Homework Assignment 3

EE32b Signals, Systems, and Transforms
January 25, 2002
Homework Assignment 3
Due (in class) 9am January 30 , 2002
Reading: OW2, Sections 7.1–7.3, pp. 520–533.
R. J. McEliece
162 Moore
Problems to Hand In:
Problem 1. Suppose that a continuous-time signal of the form
x c (t) =A cos(2πf 0 t + φ)
is sampled at a rate of 6000 samples/sec, and that the resulting discrete-time signal has
the form
x d [n] =2.2 cos(0.3πn − π/3).
Find three different x c (t)’s, all with a maximum frequency less than 8 kHz, which could
have produced x d [n].
Problem 2. In the rotating disk and strobe demo shown in class on Jan. 23, we saw that
different flashing rates of the strobe light would make the spot on the disk appear to stand
still or move in different directions at different speeds.
(a) Assume the disk is rotating clockwise at 13 rev/sec. If the flashing rate is 15
times/sec, express the position of the spot on the disk at the nth flash as a complex
phasor z[n], i.e.,
z[n] =exp(jθ[n]),
where θ[n] isthe angular position of the spot at the nth flash. (Assume z[0] = 1.)
(b) For the conditions given in part (a), determine the apparent angular velocity (in
cycles/sec) and direction of movement of the strobed spot.
(c) Now assume the rotation speed of the disk is unknown. If the flashing rate is 13
times/sec, and the strobed spot moves counterclockwise by 15 deg for each flash, determine
the true rotation speed and direction of the disk. (If the answer is not unique, give all
possible answers. )
Problem 3. (Second-order hold.) In the text on p. 526, we read
“For example, the output of a second-order hold provides an interpolation of the sample
values that is continuous and has a continuous first derivative and a discontinuous second
derivative”
Using this clue, find an interpolating pulse shape p 2 (t) for second-order hold which also
has the following properties:
(1) p 2 (n) =
{
1 if n =0
0 if n = ±1, ±2,...

(2) p 2 (t) =0for|t| ≥2.
(3) p 2 (t) ispiecewise quadratic.