Convolution Part B

8/9/2019 Convolution Part B

1/39

Convolution Lecture BDr. Khurram Kamal


2/39

Linear systems, Convolution: Impulse response, input signals ascontinuum of impulses. Convolution, discrete time andcontinuous time. LTI systems and convolution

Objectives1. Show how a DT input signal can be sifted into a sum of scaled,

time shifted impulses2. Understand a DT LTI systems impulse response

3. Use superposition to show how DT convolution can be used tocalculate a systems response.

. Worked e amples !or DT LTI Systems

Discrete Time "onvolution

∑∞−∞=

−=k

k nhk xn y #$#$#$


3/39

Convolution is the process of calculating thesystem’s response signal y[n], given the system’simpulse response signal h[n]- or “ ho to solve the!i"erence e#uation ithout $no ing hat the!i"erence e#uation is%.&t is an important concept for i!entifying, !esigningor analysing the system an! D' convolution is alsouse! as a (asis for !eriving C' convolution. 'hemain aim is to un!erstan! ho D' convolution can(e applie! to input an! impulse response signals.

Discrete Time "onvolution


4/39

)evie of D' Linear system’s

Convolution is use! to calculate the system’sresponse y[n] to any input *[n], using thesystem’s impulse response signal, h[n]

'he D' system is normally represente! (y!i"erence e#uation+n! impulse response h[n] is a powersignal


5/39

ifting roperty for D' signalsBasic idea use set of time shifte! D' impulse / (asis signals0

to represent any D' signals.Consi!er a D' signal *[n] hich can (e ritten as the sum of

scale!, time shifte! impulse signals

'herefore the signal can (e e*presse! as

o any D' signal can (e e*presse! as

'he sifting property

actual value Impulse, timeshifted signal≠==−

≠==

−≠−=−=+−

1%1#1$#1$#1$

%%%#%$#$#%$

1%1#1$#1$#1$

nn xn x

nn xn x

nn xn x

δ

δ

δ #1$#1$ +− n x δ

+−+++−++−+= #1$#1$#$#%$#1$#1$#2$#2$#$ n xn xn xn xn x δ δ δ δ

∑∞

−∞= −= k k nk xn x #$#$#$ δ


6/39

'he D' signal, *[n] is additively!ecompose! into the follo ing

scaled,time shifted, impulse

components

1nly the (asis signals correspon!ing to $2-3, -4an! -5 are sho n

6*ample D' iftingroperty


7/39

D' ystem &mpulse response

+ very important ay to analyse a D' L'& systemis to stu!y the impulse response signal, h[n]/input is an impulse signal0

Loosely spea$ing this correspon!s to giving thesystem a unit $ic$ at n27, an! then seeing hathappens.

8or a D' L'&, the impulse response, h[n] has all theinformation that e*ists in the !i"erence e#uation


8/39

6*ample &mpulse response

'he D' impulse response, h[n]!etermine system’s properties

causalitysta(ility


9/39

'he D' Convolution um

8or any L'& !iscrete time system, the response toan input signal *[n] is given as follo s.

5. 9sing the sifting property

4. h[n] is the systems impulse response to δ [n]3. Because the system is time invariant

:. 9sing the superposition property /linear0

'his is the convolution sum hich sho s that L'&system response can (e e*presse! as a sum ofscale!, time shifte!, impulse responses.


10/39

&nterpreting the convolution sum 'he convolution sum can (e interprete! in

t o ays5. +s a sum/over $0 of scale! /(y *[$]0

shifte! impulse response signal h[n-$],here n is a free signal in!e*.

4. 8or each n/time in!e*0, sum over $ theinput signal *[$] multiplie! (y time shifte!an! reverse! impulse response h[n-$]


11/39

6*ample 5a D' L'& Convolution

Consi!er a D' L'& system an impulse responseh[n]2 [7 7 5 5 5 7 7] n2[-4 -5 7 5 4 3 :]

8or input se#uence*[n]2 [7 7 7.; 4 7 7 7]

'he result is / sum of scale!, shifte! impulse responses0

y[n]2 ...


