数字信号处理邵曦lecture18

1、本次课内容：本次课内容： modulo-n reduction 1. x(n) (length-l) )(nx (length-n) n-point dft n-point dft x(k) )()(kxkx 2. idftxxxlnnmodn reductiondftdftidft9.5 modulo-n reduction modulo-n reduction or wrapping: fig.9.5.1 when l4n, 43211xxxxxl, the wrapped block 43211xxxxxn (9.5.1) nnnn3210 xxxxx0 x1x2x3xx1x2x3xex

2、pression of )(nx: 0)()(mnmnxnx （n=0n-1） (9.5.4) 0 x1x2x3xnnnn)(nx0 x1x2x3x0 x1x2x3x0 x1x2x3x)(nnx)2(nnx)3(nnx)(nxxfig.9.5.2:more generally, 0)()(mnmnxnx （n=0n-1） 是以下周期序列的主值周期：是以下周期序列的主值周期： mnmnxnx)()( （n） （9.5.5） periodic extension ( 周期延拓周期延拓) of the signal x(n) with period n. (p.499)） theorem conne

3、cting mod-n reduction to dft: the length-n wrapped signal xhas the same n-point dft as the original unwrapped signal x. )()(kxkx or )()(kkxx k=0n-1 (9.5.6) proof: xxx1010) 1()0() 1()0(1nnnkwnkkannnknnnnn xxxxxxxxxxxxxx)(|21212110101010)(12101010)1()0()1()2()12()(1()0(1aaaaaaaaammmmnlnnnkknnwnnnknnkn

4、wnnnnkknnwnnnkknnwnkklnnnnnnnnnnlln comparison of the computational cost(计算代价计算代价): (p.503) assuming l=mn and computing n-point dft of a length-l signal： the cost using a direct dft: 2mnln macs. the cost using a prewrapped dft: the cost of n-point dft of xis 2n macs. the cost of modulo-n reduction i

5、s n(m-1) macs. total cost mnn 2 macs example 9.5.4: the series that has the same modulo-n reduction is not unique. (p.504)suppose length-l x and y have equal yx,then )()1 ()()(zqzzyzxn (9.5.13) q(z): arbitrary polynomial of degree l-1-n which is the inverse dft sequence (should be unique ) of the n-

6、point x(k) ? 9.6 inverse dft if 111nlnaaxxx (9.6.1) we define the inverse dft by: xxx1)(aidft (9.6.2) a：nn square matrix、unitary matrix （酉酉阵阵） niaan*1 (9.6.4) niaa1 *11ana （9.6.5） xx*1an （idft in matrix form） （9.6.6） *)(1xan *)(1xdftn (9.6.7) (computing idft using dft module) 10)(1)(nknknkxwnnx （n0n-1） (9.6.9) ( idft in component-wise form) 10)(1nkknjxenk (9.6.10) relation of forward and inverse n-point dft: (fig.9.61.)xxxlnnmodn reductiondftdftidftx(n) (length-l) dft x(k) (length-n) idftmnmnxnx)()(n=0n-1) (length-n) whe