Operation in System

Hany FerdinandoDept. of Electrical EngineeringPetra Christian University

Operation in system 2

General Overview

Convolution both in continuous- and discrete-time system

De-convolution in discrete-time system

Operation in system 3

Convolution (discrete)

The output of a system can be analyzed with impulse response sequence

Impulse response is response of a system due to an impulse sequence as input

n hn means ‘input n gives output hn‘ c{n} c{hn} c{n±m} c{hn±m}

Operation in system 4

Convolution (discrete)

For a signal uk, we can write

uj{dk-j} gives output uj{hk-j} Applying the superposition property,

the total response is merely the sum of the individual response of the uj{hk-j}

jjkjk

kkokkk

uu

uuuuu

...... 111122

Operation in system 5

Convolution (discrete)

Thus the output sequence is

If we let m = k-j, then

kkj

jkjk huhuy *

kkm

mkmm

mmkk uhuhhuy *

Operation in system 6

Convolution Operation

Calculate the sequence for every k

To find y1, we need h1-n, then the shifted h1-n is multiplied with u

For others k, the procedures are similar

,...,, 332211

j jjjjj

jjj huyhuyhuy

Operation in system 7

Example and Exercise

h(n) = (½)n for n ≥ 0 (even) and 0 for n is odd, u(n) = {1,2} do it with u*h and h*u

h(n) = {1,2,1}, u(n) = {1,2,1} h(n) = (½)n for n ≥ 0, u(n) = (¼)n for n ≥

0 … (get other exercises from the books)

Operation in system 8

Convolution (discrete)

If both signal is positive semi infinite, then we can use table matrix to calculate the convolution

Place the values of one signal at the top row and the other at the left most column

Be careful!! You have to verify the first result…!!

Operation in system 9

Convolution (continuous)

To derive procedure for convolution in continuous-time system is similar to that of discrete-time system

The input is decomposed into a sum of impulse function, then express the output as a sum of the response resulting from individual impulses

Operation in system 10

Convolution Operation

The formula is Remark:

Inside the integral, ‘t’ is transformed into ‘’

h(t) h() and u(t) u() Get h(-) and shift it to the right to get

h(t-)

dtthuty )()()(

Operation in system 11

Convolution Operation

Always draw the signals before convolving them, this is important to get the limit for integration

It is calculated based on the range, e.g. 0<t<1, 1<t<2, 2<t<3, etc.

Operation in system 12

Examples and Exercises

Convolve h(t) = 1 for 0≤t≤2 with u(t) = t for t≥0

h(t) = 1 for 0≤t≤1 and -1 for 1≤t≤2, u(t) = t for 0≤t≤2

Operation in system 13

Deconvolution (discrete only)

It is how to ‘undo’ convolution From the output y and input u

relationship, one can derive the impulse response h, etc.

Application: To find transfer function of a system To measure the linearity of unknown

system

Operation in system 14

Deconvolution formulation

To find u:

To find h:0

1

0

h

huyu

k

mmkmk

k

0

1

0

u

uhyh

k

mmkmk

k

Operation in system 15

Exercise

Input u = {1,½} and output y = (½)k for k ≥ 0. Find h!

Operation in system 16

Next…

The operations in system are discussed! Students have to exercise themselves in order to understand those operation well.

The next topic is Fourier analysis. Read the Signals and Linear Systems by Robert A. Gabel (p. 239-255) or Modern Signals and System by Huibert Kwakernaak (p. 330-367) or Signals and System by Alan V. Oppenheim (p.161-179)