2006 Fall

Signals and SystemsSignals and SystemsLecture 4Lecture 4

Representation of signalsConvolutionExamples

2006 Fall

Chapter 2 LTI Systems

Example 1 an LTI system

0 2 t

tf11

0 1 2 t

ty1

1L

211 tftf 211 tyty

0 2 4 t

1

-10 2 4 t

tf2

1 L

ty2

1

2006 Fall

Chapter 2 LTI Systems

§2.1 Discrete-time LTI Systems : The Convolution Sum（卷积和）

§2.1.1 The Representation of Discrete-Time Signalsin Terms of impulses

11011 nxnxnxnx

knkxnxk

knkx

Example 2

1 0 1 2

1

2

3 nx

n

2006 Fall

Representing DT Signals with Sums of Unit Samples

Property of Unit Sample

Examples][][][][ 000 nnnxnnnx

10

1]1[]1[]1[

n

nxnx

00

0]0[][]0[

n

nxnx

10

1]1[]1[]1[

n

nxnx

2006 Fall

Written Analytically

Coefficients Basic Signals

]1[]1[][]0[]1[]1[]2[]2[

][][

nxnxnxnx

inxnxi

k

knkxnx ][][][

Note the Sifting Property of the Unit Sample

Important to note

the “-” sign

2006 Fall

Chapter 2 LTI Systems

§2.1.2 The Discrete-Time Unit Impulse Responses and the Convolution-Sum Representation of LTI Systems

1. The Unit Impulse Responses单位冲激响应

0 ,h n L n

2. Convolution-Sum （卷积和）

knhkxnyk

系统在 n时刻的输出包含所有时刻输入脉冲的影响

k时刻的脉冲在 n时刻的响应

nhnx

2006 Fall

Derivation of Superposition SumDerivation of Superposition Sum

Now suppose the system is LTI, and define the unit sample response h[n]:

– From Time-Invariance:

– From Linearity:

][][ nhn

][][ knhkn

][*][][][][][][][ nhnxknhkxnyknkxnxkk

convolution sum

2006 Fall

The Superposition Sum for DT SystemsThe Superposition Sum for DT Systems

Graphic View of Superposition Sum

2006 Fall

Hence a Very Important Property of LTI Systems

The output of any DT LTI System is a convolution of the input signal with the unit-sample response, i.e.

As a result, any DT LTI Systems are completely characterized by its unit sample response

k

knhkx

nhnxnyLTIDTAny

][][

][*][][

2006 Fall

Calculation of Convolution SumCalculation of Convolution SumChoose the value of n and consider it fixed

k

knhkxny ][][][

View as functions of k with n fixedFrom x[n] and h[n] to x[k] and h[n-k]Note, h[n-k]–k is the mirror image of h[n]–n with the origin shifted to n

2006 Fall

Calculating Successive Values: Shift, Multiply,

Sum

y[n] = 0 for n <y[-1] =y[0] =y[1] =y[2] =y[3] =y[4] =y[n] = 0 for n >

k

knhkxny ][][][-1

1

2

-2

-3

1

1

4

2006 Fall

Calculation of Convolution SumCalculation of Convolution SumUse of Analytical FormUse of Analytical Form

Suppose that and

then

][][ nuanx n ][][ nubnh n][*][][ nhnxny

k

knhkx ][][

k

knk knubkua ][][

n

k

knkba0

ba

ba

nuab

abnuna

nn

n

][

][)1(11

knknu

kku

][

,0][

00

,0

时为

n

n

2006 Fall

Calculation of Convolution SumCalculation of Convolution SumUse of Array MethodUse of Array Method

2006 Fall

Calculation of Convolution SumCalculation of Convolution SumExample of Array MethodExample of Array Method

If x[n]<0 for n<-1,x[-1]=1,x[0]=2,x[1]=3, x[4]=5,…,and h[n]<0 for n<-2,h[-2]=-1, h[-1]=5,h[0]=3,h[1]=-2,h[2]=1,….In this case ,N=-1,M=-2,and the array is as follows

So,y[-3]=-1, y[-2]=3, y[-1]=10, y[0]=15, y[1]=21,…, and y[n]=0 for n<-3

2006 Fall

Calculation of Convolution SumCalculation of Convolution SumUsing MatlabUsing Matlab

The convolution of two discrete-time signals can be carried out using the Matlab M-file conv.

