worksheet 8 - github pages · an annotatable copy of the notes for this presentation will be...

29/07/2020, 10:22worksheet8

of 16http://localhost:8888/nbconvert/html/5/worksheet8.md?download=false

Worksheet 8

To accompany Chapter 3.5 Convolution and the ImpulseResponse

ColophonThis worksheet can be downloaded as a PDF file (https://cpjobling.github.io/eg-247-textbook/worksheets/worksheet8.pdf). We will step through this worksheet in class.

An annotatable copy of the notes for this presentation will be distributed before the second class meetingas Worksheet 7 in the Week 3: Classroom Activities section of the Canvas site. I will also distribute acopy to your personal Worksheets section of the OneNote Class Notebook so that you can add yourown notes using OneNote.

You are expected to have at least watched the video presentation of Chapter 3.5(https://cpjobling.github.io/eg-247-textbook/laplace_transform/5/convolution) of the notes(https://cpjobling.github.io/eg-247-textbook) before coming to class. If you haven't watch it afterwards!

After class, the lecture recording and the annotated version of the worksheets will be made availablethrough Canvas.

AgendaThe material to be presented is:

First Hour

Even and Odd Functions of TimeTime Convolution

Even and Odd Functions of TimeFill in the Blanks Quiz.

https://cpjobling.github.io/eg-247-textbook/worksheets/worksheet8.pdf

https://cpjobling.github.io/eg-247-textbook/laplace_transform/5/convolution

https://cpjobling.github.io/eg-247-textbook

29/07/2020, 10:22worksheet8


Even Functions of TimeA function is said to be an even function of time if the following relation holds

Polynomials with even exponents only, and with or without constants, are even functions. For example:

Write down the Taylor-series polynomial expansion of . Is even or odd?

Odd/Even?

!(")

cos " cos "

Odd Functions of TimeA function is said to be an odd function of time if the following relation holds

Polynomials with even exponents only, and with or without constants, are even functions.

Write down the Taylor-series polynomial expansion of . Is even or odd?

Odd/Even?

!(")

cos " cos "

29/07/2020, 10:22worksheet8


ObservationsFor odd functions

</pre>

The product of two even or two odd functions is an [Even/Odd] function.The product of an even and an odd function, is an [Even/Odd] function.

!(0) =

In the following will denote an even function and an odd function.(")!# (")!$

Time integrals of even and odd functionsFor an even function (")!#

(")%" = 2 (")%"!&

!&!# !

&

0!#

For an odd function (")!$

(")%" = 0!&

!&!$

Even/Odd Representation of an Arbitrary FunctionA function that is neither even nor odd can be represented as an even function by use of:!(")

(") = [!(") + !(!")]!#12

or as an odd function by use of:

(") = [!(") ! !(!")]!$12

Adding these together, an abitrary signal can be represented as

That is, any function of time can be expressed as the sum of an even and an odd function.

!(") = (") + (")!# !$

29/07/2020, 10:22worksheet8


Example 1Is the Dirac delta an even or an odd function of time?'(")

SolutionLet be an arbitrary function of time that is continous at . Then by the sifting property of thedelta function

!(") " = "0

and for = 0"0

Also for an even function (")!#

and for an odd function (")!$

29/07/2020, 10:22worksheet8


Even or odd?An odd function evaluated at is zero, that is .(")!$ " = 0 (0) = 0!$

Hence

(")'(")%" = (0) = 0!"

!"!$ !$

Hence the product is odd function of .(")'(")!$ "

Since is odd, must be even because only an even function multiplied by an odd function canresult in an odd function.

(")!$ '(")

(Even times even or odd times odd produces an even function. See earlier slide)

Time ConvolutionConsider a system whose input is the Dirac delta ( ), and its output is the impulse response . Wecan represent the input-output relationship as a block diagram

'(") ((")

In general

29/07/2020, 10:22worksheet8


Add an arbitrary inputLet be any input whose value at is , Then because of the sampling property of the deltafunction

(output is )

)(") " = * )(*)

)(*)((" ! *)

Integrate both sidesIntegrating both sides over all values of ( ) and making use of the fact that the deltafunction is even, i.e.

we have:

* !" < * < "

'(" ! *) = '(* ! ")

29/07/2020, 10:22worksheet8


Use the sifting property of deltaThe second integral on the left side reduces to )(")

The Convolution IntegralThe integral

)(*)((" ! *)%*!"

!"

or

is known as the convolution integral; it states that if we know the impulse response of a system, we cancompute its time response to any input by using either of the integrals.

)(" ! *)((*)%*!"

!"

The convolution integral is usually written or where the asterisk ( ) denotesconvolution.

)(") # ((") ((") # )(") #

Second Hour

Graphical Evaluation of the Convolution IntegralSystem Response by Laplace

29/07/2020, 10:22worksheet8


Graphical Evaluation of the Convolution IntegralThe convolution integral is most conveniently evaluated by a graphical evaluation. The text book givesthree examples (6.4-6.6) which we will demonstrate using a graphical visualization tool(https://uk.mathworks.commatlabcentral/fileexchange/25199-graphical-demonstration-of-convolution)developed by Teja Muppirala of the Mathworks.

The tool: convolutiondemo.m (https://github.com/cpjobling/eg-247-textbook/blob/master/laplace_transform/matlab/convolution_demo/convolutiondemo.m) (see license.txt(https://github.com/cpjobling/eg-247-textbook/blob/master/laplace_transform/matlab/convolution_demo/license.txt)).

