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E85.2607: Lecture 2 – Filters 1 Basic IIR filters 2 Applications E85.2607: Lecture 2 – Filters 2010-01-28 1 / 15
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E85.2607: Lecture 2 -- Filtersronw/adst-spring2010/lectures/lecture… · E85.2607: Lecture 2 { Filters 2010-01-28 12 / 15. Time-varying lters: Wah x(n) y(n) 1-mix mix x(n) 1-mix

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  • 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)− 1tan(πfc/fs) + 1

    Flat magnitude response, butintroduces phase distortion

    Group delay = − ∂∂ω∠H(ejω)

    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)

    ½BP/BR-/+

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

    ½BP/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

    https://ccrma.stanford.edu/~jos/filters/filters.htmlhttps://ccrma.stanford.edu/~jos/filters/Elementary_Audio_Digital_Filters.html

    Basic IIR filtersApplications