EE 313 Linear Systems and Signals
Fall 2010
Signals
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Initial conversion of content to PowerPoint
by Dr. Wade C. Schwartzkopf
Course Outline
• Time domain analysis (lectures 1-10)
Roberts, ch. 1-3
Signals and systems in continuous and discrete time
Convolution: finding system response in time domain
• Frequency domain analysis (lectures 11-16)
Fourier series
Fourier transforms
Frequency responses of systems
Roberts, ch. 4-7
• Generalized frequency domain analysis (lectures 17-26)
Laplace and z transforms of signals
Roberts, ch. 9-12
Tests for system stability
Transfer functions of linear time-invariant systems
2-2
Signals
• A function, e.g. sin(t) in continuous-time or
sin(2 p n / 10) in discrete-time, useful in analysis
• A sequence of numbers, e.g. {1,2,3,2,1} which is a
sampled triangle function, useful in simulation
• A collection of properties, e.g. even
 1 for t  0

symmetric about origin, useful in
1
u t    for t  0
reasoning about behavior
2
 0 for t  0
• A piecewise representation, e.g.

• A functional, e.g. the Dirac
1 for n  0

u
[
n
]


delta functional d(t)
2-3
0 otherwise
Exponential Signals
• Solutions to linear constant-coefficient differential
equations, and hence, very common
e-t
et
t
t = -1 : 0.01 : 1;
e1 = exp(t);
plot(t, e1)
t
t = -1 : 0.01 : 1;
e2 = exp(-t);
plot(t, e2)
2-4
Exponential Signal Properties
• Real-valued exponential signals
Amplitude values are always non-negative
Might decay or not as t goes to infinity
 0 if a  0

lim e at   1 if a  0
t 
 if a  0

• Complex-valued exponential signals
e j  cos( )  j sin(  )
e  j  cos( )  j sin(  )
e j  e  j  2 cos( )
e j  e j  2 j sin(  )
We’ll need these properties throughout the semester
2-5
Piecewise Functions
• Unit area rectangular pulse
rect(t)
1
-1/2
0
1/2
t

1
 1
rect t   
2
0

• What does rect(x / a) look like?
• Unit triangle function
tri(t)
1
-1
0
1
t
1  t
tri t   
 0
1
2
1
t 
2
1
t 
2
t 
Math commands
rectpuls(t)
tripuls(0.5*t)
t 1
t 1
Both functions are even symmetric about origin.
2-6
Dirac Delta Functional
• Mathematical idealism for
an instantaneous event
• Dirac delta as generalized
function (a.k.a. functional)
Unit area:
Sifting






d (t ) dt  1
g (t )d (t ) dt  g (0)
P (t ) 
1
 t 
rect 

2
 2 
d t   lim P t 
 0

2
Area  lim
1
 0 2
P (t ) 

t
tri  
  
t
1
provided g(t) is defined at t = 0
d t   lim P t 

1
 0
if a  0
Scaling:  d (at ) dt 
a
• Note that d(0) is undefined
1
2

1
 0 
Area  lim
1



t
2-7
Dirac Delta Functional
• Generalized sifting, assuming that a > 0
1 if  a  T  a

d
(
t

T
)
dt


a
0 if T  a or T  a
a
• By convention, plot Dirac delta as arrow at origin
Undefined amplitude at origin
Denote area at origin as (area)
Height of arrow is irrelevant
Direction of arrow indicates sign of area
d t 
(1)
t
0
• Simplify Dirac delta terms only under integration
2-8
Dirac Delta Functional
• We can simplify d(t)
under integration

  t d t dt   0

• What about?
1
  t d t dt  ?

Answer: 0
• What about?

  t d t  T dt  ?

By substitution of variables,



 t  T d t dt   T 
• Other examples

 j t


d
t
e
dt  1


p t 


d
t

2
cos
  dt  0





 4 
e 2 x t  d 2  t  dt  e 2 x 2 
• What about at origin?
0
 d t  dt  ?
0
 d t  dt  0

0
 d t  dt  1

2-9
Unit Step Function
• Models event that turns on and stays on
• Definition
0 t  0
t

du
u (t )   d   d  ? t  0
 d t 


1
t 0
dt
• What happens at the origin for u(t)?
u(0-) = 0 and u(0+) = 1, but u(0) can take any value
Textbook uses u(0) = ½ to average left and right hand limits
Impulse invariance filter design uses u(0) = ½
L. B. Jackson, “A correction to impulse invariance,” IEEE Signal
Processing Letters, vol. 7, no. 10, Oct. 2000, pp. 273-275.
Math command stepfun(t,0) defines u(0) = 1
2-10
Other Important Functions
• Ramp
• Unit comb
ramp(t) = t u(t)
Impulse train
comb (t ) 

 d (t  n)
n  
comb t 
(1) (1) (1) (1) (1) (1)
t
t = -3 : 0.01 : 3;
r = t .* stepfun(t,0);
plot(t, r)
-2
-1
0
1
2
3
t
2-11
Sinc Function
sin p t 
pt
How to compute sinc(0)?
sinc t  
As t  0, numerator and
denominato r are both going
to 0. How to handle it?
t = -5 : 0.01 : 5;
s = sinc(t);
plot(t, s)
Even symmetric about origin
Zero crossings at t  1,  2,  3, ...
Amplitude decreases proportionally to 1/t
2-12
Sampling
• Many signals originate as continuous-time
signals, e.g. voice or conventional music
• Sample continuous-time signal at equally-spaced
points in time to obtain discrete-time signal
y[n] = y(n Ts)
n  {…, -2, -1, 0, 1, 2,…}
Ts is sampling period
• Example
y(t )  A cos2 p f 0 t   
y[n]
Ts
3
4
5
6
7
n
1
2
y(t)
y[n]  y(n Ts )  A cos2 p f 0 Ts n   
2-13
Audio CD Samples at 44.1 kHz
• Human hearing is from about 20 Hz to 20 kHz
• Sampling theorem (covered at mid-semester):
sample continuous-time signal at rate of more
than twice highest frequency in signal
• Analog-to-digital conversion for audio CD
First, apply a filter to pass frequencies up to 20 kHz
(called a lowpass filter) and reject high frequencies.
Lowpass filter needs 10% of cutoff frequency to roll off
to zero (filter can reject frequencies above 22 kHz)
Second, sample at 44.1 kHz captures analog frequencies
of up to but not including 22.05 kHz
Third, quantize to 16 bits per sample
2-14
Discrete-Time Impulse and Step
• Impulse function
d[n]
1 n  0
d n  
0 n  0
Also called Kronecker Delta
Even symmetric about origin
1
-2
-1
0
1
2
3
n
• Unit step (unit sequence)
1 n  0
un  
0 n  0
u[n]
n = -2 : 3;
u = stepfun(n,0);
stem(n, u);
1
-2
-1
0
1
2
3
n
2-15
Discrete-Time Sinusoidal Signals
Sinusoidal signal in
continuous time
y(t )  A cos2 p f 0 t   
• Discrete-time frequency
 0  2p
Sample using sampling
period Ts
y[n]  y(n Ts )  A cos2 p f 0 Ts n   
Substitute Ts = 1 / fs,
fs is sampling rate,


f0

y[n]  A cos 2 p
n   
fs


f0
in rad/sample
fs
Given integers N and L with
common factors removed,
discrete-time
f0
N
sinusoid has

fs
L
period L if
• Example: singing a tone
during cell phone call
f 0 1000 Hz 1


f s 8000 Hz 8
2-16