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EE 120, Midterm #2, Fall 1994

**EE 120 Fall 1994
Midterm #2
**

**Professor Fearing
**

**Problem #1 (10 points)
**

A modulation scheme is described by:

*x(t) = cos(omegac * t + phiDELTA * m (t))*

where

*omegac = 2 * pi * 103*

*phiDELTA = phi*

*m(t) = PI(t) = u(t + 1/2) - u(t - 1/2)*

**[2 pts.] a)** Sketch *x(t)*.

**[8 pts.] b)** Sketch Re{*X(omega)*}, noting maximum amplitudes, center frequencies, and frequency of
first zero crossing.

**Problem #2 (10 points)
**

A square wave *x(t)* is passed through an ideal diode. Sketch the spectrum at the output of the ideal diode
*Y(omega)*, labelling important frequencies and amplitudes. Recall for an ideal diode that *vout* = { 0, *vin* <

0 and *vin*, *vin* >= 0 }.

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EE 120, Midterm #2, Fall 1994

Sketch *Y(omega)*.

**Problem #3 (5 points)
**

The signal *x(t)* is passed through a lowpass filter with frequency response *H(omega)*. The signal *x(t)*
contains a sinusoidal component at 100 KHz. Sketch approximately *y(t)*, the output in time of the
lowpass filter for the input *x(t)*.

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EE 120, Midterm #2, Fall 1994

**Problem #4 (25 points)
**

You are given the following modulation scheme:

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EE 120, Midterm #2, Fall 1994

For each signal *x1(t), x2(t), ..., x6(t), y(t)* select one of the following sketches, specifying amplitude *A0*

and frequency *omega1*. (Hint: Amplitude *A0* may be complex.)

**letter of sketch ***A0 omega1
*

*X1(omega) *

*X2(omega) *

*X3(omega) *

*X4(omega) *

*X5(omega) *

*X6(omega) *

*Y(omega) *

The following sketches represent spectra of the signals *x1(t) ... x6(t)*, and *y(t)*. The horizontal and vertical

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EE 120, Midterm #2, Fall 1994

scale in each sketch are arbitrary, and should be considered independently.

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EE 120, Midterm #2, Fall 1994

**Problem #5 (7 points)
**

A causal system is described by the following differential equation (with input *x(t)* and output *y(t)*) :

*dy/dt = d2x/dt2 + 3 dx/dt + 2 x*

Assuming zero initial conditions,
**a)** Is this system BIBO stable?
**b)** Find *Y(s)* and *y(t)* for *x(t)* = 0 and *y(0-)* = -5.

**Problem #6 (3 points)
**

A system has Laplace Transform *X(s)* with ROC *sigma* < 2.

The system is (circle one) :
**a)** stable but not causal
**b)** causal but not stable
**c)** stable and causal
**d)** neither stable nor causal

**Problem #7 (15 points)
**

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EE 120, Midterm #2, Fall 1994

**a)** With *d(t)* = 0, compute *Y(s)/X(s)*.
**b)** For which values of *k1* and *k2* is the system stable?

**c)** Let *d(t) = u(t)* and *x(t)* = 0, with *k1* = 1 and *k2* = 1. What is the limit of *y(t)* as *t* approaches infinity?

(answer should be a **number**)
**d)** Let *d(t)* = 0 and *x(t) = u(t)*, with *k1* = 1 and *k2* = 1. What is the limit of *y(t)* as *t* approaches infinity?

(answer should be a **number**)

**Problem #8 (25 points)
**

For each pole-zero diagram below, fill in the box with the letter of the corresponding frequency response and impulse response that follow. All diagrams represent causal systems.

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EE 120, Midterm #2, Fall 1994

Sketches to be used as answers for problem #8.

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EE 120, Midterm #2, Fall 1994

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EE 120, Midterm #2, Fall 1994

**Solutions (page 1)****Solutions (page 2)**

**Posted by HKN (Electrical Engineering and Computer Science Honor Society)
University of California at Berkeley
**

**If you have any questions about these online exams
please contact mailto:[email protected]
**

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