5,6. DSP Theory

System Analysis, Filters, IIR, FIR

Rahil Mahdian 24.03.2016

2

continuous DiscreteP

erio

dic

aPer

iodi

c

FS

CTFT

DTFTZ

DFSDFT

FFT

Signal Transforms - review

3

Convolution in DFT

DFT

In DFT convolutionproperty holds, but incyclic convolution sense.

Q. How to get Linearconvolution result ofLTI systems, usingCircular convolution?

4

(length N)

(length M)

Zeropadding(M-1) 0s

Zeropadding(N-1) 0s

N+M-1point DFT

N+M-1point DFT

N+M-1point IDFT

Linear Convolution from Circularprocessing in DFT world

5

Upsampling - interpolation

xi[n] in a low-pass filtered version of x[n]

The low-pass filter impulse response is

Hence the interpolated signal is written as

L/n

L/nsinnhi

ki

L/kLn

L/kLnsinkxnx

To create an upsampled signal, from the zeropaddedexpanded version, pass it through a LPF. (Gain something!)

6

Useful Noble Identities

M H(Z)

H(ZM) M

X[n]

X[n]

Xa[n]

Xb[n]

ya[n]

yb[n]

H(Z) LX[n] Xa[n] ya[n]

X[n] Xb[n] yb[n]L H(ZL)

7

Question

Problem. Explore the condition by which the changing the sequence ofupsampler and downsampler blocks, does not make a difference on the outputof the system?

M MN NX(n) X(n)y(n) y(n)

?

8

Polyphase filtering

=

Goal: Less number of multiplications per time unit Simpler structure of implementing a filter Used in filterbank

9

Polyphase sequence decomposition

10

Polyphase components

11

PolyPhase Decimation System

12

PolyPhase Interpolation System

13

Ideal Filters

14

FIR filter - basics

No phase shift No distortion

Ideal delay filter

Linear phase shift is still good. (desirable)

15

Design of a Digital Filter

DigitalFilters:

IIR

FIR

Design ananalogequivalent

Convert it toDigital

Direct designusing computer

e.g., Butterworth, Chebyshev, etc.

Windowing approach

Frequency samplingapproach

Computer-basedoptimization methods

Filter Design Approaches

16

Filter attributes

FIRInherently BIBO stableEasy to implementCan be designed to have linear phase property

IIRSometimes unstableLower filter order than a corresponding FIR filterUsually have nonlinear phase property

17

M

k

kk

N

k

kk

MM

NN

za

zb

zaza

zbzbbzH

1

0

11

110

1

1

zXzY

pzpzpz

zzzzzzkzH

N

N

21

21

M

kk

N

kk

k

knyaknxb

knxkhny

10

0

IIR filter System Transfer function

18

Ideal Filters

19

Realistic vs. Ideal Filter Response

20

Impulse Invariance method

Design a continous time filter and take samples from it, using the followingprocedure:

(t) sampling

h[n]=

= )

+

2

)

=

, ||

21

The system function and are related by)( sH a)( zH

k

aezk

TjsH

TzH sT )

2(

1)(

]Re[z

]Im[zjj

0

s-plane

22

Butterworth Lowpass Filters

Passband is designed to be maximally flat

The magnitude-squared function is of the form

N2c

2

cj/j1

1jH

N2c

2

cj/s1

1sH

1-0,1,...,2Nkforej1s 1Nk2N2/jccN2/1

k

23

Chebyshev Filters

Equiripple in the passband and monotonic in the stopband

Or equiripple in the stopband and monotonic in the passband

xcosNcosxV/V1

1jH 1N

c2N

2

2

c

24

Impulse invariance applied to Butterworth

Since sampling rate Td cancels out we can assume Td=1

Map spec to continuous time

Butterworth filter is monotonic so spec will be satisfied if

Determine N and c to satisfy these conditions

3.00.17783eH

2.001eH89125.0

j

j

3.00.17783jH

2.001jH89125.0

0.177833.0jHand89125.02.0jH cc

N2c

2

cj/j1

1jH

Example

25

Satisfy both constrains

Solve these equations to get

N must be an integer so we round it up to meet the spec

Poles of transfer function

The transfer function

Mapping to z-domain

2N2

c

2N2

c 17783.0

13.01and

89125.0

12.01

70474.0and68858.5N c

0,1,...,11kforej1s 11k212/jcc12/1

k

21

1

21

1

21

1

z257.0z9972.01

z6303.08557.1

z3699.0z0691.11

z1455.11428.2

z6949.0z2971.11

z4466.02871.0zH

4945.0s3585.1s4945.0s9945.0s4945.0s364.0s12093.0

sH222

Example Contd

26

-1 -0.5 0 0.5 1-20

-15

-10

-5

0

5

10

15

20

, x

=

tan

(/2

)

Frequency Warping

)(

)(

cos

sin

2tan

2/2/21

2/2/21

2

2

jj

jjj

ee

ee

.1

1

.1

11

j

j

j

j

e

ej

e

e

j

Since s=jW and z=ejw, wehave

.1

11

1

z

zs

Bilinear Transform

27

Example

Bilinear transform applied to Butterworth

Apply bilinear transformation to specifications

We can assume Td=1 and apply the specifications to

To get

3.00.17783eH

2.001eH89125.0

j

j

2

3.0tan

T

20.17783jH

2

2.0tan

T

201jH89125.0

d

d

N2c

2

c/1

1jH

2N2

c

2N2

c 17783.0

115.0tan21and

89125.0

11.0tan21

28

Example Contd

Solve N and c

The resulting transfer function has the following poles

Resulting in

Applying the bilinear transform yields

6305.5

1.0tan15.0tanlog2

189125.0

11

17783.0

1log

N

22

766.0c

0,1,...,11kforej1s 11k212/jcc12/1

k

5871.0s4802.1s5871.0s0836.1s5871.0s3996.0s20238.0

sH222c

21

2121

61

z2155.0z9044.01

1

z3583.0z0106.11z7051.0z2686.11

z10007378.0zH

29

FIR - constraints

Filter to have a limited number of the taps(components)

