In order to interpret the result of the system accurately,we need to understand the artifact introduced in each step.In this lecture,we will focus on the effects of windowing and frequency sampling The Effect of Windowing As discussed in the previous page,the Fourier transform of a windowed signal corresponds to the periodic convolution of the Fourier transform of the window with the Fourier transform of the original signal.Due to the side lobe structure of a typical window,the convolution results in leakage of one frequency component into another component.In addition,the main lobe of a window is responsible for smearing frequency components and thus reducing frequency resolution.This effect of windowing is illustrated in OSB Example 10.3. In order to increase frequency resolution,we need to make the main lobe of the window as narrow as possible,and to reduce the leakage,the side lobe height as small as possible.OSB Figure 10.4(a)shows Kaiser windows for different values of B,and OSB Figure 10.4(b)shows the Fourier transforms corresponding to the windows in (a).We can see that as B increases, the main-lobe width increases and the side-lobe amplitude reduces.If we fix the value of B,and increase the length of the window,then both the main-lobe width and the side-lobe amplitude decreases,as illustrated in OSB Figure 10.4(c).However,increasing the window length requires a longer DFT,which results in more computation. The Effect of Spectral Sampling The DFT V[k]are samples of V(e),and it can sometimes produce misleading results.One common illusion from spectral sampling is the picket fence effect illustrated in OSB Example 10.4and10.5. Notice that from the OSB Figure 10.5(d),it is hard to interpret the spectrum of v[n]and this can lead to the illusion that the two frequency components are relatively same. However,this does not mean that the DTFT plot in OSB Figure 10.5(f)has higher frequency resolution than the DFT plot in OSB Figure 10.5(d).Since we can recover v[n]perfectly from VIk]using the IDFT,we didn't lose any information by taking the 64-point DFT of vn].In fact,the DTFT plot in OSB Figure 10.5(f)is not exactly drawn from the computation of the DTFT,because as we learned in the last lecture.the DTFT has values for an infinite number of frequency components,and thus can not be computed.As illustrated in OSB Example 10.7, a DTFT plot can be drawn by zero padding vn]and increasing the length of the DFT. Example: Consider the sum of two sinusoids given in OSB Example 10.5.The 64-point DFT shown in OSB Figure 10.6(b)has strong spectral lines at the frequencies of the two sinusoidal 2In order to interpret the result of the system accurately, we need to understand the artifact introduced in each step. In this lecture, we will focus on the effects of windowing and frequency sampling. The Effect of Windowing As discussed in the previous page, the Fourier transform of a windowed signal corresponds to the periodic convolution of the Fourier transform of the window with the Fourier transform of the original signal. Due to the side lobe structure of a typical window, the convolution results in leakage of one frequency component into another component. In addition, the main lobe of a window is responsible for smearing frequency components and thus reducing frequency resolution. This effect of windowing is illustrated in OSB Example 10.3. In order to increase frequency resolution, we need to make the main lobe of the window as narrow as possible, and to reduce the leakage, the side lobe height as small as possible. OSB Figure 10.4(a) shows Kaiser windows for different values of β, and OSB Figure 10.4(b) shows the Fourier transforms corresponding to the windows in (a). We can see that as β increases, the main-lobe width increases and the side-lobe amplitude reduces. If we fix the value of β, and increase the length of the window, then both the main-lobe width and the side-lobe amplitude decreases, as illustrated in OSB Figure 10.4(c). However, increasing the window length requires a longer DFT, which results in more computation. The Effect of Spectral Sampling The DFT V [k] are samples of V (ejω), and it can sometimes produce misleading results. One common illusion from spectral sampling is the picket fence effect illustrated in OSB Example 10.4 and 10.5. Notice that from the OSB Figure 10.5(d), it is hard to interpret the spectrum of v[n] and this can lead to the illusion that the two frequency components are relatively same. However, this does not mean that the DTFT plot in OSB Figure 10.5(f) has higher frequency resolution than the DFT plot in OSB Figure 10.5(d). Since we can recover v[n] perfectly from V [k] using the IDFT, we didn’t lose any information by taking the 64-point DFT of v[n]. In fact, the DTFT plot in OSB Figure 10.5(f) is not exactly drawn from the computation of the DTFT, because as we learned in the last lecture, the DTFT has values for an infinite number of frequency components, and thus can not be computed. As illustrated in OSB Example 10.7, a DTFT plot can be drawn by zero padding v[n] and increasing the length of the DFT. Example: Consider the sum of two sinusoids given in OSB Example 10.5. The 64-point DFT shown in OSB Figure 10.6(b) has strong spectral lines at the frequencies of the two sinusoidal 2