Example:– p=[0 ones(1,10) zeros(1,5)]; – x=p; h=p;– y=conv(x,h);– n=-1:14;– subplot(2,1,1),Stem(n, x(1:length(n)))– n=-2:24;– subplot(2,1,2),Stem(n, y(1:length(n)))

2006 Fall

Calculation of Convolution SumCalculation of Convolution SumUsing Matlab ResultUsing Matlab Result

2006 Fall

ConclusionConclusionAny DT LTI Systems are completely

characterized by its unit sample response.

Calculation of convolution sum:– Step1:plot x and h vs k, since the convolution sum is

on k;– Step2:Flip h[k] around vertical axis to obtain h[-k];– Step3:Shift h[-k] by n to obtain h[n-k] ;– Step4:Multiply to obtain x[k]h[n-k];– Step5:Sum on k to compute – Step6:Index n and repeat step 3 to 6.

][*][][ nhnxny

k

knhkx ][][

2006 Fall

ConclusionConclusion

Calculation Methods of Convolution Sum– Using graphical representations;– Compute analytically;– Using an array;– Using Matlab.

2006 Fall

Chapter 2 LTI Systems

§2.2 Continuous-Time LTI Systems : The Convolution Integral （卷积积分）

§2.2.1 The Representation of Continuous-Time Signalsin Terms of impulses

dtxtx

——Sifting Property

§2.2.2 The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems

y t x t h t x h t d

2006 Fall

Representation of CT Signals

Approximate any input x(t) as a sum of shifted, scaled pulses (in fact, that is how we do integration)

tkttktkxtx )1(),()(

2006 Fall

Representation of CT Signals (cont.)

areaunitahast)(

)()( ktkx

k

ktkxtx )()()(

dtxtx )()()(

↓limit as →0

⇓

Sifting

property

of the unit

impulse

2006 Fall

Response of a CT LTI System

Now suppose the system is LTI, and define the unit impulse response h(t):

(t) →h(t)– From Time-Invariance:

(t −) →h(t −)– From Linearity:

)(*)()()()(

)()()(

thtxdthxty

dtxtx

2006 Fall

Superposition Integral for CT SystemsSuperposition Integral for CT SystemsGraphic View of Staircase Approximation

2006 Fall

CT Convolution MechanicsCT Convolution MechanicsTo compute superposition integral

– Step1:plot x and h vs , since the convolution integral is on ;

– Step2:Flip h( around vertical axis to obtain h(-;– Step3:Shift h(-) by n to obtain h(n-) ;– Step4:Multiply to obtain x(h(n-;– Step5:Integral on to compute – Step6:Increase t and repeat step 3 to 6.

dthxthtxty )()()(*)()(

dthx )()(

2006 Fall

Basic Properties of Convolution

Commutativity: x(t)∗h(t) h(t) ∗x(t)Distributivity :

Associativity:

x(t)∗(t −to ) x(t −to ) (Sifting property: x(t) ∗(t) x(t))

An integrator:

)(*)()(*)()]()([*)( 2121 thtxthtxththtx

)(*)](*)([)](*)([*)( 2121 ththtxththtx

t

dxtutx )()(*)(

2006 Fall

Convolution with Singularity FunctionsConvolution with Singularity Functions

)()(*)( tfttf

)()(*)( tfttf

t

dftutf )()(*)(

)()(*)( )()( tfttf kk

)()(*)( )()( tfttf kk

2006 Fall

More about Response of LTI SystemsMore about Response of LTI SystemsHow to get h(t) or h[n]:

– By experiment;– May be computable from some known

mathematical representation of the given system.

Step response:

t

dhts )()(

n

k

khns ][][

2006 Fall

SummarySummaryWhat we have learned ?

– The representation of DT and CT signals;– Convolution sum and convolution integral

Definition; Mechanics;

– Basic properties of convolution;What was the most important point in the lecture?What was the muddiest point?What would you like to hear more about?

2006 Fall

ReadlistReadlist

Signals and Systems:– 2.3,2.4– P103~126

Question: The solution of LCCDE

(Linear Constant Coefficient Differential or Difference Equations)

2006 Fall

Problem SetProblem Set

2.21(a),(c),(d)2.22(a),(b),(c)