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Therefore, we will need to quantize :nzml to integer values through the use of the round function. ( 3 ) Compute &&-point FFTs, one fbr each column in the 7 ~ 2direction. R. 4 The S d t b foran and the DFT First;, let us begin with the mul$idirnensionaI discrete Fourier trrtrrsfornn, that is, F(k) = f (n) exp C DE~M Replace M with its Smith form. decornpositian, that is, M = TJAV. 2, LA'I"(UAV)=: LAT(UA). Novv, using elementary linear dgebraic operations, let us simplify the exyorrent of the exponentid = exp [- j 2 ?

L N " 4 ,where hi is the i tlx column of L, the^, the vector r r&~edto the matrix L pawer is a row v e e t ~ and r if;is defined to be . 3. 1. Let L he an integer-valued matrix where Zi, z' = 0,. . 2, vve can write where, Li is the ith column of L. nth comporlent of Lie a wctor power gives a scalar. Hence, k-l /k-X By definition, a vector raised ', to 46 Chapter 2 Muitidimensiond Multirate Sip& Processing Sirrce the product of terms with the same base equals the base to the sum of the exponents, the last equatkn becomes I rn If the z-transform converges for all a;, of the form z, =; exp(jwm), = OO?.

Therefore, A-I is an integer-valued unimodular matrix. II With this background on integer-vaIued unimodular matrices, we are ready to discuss the isaue of nonurriqueness of sampling 1;jtetices. 2. Given a nonsingular maLrix M and an iateger-vdued unimodular matrix V, then LATWV) -- LATW), Proof: If x E LAT(M), then there exists an n E N,such that x = Mn = . 1. Since V-' is integer-valued, then m = V-'n i s integervalued. Therefore, x E LAT(MV). Therefore, LAT(M) E LAT(MV). Obviously; LAT(MV) C LACP(M).

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