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

w0(k) w7(k) w8(k) w9(k) w10(k) w11(k)w2(k) w3(k) w4(k) w5(k) w6(k)w1(k)

W(k)=exp(-j2k/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 foreach 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.