signals & systems - concordia universityamer/teach/elec264/notes/topic2... · •a. oppenheim,...

CT Systems:

Impulse response

Convolution integral

Block diagram of systems

Properties using the impulse response

Systems characterized by Differential Equations

oDT Systems

Impulse response

Convolution sum

Block diagram of systems

Properties using the impulse response

Systems characterized by Difference Equations

Summary

ELEC264: Signals And Systems

Topic 2: LTI Systems and Convolution

Aishy Amer

Concordia University

Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

•M.J. Roberts, Signals and Systems, McGraw Hill, 2004

•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

Properties of the Unit Impulse

The area under the unit impulse

Sampling (Sifting) Property:

Scaling Property

Equivalence Property

Relation to u(t):

1)( dtt

)/()( ;||

1 abtbata

Impulse Response of Linear

Time-Invariant (LTI) Systems

Linearity

Time Invariance

)()()()(

)()(then

)()( if

00 ttyttxtytx

Impulse Response of LTI

systems The response of a Linear Time-Invariant (LTI) system

to the unit impulse δ(t) is called "The impulse

response" h(t)

h(t) = LTI-SYS { δ(t) }

The impulse response h(t) completely characterizes an

LTI system

y(t) = FUNCTION { x(t),h(t) }

CONVOLUTION

Impulse response of Basic

Systems

???????)(;)]([)(:LTI)(not Squarer

)()( ;

)()( :ator(LTI)Differenti

)()( ;)(y(t) :(LTI) Integrator

)()(x(y(t) :(shifter)delay timeIdeal

)()( x(t);y(t) :systemIdentity

thtxty

tuthdx

ttthtt

Basic Systems: Differentiators &

Integrators

Differentiators are:

Difficult to implement

Sensitive to noise and errors

Alternatives : Integrators

Integrators : amplifiersfinite) (to )(2)()()(

)(2)()(

)(2)()()()(

0)( assume

)(2)()(

dyxtyty

dyxtytydtdt

tytxdt

0;)()(

Relationship: CT impulse response and

step response

In any CT LTI system let an excitation, x(t), produce the response, y(t). Then the excitation

will produce the response

It follows then that the unit impulse response is the first derivative of the unit step response and, conversely that the unit step response is the integral of the unit impulse response

))(( txdt

))(( tydt

Outline

CT Systems

Impulse response

Block Diagrams of Systems

System properties using the impulse response

DT systems

DT Impulse response

Convolution sum

Summary

The Convolution Integral

1) An arbitrary input x(t) can be expressed as a weighted

sum of time-shifted impulses

2) An LTI is described by an impulse response h(t)

The Convolution Integral

The output y(t) must be a weighted sum of time-shifted

impulse responses

The Convolution Integral:

dtSxtxSty

dtxStxSty

tSthdtxtx

)}({)()}({)(

mswitch thecan We

opeartorlinear a also is

operator linear a is

} )()( {} )( { )(

} )( {)( )()()(

Assuming S is LTI

)()}({

invariant- timeis

tth parameter with ofFunction :)(

over on Intergrati:Important

)(*)()()()( thtxdthxty

} )( {)( tSth

CONVOLUTION

Interpretation

Interpretation:

replacing each signal amplitude at time t by a

weighted sum of its neighbors

dhtxtxthty

dthxthtxty

)()()(*)()(

Convolution Integral:

Graphical Illustration

Let the excitation, x(t), and the impulse

response, h(t), be

The convolution integral:

In the convolution integral there is a factor

We visualize this quantity

dthxty )()()(

The functional transformation in going

from to is

The convolution value is the area under the

product of x(t) and

This area depends on what t is

First, as an example, let t = 5

For this choice of t the area under the product is

So 0y(5) ),()()(with thtxty

Now let t=0

Therefore the area under the product is 2, i.e., y(0) = 2

The process of convolving to find y(t) is

illustrated below

Convolution Integral: Graphical

Illustration

Interpretation: replacing each signal amplitude

at time t by a weighted sum of its neighbors

Smoothing of sharp transitions of x(t)

Filtering out some content from x(t)

Removing some content

Steps for graphical convolution

1. Sketch the waveform for input x(τ) by changing the independent variable

from t to τ and keep the waveform for x(τ) fixed during convolution.