To have a linear phase shift Causal, stable and implementable As close as possible to the ideal desired filter

= k1+k2

N

n

nznhzH0

][)(

][][ nNhnh

30

FIR filter - basics

No phase shift No distortion

Ideal delay filter

Linear phase shift is still good. (desirable)

31

FIR filters

32

The frequency response of FIR filters

)()(

1

0

)()(

)()(

jjj

M

n

njj

eHeeH

enheH

)( jeH Magnitude response function

Amplitude response function)(H

jjjn

njj

eee

enheH

)cos21(1

)()(

2

2

0

3/2

3/20)(

0,cos21)( jeH

0

)(

cos21)(H

Example

33

FIR types (linear phase)

34

For Linear Phase t.f. (order N-1)

so that for N even:)1()( nNhnh

1

2

12

0).().()(

N

Nn

nN

n

n znhznhzH

1

2

0

)1(1

2

0).1().(

N

n

nNN

n

n znNhznh

1

2

0)(

N

n

mn zznh nNm 1 for N odd:

12

1

0

2

1

2

1).()(

N

n

N

mn zN

hzznhzH

FIR types cont.

35

Type-1 Type-2 Type-3 Type-4

N=2Lo=even N=2Lo+1=odd N=2Lo=even N=2Lo+1=odd

symmetric symmetric anti-symmetric anti-symmetric

h[k]=h[N-k] h[k]=h[N-k] h[k]=-h[N-k] h[k]=-h[N-k]

FIR types (linear phase)

36

Rectangular

Bartlett

Hann

Hamming

Blackman

Kaiser

2

1

NnN

n21

N

n2cos1

N

n2cos46.054.0

N

n

N

n 4cos08.0

2cos5.042.0

)(1

21 0

2

0 JN

nJ

Commonly used windows

37

Transition width (Hz)

Min. stopattn dB

2.12 1.5/N 30

4.54 2.9/N 50

6.76 4.3/N 70

8.96 5.7/N 90

Kaiser window

38

Windowname

Window function Filter

Peak valueof side lobe

The width ofmain lobe

Transitionwidth

Min.stopband

attenuation

Rectangular -13 dB -21 dB

Bartlett -25 dB -25 dB

Hanning -31 dB -44 dB

Hamming -41 dB -53 dB

Blackman -57 dB -74 dB

N4

N8

N8

N8

N12

N8.1

N2.4

N2.6

N6.6

N11

39

0 10 20 300

0.2

0.4

0.6

0.8

1

Triangular window: N=35

n-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response

frequency in pi units

dB

-1 -0.5 0 0.5 1

0

5

10

15

20Amplitude response

frequency in pi units-1 -0.5 0 0.5 1

-40

-30

-20

-10

0

Magnitude response of filter: wc=0.5pi

frequency in pi units

dB

40

0 10 20 300

0.2

0.4

0.6

0.8

1

Hanning window: N=35

n-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response

frequency in pi units

dB

-1 -0.5 0 0.5 1

0

5

10

15

Amplitude response

frequency in pi units-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response of filter: wc=0.5pi

frequency in pi units

dB

41

0 10 20 300

0.2

0.4

0.6

0.8

1

Hamming window: N=35

n-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response

frequency in pi units

dB

-1 -0.5 0 0.5 1

0

5

10

15

20Amplitude response

frequency in pi units-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response of filter: wc=0.5pi

frequency in pi units

dB

42

0 10 20 300

0.2

0.4

0.6

0.8

1

Blackman window: N=35

n-1 -0.5 0 0.5 1

-80

-60

-40

-20

0

Magnitude response

frequency in pi units

dB

-1 -0.5 0 0.5 1

0

5

10

15

Amplitude response

frequency in pi units-1 -0.5 0 0.5 1

-80

-60

-40

-20

0

Magnitude response of filter: wc=0.5pi

frequency in pi units

dB

43

-10 0 100

0.2

0.4

0.6

0.8

1

Kaiser window: N=35,beta=7.865

n-1 -0.5 0 0.5 1

-100

-80

-60

-40

-20

0

Magnitude response

frequency in pi units

dB

-1 -0.5 0 0.5 1

0

5

10

15

20Amplitude response

frequency in pi units-1 -0.5 0 0.5 1

-100

-80

-60

-40

-20

0

Magnitude response of filter: wc=0.5pi

frequency in pi units

dB

44

-10 0 100

0.2

0.4

0.6

0.8

1

Kaiser window: N=35,beta=9.5

n-1 -0.5 0 0.5 1

-100

-80

-60

-40

-20

0

Magnitude response

frequency in pi units

dB

-1 -0.5 0 0.5 1

0

5

10

15

20Amplitude response

frequency in pi units-1 -0.5 0 0.5 1

-100

-80

-60

-40

-20

0

Magnitude response of filter: wc=0.5pi

frequency in pi units

dB

45

0 5 10 15 20 25 30

0

0.1

0.2

0.3

Ideal Impulse Response

n 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Hamming Window

n

0 5 10 15 20 25 30

0

0.1

0.2

0.3

Actual Impulse Response

n 0 0.2 0.4 0.6 0.8 1-100

-80

-60

-40

-20

0Magnitude Response in dB

pi

dB

34N

46

Filter toolbox - MATLAB

Y = resample(X,P,Q) resamples the sequence in vector X atP/Q times the original sample rate using a polyphaseimplementation.

fdatool