S&S Combined

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Homework #1, ECE 3793, Fall 2012

Due Thursday, August 30 at the beginning of class

1. Given the signals depicted above:a. Compute the energy of both signalsb. Compute the energy of ( ) ( ) ( )1 2x t x t x t= + c. Compute the energy of ( ) ( ) ( )1 22x t x t x t= +

2. Express ( )1x t , ( )2x t , and ( ) ( ) ( )1 22x t x t x t= + from Problem #1 above using only scaledand shifted step functions

3. Given ( ) ( ) ( )1 0 010 cos 3 3 sin 1.5x t t t = + + , ( ) ( ) ( )2 0 010 sin cos 2x t t t = , and( ) 0 03

j t j tx t e e

= +

a. Compute the average power of all three signalsb. How does the total average power of ( )3x t relate to the average power of its two terms?

Explain why you think this relationship holds?

4. Let ( ) 21 1E x t dt= and ( ) 22 2E x t dt= :a. Prove that the energy of ( )1ax t (where ais a constant) is a2E1b. Prove that the energy of ( )1x at (where ais a real constant) is 1E a c. Prove that if ( ) ( )1 2 0x t x t dt= , then the energy of ( ) ( )1 2 1 2x t x t E E+ = +

5. For the signal ( )2x t shown in the figure for Problem #1, accurately sketch:a. ( )2 3x t b. ( )2 3x t (careful!)c. ( )2 2 2x t

6. Textbook #1.97. Textbook #1.27


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Due Thursday, September 13 at the beginning of class

1. Given the signals depicted above, find 1 2x t x t .2. Let an LTI system have an impulse response described by 3 22 t th t e e u t . Use

convolution to find the output of the system for the following inputs:

a. x t u t b. tx t e u t

3. For an LTI system with impulse response described by th t e u t , find the system outputdue to 1x t in the figure above.

4. Textbook #2.22 (Part c only)5. Textbook #2.216. Textbook #2.287. Textbook #2.298. For a discrete-time system with impulse response described by 0.5 nh n u n , use

convolution to find the system output for the following inputs:

a. 2nx n u n b. 12nx n u n c.

2 2n

x n u n

9. Textbook #2.4210.Textbook #2.43 (Part c only)


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No Due DateFor exam study only

1. Textbook #3.12. Textbook #3.33.

Textbook #3.44. Textbook #3.6

5. Textbook #3.136. Textbook #3.16 (a) and (b)7. Textbook #3.22(a) for parts (b) and (e) in Figure P3.228. Textbook #3.28: part (b) and Figure P3.28(b)


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Due Tuesday, October 16, 2012 at the beginning of class

Figure 1

Figure 2

Figure 3

1. Calculate the Fourier transforms of the signals shown in Figures 1 and 2 using the definition ofthe Fourier transform.

2. Calculate the inverse Fourier transform of the signal shown in Figure 3.

3. Textbook #4.21, parts a, b, d, g4. Textbook #4.22, parts b,c,d,e

5. Textbook #4.23, part a only

6. Textbook #4.24, part a7. Textbook #4.28, part a, and part b-vi

8. Textbook #4.32

9. Textbook #4.33 parts a and b10. Use Parsevals relation to determine the energy in the signal ( ) ( ) ( )sinc 2sinc 2x t t t= + .11. Use Fourier transform pairs and properties to determine the Fourier transform of

( ) ( ) ( )rect rectt tx t

=

-0 0


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No Due DateFor exam study only

1. Textbook #4.262. Textbook #9.43.

Textbook #9.54. Textbook #9.6

5. Textbook #9.76. Textbook #9.87. Textbook #9.198. Textbook #9.21 (a-d)9. Textbook #9.22 parts (a), (c), and (e)10.Textbook #9.23 parts (1) and (3)11.Textbook #9.2612.Textbook #9.2913.Textbook #9.3014.

Textbook #9.39 (a)15.Textbook #9.40


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1. Convolve the signal in Figure 1 below with the unit step function u(t).

Figure 1

2. Convolve the sequence in Figure 2 below with a shifted unit impulse sequence [n3].

Figure 2

3. Convolve the sequence in Figure 2 above with the unit step sequence u[n].

4. Find 5 6 3 5u t u t u t u t

5. Find 1

0.1 n

u n u n

6. Without using your notes, re-derive the Fourier transform of the rectangular function rect tx t A T

where rect tT is a square function centered at t= 0 and having with T.

7. Find the Fourier Transform of the signal in Figure 3 using a) the integral definition of the Fourier

Transform, and b) Fourier Transform pairs and properties.

Figure 3


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8. Find the Discrete-Time Fourier Transform of the sequence shown in Figure 4. Try using the definition

of the DTFT and also using DTFT pairs and properties.

For problems 9-12, use the signal definitions depicted in the figures below.

9. Suppose the periodic square wave pictured above is passed through an ideal lowpass filter with cutoff

frequency Fc= 1.25 Hz. Find the output of the filter.

10. Now suppose the same signal x(t) above is passed through a causal LTI filter with the impulse

response shown above. Find the output of the filter using convolution and the Fourier Series of x(t).

11. Now define a new signal 1x t A x t where Ais a constant. What is the output of the filter in

Problem #10 due to this new signal?

12. Now let h(t) have a width of 1 second (rather than the 2 seconds as depicted above), what is the output

of this new filter in response to x(t) as pictured above?


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13. Find the Fourier series for the periodic signal below.

14. One period of a discrete-time periodic signal is shown below. Find the signals Fourier Series

15. Textbook Problems #4.1, #4.3, & #4.4, and #4.19. For problem #4.3, sketch the magnitude and phase

diagrams of the Fourier Transform.

16. For the signals in problems 13 and 14, express the signals average power in both the time domain (t

or n) and in the frequency domain using Parsevals relation.

17. True or False: All discrete-time signals are energy signals.

18. Textbook #9.9.

19. Textbook #9.6, #9.21, #9.22, #9.26, #9.29, and #9.41 (repeats of study problems for midterm #2)

20. Textbook #10.2, #10.4, #10.6, #10.7, #10.8, #10.9, #10.21, #10.23 (pick a couple and use partial

fraction expansion), #10.30, #10.32, and #10.33. (Note similarities in many of these problems with the

ones selected from Chapter 9).

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