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Course: EENG 320, Spring 2010
School: American University in...
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and Signals Systems (EENG 320) Assignment (5) 1. Use either the Fourier transform analysis and synthesis equations or the Fourier transform properties to: a. Compute the Fourier transform of each of the following signals: i. [e t cos(o t )]u (t ), &gt; 0 ii. x(t ) as shown in figure (1). x(t) 2 1 . -6 -5 -4 -3 -1 -2 0 1 . 2 3 4 5 6 7 t Figure 1 b. Determine the continuous-time signal...

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and Signals Systems (EENG 320) Assignment (5) 1. Use either the Fourier transform analysis and synthesis equations or the Fourier transform properties to: a. Compute the Fourier transform of each of the following signals: i. [e t cos(o t )]u (t ), > 0 ii. x(t ) as shown in figure (1). x(t) 2 1 . -6 -5 -4 -3 -1 -2 0 1 . 2 3 4 5 6 7 t Figure 1 b. Determine the continuous-time signal corresponding to each of the following transforms: 2 sin [3( 2 )] ( 2 ) ii. X ( j ) = cos(4 + / 3) i. X ( j ) = 2. Determine which, if any, of the real signals depicted in Figures (2-4) have Fourier transforms that satisfy each of the following conditions: i. e{X ( j )} = 0 ii. m{X ( j )} = 0 x(t) 1 -2 2 -1 Figure (2) 1 t x(t ) = e t 2 /2 t Figure (3) x (t ) = t 2 e t t Figure (4) 3. If X ( j ) denotes the Fourier transform of the signal x(t ) depicted in figure (5), x(t) 2 1 -1 0 1 2 3 t Figure (5) i. Find X ( j ) ii. Find X ( j 0) + iii. Find X ( j ) d + iv. Evaluate X ( j ) d 2 v. Sketch the inverse Fourier transform of e{X ( j )} Note: You should perform all these calculations without explicitly evaluating X ( j ) . 2 4. a. Let x(t ) have the Fourier transform X ( j ) , and let p (t ) be periodic with fundamental frequency o and the Fourier series representation p (t ) = ae jno t n n = Determine an expression for the Fourier transform of y (t ) = x(t ) p(t ) b. Suppose that X ( j ) is depicted in Figure 6. Sketch the spectrum of y (t ) for each of the following choices of p(t ) . i. p (t ) = cos(t / 2) ii. p(t ) = n = (t n) + X( j ) 1 -1 1 Figure (6) 5. a Write Matlab code to plot the magnitude and phase of the spectrum for the following signal: t <2 1, x(t ) = otherwise 0, 6. Write a Matlab code to plot the continuous time domain signal for the following spectrum: 2 sin( ) X ( j ) = 7. Suppose that you have an LTI system whose impulse response h(t ) = e t [u (t ) u (t 10)] and let 1, 1 < t < 3 x(t ) = 0, otherwise be the input to that system and y (t ) be the output. Write a Matlab program to: i. Plot the output y (t ) using time convolution. ii. Plot the output y (t ) using frequency domain analysis. iii. Compare the results from i) and ii) 3 8. A signal x(t ) with Fourier transform X ( j ) undergoes impulse-train sampling to generate x p (t ) = + x(nT ) (t nT ) n = Where T = 104 . For each of the following sets of constrains on x(t ) and/or X ( j ) . Does the sampling theorem guarantee that x(t ) can be recovered exactly from x p (t ) ? a. X ( j ) = 0 for > 5000 b. X ( j ) = 0 for > 15000 c. e { X ( j )} = 0 for > 5000 d. x(t ) is real and X ( j ) = 0 for > 5000 e. x(t ) is real and X ( j ) = 0 for < 15000 f. X ( j ) * X ( j ) = 0 for > 15000 g. X ( j ) = 0 for > 5000 9. The signal y (t ) is generated by convolving a band-limited signal x1 (t ) with another band-limited signal x2 (t ) , that is, y (t ) = x1 (t ) * x2 (t ) where X 1 ( j ) = 0 for > 1000 X 2 ( j ) = 0 for > 2000 Impulse-train sampling is performed on y (t ) to obtain y p (t ) = + y(nT ) (t nT ) n = Specify the range of values for the sampling period T which ensures that y (t ) is recoverable from y p (t ) 4
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