PPT: Discrete Time Convolution | Signals and Systems - Electrical Engineering (EE) PDF Download

 Page 1


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Representation of DT Signals
Response of DT LTI Systems
Convolution
Examples
Properties
CONVOLUTION OF
DISCRETE-TIME SIGNALS
URL:
Page 2


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Representation of DT Signals
Response of DT LTI Systems
Convolution
Examples
Properties
CONVOLUTION OF
DISCRETE-TIME SIGNALS
URL:
EE 3512: Lecture 14, Slide 1
 Are there sets of “basic” signals, x
k
[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:
§ 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.
Exploiting Superposition and Time-Invariance
DT LTI
System
å
=
k
k k
n x a n x ] [ ] [
å
=
k
k k
n y b n y ] [ ] [
§ DT Systems:
(unit pulse)
§ CT Systems:
(impulse)
( )
0
t t - d
[ ]
0
n n - d
Page 3


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Representation of DT Signals
Response of DT LTI Systems
Convolution
Examples
Properties
CONVOLUTION OF
DISCRETE-TIME SIGNALS
URL:
EE 3512: Lecture 14, Slide 1
 Are there sets of “basic” signals, x
k
[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:
§ 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.
Exploiting Superposition and Time-Invariance
DT LTI
System
å
=
k
k k
n x a n x ] [ ] [
å
=
k
k k
n y b n y ] [ ] [
§ DT Systems:
(unit pulse)
§ CT Systems:
(impulse)
( )
0
t t - d
[ ]
0
n n - d
EE 3512: Lecture 14, Slide 2
Representation of DT Signals Using Unit Pulses
Page 4


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Representation of DT Signals
Response of DT LTI Systems
Convolution
Examples
Properties
CONVOLUTION OF
DISCRETE-TIME SIGNALS
URL:
EE 3512: Lecture 14, Slide 1
 Are there sets of “basic” signals, x
k
[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:
§ 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.
Exploiting Superposition and Time-Invariance
DT LTI
System
å
=
k
k k
n x a n x ] [ ] [
å
=
k
k k
n y b n y ] [ ] [
§ DT Systems:
(unit pulse)
§ CT Systems:
(impulse)
( )
0
t t - d
[ ]
0
n n - d
EE 3512: Lecture 14, Slide 2
Representation of DT Signals Using Unit Pulses
EE 3512: Lecture 14, Slide 3
Response of a DT LTI Systems – Convolution
 Define the unit pulse response, h[n], as the response of a DT LTI system to a
unit pulse function, d[n].
 Using the principle of time-invariance:
 Using the principle of linearity:
 Comments:
§ Recall that linearity implies the weighted sum of input signals will produce a
similar weighted sum of output signals.
§ Each unit pulse function, d[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.
DT LTI
å
=
k
k k
n x a n x ] [ ] [
å
=
k
k k
n y b n y ] [ ] [
[ ] n h
] [ ] [ ] [ ] [ k n h k n n h n - ® - Þ ® d d
] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ n h n x k n h k x n y k n k x n x
k k
* = - = ® - =
å å
¥
-¥ =
¥
-¥ =
d
convolution sum
convolution operator
Page 5


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Representation of DT Signals
Response of DT LTI Systems
Convolution
Examples
Properties
CONVOLUTION OF
DISCRETE-TIME SIGNALS
URL:
EE 3512: Lecture 14, Slide 1
 Are there sets of “basic” signals, x
k
[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:
§ 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.
Exploiting Superposition and Time-Invariance
DT LTI
System
å
=
k
k k
n x a n x ] [ ] [
å
=
k
k k
n y b n y ] [ ] [
§ DT Systems:
(unit pulse)
§ CT Systems:
(impulse)
( )
0
t t - d
[ ]
0
n n - d
EE 3512: Lecture 14, Slide 2
Representation of DT Signals Using Unit Pulses
EE 3512: Lecture 14, Slide 3
Response of a DT LTI Systems – Convolution
 Define the unit pulse response, h[n], as the response of a DT LTI system to a
unit pulse function, d[n].
 Using the principle of time-invariance:
 Using the principle of linearity:
 Comments:
§ Recall that linearity implies the weighted sum of input signals will produce a
similar weighted sum of output signals.
§ Each unit pulse function, d[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.
DT LTI
å
=
k
k k
n x a n x ] [ ] [
å
=
k
k k
n y b n y ] [ ] [
[ ] n h
] [ ] [ ] [ ] [ k n h k n n h n - ® - Þ ® d d
] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ n h n x k n h k x n y k n k x n x
k k
* = - = ® - =
å å
¥
-¥ =
¥
-¥ =
d
convolution sum
convolution operator
EE 3512: Lecture 14, Slide 4
LTI Systems and Impulse Response
 The output of any DT LTI is a convolution of the input signal with the unit
pulse response:
 Any DT LTI system is completely characterized by its unit pulse response.
 Convolution has a simple graphical interpretation:
DT LTI
] [n x ] [ * ] [ ] [ n h n x n y =
[ ] n h
] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ n h n x k n h k x n y k n k x n x
k k
* = - = ® - =
å å
¥
-¥ =
¥
-¥ =
d