E85.2607: Lecture 2 – Filters

1 Basic IIR filters

2 Applications

E85.2607: Lecture 2 – Filters 2010-01-28 1 / 15

Basic filters

LP BP

HP BR

H(f)

f/Hzfc0

H(f)

f/Hzfc0

H(f)

f/Hzfch0

H(f)

f/Hz0

fcl

fchfcl

Resonator

Notch

H(f)

f/Hz0

H(f)

f/Hz0 fc

fc

fc cutoff or center frequency

fb bandwidth

Q “quality factor” Q = fbfc

E85.2607: Lecture 2 – Filters 2010-01-28 2 / 15

IIR filter structures: Digital biquad

H(z) =B(z)

A(z)=

b0 + b1 z−1 + b2 z−2

1 + a1 z−1 + a2 z−2

Direct form 1 Direct form 2

zeros on left, poles on right canonical (minimum delays)

E85.2607: Lecture 2 – Filters 2010-01-28 3 / 15

Building blocks: Allpass filter

H(z) =B(z)

z−NB(z−1)

=b0 + b1 z−1 + . . .+ bN−1 zN−1 + bN z−N

bN + bN−1 z−1 + . . .+ b1 zN−1 + b0 zN

Where are the poles and zeros?

What about the frequency response?

E85.2607: Lecture 2 – Filters 2010-01-28 4 / 15

Building blocks: Allpass filter

H(z) =B(z)

z−NB(z−1)

=b0 + b1 z−1 + . . .+ bN−1 zN−1 + bN z−N

bN + bN−1 z−1 + . . .+ b1 zN−1 + b0 zN

Where are the poles and zeros?

What about the frequency response?

E85.2607: Lecture 2 – Filters 2010-01-28 4 / 15

Parametric first order allpass filter

A(z) =c + z−1

1 + c z−1

c =tan(πfc/fs)− 1

tan(πfc/fs) + 1

Flat magnitude response, butintroduces phase distortion

Group delay = − ∂∂ω∠H(e jω)

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group Delay

Mag

nitu

de in

dB

0 0.1 0.2 0.3 0.4 0.5−200

−150

−100

−50

0

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

f/fS !

Gro

up d

elay

in s

ampl

es

E85.2607: Lecture 2 – Filters 2010-01-28 5 / 15

Tunable lowpass/highpass filters

x(n-1)x(n)

y(n-1)

y(n)

-c

c 1

T

T

-c

x(n) y(n)T

c

Direct-form structure Allpass structure

x (n)h x (n-1)h

A(z)x(n) y(n)1 /2

LP/HP+/-

x(n-1)x(n)

y(n-1)

y(n)

-c

c 1

T

T

-c

x(n) y(n)T

c

Direct-form structure Allpass structure

x (n)h x (n-1)h

A(z)x(n) y(n)1 /2

LP/HP+/-

Why does this work?

E85.2607: Lecture 2 – Filters 2010-01-28 6 / 15

LP/HP frequency response

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group DelayM

agni

tude

in d

B

0 0.1 0.2 0.3 0.4 0.5−100

−50

0

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

f/fS !

Gro

up d

elay

in s

ampl

es

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group Delay

Mag

nitu

de in

dB

0 0.1 0.2 0.3 0.4 0.50

50

100

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

f/fS !

Gro

up d

elay

in s

ampl

es

E85.2607: Lecture 2 – Filters 2010-01-28 7 / 15

Second order allpass

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group Delay

Mag

nitu

de in

dB

0 0.1 0.2 0.3 0.4 0.5−100

−50

0

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

f/fS !

Gro

up d

elay

in s

ampl

es

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group Delay

Mag

nitu

de in

dB

0 0.1 0.2 0.3 0.4 0.50

50

100

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

f/fS !

Gro

up d

elay

in s

ampl

es

c controls bandwidth

d controls cut-off

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group Delay

Mag

nitu

de in

dB

0 0.1 0.2 0.3 0.4 0.5−400

−300

−200

−100

0

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.50

10

20

30

f/fS !

Gro

up d

elay

in s

ampl

es

x(n)

y(n)

Tx(n-1)

y(n-2)

1d(1-c)-c

T

T Ty(n-1)

x(n-2)

-d(1-c)c

E85.2607: Lecture 2 – Filters 2010-01-28 8 / 15

Tunable bandpass/bandreject filters

A(z)x(n) y(n)

!BP/BR-/+

A(z)x(n) y(n)

!BP/BR-/+

typoE85.2607: Lecture 2 – Filters 2010-01-28 9 / 15

BP/BR frequency response

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group DelayM

agni

tude

in d

B

0 0.1 0.2 0.3 0.4 0.5−100

−50

0

50

100

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.50

5

10

15

f/fS !

Gro

up d

elay

in s

ampl

es

0 0.1 0.2 0.3 0.4 0.5−10

−5

0

Magnitude Response, Phase Response, Group Delay

Mag

nitu

de in

dB

0 0.1 0.2 0.3 0.4 0.5−100

−50

0

50

100

Phas

e in

deg

rees

0 0.1 0.2 0.3 0.4 0.5−6

−4

−2

0

2x 1014

f/fS !

Gro

up d

elay

in s

ampl

es

E85.2607: Lecture 2 – Filters 2010-01-28 10 / 15

Cascading filters

Make the frequency response sharper by passing the signal through the samefilter multiple times

2008-10-14Dan Ellis 17

Cascading Filters! Repeating a filter (cascade connection)

makes its characteristics more abrupt:

! Repeated roots in z-plane:

H(ejω)

H(ejω) H(ejω) H(ejω)

ω

|H(ejω)|

ω

|H(ejω)|3

1

ZP

h

h3

E85.2607: Lecture 2 – Filters 2010-01-28 11 / 15

Equalizers

MFPeakx(n) y(n)

LFShelving

MFPeak

HFShelving

Cut-off frequency fBandwidth fGain G in dB

c

b

Cut-off frequency fBandwidth fGain G in dB

c

b

+12+6

0-6

-12

Cut-off frequency fGain G in dB

c Cut-off frequency fGain G in dB

c

f

fc fc fc fc

fb fb

G/dB

A(z)x(n) y(n)

LF/HF+/-

x(n-1)x(n)

y (n-1)1

aB/C 1

T

T

y (n)1

y (n)1

-aB/C

H /20

Chain of simple filters to shape spectrum

Low/high shelf H(z) = 1 + (10G/20 − 1) HLP/HP(z)

Peak H(z) = 1 + (10G/20 − 1) HBP(z)

Parameters: Gain, center frequency, bandwidth (Q)

Applications: mixing, compensate for room acoustics, genre controls

E85.2607: Lecture 2 – Filters 2010-01-28 12 / 15

Time-varying filters: Wah

x(n) y(n)1-mix

mix

x(n) y(n)1-mix

mix

Bandpass filter with time varying center frequency

Mimics formant resonances in speech

E85.2607: Lecture 2 – Filters 2010-01-28 13 / 15

Time-varying filters: Phaser

Time

Freq

uenc

y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.5

1

1.5

2

x 104

x(n) y(n)d

e

AP APAP

x(n) y(n)d

e

fb

x(n) y(n)

e1

e2

e3

e4

d

Notches with time varying center frequency

Controlled by a low frequency oscillator

E85.2607: Lecture 2 – Filters 2010-01-28 14 / 15

Reading

Introduction to Digital Filters

Elementary Audio Digital Filters

DAFX, Chapter 2 (if you have it)

E85.2607: Lecture 2 – Filters 2010-01-28 15 / 15