Digital Speech Processing— Lecture 10 Short-Time speech...Short-Time Fourier Analysis Methods...

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Transcript of Digital Speech Processing— Lecture 10 Short-Time speech...Short-Time Fourier Analysis Methods...

  • 1

    Digital Speech ProcessingLecture 10

    Short-Time Fourier Analysis Methods - Filter

    Bank Design

  • 2

    Review of STFT1

    1 2 3 0

    2

    1

    . ( ) [ ] [ ]

    function of (looks like a time sequence) function of (looks like a transform)

    ( ) defined for , , ,...;

    . Interpretations of ( ). fix

    =

    =

    =

    j j mnm

    jn

    jn

    X e x m w n m e

    n

    X e n

    X e

    n

    ed, variable; ( ) DTFT [ ] [ ]

    DFT View OLA implementation 2. variable, fixed; ( ) [ ] [ ]

    Linear Filtering view filter bank implementation FBS i

    = =

    = =

    jn

    j j nn

    X e x m w n m

    n n X e x n e w n

    mplementation

  • 3

    Review of STFT

    3

    22

    . Sampling Rates in Time and Frequency recover [ ] from ( )

    1. time: ( ) has bandwidth of Hertz samples/sec rate

    Hamming Window: (Hz) sample

    =

    jn

    j

    S

    x n X e

    W e BB

    FBL

    exactly

    4

    at

    (Hz) or every L/4 samples

    2. frequency: [ ] is time limited to samples need at least frequency samples to avoid time aliasing

    SFL

    w n LL

  • 4

    Review of STFT with OLA method can recover ( ) using lower

    sampling rates in either time or frequency, e.g., can sample every samples (and divide by window), or can use fewer than frequency sam

    x n

    LL

    exactly

    ples (filter bank channels); but these methods are highly subject to aliasing errors with any modifications to STFT can use windows (LPF) that are longer than samples and

    still recover with L

    frequency channels; e.g., ideal LPF is infinite in time duration, but with zeros spaced samples apart where / is the BW of the ideal LPF

    <

    S

    N LN

    F N

  • 5

    Review of STFTH0(ej)

    H1(ej)

    HN-1(ej)

    +

    X(ej) Y(ej)

    1

    0

    1

    0

    1

    0

    [ ] [ ]

    ( ) ( )

    [ ] [ ] [ ] [ ] [ ]

    [ ] [ ]

    [ ] [ ] [ ] [ ] [ ] ...

    need to design digital filters that match criteria for exact reconstr

    =

    =

    =

    =

    =

    = =

    = = =

    =

    = = + +

    %

    %

    %

    kj nk

    Nj j

    kk

    N

    kk

    r

    r

    h n w n e

    H e H e

    h n h n n w n p n

    p n N n rN

    h n N w rN n rN w w N

    uction of [ ] and which still work with modifications to STFTx n

  • Tree-Decimated Filter Banks

    6

  • Tree-Decimated Filter Banks can sample STFT in time and frequency using lowpass

    filter (window) which is moved in jumps of R

  • Filter Bank Channels

    8

    Fully decimated and interpolated filter bank channels; (a) analysis with bandpass filter, down-shifting frequency and down-sampling; (b) synthesis with up-sampler followed by lowpassinterpolation filter and frequency up-shift; (c) analysis with frequency down-shift followed by lowpass filter followed by down-sampling; (d) synthesis with up-sampling followed by frequency up-shift followed by bandpass filter.

  • Full Implementation of Analysis-Synthesis System

    9

    Full implementation of STFT analysis/synthesis system with channel decimation by a factor of R, and channel interpolation by the same factor R (including a box for short-time modifications.

  • 10

    Review of Down and Up-Sampling

    R][nx[ ] [ ]dx n x nR=

    11 2

    0

    ( )/( ) ( )

    aliasing addition

    =

    = R

    j j r Rd R

    rX e X e

    R][nx[ / ] 0, 1, 2,...

