Revision of Chapter IV Three forms of transformations z transform DTFT: a special case of ZT DFT: numerical implementation of DTFT DTFT X(w)= X (z)|Z=exp(jw)

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Revision of Chapter IV Three forms of transformations z transform DTFT: a special case of ZT DFT: numerical implementation of DTFT DTFT X(w)= X (z)|Z=exp(jw) DFT: Truncated the DTFT to finite terms and use w = 2p k / N, where N is the total Length of the data

x(1)x(8)x(9)x(10)x(11)x(12)x(3)x(4)x(5)x(6)x(7)x(2) DATA w 0 (k)w 7 (k)w 8 (k)w 9 (k)w 10 (k)w 11 (k)w 2 (k)w 3 (k)w 4 (k)w 5 (k)w 6 (k)w 1 (k) W(k)=exp(-j2 k/N) x X(1)X(8)X(9)X(10)X(11)X(12)X(3)X(4)X(5)X(6)X(7)X(2) = (k=1,2,3,4,…,12) This is a sequence of complex number so we have magnitude and phase for each number above X(i) = r(i) exp( j p(i) ) r(1)r(8)r(9)r(10)r(11)r(12)r(3)r(4)r(5)r(6)r(7)r(2)p(1)p(8)p(9)p(10)p(11)p(12)p(3)p(4)p(5)p(6)p(7)p(2)

DFT is a windowed version of DTFT. When we use DFT to estimate spectrum, there are two effectors: loss of resolution and leakage of energy Sampling theorem tells us that if we sample an analogous signal fast enough (double the bandwidth of it), we could recover the analog signal completely.