The downsampler with rate R removes time redundancies in yn]to obtain v[n]=X[nR,k]. Because of the existence of the downsampler,a polyphase implementation for this filter bank is possible,we will explore in detail how this is carried out in the next lecture. In typical spectral analysis of filter banks,the filter frequency spacing wk is chosen to be approximately equal to the filter bandwidth.Also,we typically look at the magnitude of the outputs only,although the filter impulse responses are complex.OSB Figure 10.16 shows the frequency response for sets of bandpass filters obtained from rectangular and Kaiser windows. The Kaiser window has narrower sidelobes but a wider mainlobe.In either case,the filter responses overlap significantly and are of infinitely length.Nonetheless,it can be shown that time aliasing of the reconstructed signal does not occur when inverting the time and frequency sampled TDFT,hence exact reconstruction is possible.Problem 10.40 in OSB illustrates this point. In addition to using filter banks,the short-time Fourier analysis can also be performed with only one filter holn]by sliding the signal n]in frequency,i.e.,modulate xIn]with ejwkn for different we values and filter the modulated signal with hon]to obtain y[n]. The short-time Fourier transform plays a significant role in discrete-time processing of speech.For a much more detailed discussion of this topic,see OSB Section 10.5 and Chapter 3 Digital processing of Speech,A.V.Oppenheim,Applications of Digital Signal Processing, downloadable through the MIT RLE Digital Signal Processing Group web site at http://www.rle.mit.edu/dspg/documents/DigitalSpeech.pdf 5The downsampler with rate R removes time redundancies in yk[n] to obtain vk[n] = X[nR, k]. Because of the existence of the downsampler, a polyphase implementation for this filter bank is possible, we will explore in detail how this is carried out in the next lecture. In typical spectral analysis of filter banks, the filter frequency spacing ωk is chosen to be approximately equal to the filter bandwidth. Also, we typically look at the magnitude of the outputs only, although the filter impulse responses are complex. OSB Figure 10.16 shows the frequency response for sets of bandpass filters obtained from rectangular and Kaiser windows. The Kaiser window has narrower sidelobes but a wider mainlobe. In either case, the filter responses overlap significantly and are of infinitely length. Nonetheless, it can be shown that time aliasing of the reconstructed signal does not occur when inverting the time and frequency sampled TDFT, hence exact reconstruction is possible. Problem 10.40 in OSB illustrates this point. In addition to using filter banks, the short-time Fourier analysis can also be performed with only one filter h0[n] by sliding the signal x[n] in frequency, i.e., modulate x[n] with ejωkn for different ωk values and filter the modulated signal with h0[n] to obtain yk[n]. The short-time Fourier transform plays a significant role in discrete-time processing of speech. For a much more detailed discussion of this topic, see OSB Section 10.5 and Chapter 3 Digital processing of Speech, A. V. Oppenheim, Applications of Digital Signal Processing, downloadable through the MIT RLE Digital Signal Processing Group web site at http://www.rle.mit.edu/dspg/documents/DigitalSpeech.pdf 5