S. C. Chan, K. L. Ho, and C. W. Kok. Abstruct- Discrete interpolation between successive samples of a se- quence is often required in digital signal processing.
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the butterfly modules is used to discover faults other than the cell faults, e.g., interconnection faults. This method also can detect all possible combinational module faults pseudoexhaustively . ACKNOWLEDGMENT
The authors are grateful for the anonymous reviewers for their careful reading of the manuscript and constructive suggestions which greatly improve the readability of the paper. REFERENCES
[ l ] J.-Y. Jou and J. A. Abraham, “Fault-tolerant F I T networks,” IEEE Trans. Computers, vol. C-37, pp. 548-561, May 1988. [2] Y.-H. Choi and M. Malek, “A fault-tolerant FFT processor,” IEEE Trans. Computers, vol. C-37, pp. 617-621, May 1988. [3] D. L. Tao, C. R. P. Hartmann, and Y. S . Chen, “A novel concurrent error detection scheme for FFI’ networks,” in Proc. Int. Symp. Fault-Tolerant Computing, pp. 114-121, June 1990. 141 K. Yamashita, A. Kanasugi, S . Hijiya, G. Goto, N. Matsumura, and T. Shirato, “A wafer-scale 170000-gate FFT processor with built-in test circuits,” IEEE J. Solid-State Circuits, vol. 23, pp. 336-342, Apr. 1988. [SI S . A. AI-Arian, B. A. Alhalabi, and H. Y. Abujbara, “Partitioning scheme and built-in-self-test for WSI FFT MSA,” in Proc. IEEE Test Symp., Apr. 1990. [6] V. K. Jain, S . A. AI-Arian, D. L. Landis, and H. A. Nienhaus, “Fully parallel and testable WSI architecture for an FFT processor,” Int. J. Computer-Aided V U 1 Design, vol. 3, pp. 113-135, 1991. [7] C.-W. Wu and P. R. Cappello, “Easily testable iterative logic arrays,” IEEE Trans. Computers, vol. C-39, pp. 640-652, May 1990. [8] C.-W. Wu and S.-K. Lu, “Designing self-testable cellular arrays,” in Proc. IEEEInt. Conh ComputerDesign (ICCD), pp. 110-1 13, Oct. 1991. [9] A. D. Friedman, “Easily testable iterative systems,” IEEE Trans. Computers, vol. C-22, pp. 1061-1064, Dec. 1973. 1101 W. H. Kautz, “Testing for faults in combinational cellular logic arrays,’’ in Proc. 81hAnnual Symp. Switching andAutomata Theory).,pp. 161-174, 1967. [ 111 P. R. Menon and A. D. Friedman, “Fault detection in iterative arrays,” IEEE Trans. Computers, vol. C-20, pp. 524-535, May 1971. 1121 T. Sridhar and J. P. Hayes, “Design of easily testable bit-sliced systems,” IEEE Trans. Computers, vol. C-30, pp. 842-854, Nov. 1981. 1131 R. Parthasarathy and S . M. Reddy, “A testable design of iterative logic arrays,’’ IEEE Trans. Computers, vol. C-30, pp. 833-841, Nov. 1981. [I41 J. P. Shen and F. J. Ferguson, “The design of easily testable VLSI array multipliers,” IEEE Trans. Computers, vol. C-33, pp. 554-560, June 1984. [IS] H. Elhuni, A. Vergis, and L. Kinney, “C-testability of two-dimensional iterative arrays,” IEEE Trans. Computer-Aided Design, vol. CAD-5, pp, 573-581, Oct. 1986. [16] W.-T. Cheng and J. H. Patel, “Testing in two-dimensional iterative logic arrays,” in Proc. Int. Symp. Fault Tolerant Computing, pp. 7 6 8 1 , 1986. [I71 W.-K. Huang and F. Lombardi, “On an improved design approach for C-testable orthogonal iterative arrays,’’ IEEE Trans. Computer-Aided Design, vol. 7, pp. 609-615, May 1988. [18] A. Chatterjee and J. A. Abraham, “NCUBE: An automatic test generation program for iterative logic arrays,” in Proc. IEEE Inr. Con5 Computer-Aided Design, pp. 428-43 1, Nov. 1988.
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Interpolation of 2-D Signal by Subsequence FFT S. C. Chan, K. L. Ho, and C. W. Kok Abstruct- Discrete interpolation between successive samples of a sequence is often required in digital signal processing. In this paper, the subsequence approach for 1-D interpolation is extended to two dimensions to avoid redundant operations. Also an improved intermediate sequence is proposed to preserve the Hermitian symmetry in interpolating realvalued signals. The resulting algorithm is very efficient and convenient because it permits the use of IFFT’s with sizes that are the same as the corresponding forward transforms.
