Quarter 1
EE102
Fall 2011
Systems and Signals
Jin Hyung Lee
Homework #2
Due: Wednesday, Oct 12, 2011 at 5 PM.
1. State whether the following systems are linear or nonlinear; time invariant or time variant;
and why.
(a) y (t) = x(t) sin(t + )
(b) y (t) = x(t)x(t 1)
(c) y (t) = 1 + x(t)
(d) y (t) = cos(t + x(t))
t
(e) y (t) =
(f) y (t) =
x( )d
t/2
x( )d
2. A periodic signal x(t), with a period T , is applied to a linear, time-invariant system H .
Show that the output y (t)
y (t) = H (x(t))
is also periodic, with period T .
3. Sample and hold system. A sample and hold (S/H) system, with sample time h, is described
by y (t) = x(h t/h ), where a denotes the largest integer that is less than or equal to a.
Sketch an input and corresponding signal output for a S/H, to illustrate that you understand what it does.
Is a S/H system linear?
4. Consider a system that takes a signal x(t) and returns the even part of x(t) as its output
xe (t) = H (x(t))
where xe (t) is the even part of x(t). Is this system linear? Is it time invariant?
2
5. System Equations from Block Diagrams Find the differential equation that correspond to this
block diagram. Hint: Label all of the intermediate signals, and write equations for each.
b
x
+
+
Z
+
+
Z
y
a
6. It is often useful to represent operations on signals as convolutions. For each of the following, nd a function h(t) such that y (t) = (x h)(t).
t
(a) y (t) =
(b) y (t) =
x( ) d
t
x( ) d
tT
(c) y (t) = x(t)
(d) y (t) = x(t 1)