class_03-convolution-discrete.pdf

Chengbin Ma UM-SJTU Joint Institute

Class#3

- The convolution sum (2.2)

- Convolution sum evaluation procedure (2.3)

Slide 1


Review of Previous Lecture Elementary Signals:

Exponential signals: the 1st-order dynamics (RC), time constant

Sinusoidal signals: the 2nd-order dynamics (LC), natural freq.

Exponentially damped sinusoidal signals (RLC): Time constant and

natural frequency

Step function: construct discontinuous waveforms, speed of response

Impulse function: sampling/shift/scaling properties

Ramp function: test tracking performance

Properties of Systems:

Stability, Memory, Causality, Invertibility, Time Invariance, Linearity.

Slide 2


This Lecture

Chapter 2: examine several methods for describing

the relationship between the input and output signals

of linear time-invariant (LTI) systems in time

domain.

Convolution sum/integral

Linear constant-coefficient difference/differential equation

The convolution sum: the output of an discrete-time

LTI system is the convolution sum of the input to the

system and the impulse response of the system.

Convolution sum evaluation procedure:

LTI Forms, Convolution Table, Reflection and Shift, Direct

Form

Slide 3


Class#3



Slide 4


Convolution

Slide 5

In mathematics and, in particular, functional analysis,

convolution is a mathematical operation on two

functions f and g, producing a third function that is

typically viewed as a modified version of one of the

original functions.

Convolution is similar to cross-correlation. It has

applications that include probability, statistics,

computer vision, image and signal processing,

electrical engineering, and differential equations.


LTI Systems

Slide 6

Convolution Sum Convolution Integral


Representation of Discrete-time Signals

Slide 7

How to represent discrete-time signals?


Sampling Property

Sample the value of t=0:

Slide 8

)()()()(

)()0()()(

000 tttxtttx

txttx

][][][][

][]0[][][

knkxknnx

nxnnx

0,0

0,1][ where

n

nn

A time-shifted impulse with

amplitude given by the value of

the signal at the time the impulse

occurs.

Still a function. only after integration will it become a value.
MichaelMichaelLineMichaelLine


Sum of Time-shifted Impulses

x[n] can be represented as weighted sum of

time-shifted impulses:

Slide 9

[ ] [ 1] [ 1] [0] [ ] [1] [ 1]

[ ] [ ]

[ ]: the entire signal

[ ]: a specific value of the signal [ ] at time .

k

x n x n x n x n

x k n k

x n

x k x n k

k

knkxnx ][][][ Weighted sum of basis functions Again, x[k]\delta[n-k] is still a function.
MichaelMichaelMichaelRectangleMichaelOval


Graphical Example

Slide 10

The representation of a signal x[n] as

a weighted sum of time-shifted

impulses.

k

knkxnx ][][][

impulse function.


Output of An LTI System

Slide 11

The output of an LTI system is the convolution

sum of the input to the (Linear) system and the

impulse response of the (Linear) system:


Convolution Sum

Convolution sum is denoted by the symbol *:

Slide 12

[ ] [ ]* [ ] [ ] [ ]

Derivation: [ ] [ ] (impulse response)

[ ] [ ] (time invariance)

[ ] [ ] [ ] [ ] (homogeneity)

[ ] [ ] [ ] [ ] [ ] [ ]

(superposition)

k

k k

y n x n h n x k h n k

n h n

n k h n k

x k n k x k h n k

x n x k n k y n x k h n k

LTI
Michael


Class#3



Slide 13


Evaluation Procedure (1)

#Example 2.1 in Page 101 (textbook)

For a discrete-time LTI system:

The output of the system in response to the input?

Slide 14

otherwise ,0

2 n ,2

1 n ,4

0n ,2

][nx

]1[2

1][][ nxnxny

k

knhkxnhnxny ][][][*][][


Evaluation Procedure (2)

Find impulse response first:

Slide 15

]1[2

1][][ nxnxny

otherwise ,0

1 n ,2

1 0n ,1

][nh

][][Let nnx

has memory

KEY STEP!
MichaelMichaelHighlightMichaelLine


Direct Form (1)

Slide 16

otherwise ,0

2 n ,2

1 n ,4

0n ,2

][nx

]2[2]1[4][2][ nnnnx

A weighted sum of time-shifted impulses

k

knhkxnhnxny ][][][*][][


Direct Form (2)

Slide 17

]2[2]1[4][2][ nnnnx

]2[2]1[4][2][ nhnhnhny

A weighted sum of time-shifted impulse

response outputs

Sum the weighted and time-shifted

impulse responses

],0 ,1 ,0 ,5 ,2 ,0 ,[][ ny


LTI Form

Slide 18

]1[]1[][]0[

][][][][][

nhxnhx

knhkxnhnxnyk

otherwise ,0

1 n ,2

1 0n ,1

][nh

otherwise ,0

2 n ,2

1 n ,4

0n ,2

][nx

Time invariance, linearity, memory


Convolution Table

Slide 19


Reflection and Shift (a convenient method)

Slide 20

signalproduct The :][][][

][

][][][][][

knhkxkw

kw

knhkxnhnxny

n

k

n

k

otherwise ,0

1 n ,2

1 0n ,1

][nh


Homework

Problem 2.33(d)(h)

Problem 2.34(a)(j)

- Due: before 2:00PM, Thursday in next week

Slide 21

class_03-convolution-discrete.pdf

Documents

weighted sum of time

time signals slide

convolution table

lti system slide

discretetime signals

discretetime lti system

time invariance

time domain