I. INTRODUCTION Discrete interpolation between successive samples of a sequence is often required in digital signal processing [I], [2]. To increase the accuracy as well as the efficiency of the process, the fast Fourier transform (FFT) is often used [2]-[6]. Prasad and Satyanarayana [5] have introduced a zero-filling fast algorithm for the interpolation of I-D signals using the fast Fourier transform (FFT). The algorithm begins by taking the FFI of the original sequence. This is then zerofilled and properly scaled before the application of an inverse fast Fourier transform (IFFT) to obtain the interpolated sequence. Since the input to the IFFT consists mostly of zeros, additional savings can be obtained by eliminating operations with zeros. Adams [6] has proposed a subsequence approach that permits the use of an IFFT with the same size as the forward transform. A l s o , the intermediate sequence employed is different from that in [5] and preserves the Hermitian symmetry in interpolating real-valued signals. Recently, Sathyanarayana et al. [7] have introduced a fast algorithm for the interpolation of 2-D signals. The algorithm is very similar to its 1-D counterpart and begins by computing the 2-D DFT of the original sequence. The transformed sequence is then zero-filled and properly scaled to obtain an intermediate sequence. A 2-D inverse DFT is then performed to obtain the desired interpolated sequence. Unfortunately, no measures are taken to eliminate the operations of zeros during the inverse transform resulting in an inefficient algorithm for moderate to large values of interpolation ratio. Smith e t al. [8] used the pruning FFT to reduce the redundant operations in interpolating 2-D images. However, the pruning FFT’s are applied in a row-column manner and other efficient 2-D FFT algorithms cannot be used. In this paper, the subsequence approach for 1-D interpolation [6] is extended to two dimensions to avoid redundant operations. Instead of using the intermediate sequence obtained in [7], a better sequence is chosen that preserves the Hermitian symmetry in interpolating real-valued signals. The resulting algorithm is significantly more efficient than the 2-D FFT method in [7]. In addition, it is also more convenient because since it permits the use of IFFT’s with sizes that are the same as the original FFT’s. If infinite precision is assumed in computation, then the interpolation of a sequence by DFT is perfect if the sequence represents S equispaced samples of a continuous signal that is band-limited below the Nyquist limit and provided that the signal has a discrete Manuscript received December 20, 1991; revised August 3, 1992 and January 14, 1993. This paper was recommended by Associate Editor Y. C. Lim. S . C. Chan and C. W. Kok are with the Department of Electronic Engineering, City Polytechnic of Hong Kong, Kowloon, Hong Kong. K. L. Ho is with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong. IEEE Log Number 9207961,
1057-7130/93$03.00 0 1993 IEEE
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spectral density distribution consisting of only the frequencies f = 0, 1, 2,. . . ,AJ/2. In practice, while a continuous signal may be band-limited to below the Nyquist limit, it will, in general, have a continuous rather than a discrete spectral density distribution. In this general case, the aliased spectral leakage that results from a finite number of samples entails that interpolation by the D I T is only approximate, although the resulting error may be very small, and may be further reduced with even quite limited windowing of the sequence ends with cosine half-bells. Interested readers are referred to [IO] for various issues in the FFT interpolation. 11. INTERPOLATION OF 2-D SIGNALS Let c,(tl, t 2 ) be a bandlimited continuous signal and x ( r i 1 , 722) be the sampled version of ~ ( twith ) a uniform sampling rate of 1/T for both the variables. If the sampling rate is sufficiently high, we can recover the original analog signal from the following formula: m
o
ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 40, NO. 2, FEBRUARY 1993
kl
= 0. l;..,Nl
- 1.
kp = 0. l;.., Np - 1,
I/l.’2v= exp ( - 2 7 r j / N ) . Step 2: Construct an intermediate sequence Y(k1, k 2 ) of length ( LAV1) x ( LlV2) from X ( k l , k2 ) as shown at the bottom of the page. Step 3: Perform inverse 2-D FIT on l r ( n l , 7 1 2 ) to obtain the interpolated sequence:
(3c) Careful examination reveals that the intermediate sequence Y ( h ,k g ) does not preserve the Hermitian symmetry found in interpolating real-valued signal. If the original sequence z(n1, n 2 ) is real-valued, then its DFT must satisfy the following complex conjugate symmetry:
X ( k 1 . k 2 ) = X * ( 5 1 - kl.
o
IVZ - Icy).
