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# HW3 - ω = 2 π 5 How many distinct CT HRCEs do we have...

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ECE 301, Homework #3, due date: 9/14/2011 http://cobweb.ecn.purdue.edu/ chihw/11ECE301F/11ECE301F.html Question 1: [Advanced] Consider two functions f ( t ) and g ( t ) described as follows. f ( t ) = ( 1 if - 2 t < 0 0 otherwise (1) g ( t ) = ( e πt if t < 3 0 otherwise . (2) Define a new function h ( t ) = R -∞ f ( s ) g ( t - s ) ds . Plot h ( t ) as a function of t . Question 2: [Basic] Following the previous question, what are the “total energies” of the three signals f ( t ), g ( t ), and h ( t )? Are they of finite energies? What are the “(overall) average powers” of f ( t ) and h ( t )? Are f ( t ) and h ( t ) of finite powers? Question 3: [Basic] Let x 1 ( t ) = e -| t |- 2 j , x 2 ( t ) = sin( t + 3 π/ 4), and x 3 [ n ] = e -| n - 1 | . Find the total energy and the average power of x 1 ( t ), x 2 ( t ), and x 3 [ n ]. Question 4: [Basic] p. 59, Problem 1.22 (g) and p. 60, Problem 1.24 (a) and (b). Question 5: [Basic] Find out the expression of the even part of sin( t + π/ 4). Plot it. Question 6: [Basic] p. 61, Problem 1.26. Question 7: [Basic] Write down the expression of the continuous-time harmonically related complex (HRCE) for a given fundamental frequency

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Unformatted text preview: ω = 2 π 5 . How many distinct CT HRCEs do we have? What is the common period of all HRCEs? Write down the expression of the discrete-time harmonically related complex (HRCE) for a given fundamental frequency ω = 2 π 5 . How many distinct DT HRCEs do we have? What is the common period of all HRCEs? Run the following matlab program. t=linspace(-2,6); omega=2*pi/5; k=4; fourth=cos(k*omega*t); k=-1; negative_first=cos(k*omega*t); plot(t,fourth,t,negative_first) grid on; You will see two curves: One is the CT HRCE with k = 4 and the other is the CT HRCE with k =-1. As seen in the ﬁgure, for k = 4 and for k =-1, the two CT HRCEs: cos ( 4 2 π 5 t ) and cos ( (-1) 2 π 5 t ) are obviously diﬀerent. Use the ﬁgure and explain why for the discrete-time (DT) HRCE, we have cos ± 4 2 π 5 n ¶ = cos ± (-1) 2 π 5 n ¶ . (3)...
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HW3 - ω = 2 π 5 How many distinct CT HRCEs do we have...

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