Sampling in Frequency and Time Since A is continuous,explicitly computing X[n,A)can be done only for a finite set of sample values.Sampling X[n,A)at N equally spaced frequencies gives the DFT of the windowed sequence: Xn,月=Xln,l=装=N-point DFT{rn+mjwm]}. If the window is of length L and starts at m =0,i.e.,w[m]=0 except for 0<M<L-1,by properties of the inverse DFT,the windowed sequence can be recovered as long as N>L: N-1 ln+mjwm=六∑xn,e3特m0≤MsL-1. k=0 The sequence values x[n]within the interval [n,n+L-1]can be subsequently computed. Given the STFT of a sequence at time no and exact form of the window,we are able to synthesize x[n]for no n <no+L-1.This suggests that the frequency sampled STFT is redundant,thus can be further downsampled in time.Such a sampled STFT is defined as L-1 XrR=XrR21-R+ome号， m=0 where-oo<r<oo,0<k<N-1,and R is the sampling interval in the time domain.Exact reconstruction of the signal is possible if R<L<N.An alternative notation for XrR,k]is X,[k].OSB Figure 10.15 shows the region of support for X[n,A)in the [n,A)-plane as lines and for Xrk]=XIrR,k]as discrete grid points.The amount R by which the window slides from block to block depends on how fast the frequency of the signal changes,and the length L of the window should be chosen according to the desired frequency resolution (long window), or time resolution (short window). Modulated Filter Banks for Short-Time Fourier Analysis A common method for short-time Fourier analysis before the development of the FFT is to use a filter bank in which each filter has the same frequency response shape,but different center frequencies.An example is shown below where the prototype lowpass filter is shifted N-1 times to obtain a bank of N filters with complex impulse responses. 3Sampling in Frequency and Time Since λ is continuous, explicitly computing X[n, λ) can be done only for a finite set of sample values. Sampling X[n, λ) at N equally spaced frequencies gives the DFT of the windowed sequence: X[n, k] = X[n, λ) λ= 2πk = N-point DFT{x[n + m]w[m]} . N | If the window is of length L and starts at m = 0, i.e., w[m] = 0 except for 0 ≤ M ≤ L − 1, by properties of the inverse DFT, the windowed sequence can be recovered as long as N ≥ L: N−1 1 j 2π km x[n + m]w[m] = � X[n, k]e N 0 ≤ M ≤ L − 1 . N k=0 The sequence values x[n] within the interval [n, n + L − 1] can be subsequently computed. Given the STFT of a sequence at time no and exact form of the window, we are able to synthesize x[n] for no ≤ n ≤ no + L − 1. This suggests that the frequency sampled STFT is redundant, thus can be further downsampled in time. Such a sampled STFT is defined as 2πk L−1 e−j 2π X km [rR, k] = X[rR, ] = � x[rR + m]w[m] N , N m=0 where −∞ < r < ∞, 0 ≤ k ≤ N − 1, and R is the sampling interval in the time domain. Exact reconstruction of the signal is possible if R ≤ L ≤ N. An alternative notation for X[rR, k] is Xr[k]. OSB Figure 10.15 shows the region of support for X[n, λ) in the [n, λ)-plane as lines and for Xr[k] = X[rR, k] as discrete grid points. The amount R by which the window slides from block to block depends on how fast the frequency of the signal changes, and the length L of the window should be chosen according to the desired frequency resolution (long window), or time resolution (short window). Modulated Filter Banks for Short-Time Fourier Analysis A common method for short-time Fourier analysis before the development of the FFT is to use a filter bank in which each filter has the same frequency response shape, but different center frequencies. An example is shown below where the prototype lowpass filter is shifted N − 1 times to obtain a bank of N filters with complex impulse responses. 3