2. Sketch the waveform for the impulse response h(τ) by changing the

independent variable t to τ.

3. Reflect h(τ) about the vertical axis to obtain the time-inverted impulse

response h(-τ).

4. Shift the time-inverted impulse function h(-τ) by a selected value of "t".

The resulting function represents h(t-τ).

5. Multiply function x(τ) by h(t- τ) and plot the product function x(τ)h(t-τ).

6. Calculate the total area under the product function x(τ)h(t-τ) by integrating

it over τ =[-∞,∞].

7. Repeat steps 4-6 for different values of t to obtain y(t) for all times, -

∞≤t≤∞.

01&0:case3

01&0:case2

01&0:1 case

)()()(

dthxty

Convolution: Example 1 infinite-duration signals

Convolution: example 2 finite & infinite-duration signals

Convolution: example 3 with finite-duration signals

Convolution: example 4

Exam question

only. itiesdiscontinu threehas )(

if ? (b)

?)( (a)

Solution: Step 1

(a) Write down x(t) and h(t) functionally and graphically

Note that h(t) is a scaled version of x(t)

dthxthtxty

)()()()()(

Solution: Step 2

Sketch h(-τ) and h(t-τ)

h(-τ)

Rreflection around y-axis

Chage t to τ

h(t-τ) = h(-τ+t)

Add t to all axis points

Move the graph away to the left

Solution: Step 3

Slide h(t-τ) to the right and collect the overlap

As you go, find

Limits for y(t)

Limits for integration

Solution: Step 3a

tORtandtfor

][1)-)h(tx(

graph) from (findn integratiofor Limits

)(for Limits

Solution: Step 3b

tORtandtfor

][1)-)h(tx(

)(for Limits

Solution: Step 3c

tORtandtfor

t 1][1)-)h(tx(

)(for Limits

Solution: Step 4

Put the limits together to make y(t)

Solution: Step 5

(b) find the first derivative of y with respect to t both

functionally and graphically

This function has 4 discontinuities

Only when = 1, it has 3 discontinuities

• (two discontinuities become one)

Note that we know 0< 1

Outline

CT Systems

Impulse response

Block Diagram of Systems

DT systems

DT Impulse response

Convolution sum (DT)

DT properties using the impulse response

DT Systems characterized by Difference Equations

Summary

System Block Diagrams

LTI Systems can be described using the impulse

response h(t) which completely characterizes an LTI system

LTI systems can also be described

mathematically by differential equations

)()()()( 0012 txbtyatyatya

Block Diagram ElementsBlock diagram: A very useful method for describing and analyzing systems is the block diagram

A block diagram can be drawn directly from the differential equation which describes the system

For example, if the system is described by

It can also be described by this block diagram …

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Summary

System properties via the

convolution properties Convolution properties help to solve convolution of

complex signals in term of operations on another

signal for which the convolution is known

Example:

)(*)()(*)(

)(*)]()([)(*)()(y

:property edistibutiv theUsing

)()()()()(

thtxthtx

thtxtxthtxt

txtxtuetuetx tt

convolution properties

)(*)()(*)(

)(*)]()([)(y

:Example

thtuethtue

thtuetuet

systems two thecascade

i.e., ity,associativ theusecannot we

LTInot )(

)()( );(2)(:Example

txtytxty

System properties via the convolution

properties: System Interconnections

Example:

o Since the integrator and differentiator are both LTI system operations, when

used in combination with another system having impulse response h(t), we find

that the cascade property holds

Performing differentiation or integration before a signal enters an LTI system,

gives the same result as performing the differentiation or integration after the

signal passes through the system

• “Convolution” property:

