Time-Domain System Analysis M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1

Continuous Time M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl2

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl3

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl4

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl5

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl6

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl7

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl8

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl9

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl10

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl11

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl12

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl13

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl14

The Convolution Integral Exact Excitation Approximate Excitation M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl15

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl16

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl17

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl18

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl19

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl20

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl21

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl22

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl23

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl24

The Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl25

A Graphical Illustration of the Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl26

A Graphical Illustration of the Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl27

A Graphical Illustration of the Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl28

A Graphical Illustration of the Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl29

A Graphical Illustration of the Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl30

A Graphical Illustration of the Convolution Integral The process of convolving to find y(t) is illustrated below. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl31

A Graphical Illustration of the Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl32

A Graphical Illustration of the Convolution Integral M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl33

Convolution Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl34

Convolution Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl35

Convolution Integral Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl36

The Unit Triangle Function The unit triangle, is the convolution of a unit rectangle with Itself. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl37

System Interconnections If the output signal from a system is the input signal to a second system the systems are said to be cascade connected. It follows from the associative property of convolution that the impulse response of a cascade connection of LTI systems is the convolution of the individual impulse responses of those systems. Cascade Connection M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl38

System Interconnections If two systems are excited by the same signal and their responses are added they are said to be parallel connected. It follows from the distributive property of convolution that the impulse response of a parallel connection of LTI systems is the sum of the individual impulse responses. Parallel Connection M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl39

Unit Impulse Response and Unit Step Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl40

Stability and Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl41

Systems Described by Differential Equations M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl42

Systems Described by Differential Equations M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl43

Systems Described by Differential Equations M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl44

Systems Described by Differential Equations M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl45

MATLAB System Objects M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl46

MATLAB System Objects M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl47

Discrete Time M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl48

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl49

Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl50

Impulse Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl51

Impulse Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl52

Impulse Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl53

Impulse Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl54

Impulse Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl55

Impulse Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl56

System Response Once the response to a unit impulse is known, the response of any LTI system to any arbitrary excitation can be found Any arbitrary excitation is simply a sequence of amplitude-scaled and time-shifted impulses Therefore the response is simply a sequence of amplitude-scaled and time-shifted impulse responses M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl57

Simple System Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl58

More Complicated System Response Example System Excitation System Impulse Response System Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl59

The Convolution Sum M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl60

A Convolution Sum Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl61

A Convolution Sum Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl62

A Convolution Sum Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl63

A Convolution Sum Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl64

Convolution Sum Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl65

Convolution Sum Properties (continued) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl66

Numerical Convolution M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl67

Numerical Convolution M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl68

Numerical Convolution M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl69

Numerical Convolution M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl70

Numerical Convolution M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl71

Stability and Impulse Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl72

System Interconnections 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. This is a direct consequence of the associativity property of convolution. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl73

System Interconnections 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. This is a direct consequence of the distributivity property of convolution. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl74

Unit Impulse Response and Unit Sequence Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl75

Systems Described by Difference Equations M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl76

Systems Described by Difference Equations M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl77

Frequency Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl78

Frequency Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl79

Frequency Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl80

Frequency Response Example M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl81