Systems: Definition

][nx ][ny

S

A system is a transformation from an input signal into an output signal .

][nx ][ny

Example: a filter

][ns

][nv

][nx ][][ nsny Filter

SIGNAL

NOISE

Systems and Properties: Linearity

Linearity:

S][][][ 2211 nxanxanx ][][][ 2211 nyanyany

][1 nx ][1 ny

][2 nx ][2 nyS

S

Systems and Properties: Time Invariance

if

then

][nx ][nyS

S

][ Dnx ][ Dny

D D

time

time

time

time

Systems and Properties: Stability

S][nx ][ny

Bounded InputBounded Output

Systems and Properties: Causality

the effect comes after the cause.

Examples:

S][nx ][ny

]3[4]2[2]1[3][ nxnxnxny Causal

]3[4]2[2]1[3][ nxnxnxny Non Causal

Finite Impulse Response (FIR) Filters

][nx ][nyFilter

N

nxhnxnhny0

][][][*][][

][][...]1[]1[][]0[][ NnxNhnxhnxhny

Filter Coefficients

General response of a Linear Filter is Convolution:

Written more explicitly:

Example: Simple Averaging

][nx ][nyFilter

]9[...]1[][10

1][ nxnxnxny

Each sample of the output is the average of the last ten samples of the input.

It reduces the effect of noise by averaging.

FIR Filter Response to an Exponential

njenx 0][ njeHny 00][ Filter

njN

jN

nj eehehny 000

0

)

0

)( ][][][

Let the input be a complex exponential

Then the output is

njenx 0][

Example

njenx 0][ njeHny 00][ Filter

Consider the filter

]9[...]1[][10

1][ nxnxnxny

with inputnjenx 1.0][

Then 4137.11.0

101.09

0

1.0 6392.01

1

10

1

10

11.0 j

j

jj e

e

eeH

and the output njj eeny 1.04137.16392.0][

Frequency Response of an FIR Filter

njenx 0][ njeHny 00][ Filter

N

n

njenhH0

][)(

is the Frequency Response of the Filter

Significance of the Frequency Response

k

njk

keXnx ][Filter

If the input signal is a sum of complex exponentials…

k

njk

keYny ][

… the output is a sum is a sum of complex exponential.

Each coefficient is multiplied by the corresponding frequency response:

kX kkk XHY )(

Example

Consider the Filter

Filter][nx ][ny

]4[...]1[][5

1][ nxnxnxnydefined as

Let the input be:

)7.03.0cos(2)2.01.0cos(3][ nnnx

Expand in terms of complex exponentials:

njjnjj

njjnjj

eeee

eeeenx

3.07.03.07.0

1.02.01.02.0

0.10.1

5.15.1][

Example (continued)

The frequency response of the filter is (use geometric sum)

j

jjj

e

eeeH

1

1

5

1...1

5

1)(

54

njjnjj

njjnjj

eeHeeH

eeHeeHny

3.07.03.07.0

1.02.01.02.0

0.12.00.12.0

5.11.05.11.0][

Then

2566.12566.1

6283.06283.0

647.0)1.0(,647.0)2.0(

904.0)1.0(,904.0)1.0(jj

jj

eHeH

eHeH

with

)956.13.0cos(294.1)428.01.0cos(712.2][ nnny Just do the algebra to obtain:

The Discrete Time Fourier Transform (DTFT)

Given a signal of infinite duration with

define the DTFT and the Inverse DTFT

][nx n

n

njenxnxDTFTX ][][)(

deXXIDTFTnx nj)(2

1)(][

Periodic with period 2

)()2( XX

)(rad

|)(| X

0

)(X

)(HzF2SF

2SF 0

If the signal is real, then ][nx )()( * XX

General Frequency Spectrum for a Discrete Time Signal

Since is periodic we consider only the frequencies in the interval

)(1

1)(

1

0

Nj

NjN

n

nj We

eeX

Then

Example: DTFT of a rectangular pulse …

Consider a rectangular pulse of length N

0 1N

1

][nx

n

2/sin

2/sin )( 2/)1(

N

eW NjN

where

0 1N

1

][nx

n

DTFT

Example of DTFT (continued)

-3 -2 -1 0 1 2 30

2

4

6

8

10

12

( )NW

N

2

N

2

N

Why this is Important

][nx ][nyFilter

Recall from the DTFT

deXnx nj)(2

1][

Then the output

deHXny nj)()(2

1][

Which Implies )()(][)( XHnyDTFTY

Summary Linear FIR Filter and Freq. Resp.

][nx ][nyFilter

1

0

][][][*][][N

nxhnxnhny

Filter Definition:

Frequency Response: ,][)(1

0

N

n

njenhH

DTFT of output )()()( XHY

Frequency Response of the Filter

][nx ][nyFilter

Frequency Response:

,][)(1

0

N

n

njenhH

We can plot it as magnitude and phase. Usually the magnitude is in dB’s and the phase in radians.

Example of Frequency Response

Again consider FIR Filter

The impulse response can be represented as a vector of length 10

]9[...]1[][10

1][ nxnxnxny

1.0...1.0,1.0h

Then use “freqz” in matlab

freqz(h,1)

to obtain the plot of magnitude and phase.

Example of Frequency Response (continued)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-100

0

100

Normalized Frequency ( rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80

-60

-40

-20

0

Normalized Frequency ( rad/sample)

Mag

nitu

de (

dB)