Transcript lecture_14
ECE
8443
– PatternContinuous
Recognition
EE
3512
– Signals:
and Discrete
LECTURE 14: CONVOLUTION OF
DISCRETE-TIME SIGNALS
• Objectives:
Representation of DT Signals
Response of DT LTI Systems
Convolution
Examples
Properties
• Resources:
MIT 6.003: Lecture 3
Wiki: Convolution
CNX: Discrete-Time Convolution
JHU: Convolution
ISIP: Convolution Java Applet
URL:
Exploiting Superposition and Time-Invariance
x[n] ak xk [n]
k
DT LTI
System
y[n] bk y k [n]
k
• Are there sets of “basic” signals, xk[n], such that:
We can represent any signal as a linear combination (e.g, weighted sum) of
these building blocks? (Hint: Recall Fourier Series.)
The response of an LTI system to these basic signals is easy to compute
and provides significant insight.
• For LTI Systems (CT or DT) there are two natural choices for these building
blocks:
DT Systems: n n0
(unit pulse)
CT Systems: t t 0
(impulse)
Later we will learn that there are many families of such functions: sinusoids,
exponentials, and even data-dependent functions. The latter are extremely
useful in compression and pattern recognition applications.
EE 3512: Lecture 14, Slide 1
Representation of DT Signals Using Unit Pulses
EE 3512: Lecture 14, Slide 2
Response of a DT LTI Systems – Convolution
x[n] ak xk [n]
DT LTI
hn
k
y[n] bk y k [n]
k
• Define the unit pulse response, h[n], as the response of a DT LTI system to a
unit pulse function, [n].
• Using the principle of time-invariance:
[n] h[n] [n k ] h[n k ]
convolution operator
• Using the principle of linearity:
x[n]
x[k ] [n k ]
k
• Comments:
y[n]
x[k ] h[n k ] x[n] h[n]
k
convolution sum
Recall that linearity implies the weighted sum of input signals will produce a
similar weighted sum of output signals.
Each unit pulse function, [n-k], produces a corresponding time-delayed
version of the system impulse response function (h[n-k]).
The summation is referred to as the convolution sum.
The symbol “*” is used to denote the convolution operation.
EE 3512: Lecture 14, Slide 3
LTI Systems and Impulse Response
• The output of any DT LTI is a convolution of the input signal with the unit
pulse response:
x[n]
x[n]
DT LTI
hn
x[k ] [n k ]
k
y[n]
y[n] x[n] * h[n]
x[k ] h[n k ] x[n] h[n]
k
• Any DT LTI system is completely characterized by its unit pulse response.
• Convolution has a simple graphical interpretation:
EE 3512: Lecture 14, Slide 4
Visualizing Convolution
• There are four basic steps to the
calculation:
• The operation has a simple graphical
interpretation:
EE 3512: Lecture 14, Slide 5
Calculating Successive Values
• We can calculate each output point by
shifting the unit pulse response one
sample at a time:
y[n]
x[k ] h[n k ]
k
• y[n] = 0 for n < ???
y[-1] =
y[0] =
y[1] =
…
y[n] = 0 for n > ???
• Can we generalize this result?
EE 3512: Lecture 14, Slide 6
Graphical Convolution
2
h(k )
1
-1
x(k )
-1
1
-1
h(3 k )
h(2 k )
h(1 k )
h(0 k )
k = -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
EE 3512: Lecture 14, Slide 7
y (3)
x(k )h(3 k ) 0
k
y(2)
x(k )h(2 k ) 0
k
y(1) (1)(1) 1
y(0) (1)(0) (2)(1) 2
Graphical Convolution (Cont.)
2
h(k )
1
-1
x(k )
-1
1
-1
h(1 k )
y(1) (1)(1) (2)(0) (1)(1) 2
h( 2 k )
y(2) (1)(0) (2)(1)
(1)(0) (1)(1) 2
h(3 k )
y (3) (1)(0) (2)(0)
(1)(1) (1)(0) 1
h( 4 k )
y(4) (1)(1) 1
k = -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
EE 3512: Lecture 14, Slide 8
Graphical Convolution (Cont.)
• Observations:
y[n] = 0 for n > 4
If we define the duration of h[n] as the difference in time from the first
nonzero sample to the last nonzero sample, the duration of h[n], Lh, is
4 samples.
Similarly, Lx = 3.
The duration of y[n] is: Ly = Lx + Lh – 1. This is a good sanity check.
• The fact that the output has a duration longer than the input indicates that
convolution often acts like a low pass filter and smoothes the signal.
EE 3512: Lecture 14, Slide 9
Examples of DT Convolution
• Example: delayed unit-pulse
• Example: unit-pulse
h[n] [n n0 ]
h[n] [n]
y[n]
x[k ] h[n k ]
y[n]
k
x[k ] [n k ] x[n]
• Example: unit step
k ] x[n n0 ]
h[n] a n u[n] a 1
x[k ] h[n k ]
k
n
x[k ] u[n k ] x[k ]
k
k
0
x[n] u[n]
x[k ] [n n
• Example: integration
h[n] u[n]
x[k ] h[n k ]
k
k
y[n]
k
y[n]
x[k ] h[n k ]
k
u[n]a u[n]
n
k
(1) [n] (1 a) [n 1] ...
1
1 a n 1
1 a
EE 3512: Lecture 14, Slide 10
n0
n0
Properties of Convolution
• Commutative:
x[n] * h[n] h[n] * x[n]
• Distributive:
x[n] * (h1[n] h2 [n])
( x[n] * h1[n]) ( x[n] * h2 [n])
• Associative:
x[n] * h1[n] * h2 [n]
( x[n] * h1[n]) * h2 [n]
( x[n] * h2 [n]) * h1[n]
EE 3512: Lecture 14, Slide 11
• Implications
Useful Properties of (DT) LTI Systems
• Causality: h[n] 0
• Stability:
n0
h[k ]
k
Bounded Input ↔ Bounded Output
Sufficient Condition:
for x[n] x max
y[n]
x[k ]h[n k ] x
k
max
h[n k ]
k
Necessary Condition:
if
h[n k ]
k
Let x[n] h * [n] / h[n] , then x[n] 1 (bounded)
But y[0]
x[k ]h[0 k ] h [k ]h[k ] / h[k ] h[k ]
*
k
EE 3512: Lecture 14, Slide 12
k
k
Summary
• We introduced a method for computing the output of a discrete-time (DT)
linear time-invariant (LTI) system known as convolution.
• We demonstrated how this operation can be performed analytically and
graphically.
• We discussed three important properties: commutative, associative and
distributive.
• Question: can we determine key properties of a system, such as causality
and stability, by examining the system impulse response?
• There are several interactive tools available that demonstrate graphical
convolution: ISIP: Convolution Java Applet.
EE 3512: Lecture 14, Slide 13