Convolution Graphical

7/31/2019 Convolution Graphical

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EECE 301

Signals & Systems

Prof. Mark Fowler

Note Set #11

C-T Systems: Computing Convolution Reading Assignment: Section 2.6 of Kamen and Heck


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Ch. 1 Intro

C-T Signal Model

Functions on Real Line

D-T Signal Model

Functions on Integers

System Properties

LTICausal

Etc

Ch. 2 Diff EqsC-T System Model

Differential Equations

D-T Signal ModelDifference Equations

Zero-State Response

Zero-Input Response

Characteristic Eq.

Ch. 2 Convolution

C-T System Model

Convolution Integral

D-T System Model

Convolution Sum

Ch. 3: CT FourierSignalModels

Fourier Series

Periodic Signals

Fourier Transform (CTFT)

Non-Periodic Signals

New System Model

New Signal

Models

Ch. 5: CT FourierSystem Models

Frequency Response

Based on Fourier Transform

New System Model

Ch. 4: DT Fourier

SignalModels

DTFT

(for Hand Analysis)DFT & FFT

(for Computer Analysis)

New Signal

Model

Powerful

Analysis Tool

Ch. 6 & 8: LaplaceModels for CT

Signals & Systems

Transfer Function

New System Model

Ch. 7: Z Trans.

Models for DT

Signals & Systems

Transfer Function

New System

Model

Ch. 5: DT Fourier

System Models

Freq. Response for DT

Based on DTFT

New System Model

Course Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between

the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).


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)()(

)()()]()([

tvtx

tvtxtvtx

dt

d

=

=

C-T convolution properties

Many of these are the same as for DT convolution.

We only discuss the new ones here.

See the next slide for the others

derivative

=

=

ttt

dhtxthdxdy )()()()()(

The properties of convolution help perform analysis and design tasks that

involve convolution. For example, the associative property says that (in

theory) we can interchange to order of two linear systems in practice,

before we can switch the order we need to check what impact that mighthave on the physical interface conditions.

1. Derivative Property:

2. Integration Property Let y(t) =x(t)*h(t), then


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Convolution Properties

These are things you can exploit to make it easier to solve convolution problems

1.Commutativity

You can choose which signal to flip)()()()( txththtx =

2. Associativity

Can change order sometimes one order is easier than another

)())()(())()(()( twtvtxtwtvtx =

3. Distributivity

may be easier to split complicated system h[n] into sum of simple ones

we can split complicated input into sum of simple ones

(nothing more than linearity)

OR

4. Convolution with impulses

)()()( = txttx

)()()()())()(()( 2121 thtxthtxththtx +=+


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Computing CT Convolution

-For D-T systems, convolution is something we do for analysis and forimplementation (either via H/W or S/W).

-For C-T systems, we do convolution for analysis

nature does convolution for implementation.

If we are analyzing a given system (e.g., a circuit) we may need to compute a

convolution to determine how it behaves in response to various different input

signals

If we are designing a system (e.g., a circuit) we may need to be able to visualize

how convolution works in order to choose the correct type of system impulse

response to make the system work the way we want it to.

Well learn how to perform Graphical Convolution, which is nothing more

than steps that help you use graphical insight to evaluate the convolution integral.


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Steps for Graphical Convolutionx(t)*h(t)

1. Re-Write the signals as functions of : x() and h()

2. Flipjust one of the signals around t= 0 to get eitherx(-) or h(-)a. It is usually best to flip the signal with shorter duration

b. For notational purposes here: well flip h() to get h(-)

3. Find Edges of the flipped signal

a. Find the left-hand-edge -value ofh(-): call it L,0

b. Find the right-hand-edge -value ofh(-): call it R,0

4. Shift h(-) by an arbitrary value oftto get h(t- ) and get its edgesa. Find the left-hand-edge -value ofh(t- ) as a function of t: call it L,t

Important: It will always beL,t

= t +L,0

b. Find the right-hand-edge -value ofh(t- ) as a function of t: call it R,t Important: It will always be R,t = t + R,0

Note: I use for

what the book

uses ... It is not

a big deal as they

are just dummy

variables!!!

dthxty )()()( =

Note: If the signal you flipped is NOT finite duration,one or both ofL,tand R,twill be infinite (L,t= and/or R,t= )


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5. Find Regions of -Overlap

a. What you are trying to do here is find intervals oftover which the product

x() h(t-

) has a single mathematical form in terms of

b. In each region find: Interval oftthat makes the identified overlap happen

c. Working examples is the best way to learn how this is done

Tips: Regions should be contiguous with no gaps!!!