In [ ]: clear allcd ../matlab/convolution_demopwdformat compact

In [ ]: convolutiondemo % ignore warnings

Convolution by Graphical Method - Summary of Steps

For simplicity, we give the rules for , but the procedure is the same if we reflect and slide )(") ((")

1. Substitute with – this is a simple change of variable. It doesn't change the definition of .

)(") )(*))(")

1. Reflect about the vertical axis to form )(*) )(!*)

1. Slide to the right a distance to obtain )(!*) " )(" ! *)

1. Multiply the two signals to obtain the product )(" ! *)((*)

1. Integrate the product over all from to .* !" "

https://uk.mathworks.commatlabcentral/fileexchange/25199-graphical-demonstration-of-convolution

https://github.com/cpjobling/eg-247-textbook/blob/master/laplace_transform/matlab/convolution_demo/convolutiondemo.m

https://github.com/cpjobling/eg-247-textbook/blob/master/laplace_transform/matlab/convolution_demo/license.txt

29/07/2020, 10:22worksheet8


Example 2(This is example 6.4 in the Karris)

The signals and are shown below. Compute using the graphical technique.((") )(") ((") # )(")

29/07/2020, 10:22worksheet8


h(t)The signal is the straight line but this is defined only between and . Wethus need to gate the function by multiplying it by as illustrated below:

Thus

u(t)The input is the gating function:

so

((") !(") = !" + 1 " = 0 " = 1(") ! (" ! 1))0 )0

((") $ +(,)((") = (!" + 1)( (") ! (" ! 1)) = (!" + 1) (") ! (!(" ! 1) (" ! 1)) = !" (") + (") + (" ! 1))0 )0 )0 )0 )0 )0

!" (") + (") + (" ! 1) (" ! 1) $ ! + +)0 )0 )01,2

1,

#!,

,2

+(,) = , + ! 1#!,

,2

)("))(") = (") ! (" ! 1))0 )0

-(,) = ! =1,

#!,

,1 ! #!,

,

Prepare for convolutiondemoTo prepare this problem for evaluation in the convolutiondemo tool, we need to determine theLaplace Transforms of and .((") )(")

convolutiondemo settingsLet g = (1 - exp(-s))/sLet h = (s + exp(-s) - 1)/s^2Set range !2 < * < !2

29/07/2020, 10:22worksheet8


Summary of result1. For : 2. For : and 3. For : 4. For : 5. For :

" < 0 )(" ! *)((*) = 0" = 0 )(" ! *) = )(!*) )(!*)((*) = 00 < " % 1 ( # ) = (1)(!* + 1)%* = = " ! /2& "

0 * ! /2*2 ''"0 "2

1 < " % 2 ( # ) = (!* + 1)%* = = /2 ! 2" + 2& 1"!1 * ! /2*2 ''

1"!1 "2

2 % " )(" ! *)((*) = 0

Example 3This is example 6.5 from Karris.

((") = #!"

)(") = (") ! (" ! 1))0 )0

Answer 3

.(") =!

"#$$

0 : " % 01 ! : 0 < " % 1#!"

(# ! 1) : 1 < " < "#!"

29/07/2020, 10:22worksheet8


Check with MATLAB

In [ ]: syms t taux1=int(exp(-tau),tau,0,t)

In [ ]: x2=int(exp(-tau),tau,t-1,t)

Example 4This is example 6.6 from the text book.

((") = 2( (") ! (" ! 1)))0 )0)(") = (") ! (" ! 2))0 )0

Answer 4

.(") =

!

"

#$$$$

0 : " % 02" : 0 < " % 12 : 1 < " % 2!2" + 6 : 2 < " % 30 : 3 % "

29/07/2020, 10:22worksheet8


System Response by LaplaceIn the discussion of Laplace, we stated that

We can use this property to make the solution of convolution problems even simpler.

! {!(") # /(")} = 0 (,)1(,)

Impulse Response and Transfer FunctionsReturning to the example we started with

Then the impulse response of the system will be given by:

Where be the laplace transform of the impulse response of the system . From properties of theLaplace transform we know that

so that and

A consequence of this is that the transform of the impulse response of a system with transferfunction is completely defined by the transfer function itself.

Previously we argued that the response of system with impulse response was given by theconvolution integrals:

Thus the Laplace transform of any system subject to an input is simply

and

Using tables, solution of a convolution problem by Laplace is usually simpler than using convolutiondirectly.

((")! {((") # '(")} = +(,)((,)

+(,) ((")

'(") $ 1

((,) = 1((") # '(") $ +(,).1 = +(,)

((")+(,)

((")

((") # )(") = )(*)((" ! *)%* = )(" ! *)((*)%*!"

!" !"

!"

)(")2 (,) = +(,)-(,)

.(") = {1(,)-(,)}!!1

29/07/2020, 10:22worksheet8


Example 5This is example 6.7 from Karris.

For the circuit shown above, show that the transfer function of the circuit is:

+(,) = =(,)34(,)3,

1/56, + 1/56

Determine the impulse respone of the circuit and the response of the capacitor voltage when theinput is the unit step function and .

((")("))0 ( ) = 074 0!

Assume and .6 = 1 F 5 = 1 )

29/07/2020, 10:22worksheet8


Solution 5a - Impulse response

Solution 5b - Step response

29/07/2020, 10:22worksheet8


HomeworkVerify this result using the convolution integral

((") # )(") = )(*)((" ! *)%*!"

!"

worksheet 8 - github pages · an annotatable copy of the notes for this presentation will be...

Documents