Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341:DISCRETE-TIME SIGNAL PROCESSING OpenCourseWare 2006 Lecture 17 Spectral Analysis with the DFT Reading:Sections 10.1 and 10.2 in Oppenheim,Schafer Buck (OSB). One of the important tools in signal processing is spectral analysis,i.e.examining the frequency content in a signal.For example,in order to design the whitening filters for matched filtering or Wiener filtering,we need to measure the power spectral density of a noise process. The two major applications of the DFT are implementing linear convolution,which we covered in the last lecture,and performing spectral analysis of a signal.OSB Figure 10.1 shows the basic steps of applying the DFT to continuous-time signals. First,an anti-aliasing lowpass filter is required to minimize the effect of aliasing because a signal is not perfectly band-limited,as shown in OSB Figure 10.2(a).The frequency response of an anti-aliasing filter and output of the filter are illustrated in OSB Figure 10.2(b)and(c), respectively.The conversion of a continuous signal to a discrete-time sequence corresponds to periodic replication and frequency normalization in the frequency domain as shown in OSB Figure 10.2(d). Since the DFT can not be applied to an infinite length signal,we need to multiply the sequence x[n]with a finite-duration window wn]as indicated in OSB Figure 10.1.This procedure results in a periodic convolution in the frequency domain,i.e. l=v)=云厂Xewe- OSB Figure 10.2(e)illustrates the Fourier transform of a typical window sequence,and the continuous curve in OSB Figure 10.2(f)shows the periodic convolution of W(ej)with X(ej). Notice that the convolution process smooth sharp peaks and discontinuities in X(e).The last step is computing the DFT of the finite-length sequence vn,which corresponds to sampling the DTFT.OSB Figure 10.2(f)also shows V[k]as samples of V(eiw).Since the spacing between the samples is 2/N and the relationship between the continuous-time frequency n and the normalized discrete-time frequency w is w=OT,the DFT corresponds to the continuous-time frequencies 2mk/NT. 1Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture 17 Spectral Analysis with the DFT Reading: Sections 10.1 and 10.2 in Oppenheim, Schafer & Buck (OSB). One of the important tools in signal processing is spectral analysis, i.e. examining the frequency content in a signal. For example, in order to design the whitening filters for matched filtering or Wiener filtering, we need to measure the power spectral density of a noise process. The two major applications of the DFT are implementing linear convolution, which we covered in the last lecture, and performing spectral analysis of a signal. OSB Figure 10.1 shows the basic steps of applying the DFT to continuous-time signals. First, an anti-aliasing lowpass filter is required to minimize the effect of aliasing because a signal is not perfectly band-limited, as shown in OSB Figure 10.2(a). The frequency response of an anti-aliasing filter and output of the filter are illustrated in OSB Figure 10.2(b) and (c), respectively. The conversion of a continuous signal to a discrete-time sequence corresponds to periodic replication and frequency normalization in the frequency domain as shown in OSB Figure 10.2(d). Since the DFT can not be applied to an infinite length signal, we need to multiply the sequence x[n] with a finite-duration window w[n] as indicated in OSB Figure 10.1. This procedure results in a periodic convolution in the frequency domain, i.e. v[n] = x[n]w[n] V (ejω) = 1 � π X(ejθ)W(ej(ω−θ) ↔ )dθ. 2π −π OSB Figure 10.2(e) illustrates the Fourier transform of a typical window sequence, and the continuous curve in OSB Figure 10.2(f) shows the periodic convolution of W(ejω) with X(ejω). Notice that the convolution process smooth sharp peaks and discontinuities in X(ejω). The last step is computing the DFT of the finite-length sequence v[n], which corresponds to sampling the DTFT. OSB Figure 10.2(f) also shows V [k] as samples of V (ejω). Since the spacing between the samples is 2π/N and the relationship between the continuous-time frequency Ω and the normalized discrete-time frequency ω is ω = ΩT, the DFT corresponds to the continuous-time frequencies Ωk = 2πk/NT. 1