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 Roberts, ch. 4-7
Fourier transforms
Frequency responses of systems
• 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
• A piecewise representation, e.g. 0 for t 0
• A functional, e.g. the Dirac 1 for n 0
delta functional d(t) u[n]
0 otherwise
2-3
Exponential Signals
• Solutions to linear constant-coefficient differential
equations, and hence, very common
e-t
et
t t
t = -1 : 0.01 : 1; t = -1 : 0.01 : 1;
e1 = exp(t); e2 = exp(-t);
plot(t, e1) 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 e j 2 cos( )
e j cos( ) j sin( ) e j e j 2 j sin( )
We’ll need these properties throughout the semester
2-5
Piecewise Functions
• Unit area rectangular pulse
t
1
rect(t) 1 2
1
rectt
1 1
t
2 2
0 1
t
-1/2 0 1/2
t
2
• What does rect(x / a) look like? Math commands
rectpuls(t)
• Unit triangle function tripuls(0.5*t)
tri(t)
1
1 t t 1
trit
0 t 1
t
-1 0 1
2-6
Both functions are even symmetric about origin.
Dirac Delta Functional
t
• Mathematical idealism for P (t )
1
2
rect
2
an instantaneous event 1
d t lim P t 2
• Dirac delta as generalized 0
function (a.k.a. functional) t
2
Area lim 1
Unit area:
d (t ) dt 1 0 2
Sifting
g (t )d (t ) dt g (0) P (t )
1 t
tri
provided g(t) is defined at t = 0 1
1 d t lim P t
Scaling: d (at ) dt
a
if a 0 0
t
• Note that d(0) is undefined Area lim
1
0 2-7
Dirac Delta Functional
• Generalized sifting, assuming that a > 0
1 if a T a
a
a d (t T ) dt
0 if T a or T a
• By convention, plot Dirac delta as arrow at origin
Undefined amplitude at origin d t
(1)
Denote area at origin as (area)
Height of arrow is irrelevant
Direction of arrow indicates sign of area 0 t
• Simplify Dirac delta terms only under integration
2-8
Dirac Delta Functional
• We can simplify d(t) • Other examples
under integration
d t e j t dt 1
t d t dt 0 p t
d t 2 cos dt 0
• What about?
4
1
t d t dt ?
e 2 x t d 2 t dt e 2 x 2
Answer: 0
• What about at origin?
• What about?
d t dt ?
0
t d t T dt ?
0
By substitution of variables, d t dt 0
t T d t dt T 0
d t dt 1
2-9
Unit Step Function
• Models event that turns on and stays on
• Definition 0 t 0
u (t ) d d ? t 0
t
d t
du
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
-2 -1 0 1 2 3
t = -3 : 0.01 : 3;
r = t .* stepfun(t,0);
plot(t, r) 2-11
Sinc Function
sin p t
sinct
pt
How to computesinc(0)?
As t 0, numerator and
denominator are both going
to 0. How to handle it?
t = -5 : 0.01 : 5; Even symmetric about origin
s = sinc(t); Zero crossings at t 1, 2, 3, ...
plot(t, s) 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) y[n]
Ts
n {…, -2, -1, 0, 1, 2,…}
3 4 5 6 7
Ts is sampling period n
1 2
• Example
y(t)
y (t ) A cos2 p f 0 t
y[n] y (n Ts ) A cos2 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
1
0 n 0
Also called Kronecker Delta -2 -1 0 1 2 3
n
Even symmetric about origin
• Unit step (unit sequence)
1 n 0 u[n]
un n = -2 : 3;
0 n 0 1 u = stepfun(n,0);
stem(n, u);
n
-2 -1 0 1 2 3 2-15
Discrete-Time Sinusoidal Signals
Sinusoidal signal in • Discrete-time frequency
continuous time f0
0 2p in rad/sample
y (t ) A cos2 p f 0 t
fs
Given integers N and L with
Sample using sampling common factors removed,
period Ts discrete-time
f0 N
y[n] y (n Ts ) A cos2 p f 0 Ts n sinusoid has
fs
L
period L if
Substitute Ts = 1 / fs, • Example: singing a tone
fs is sampling rate, during cell phone call
f
y[n] A cos 2 p 0 n
f 0 1000 Hz 1
fs f s 8000 Hz 8 2-16