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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
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FAQs on PPT: Discrete Time Convolution - Signals and Systems - Electrical Engineering (EE)

1. What is discrete time convolution?
Ans. Discrete time convolution is a mathematical operation that combines two discrete-time signals to produce a third signal. It involves multiplying corresponding samples of the two signals and summing the results.
2. How is discrete time convolution different from continuous time convolution?
Ans. Discrete time convolution operates on discrete-time signals, which are sequences of numbers at discrete points in time. Continuous time convolution, on the other hand, works with continuous-time signals, which are functions of time.
3. What is the significance of discrete time convolution in signal processing?
Ans. Discrete time convolution plays a crucial role in various signal processing applications. It is used to model the effects of linear time-invariant systems, filter signals, and analyze system behavior in both discrete-time and continuous-time domains.
4. Can you explain the steps involved in performing discrete time convolution?
Ans. To perform discrete time convolution, you need to align the two input signals, reverse one of them, multiply corresponding samples, and sum the results. This process is repeated for all possible alignments of the two signals.
5. Are there any properties associated with discrete time convolution?
Ans. Yes, discrete time convolution possesses several important properties. These include linearity, time shifting, time reversal, and time scaling. These properties allow for simplification and manipulation of convolution operations in signal processing applications.
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