)

if H (2l + 1) 6= 0 for l = 0; : : : ; N ? 1: 1

The algorithm requires two 2N FFT's, one inverse FFT with the length N , and N complex multiplications. Its computational complexity is therefore proportional to N (1 + log2 N ) + 4N log2 2N

 5N log

2

N

and it requires two memory arrays of 2N complex samples.

5 Study of noisy signals The noise behaviour of the algorithm was tested by adding zero-mean white noise with preselected signal-to-noise ratios to the output signal sequence. The experiments were carried out on real signals: we took them from the American Heart Association Database of standard electrocardiograms. Kernels h(n) were obtain- ed as the averages of 500 ECG periods and truncated to length N . Randomly chosen extrasystoles from the same ECG were considered the output signals y(n) of length N . To achieve statistical signi cance, 15 dierent ECGs, two leads from each, were tested and each measurement was averaged across 30 runs. As the condition (13) was not generally ful lled, we had to modify the output y(n), using rst the DFT deconvolution of Y (k) and H (k); k = 0; : : : ; N ? 1, and then the linear convolution of the deconvolved x(n) and h(n), x(n) with samples n = N +1; : : : ; N ? 1 forced to zero. Figures 1(a)-(c) show such signal sequences for h(n), x(n), and y(n), respectively, n = 0; : : : ; N ? 1. The kernel amplitude spectrum of length 2N is devided into H (2l) and H (2l + 1); l = 0; : : : ; N ? 1, in Fig. 1(d). We investigated input S/N ratios (denoted by SNRx) versus output S/N ratios (SNRy ) in four dierent situations explained by Fig. 2 for the signals of lengths N = 256 and N = 32. Input SNRs are presented with their average (circles) and the double standard deviation (). The solid lines correspond to deconvolution with interpolated samples, the dashed lines to the direct DFT deconvolution. The right, i.e. (b) parts of the gures are for the singular kernels. We made them singular synthetically by replacing with zeros all the frequency samples H (k) whose magnitudes were lower than 1.5 times the smallest magnitude value. 2

2

6 Conclusion The proposed method is applicable in the frequency domain when the kernel has no spectral inverse. Its computational complexity is proportional to the complexity of an FFT and it is therefore much faster than equivalent time-domain algorithms [2, 5]. This advantage is 5

Figure 1: Time sequences of (a) h(n), (b) x(n), and (c) y(n). Input x(n) has been derived by the frequency-domain deconvolution with interpolated samples. The kernel's magnitude spectrum jH (k)j is divided into the original part with zero-valued samples (d{top), and interpolated sequences without zeros (d{bottom). 1

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Figure 2: Input versus output SNR, circles denote averages and lines double standard deviation for 30 ECG signals of length N = 256{ left, N = 32{right. Solid lines mean deconvolution with interpolated samples, dashed lines direct DFT deconvolution: (a) nonsingular kernels, (b) singular kernels. obvious in systems performing linear convolution, but is lost in case of periodic convolution because the correction scheme (10) must be applied. Section 5 shows that in a noisy environment, the frequency-domain deconvolution with interpolated samples achieved better input SNRs than the direct DFT algorithm on average. The method performs better for shorter kernels as the comparison in Figures 2 can con rm. Namely, the kernel length N enters the interpolation formula [6], suppressing the distant spectral samples by a factor of N compared to those near the interpolation point. This phenomenon is important when a kernel's energy is concentrated in a very short frequency interval, which causes the interpolated samples distant from this region to be very small. In the deconvolution process, they then excessively magnify the corresponding frequency components in the obtained input signal. That happened in our experiment with four signals with a very high positive oset. By elimination of the kernel's zero-frequency component H (0) contributing to the energy concentration at the beginning of the spectrum a great deal, a remarkable improvement of input SNRs from 19 dB to 35:29 dB (50 dB at the output) was achieved on average.

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Acknowledgment The authors gratefully acknowledge the valuable comments provided by the Associate Editor and the reviewers.

References [1] N. S. Nahman, M. E. Guillaume, Deconvolutions of Time Domain Waveforms in the Presence of Noise. Washington: US Government Printing Oce, National Bureau of Standards, 1981. [2] Tapan K. Sarkar, Fung I. Tseng, Sadisava M. Rao, Soheil A. Dianat, Bruce Z. Hollmann, "Deconvolution of Impulse Response from Time-Limited Input and Output: Theory and Experiment," IEEE Transactions on Instrumentation and Measurement, Vol. IM-34, No. 4, pp. 541-546, December 1985. [3] Thomas G. Stockham, Thomas M. Cannon, Robert B. Ingebratsen, "Blind Deconvolution Through Digital Signal Processing," Proceedings of the IEEE, Vol. 63, No. 4, pp. 678-692, April 1975. [4] Renlong Pan, Chrysostomos L. Nikias, "The Complex Cepstrum of Higher Order Cumulants and Nonminimum Phase System Identi cation," IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 2, pp. 186-205, February 1988. [5] Wai-Kwong Yeung, Fan-Nian Kong, "Time Domain Deconvolution When the Kernel Has No Spectral Inverse," IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-34, No. 4, pp. 912-918, August 1986. [6] Alan V. Oppenheim, Ronald W. Schafer, Discrete-Time Signal Processing. Englewood Clis, NJ: Prentice-Hall, 1989. [7] Kenneth Steiglitz, Bradley Dickinson, "Phase Unwrapping by Factorization," IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-30, No. 6, pp. 984-991, December 1982.

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