System properties using the

impulse response

for causal-non is)()(

for causal is )()(

)integratoran is system (this causal is )()(

:Examples

impulse response

System stability: A CT system is BIBO stable if its impulse response is

absolutely integrable

stable is )(h(t) )2

|)(| since unstable is system

)()()()(*)()(

summer)or or (accumulat integratoran is )()( )1

:Examples

dxdtuxthtxty

impulse response

1)( with invertable is )(2)()(2)()2

)()(*)(

)((t)h with invertable is )(h(t)

)t-x(ty(t):delay Ideal )1

:Examples

tthtthtxty

impulse response

A CT LTI system is memory less if and only if

)constant );()( (

0for 0)(

KtKxty

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Summary

LTI Systems: Differential

Equations LTI Systems can be described using the impulse

response h(t)

mathematically by a differential equation

A linear combination of a function and its derivatives

LTI Systems: Differential Equations

General Nth-order linear constant-coefficient differential equation

Differential equations play a central role in describing input-output relationships in (electrical) systems

The general solution is given by: y(t) = yp(t) + yh(t)

yp(t) is a particular solution

yh(t) is the homogeneous solution satisfying

• To get yh(t), N auxiliary conditions are required

• Auxiliary conditions are the values of:

at some point in time

constants real , ;)()(

k badt

1 )(,,

Equation: Example

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Summary

Impulse response of DT LTI

Systems

Linearity

Time Invariance

][][][][

][][then

][][ if

00 nnynnxnynx

Systems

Once the response to a unit impulse is known, the response of any discrete-time LTI system to any arbitrary excitation can be found

Any arbitrary excitation is a sequence of amplitude-scaled and time-shifted DT impulses

Therefore the response is a sequence of amplitude-scaled and time-shifted DT impulse responses

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

nxnxnxnx

systems

The impulse response h[n] completely

characterizes an LTI system

DT LTI Systems: Use the unit impulse to construct any signal

A DT signal is a sequence of individual weighted impulses

The response of the system is the sum of delayed h[n]

Response of LTI Systems

][]}[{ nynxS

nhnSnxSny

of response Impulse :][

][]}[{]}[{][

][][ If

Question: if h[n] known, how to find y[n]?

y[n] = x[n]*x[n]

Response of LTI Systems:

Example

Response of LTI Systems:

Example

Relationship: DT impulse response

and step response

In any DT LTI system let an excitation, x[n], produce the response, y[n]

Then the excitation x[n] - x[n - 1] will produce the response y[n] - y[n - 1]

It follows then that the unit impulse response is the first backward

difference of the unit step response and, conversely that the unit

sequence (step) response is the accumulation of the unit impulse

response

nsnsnsnh

response step theis ][ where]1[][][

DT impulse response and step

response:Example

Suppose that the step response is given by

What is the impulse response h[n] ?

445][ nuns

]1[][][

nsnsnh

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Convolution sum

Summary

Convolution of Two Signals

A signal x[n] can be represented as linear

combination of DELAYED Impulses

If the system is LINEAR

nhkxny

nykxny

knkxnx

][][][

][ toresponse [n]][

][ tosystem theof response ][with

][][][

Convolution of Two Signals

If the system is Time Invariant

n Convolutio ][][][

][][][

0 Omit the ]][]0[][ OR[

][][then

][ toresponse ][ if

nhnxny

knhkxny

nhkxny

nhnhnh

The Convolution Sum

The response, y[n], to an arbitrary excitation, x[n], is of

the form

h[n] is the impulse response

This can be written in a more compact form,

called the convolution sum

)1()1()()0()1()1()( nhxnhxnhxny

knhkxny ][][][

Obtain the sequence h[n-k]

Reflecting h[k] about the origin to get h[-k]

Shifting the origin of the reflected sequence to k=n

Multiply x[k] and h[n-k] for

Sum the products to compute the output

sample y[n]

Computation of the convolution

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

Convolution sum: graphical

1) Sketch the waveform for input x[k] by changing the independent variable of

x[n] from n to k and keep the waveform for x[k] fixed during steps (2)-(7).

