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Unformatted text preview: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS College of Engineering Department of Electrical and Computer Engineering 332:322 Principles of Communications Systems Spring 2004 Problem Set 10 Haykin: 4.1-4.4,5.1-5.7 1. Which of the following signals are orthogonal? You must show all work. (a) cos t and sin t on (0 , π ) . SOLUTION: Antisymmetric about π/ 2 so when you integrate product you get zero. ORTHOGONAL (b) cos t and sin t on (0 , π/ 2) . SOLUTION: Product is 1 2 sin 4 πt . Only one positive hump occurs in (0 , π/ 2) so the integral is nonzero. NOT ORTHOGONAL (c) t 2 and t 3 on (- 1 , 1) . SOLUTION: Product has odd symmetry so integral on symmetric interval about zero will be zero. ORTHOGONAL (d) t cos 2 πt and cos 2 πt on (- π, π ) . SOLUTION: Same idea (product has odd symmetry). ORTHOGONAL (e) te-| t | and t 2 e- t 2 on (- 1 , 1) . SOLUTION: Same idea again (product has odd symmetry). ORTHOGONAL 2. Consider the signal s ( t ) shown in Figure P4.1 (page 300, Haykin). (a) Determine the impulse response of a filter matched to this signal and sketch it as a function of time. SOLUTION: The impulse response of the matched filter is h ( t ) = s ( T- t ) Both waveforms are plotted in FIGURE 1 (b) Plot the matched filter output as a function of time. SOLUTION: Output of the matched filter is obtained by convolving h ( t ) with s ( t ) . The result is shown in FIGURE 2 (c) What is the peak value of output. SOLUTION: From FIGURE 2 it is clear that peak value of filter output is equal to A 2 T/ 4 and occurs at t = T . 1 Figure 1: waveforms Figure 2: Matched filter output waveform 3. Figure P4.2a (page 301, Haykin) shows a pair of pulses that are orthogonal to each other over the interval [0,T]. In this problem we investigate the use of this pulse pair to study a two-dimensional matched filter. (a) Determine the matched filters for pulses s 1 ( t ) and s 2 ( t ) considered individually. SOLUTION: The matched filter h 1 ( t ) for pulse s 1 ( t ) is given in solution to previous problem. For h 2 ( t ) , which is matched to s 2 ( t ) , h 2 ( t ) = s 2 ( T- t ) whis is represented by FIGURE 3 (b) Form a two dimensional matched filter by connecting two of the matched filters of part 1 in parallel, as shown in figure P4.2b (page 301, Haykin). Hence, demonstrate the following: i. When the pulse s 1 ( t ) is applied to the two-dimensional matched filter, the response of the lower matched filter (sampled at time T ) is zero....
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This homework help was uploaded on 04/03/2008 for the course ECE ECE332 taught by Professor Rose during the Spring '08 term at Rutgers.
- Spring '08