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(solution) There is a set of questions on signals and systems. Could you

There is a set of questions on signals and systems. Could you help me with these questions (mainly questions from 5 and on) ? Detailed solutions are needed. Thank you!

Fundamentals Review Homework for EECS 562

1. For

x (t ) = 2sin(100? t ) ? 4cos(200? t )

a) Find the complex Fourier series for x(t)

b) Plot the double sided phase and amplitude spectral density for x(t).

t +1

t ?1

2. a) Plot x (t ) = 10(rect (

) ? rect (

))

0.125

0.125

b) Find the energy and power in x(t).

Ex=________

Px=__________

c) Find X(f) and plot its amplitude spectrum.

3. An input signal x(t) is processed by a filter with an amplitude |H(f)|

and phase ?(f) response given below. a) For x (t ) = 10cos(2? 500t ) + 7.5cos(2? 750t ) find output system y(t).

b) For x (t ) = 10cos(2? 500t ) + 7.5cos(2? 1500t ) find output system

y(t).

c) For x (t ) = 10cos(2? 500t ) + 7.5cos(2? 3500t ) find output system

y(t).

d) An input signal x(t) with a bandwidth B is processed by a filter

with an amplitude |H(f)| and phase ?(f) response given above.

What is the maximum value of B that will result in distortionless transmission of x(t) through the filter, H(f). 1 4. Find and plot the amplitude spectrum for x(t)=10sinc(2Kt )

a) For K=0.1

b) For K=1

c) For K=10

d) As K increases the bandwidth of x(t) increases, TRUE or FALSE.

e) Find the energy and power in x(t).

f) Is x(t) and energy or power signal?

5. The spectrum of x(t) is given below: a) The signal x(t) is sampled at 8000 samples/sec to form xs(t). Plot the

spectrum of xs(t). b) For x(t) given above, what is the minimum sample rate required to

prevent aliasing? c) If no aliasing is present, describe how x(t) is recovered from xs(t). 2 6. Two linear time invariant systems have transfer functions of H1 and

H2 are configured as: x(t) z(t) H1(f) y(t) H2(f) System 1 x(t) H(f) y(t) System 2 H1 and H2 have the following transfer functions H 1 (f ) = e ? j 2? (0.1)f H 2 (f ) = 1

1

+ j 2? f

4 a) Find H(f) such that the two systems above (System 1 and System 2)

are the same, i.e., for the same input x(t) find H(f) such that System 1

and System 2 produce the same output.

b) Plot |H2(f)|

c) Find h2(t).

d) Find the output signal, y(t), when the input signal is

x (t ) = cos(2?t ) .

e) Is the system H(f) casual, Circle YES or NO, Justify your answer.

7. An ideal bandpass filter H(f) has center frequency of 200 kHz and

bandwidth Bh=50 kHz. The input to H(f) is x(t), where

? t ? kT0 x (t ) = ? rect where ? = 2µs and T0 = 10µs ? k =?? a) Sketch |H(f)|.

b) Sketch |X(f)|. c) Find the power at the fundamental frequency f0. d) For the x(t) and H(f) given above find the system output y(t). 3 8. Consider a linear time invariant system with a impulse response of

h(t), and input signal x(t) given below. The input signal x(t) given

below produces and output of y(t). x(t) 1

exp(-t) -1 time h(t) 1 -.5 .5 time 0

a) What is y(-10)?

b) What is y(-0.5)?

c) What is y(0.5)? 9. The signal x[n] is input to a LTI system with impulse response h[n].

x[n] 4

1 1 2 3 4 n 3 h[n] 3 5 n a) Find X(z) and H(z).

b) Find the discrete time convolution of x[n]*h[n]=y[n]. 4 10. A video signal has a bandwidth of about 5 MHz. A DFT is use to

analyze the frequency content of a video signal with a frequency

resolution of 1 kHz.

a) To achieve this frequency resolution what is the required record

length in seconds?

b) How many samples are in the record, state any assumptions?

11. Properties of the DFT.

a) Let X1[n] = cos(n?/2), n=1...16. Plot the magnitude of the DFT of

X1[n]. (Use Matlab or other tool)

b) Let X2[n]= 0, n=1..4, cos(n?/2), n=5..16. Plot the magnitude of

the DFT of X2[n]. (Use Matlab or other tool)

c) Explain the difference between the results of part a) and part b).

12. Let of sI(t)= x(t)cos(2?fot) and sq(t)= z(t)sin(2?fot), and

s(t) = sI(t) + sq(t) =x(t)cos(2?fot) + z(t)sin(2?fot), assume that both x(t) and

z(t) have a 10KHz bandwidth and that fo=1 MHz. Assume

f

f

) and Z (f ) = tri (

).

X (f ) = rect (

20000 10000 a) Sketch the amplitude spectrum of sI(t)= x(t)cos(2?fot) and

sq(t)= z(t)sin(2?fot).

b) Sketch the amplitude spectrum of

s(t) = sI(t) + sq(t) = x(t)cos(2?fot) + z(t)sin(2?fot).

c) Find the output y(t) of the following system in terms of x(t)

and z(t): d) Discuss the spectral occupancy of s(t), sI(t), and sq(t)? What property

of signals can be used to explain the result of part c) assuming that an

ideal low pass filter is an approximation for an integrator? 5

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