Transcript Linearity

Spring 2008
Linear Systems and Signals
Lecture 8
Discrete-Time Convolution
Linear Time-Invariant System
• Any linear time-invariant system (LTI)
system, continuous-time or discrete-time,
can be uniquely characterized by its
– Impulse response: response of system to an impulse
– Frequency response: response of system to a
complex exponential e j 2 p f for all possible
frequencies f
– Transfer function: Laplace transform of impulse
response
• Given one of the three, we can find other
two provided that they exist
8-2
Example Frequency Response
• System response to complex exponential e j w for
all possible frequencies w where w = 2 p f
 H(w)
|H(w)|
passband
stopband
-ws -wp
stopband
wp ws
w
w
• Passes low frequencies, a.k.a. lowpass filter
8-3
Kronecker Impulse (Function)
• Let d[n] be a discrete-time impulse function,
a.k.a. the Kronecker delta function:
1 n  0

d n  
0 n  0
d[n]
1
n
• Impulse response h[n]: response of a discretetime LTI system to a discrete impulse function
8-4
Discrete-time Convolution
• Output y[n] for input x[n]
yn  T xn
 

yn  T   xmd n - m
m-

• Any signal can be decomposed
into sum of discrete impulses
yn 
• Apply linear properties
yn 
• Apply shift-invariance
• Apply change of variables
h[n]
1
2
Averaging filter
impulse response
yn 

 xmT d n - m
m  -

 xm hn - m
m  -

 hm xn - m
m  -
y[n] = h[0] x[n] + h[1] x[n-1]
0
1
2
3
n
= ( x[n] + x[n-1] ) / 2
8-5
Comparison to Continuous Time
• Continuous-time convolution of x(t) and h(t)


-
-
yt   xt  ht    x  ht -   d   h  xt -   d
– For each value of t, we compute a different (possibly)
infinite integral.
– Discrete-time definition is the continuous-time
definition with integral replaced by summation
y[n]  x[n]  h[n] 
• LTI system


m  -
m  -
 x[m] h[n - m]   h[m] x[n - m]
– If we know impulse response and input, we can
determine the output
– Impulse response uniquely characterizes it
8-6
Fundamental Theorem
• The Fundamental Theorem of Linear Systems
– If one inputs a complex sinusoid into an LTI system,
then the output will be a complex sinusoid of the same
frequency that has been scaled by the frequency
response of the LTI system at that frequency
– Scaling may attenuate the signal and shift it in phase
– Example in continuous time: see handout G
– Example in discrete time. Let x[n] = e j W n,
y[n] 

e
m  -
j W n - m 
h[m]  e
jWn

 h[m] e
-jWm
 e j W n H W 
m  -


H W 
H(W) is the discrete-time Fourier transform of h[n] and
is also called the frequency response
8-7
Convolution Demos
• Johns Hopkins University Demonstrations
http://www.jhu.edu/~signals
Convolution applet to animate convolution of simple
signals and hand-sketched signals
Convolve two rectangular pulses of same width gives a
triangle
8-8
• Five-tap discrete-time (scaled) averaging FIR
filter with input x[n] and output y[n]
yn  x[n]  x[n -1]  x[n - 2]  x[n - 3]  x[n - 4]
Lowpass filter (smooth/blur input signal)
Impulse response is {1, 1, 1, 1, 1}
• First-order difference FIR filter
yn  x[n] - x[n -1]
Highpass filter (sharpens
input signal)
Impulse response is {1, -1}
h[n]
1
First-order difference
impulse response
n
-1
8-9