12/39

Lets !erive the previous result in amore mathematical /e*amina(le0manner

h[n]2 u[n] u[4-n] or /u[n] > u[n-3]0 'herefore?[n-$]2 u[n-$] u[4-n


13/39

Consi!er a D' L'& system that has an impulseresponse

h[n]2 7.@An u[n]hat is the response hen the step input signal*[n]2 u[n] is applie!

n 7 n 7

6*ample 4 D' L'& Convolution

∑∞

−∞=−=

k

k nhk xn y #$#$#$

∑∞

−∞=

−

−=k

k n

k nuk u #$&.%#$

== ∑=

−n

k

k n

%

&.%&.%

#$'&.%1(1%

&.%1&.%1

&.%

%

1

1

'1(

nun

nn

+

−

+−

−=

−

−=

=

≥


14/39

6*ample 3 D' L'& Convolution

8in! the D' L'& system response hen

By using D' L'& convolution

#$).%2.%#$

#1$*.%#$+

1

nunh

nun xn

n

=

−= −


15/39

ummary to D'

Convolution+ny D' signal can (e sifte! into time shifte! impulse basissignals

'he system is impulse response , h[n] contains the sameinformation as the e#uivalent di erence equation L'& systemrepresentation.

'he impulse response h[n] can (e use! to predict the D' L'&system’s response to an ar(itrary input signal using the

convolution operator

Because of the sifting an! superposition properties.

Convolution is a (asis for Fourier & aplace transforms andtransfer functions!

∑∞

−∞=−=

k

k nk xn x #$#$#$ δ

∑∞

−∞=−=

k

k nhk xn y #$#$#$


16/39

Continuous 'ime Convolution

inear "ystems, #onvolution$ %mpulse response,input signals as continuum of impulses!#onvolution , Discrete time an! continuous time . L'&

systems an! convolution

1(Eectives5. ho ho a C' input signal */t0 can (e sifte! into a

continuum of time shifte!, impulse (asis signals4. 9n!erstan! a C' system’s impulse response properties3. uperposition an! the C' convolution integral:. or$e! e*amples of C' convolution.

∫ ∞

∞−−= τ τ τ d t h xt y '('('(

τ τ δ d t '( −


17/39

)evie D' Convolution

+ D' signal can (e sifte! into time shifte!, impulse(asis signals

'he system’s impulse response, h[n], is the system’soutput hen an impulse input, *[n]2 is applie!.

h[n] can (e use! to pre!ict the D' L'& systemsresponse, y[n], to an input signal, *[n], using theconvolution sum.

∑∞

−∞=−=

k

k nk xn x #$#$#$ δ

#$nδ

∑∞

−∞= −= k k nhk xn y #$#$#$


18/39

ifting e*pression for C' signals

8or a C' input signal, */t0, there are aninFnite num(er of time shifte!, impulse

(asis signals

here is the constant an! t is varia(le.

5. &t is only non Gero hen4. 9ni#ue (asis signal for each value

'he C' sifting e*pression is

δ #$ τ −

t τ

τ =t

τ


19/39

+t particular time t2 only oneimpulse (asis signals,

, is non Gero an! the outputis e#ual to */t=0 (ecause the C'impulse has a unit integral

ifting e*pression for C' signals

δ #$ τ −t τ


20/39

C' ystem &mpulse )esponse

'he C' unit impulse signal /t0, provi!es a unitarea (urst of energy at t27, in an inFnitely smalltime perio!.

+ll of the information containe! in an L'&, 1D6 iscontaine! in the C' impulse response signal h/t0

h/t0 represents the impulse /unforce!0 solution to

1D6.

δ


21/39

6*amples of C' &mpulse )esponse

&mpulse responses, -Hh/t0, for simple C' L'&systems

50 5 st or!er causal, sta(le

40 5 st or!er causal, unsta(le 304 n! or!er causal, unsta(le

'( t δ


22/39

uperposition an! C' Convolution

8rom the C' sifting property e have

Because the system is time invariant, e $no that

+lso (ecause the system is linear/ therefore superpositionapplies0, the overall response is

Convolution is a linear com(ination /integral0 of the timeshifte!, impulse response signals, scale! (y the magnitu!eof the input signal at that point. ometimes e*presse! as