Dont worry about < vs. etc.

6. For Each Region: Form the Productx( )h(t - ) and Integrate

a. Form productx() h(t- )

b. Find the Limits of Integration by finding the interval of over which theproduct is nonzeroi. Found by seeing where the edges ofx() and h(t- ) lieii. Recall that the edges ofh(t- ) are L,tand R,t, which often depend on

the value oft

So the limits of integration may depend on t

c. Integrate the productx() h(t- ) over the limits found in 6bi. The result is generally a function oft, but is only valid for the interval

of t found for the current region

ii. Think of the result as a time-section of the outputy(t)

Steps Continued


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7. Assemble the output from the output time-sections for all the regions

a. Note: you do NOT add the sections togetherb. You define the output piecewise

c. Finally, if possible, look for a way to write the output in a simpler form

Steps Continued


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x(t)

t2

2

h(t)

t1

3

Example: Graphically Convolve Two Signals

=

=

dthx

dtxhty

)()(

)()()(By Properties of

Convolution

these two forms are

Equal

This is why we can

flip either signal

By Properties of

Convolution

these two forms areEqual

This is why we can

flip either signalConvolve these two signals:


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x()

2

2

h(-)3

Step #2: Fliph( ) to geth(- )

1

x()

2

2

h()

1

3

Step #1: Write as Function of

Usually Easier

to Flip theShorter Signal

0

0


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x()

2

2

h(-)3

Step #3: Find Edges of Flipped Signal

0

1 0

R,0 = 0L,0 = 1

i i S #4 S if h( ) &


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Motivating Step #4: Shift byt to geth(t- ) & Its Edges

L,t = t + L,0

R,t =t + R,0

h(t-) =h(-2-)

3

3

Fort = -2Fort = -2

2

R,t = t + R,0

R,t = t + 0

R ,-2 = 2+0

L,t= t + L,0

L,t= t 1

L,-2 = -2 1

Just looking at 2 arbitrary tvalues

In Each Case We Get

3

1

h(t-) =h(2-)

Fort = 2Fort = 2

2

R,t = t + R,0

R,t = t + 0

R,2 = 2+0

L,t= t + L,0

L,t= t 1

L,2 = 2 1

D i St #4 Shift b t t h( ) & It Ed


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Doing Step #4: Shift byt to geth(t- ) & Its Edges

3

t 1

h(t )

ForArbitrary Shift bytForArbitrary Shift byt

t

R,t = t + R,0

R,t = t + 0

L,t= t + L,0

L,t= t 1

Step #5: Find Regions of Overlap


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h(t-)3

L,t=t -1

R,t=t

Step #5: Find Regions of -Overlap

x()

2

2

Region I

No -Overlap

t < 0

Region I

No -Overlap

t < 0

h(t-)3

t -1 t

x()

2

2

Region II

Partial -Overlap

0 t 1

Region II

Partial -Overlap

0 t 1

Want R,t 0 t 0Want L,t 0 t-1 0 t 1

Want R,t< 0 t< 0

Step #5 (Continued): Find Regions of Overlap


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Step #5 (Continued): Find Regions of -Overlap

h(t-)3

t -1 t

x()

2

2

Region IIITotal -Overlap

1 0 t > 1

h(t-)3

t -1 t

x()

2

2

Region IV

Partial -Overlap

2 2Want L,t 2 t-1 2 t 3


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h(t-)3

t -1 t

x()

2

2 Region V

No -Overlap

t > 3

Region V

No -Overlap

t > 3

Want L,t> 2 t-1 > 2 t > 3

Step #5 (Continued): Find Regions of -Overlap

Step #6: Form Product & Integrate For Each Region


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Step #6: Form Product & Integrate For Each Region

Region I:t < 0Region I:t < 0

Region II: 0 t 1Region II: 0 t 1

h(t-)3

t -1 t

2

2

h(t-)x() = 0

0allfor0)(

00

)()()(


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Step #6 (Continued): Form Product & Integrate For Each Region

h(t-) 3

t -1 t

x()

2

2

Region III: 1


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Region V:t > 3Region V:t > 3

Step #6 (Continued): Form Product & Integrate For Each Region

h(t-)3

t -1 t

2

2

h(t-)x() = 0

3allfor0)(

00

)()()(

>=

==

=

tty

d

dthxty

x()

Step #7: Assemble Output Signal


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Step #7: Assemble Output Signal

Region It < 0

Region I

t < 0

0)( =ty

Region II0 t 1

Region II

0 t 1

tty 6)( =

Region III1

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Convolution Graphical