2006 Fall Signals and Systems Lecture 4 Representation of signals Convolution Examples

2006 Fall Chapter 2 LTI Systems Example 1 an LTI system 0 2 t t 1 L t t 1 L

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 Signals in Terms of impulses Example 2 n

2006 Fall Representing DT Signals with Sums of Unit Samples Property of Unit Sample Examples

2006 Fall Written Analytically CoefficientsBasic Signals 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 2. Convolution-Sum （卷积和） 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

2006 Fall Derivation of Superposition Sum Now suppose the system is LTI, and define the unit sample response h[n]: – From Time-Invariance: – From Linearity:  convolution sum

2006 Fall The 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

2006 Fall Calculation of Convolution Sum Choose the value of n and consider it fixed View as functions of k with n fixed From 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 >

2006 Fall Calculation of Convolution Sum Use of Analytical Form Suppose that and then

2006 Fall Calculation of Convolution Sum Use of Array Method

2006 Fall Calculation of Convolution Sum Example 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 Sum Using 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 Sum Using Matlab Result

2006 Fall Conclusion Any 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.

2006 Fall Conclusion 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 Signals in Terms of impulses ——Sifting Property §2.2.2 The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems

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)

2006 Fall Representation of CT Signals (cont.) ↓  limit as Δ  →  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:

2006 Fall Superposition Integral for CT Systems Graphic View of Staircase Approximation

2006 Fall CT Convolution Mechanics To 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.

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:

2006 Fall Convolution with Singularity Functions

2006 Fall More about Response of LTI Systems How to get h(t) or h[n]: – By experiment; – May be computable from some known mathematical representation of the given system. Step response:

2006 Fall Summary What 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 Readlist Signals and Systems: – 2.3,2.4 – P103~126 Question: The solution of LCCDE (Linear Constant Coefficient Differential or Difference Equations)

2006 Fall Problem Set 2.21(a),(c),(d) 2.22(a),(b),(c)