23/39

6*ample 5 C' L'& Convolution

Let */t0 (e the input to a L'& systemith unit impulse response h/t0

*/t02 u/t0

'('( t uet h t −= τ τ τ d t h xt y '('('( ∫ ∞

∞−

−=

τ

τ τ τ

τ

τ

d ee

d t ueu

t t

t

∫ =

−=

−

∞

∞−

−−∫

%

( '(''(

%#%$

%#$ %≤=

>= −

t

t ee t t τ

'('1( t ue t −

−=


24/39

Consi!er C' input an! impulseresponse signals hich are unit(loc$s

*/t02 5 2 7 other ise


'2('(2% −−=≤≤ t ut ut


25/39

∫ ∫ ∫ ∫

∫ ∫

−=

−−−−=

−−−=

−=

∞

∞−

∞

∞−

∞

∞−

∞

∞−

t t

d d

d t uud t uu

d t uuu

d t h xt y

2%

11

'(''2('('(

'(''2('((

'('('(

τ τ

τ τ τ τ τ τ

τ τ τ τ

τ τ τ

'2('2('(

2#2$

2%#$

%#%$#$#$ 2%

−−−=

>=

≤


26/39

Calculate the convolution of


'1('(

'('(

'1(

2

−=

=

−−

−

t uet h

t uet x

t

t

τ τ τ d t h xt y '('('( ∫ ∞

∞−

−=


27/39

ummary to C' Convolution+ C' signal */t0 can (e represente! via the sifting property

here there are an inFnite num(er of time shifted,impulse basis signals one for each value of+ny C' L'& system /1D60 can (e uni#uely represente! interms of its impulse response h/t0Iiven the input signal an! the impulse response, the C' L'&systems output can (e !etermine! via convolution

Jote that this is an alternative ay of calculating thesolution y/t0 compare! to an 1D6. h/t0 contains the!erivative information a(out the " of the 1D6/ naturalresponse0 an! the convolve! input signal represents the' " /force! response0

δ #$ τ −t τ

∫ ∞

∞− −= τ τ τ d t h xt y '('('(


28/39


29/39

)evie C' D' Convolution

C' an! D' L'& systems are completely !escri(e! (y theirimpulse response through convolution sum integral

'his is (ecause the impulse response signal representsthe unforced solution , an! the convolution operatorcalculates the impulse response for each impulse (asissignals (sifting) of the input signal an! aggregates thepieces / superposition) .

e can transform / ith a (it of or$0 from the!i"erential !i"erence e#uation to the impulse response

signal an! (ac$ again without loss of information .

#$+#$#$#$#$ nhn xk nhk xn yk

=−= ∑∞

−∞=

'(+'('('('( t ht xd t h xt y =−= ∫ ∞

∞−τ τ τ


30/39

Convolution is a commutative operat

Convolution is a commutative operator

'his can (e easily proven. 'he t o systems aree#uivalent

'herefore, hen calculating the response of a systemto an input signal *[n], e can imagine the input signal(eing convolve! ith the unit impulse response h[n],or vice versa, hich ever appears the easiest . &t isimportant to realise, the input an! impulse responseare both *ust signals .

∑∑ ∞

−∞=

∞

−∞=

−=−k k

k n xk hk nhk x #$#$#$#$ ∫ ∫ ∞

∞−

∞

∞−−=− τ τ τ τ τ τ d t xhd t h x '('('('(


31/39

Convolution is a Linear 1perator

Convolution is a Linear 1perator

'his can (e easily veriFe! an! also for the C' case. 'herefore, the t o systems in parallel

are e#uivalent. 'he convolve! sum of t o impulseresponses is e#uivalent to consi!ering the t o e#uivalentparallel system /e#uivalent for !iscrete-time systems0

'('('(+'('(+'(''('((+'(

#$#$#$+#$#$+#$#'$#$(+#$

212121

212121

t yt yt ht xt ht xt ht ht x

n yn ynhn xnhn xnhnhn x+=+=+

+=+=+


32/39

6*ample Linear Commutative roperties

Let y[n] !enote the output /convolution0 of thefollo ing t o signals

M[n] is non Gero for all n. Jo use linear an!commutative properties to e*press y[n] as the sum oft o simpler convolution pro(lems. Let x 5[n ] 2 7.@ n u [n ], x 4[n ] 2 7.; n u [n ], it follo s that

