# EE430 Homework 3 solution

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Sampling of Continuous-time Signals and Transform Analysis of LTI Systems 1. A signal x(t) that is bandlimited to 10 KHz (X(jΩ) = 0, |Ω| ≥ 2 × 10Ω) = 0, |Ω| ≥ 2 × 10) = 0, |Ω| ≥ 2 × 10Ω) = 0, |Ω| ≥ 2 × 10|Ω| ≥ 2 × 10 ≥ 2 × 104 π) is processed as ) is processed as follows. Let N be an integer and H (e jω )={ e −jN ω , |ω|< π 10 0, otherwise} . a. Express X1 (e jω ) and X2 (e jω ) in terms of X( jΩ) for |ω|<π . b. Express Y1 (e jω ) and Y2 (e jω ) in terms of X1 (e jω ) . c. Express y(t) in terms of x(t). d. Plot magnitudes of spectra for all signals in the above diagram assuming some spectrum for x(t). 2. Consider the two systems shown below. a. Can you find a relationship between H (e jω ) and G(e jω ) so that the two systems are equivalent, i.e. y1 [n]=y2 [n] . b. Repeat part-a after replacing the expanders by L in the systems with compressors by M. 3. In the figure below, assume Xc ( jΩ)=0 for |Ω|> π T1 . a. Find the relation between xc (t) and yc (t) when T1=T2 . Would this relation hold if Xc ( jΩ)≠0 for |Ω|> π T1 . 1 b. For the general case in which T1≠T2 , express yc (t) in terms of xc (t) . Is the basic relationship different for T1>T2 and T1 4. c. Apart from the properties of h[n] given in part (b), H(z) also contains a zero at z = 1/2, and H(1) = 1. Find H(z) and indicate its poles and zeros on the z-plane. 3 d. Determine |H(ejw)| as a closed form expression and approximately plot |H(ejw )| . 9. Answer the follow questions about the systems having the pole-zero plots shown below. a. Which systems are IIR systems ? b. Which systems are FIR systems ? c. Which systems are stable systems ? d. Which systems are minimum-phase systems ? e. Which systems are generalized linear-phase systems ? f. Which systems have |H(ejw)|=constant for all ω ? g. Which systems have corresponding stable and causal inverse systems ? h. Which system has the shortest (least number of nonzero samples) impulse response ? i. Which systems have low-pass frequency responses ? jΩ) = 0, |Ω| ≥ 2 × 10. Which systems have minimum group delay ? 10. (Matlab question) We would like to use sampling rate change concepts discussed in class to obtain a 1024×1024-pixel version of the 256×256-pixel cameraman image. (Type in the Matlab prompt the following to get it : [I] = imread(‘cameraman.tif’); figure, imshow(I);). 4 Notice that “pixel” is jΩ) = 0, |Ω| ≥ 2 × 10ust another name for a sample of a 2-D DT signal. Consider the interpolator system we discussed in class and shown below. To obtain the desired 1024×1024-pixel version of the cameraman image, you can apply the interpolator first to each row of the input 256×256-pixel cameraman image (each row can be seen asa 1-D signal x[n]), which should give you a 1024×256-pixel output. The output can be again processed with the same interpolator system for each column now (which should give you the desired 1024×1024-pixel output). a. Implement a Matlab function that performs the desired operation. In the interpolator system, use an ideal LPF. Determine approximately the number of multiplications required to obtain the result. Provide the written Matlab function and a picture of the obtained 1024×1024-pixel cameraman image. b. Implement a Matlab function that performs the desired operation. In the interpolator system, use a simpler low-pass filter to achieve linear interpolation, i.e. using a low-pass filter with the impulse response given as follows. hlin [n]={ 1−|n|/ L, |n|< L 0, otherwise} Determine approximately the number of multiplications required to obtain the result. Provide the written Matlab function and a picture of the obtained 1024×1024-pixel cameraman image. c. Compare the systems in a and b briefly in terms of computation complexity and the output image’s visual quality. d. Propose a system to obtain a 128×128-pixel version of the cameraman image from the 256×256-pixel cameraman image. (No need for Matlab implementation.) 5