(4)
To ensure that the interpolated signal y(n1, n2) is real-valued, we must also require that k’(kl, If the function is causal, (1) can further be truncated to
sin+(tl F(t1
-
n l T ) s i n ~ ( t-~n,T) . (2) / i I T ) + ( t ,- rt2T)
The problem of discrete interpolation is to compute new samples between adjacent pairs of samples of the original sequence. Direct computation using the interpolation formula is usually time consuming and an approach based on the 2-D FFT has been proposed
171. The algorithm of Sathyanarayana et al. [7] can be stated as follows. Step I: Compute the 2-D FFT of the finite size sequence .r(n1, n 2 ) of size N1 x NL: vl-1 vz-1 .r(ri1. n 2 ) ~ c ~ ~ 1 t t (~3 4~ ~ ( k 1 k. z ) = 12
E’(k1, kZ) =
1
=o
n2=o
k.1)
= t7*(L1V1- k l . LLV2- k z ) .
(5)
However, it can be seen from the definition that Y(k1, k 2 ) does not satisfy this property because I’(12T,/2, .\‘r/2) = A-(1\71/2. ’V1/2) # I-(LN1 - L\J1/2. LAr2 - 1V2/2) = 0. ( 6 ) This will produce interpolated signal with nonzero imaginary components and prevent the use of efficient real-valued FlT algorithms for performing the inverse transform. The same effect has already been observed in [6] for I-D interpolation. To avoid this shortcoming, we define the intermediate sequence (see bottom of page) where
5 , = { k , : k , = 0.. . . . 5 , / 2 - 1) and - l}. Q L= {I;,: k , = L - S , / 2 + 1, . . . , 1’iJCL It can be shown that 3.’(X1, k r ) satisfies the complex conjugate condition as depicted in ( 5 ) when the original sequence is real-valued. This permits the use of efficient real-valued FFT
~ symmetry ‘ ~
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algorithm for computing the inverse transform. Another property of the sequence Y(k1, k 2 ) is that the interpolated sequence coincides exactly with the original sequence:
(10)
where A41 = LNl, M 2 = LN,. It can be observed that the intermediate sequence, Y(k1,k 2 ) , consists of mostly zero and a lot of redundant operations on zero are found in the inverse transform. In the next section, we shall develop the subsequence FFT algorithm for interpolating 2-D signal to avoid these redundant operations. 111. SUBSEQUENCE
FIT INTERFQLATION ALGORITHM
First of all, we shall make use of the following common factor map (CFM): nt = &
+
n,L, k,=N,G+IC,,
(, = 0 , . . . ,L - 1 ;
C*=O,...,L-l;
n, = 0 , . . . , N , - 1 IC z --0 ,... , A T z - 1
to rewrite the 2-D DFT as follows: !/(El
+ n1L. + n2L) E2
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+ Y ( N 1 L - N 1 / 2 , N2/2)Wj2 + Y ( N i L - N1/2,
N2L -
+ Y ( N 1 L - N1/2,
iyZL - N 2 / 2 ) W i E 1 f E 2 ) ] .
TABLE I NUMBER OF ARITHMETIC OPERATIONS TO INTERPOLATE A ( 8 x 8 ) 2-D SIGNAL
N2/2)Wj1
Proposed Ref. [7] L Mult. Add. Mult. Add. 4 1779 6918 2760 23564 Using the definition of y ( k l , I C 2 ) , we finally obtain: 8 7395 27942 15048 113676 16 29859 112038 76488 535564 1 N1-1N2-1 It is assumed that: 1) the 3 mult. 3 add. scheme is used to implement a y c l c z ( n l ,m ) = XElE2(k1. k2)W.;:1k1W,;12k2 complex multiplication, 2) the scaling in obtaining Y ( k 1 , k z ) is not N 1N2 k l = 0 k 2 = 0 counted, 3) the 2D DFT is computed via the polynomial transform with (13) the following arithmetic complexity [9]: and the equation at the top of the page, where IP, E { k , : k , = p RDFT ( 2 ” , 2 “ ) = ,2’”-’ - (7/3)22n-1 (8/3) N , / 2 l,...,Nt - l}. cuRDFT(2n. 2 ” ) = . 5 r ~ 2 ~ “ - ’ (13/3)2’”-’ (20/3) Since yoo(n1, 7 ~ 2 ) = z ( n 1 , n z ) , it is sufficient to compute p C D F T ( 2 ” . 2 “ ) = n2’n - ( 7 / 3 ) 2 ‘ n (16/3) ~ c ~ ~ ~m ( ) n for 1
