Modulated filter banks for short-time Fourier analysis holn] ↓R voln] haln] ↓R n hN-1[n] ↓R vN-1[n] Figure 1 A闪=hu=袋or Ne hon]=lowpass prototype filter Ho(ej) Hz(eiv)=Ho(ei(w-)) 0 Figure 2 2xk For Wk= N H(2)=Ho(e) In this system,the output yn]from filter hen]to the downsampler satisfies the following equation: y(n] hoin-n=∑hol--rikn+月 三0 DTFTfIn +r]ho[-r]}lw=we. Let the window sequence be w[m]=ho[-m],then [n=X[n,入l=wk· This is the STFT of n]sampled at w=wk.If ho-r]is N-point long,and wk=,then yk[n]=X[n,k]N-point DET{x[n +r]ho[-r]}. 4In this system, the output yk[n] from filter hk[n] to the downsampler satisfies the following equation: ∞ ∞ e−jωkr yk[n] = h0 � ejωkrh0[r]x[n − r] = � [−r]x[n + r] r=−∞ r=−∞ = DTFT{x[n + r]h0[−r]}|ω=ω . k Let the window sequence be w[m] = h0[−m], then yk[n] = X[n, λ) ω=ωk | . This is the STFT of x[n] sampled at ω = ωk. If h0[−r] is N-point long, and ωk = 2πk , then N yk[n] = X[n, k] = N-point DFT{x[n + r]h0[−r]} . 4