Colorado State University, Ft. Collins
Spring 2016
ECE 312: Linear Systems Analysis II (Signal and Systems)
Homework 1
Assigned on: 02/09/ 2016, Due by: 02/25/2016
1.1
Use the definition of the unilateral Laplace transform and an integral table to verify
the following Laplace transforms, and specify their regions of convergence:
(a) L t sin bt  
(b) L cos bt  
 
(c) L te at 
(d) L t  

1
s2
s
2bs
2
 b2

2
s
s  b2
1
2
s  a  2

(e) L e  at cosbt u t  
sa
s  a 2  b 2
1.2
Consider the waveform f t  in the following figure. This waveform is one cycle of a
sinusoid for 0  t  s and is zero else where.
(a) Write a mathematical expression for f t  .
(b) Find the unilateral Laplace transform for this waveform, using any method.
1
10
8
6
4
2
/2
0

-2
-4
-6
-8
-10
Figure 1.2
1.3
Given the Laplace transform
V s  
2s  1
s2  4
 
(a) Find the initial value of vt  , v 0  , by
(i) the initial value property;
(ii) finding vt   L1 V s  .
(b) Find the final value of vt  by
(i) the final value property;
(ii) finding vt   L1 V s  .
1.4
Find the inverse (unilateral) Laplace transforms of the following functions.
2
(a) F s  
1
s s  1
s3
(b) F s  
ss  1s  2
1
(c) F s  
2
ss  1
s3
(d) F s  
2
s s  2 s  5
2
1.5
Find the inverse Laplace transforms of the functions given. Sketch the time
functions.
e 2 s
ss  1
1  e s
(b) F s  
ss  1
e 2 s  e 3s


(c) F s 
2
1  e 5 s
(d) F s  
ss  5
(a) F s  
3