2) Sketch the waveform for the impulse response h[k] by changing the

independent variable from n to k.

3) Reflect h[k] about the vertical axis to obtain the time-inverted impulse

response h[-k].

4) Shift the sequence h[-k] by a selected value of n. the resulting function

represents h[n-k].

5) Multiply the input sequence x[k] by h[n-k] and plot the product function

x[k]h[n-k].

6) Calculate the summation .

7) Repeat steps (4)-(6) for -∞≤n≤+∞ to obtain the output response y[n] over all

time n.

Forming the sequence h[n-k]

Computing a discrete convolution:

Example 1

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

][][][

nxhnxh

knxkhny

If the system LTI

Computing a discrete convolution:

Example 2

Convolution sum:

Example 3: infinite-duration signals

-„k‟ refers here to the “n” of the other

examples

-Compare to the CT equivalent example

Convolution sum:

Example 4: finite-duration signals

Convolution sum : Example 5

.1 ),1

,10 ,1

otherwise. ,0

,10 ,1

][][][

Nnununh

Convolution Sum:

Example 6

Convolution Sum:

Example 6

Convolution Sum:

Example 6

Convolution Sum:

Example 6

Convolution Sum:

Example 7 Consider an LTI system with input x[n] and unit impulse h[n] response

shown. Find the output of this system

Solution: the output of the system

Convolution Sum:

Analytical Example 8

Convolution Sum:

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Convolution sum

Summary

LTI Systems can be described using the impulse

response h[n] which completely characterizes an LTI system

mathematically by difference equation

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

A block diagram can be drawn directly from the difference equation which describes the system

For example, if the system is described by

It can also be described by

the block diagram below in

which “D” represents a delay

of one in discrete time

][]2[2]1[3][ nxnynyny

Block Diagram Elements

Discrete-Time

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Convolution sum

Summary

System properties via

Convolution properties Commutative

Distributive or Linear

The distributive property implies that the

following two LTI systems are equivalent

][*][][*][ nxnhnhnx

][*][][*][])[][(*][ 2121 nhnxnhnxnhnhnx

System properties via Convolution

Direct consequence of the distributivity property:

If two systems are excited by the same signal and their responses are added they are said to be parallel connected.

The parallel connection of two systems can be viewed as a single system whose impulse response is the sum of the two individual system impulse responses

Convolution properties

Direct consequence of the associativity property:

If the response of one system is the excitation of another system the two systems are said to be cascade connected

The cascade connection of two systems can be viewed as a single system whose impulse response is the convolution of the two individual system impulse responses

Convolution properties

The following properties can be proven from the

convolution definition:

])1[][(][]1[][

:property difference Backward

][][][][][

:propertyDelay

][][][

:property n"Convolutio"

nhnhnxnyny

nnAhnxnhnnAxnny

nnAxnnAnx

Delay property: Example

][][][][][ 000 nhnnAxnnAhnxnny

properties: Example 1

))5(exp()5()5()(

:have weproperty,n convolutio By the

ANSWER

).5()( Compute

)exp()(Let

5 nanxnnx

Problem: a discrete-time LTI system has impulse response

Find the output y[n] due to input

x[n] = u[n + 1] – u[n - 1] + 2δ[n - 2],

where u[n] is the discrete time unit step function

Suggestions: Use convolution properties

Plot the functions of h[n] and x[n]

In other problems: you may be • Given y(t) = integral (..); find h(t) analytically or graphically

• Given x(t) and h(t) ; find y(t) analytically or graphically

• Pay attention that you may need to do variable substitution, e.g.,

integral(e^(t-p) h(p-5) dp) –inf to tp' = p-5 p=p'+5integral(e^(t-p'-5) h(p') dp') -inf to t-5

Solution: the simplest way to solve for the output y[n] would be to first plot the functions of h[n] and x[n]

]1[2][3][ nnnh

System properties via Convolution sum

The sequence h[n] consists of two samples. Therefore, convolving x[n] and h[n] can be simplified by convolving

x[n] with h[n] one sample at a time.