+n! N[n]2 N5[n]


33/39

Causality for L'& ystems)emem(er , a causal system only !epen!s on presentan! past values of the input signal . &t !oes not use$no le!ge a(out future information.8or a causal D' L'& system, the impulse response musth[n]27 for n 7+s y[n] must not !epen! on *[$] for $Hn, so theimpulse response must be 0er1 (efore the impulseis applie!. imilarly for causal C' D' systems, the

convolution sum2 integral will only go up to n2t ,respectively

+ non causal/ elec mech0 system cannot (emanufacture!

#$+#$#$#$#$ nhn xk nhk xn yk

=−= ∑∞

−∞='(+'('('('( t ht xd t h xt y =−= ∫

∞

∞−τ τ τ


34/39

5 ! #ausal "ystem 'he system representing the D' impulse response

is causal (ecause h[n]27 for n 7.

4 ! 3on4causal "ystem 'he system representing the C' impulse response

h/t02 sin/t0 pi=t is not causal (ecause h/t0 is not e#ual to 7for t 7.

Jote 'here are very famous e*amples of non causalsystems, such as perfect high pass or lo pass Flter as e

shall e*amine in the course.

6*ample Causality


35/39

)emem(er + system is sta(le if every (oun!e!input pro!uces a (oun!e! output

5 ./ /% system is stable if an! only if itsimpulse response is absolutely summable$

5 #/ /% system is stable if anl! only if itsimpulse response is abslotely integratable$

/% "ystem "tability via%mpulse 'esponse


36/39

L'& ystem ta(ility via &mpulse)esponse

roof /D' systems0, consi!er a (oun!e! inputsignalO*[n]O B for all n

+pplying convolution an! ta$ing the a(solute value

9sing the triangle ine#uality /magnitu!e of a sumof a set of num(ers is no longer than the sum ofmagnitu!e of the num(ers0

∑∑ ∞=

−∞=

∞

−∞=

≤−≤ K

K k

k h Bk n xk hn y ,#$,,#$#$,,#$,


37/39

6*ample ystem ta(ility5. C' )C circuit ith a negative e*ponential impulse

response signal

4. D' system ith a step impulse response signal

'he D' accumulator integrator system is unsta(le asthe impulse response signal sums to inFnity. 'his ise#uivalent for C' systems ith h/t02u/t0

2%2% '(

1,'(

11,'(,

'('(

'(1

'(

RC e

RC dt e

RC dt t h

t xt ydt dy

RC

t ue RC

t h

RC t

RC t

RC t

=−==

=+

=

∞−∞ −∞

∞−

−

∫ ∫

∞===

=−−=

∑∑∑ ∞=

=

∞=

−∞=

∞=

−∞=

k

k

k

k

k

k

k uk h

n xn yn ynunh

%

1,#$,,#$,

#$#1$#$#$#$


38/39

ummaryL'& systems are completely characteri0ed(y their impulse response h[n], h/t0

Convolution iso Commutative

o Linear

'hese properties can (e use! to simplifyevaluating convolution, (y !ecomposing thepro(lem into simpler parts, an! then solving

them in!ivi!ually.

∑∑ ∞

−∞=

∞

−∞=

−=−k k

k n xk hk nhk x #$#$#$#$

∫ ∫ ∞∞−∞

∞−−=− τ τ τ τ τ τ d t xhd t h x '('('('(

'('('(+'('(+'(''('((+'(

#$#$#$+#$#$+#$#'$#$(+#$

212121

212121

t yt yt ht xt ht xt ht ht x

n yn ynhn xnhn xnhnhn x+=+=+

+=+=+


39/39

tan!ar! system properties of

o Causality h[n]27 for n 7, h/t027 t 7o ta(ility

Can (e interprete! using5. Di"erential !i"erence e#uation (ehaviour4. &mpulse response3. 'ransfer function

+n important part of this course is to analyse !esignsystems using /transforme!0 impulse responserepresentations.

ummary

∞

Documents

Convolution Part B