For example, we can convolved x[n] first with and then with

Finally, the convolution sum (y[n]) can be then obtained by adding the two sequences (adding sample by corresponding sample).

In doing this, the output y[n] is

The same can be achieved graphically

]1[2][3][ nnnh

][3][1 nnh

]1[2][2 nnh

]3[4]2[6]1[][]1[3][ nnnnnny

nxhnxhknxkhny ....)1()1()0()0(][][][

System Properties using impulse

response

It can be shown that a BIBO-stable DT system has an

impulse response that is absolutely summable

nhBkhknxkhknxny ][][][][][][

System properties using impulse

response

systems inverse are

]1[][][ : difference backward The

][][ :r accumulato The

:Example

nynynw

][][*][][*][ nnhnhnhnh ii

Convolution sum properties

“Finite/Infinite” Systems: reflected

in h[n]

Depending on h[n], we divide LTI systems into

Finite-duration impulse response (FIR) systems

Infinite-duration impulse response (IIR) systems

]1[][][ nxnxny

in h[n]

Finite-duration impulse response (FIR) system

The impulse response has only a finite number of nonzero samples

Ideal delay

Forward difference

Backward difference

integer positive a ],[][

nnnxny

][]1[][

nxnxny

]1[][][

nxnxny

in h[n]

Infinite-duration impulse response (IIR) system

The impulse response is infinitive in duration

Accumulator

Stability

FIR systems always are stable, if each of h[n] values is

finite in magnitude

IIR systems can be stable, e.g.

][][][

|)|1(1||

1|| with ][][

anuanh

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Convolution sum

Summary

LTI Systems: Difference

Equations An important class of LTI systems:

Input & output satisfy an Nth-order Linear Constant Coefficient Difference

Equations (LCCD) equation

An LTI System can be described by a difference

equation

recursive)-(non memoryless is System ,0 and 1 if

]}[{]}[{]}[{]}[{

0,1],[][

nbnxnany

namnxbmnya

k mnxbknya00

LTI System: Difference Equation

Example 1

]1[][]1[][

nxbnxbnyany

A first order LTI system:

LTI System: Difference Equation

Example 2

Difference equation representation of the accumulator

][]1[][

]1[][][][][

nxnyny

nynxkxnxny

k mnxbknya00

One-sample

y[n-1]

Recursive representation

LTI Systems: Solving Difference

Equations: Example 3

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Convolution sum

Summary

Outline

CT Systems

Impulse response

DT systems

DT Impulse response

Convolution sum

Summary

Appendix

& impulse response

Let a CT system be described by

Let the excitation be a unit impulse at time, t = 0

Then the response, y, is the impulse response, h.

Since the impulse occurs at time, t = 0, and nothing has excited the system before that time, the impulse response before time, t = 0, is zero

After time, t = 0, the impulse has occurred and gone away

Therefore there is no excitation and the impulse response is the homogeneous solution of the differential equation

)()()()( 012 txtyatyatya

)()()()( 012 tthathatha

& impulse response

What happens at time t = 0?

The equation must be satisfied at all times. So the left sideof the equation must be a unit impulse

We already know that the left side is zero before time, t = 0because the system has never been excited.

We know that the left side is zero after time, t = 0, becauseit is the solution of the homogeneous equation whose rightside is zero.

This is consistent with an impulse. The impulse responsemight have in it an impulse or derivatives of an impulsesince all of these occur only at time, t = 0.

What the impulse response does have in it depends on theequation.

)()()()( 012 tthathatha

& impulse response

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