    [ ]0 otherwiseu

    x n R nx n

    = =

    ( ) ( ) imaging removal

    =j j RuX e X e

  • 11

    [ ]

    1

    0

    1

    0

    1

    1

    1 ( )

    assuming no short time modifications ( ) ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) [ ] [ ]

    ( ) ( ) ( ) ( )

    k k

    rR

    k k

    k

    k

    rR

    j j mrR rR

    m W

    Nj j n

    rRkN

    j nrR

    k

    j n m

    m W r k

    X k X e w rR m x m e

    y n P k Y e eN

    P k X k f n rR eN

    x m w rR m f n rR P k eN

    =

    =

    = =

    = =

    =

    =

    =

    1

    0

    1

    0

    11 ( )

    recognizing that when ( ) ,

    ( )

    can show that the condition for ( ) ( ), is

    ( ) ( ) ( )

    k

    N

    Nj n m

    k q

    r

    P k k

    e n m qNN

    y n x n n

    w rR n qN f n rR q m n

    = =

    =

    =

    =

    =

    + = =

    Filter Bank Reconstruction

  • 12

    Filter Bank Reconstruction

    ( ) ( )( )

    1

    0

    2 21

    0

    1

    1

    1

    ( ) ( )

    ( ) ( )

    frequency domain equations, assuming: [ ] [ ] ; [ ] [ ] ;

    ( ); ( ) ;

    ( ) ( )

    ( ) ( )

    ( )

    k k

    k k k k

    k

    j n j nk k

    j j j jk k

    Njj

    kk

    R j jR R

    k

    j

    h n w n e g n f n e

    H e W e G e F e

    Y e P k G eN

    H e X eR

    X e

    =

    =

    = =

    = =

    =

    =

    l l

    l

    ( )1

    0

    2 21 1

    1 0

    1( ) ( )

    ( ) ( )

    ( ) ( ) ( ) ( )

    ( ) ( ) aliasing terms

    k

    k

    Nj j

    k kk

    R Nj jjR Rk k

    k

    j j

    P k G e H eRN

    X e P k G e H eRN

    H e X e

    =

    = =

    +

    = +

    l l

    l

    %

  • 13

    Filter Bank Reconstruction

    1

    0

    21

    0

    1 1

    1 0 1 2( )

    conditions for perfect reconstruction ( ( ) ( ))

    ( ) ( ) ( ) ( )

    and

    ( ) ( ) ( ) for , ,...,

    k

    k

    j j

    Njj j

    k kk

    N jj Rk k

    k

    Y e X e

    H e P k G e H eRN

    P k G e H e RRN

    =

    =

    =

    = =

    = =

    l

    %

    l

    Flat Gain

    Alias Cancellation

  • Filter Bank Reconstruction

    14

  • 15

    Decimated Analysis-Synthesis Filter Bank

    2

    1 0 1 14

    1 04

    ( ) ( )

    ( )

    practical solution for the case ( ) ( ), (2-band solution)where the conditions for exact reconstruction become

    ( ) ( ) ( ) ( ) ( ) ( )

    and

    ( ) ( ) (

    = = =

    + = j j j j

    j j

    w n f n N R

    P F e W e P F e W e

    P F e W e

    ( ) ( )

    2

    2 2

    1 0

    0 1 114

    ( ) ( )

    ( )

    ) ( ) ( ) ( )

    aliasing cancels exactly if ( ) ( ) (with ( ) ( ))giving

    ( ) ( )

    ( ) ( )

    + =

    = = =

    = =

    j j

    j j

    j j j M

    P F e W e

    P P F e W e

    W e W e e

    y n x n M

  • Decimated Analysis-Synthesis Filter Bank

    16

    1( )

    0

    1( ) [ ] ( )

    ( ) ( ) ( )

    when aliasing is completely eliminated (by suitable choice of filters),the overall analysis-synthesis system has frequency response:

    where

    =

    =

    =

    %

    %

    k

    Njj

    ek

    j j je

    H e P k W eRN

    W e W e F e

    h

    1

    0

    [ ] [ ] [ ]

    [ ] [ ] [ ]1[ ] [ ]

    where

    =

    =

    =

    = k

    e

    eN

    j n

    k

    n w n p n

    w n w n f n

    p n P k eRN

  • Decimated Analysis-Synthesis Filter Bank

    To design a decimated analysis/synthesis filter bank system, need to determine the following:1. the number of channels, N, and the decimation/interpolation

    ratio R. (Usually determined by the desired frequency resolution)

    2. the window, w[n], and the interpolation filter impulse response, f [n]. (These are lowpass filters that should have good frequency-selective properties such that the bandpass channel responses do not overlap into more than one band on either side of the channel).

    3. the complex gain factors, P[k], k=0,1,,N-1. (These constants are important for achieving flat overall response and alias cancellation.)

    17

  • Maximally Decimated Filter Banks

    we show here that it is possible to obtain exact reconstruction with R=N and N

  • Maximally Decimated Filter Banks

    19

  • Analysis-Synthesis Filter Bank (all modulators removed)

    20

  • Two Channel Filter Banks

    21

  • Two Channel Filter Banks

    22

    0 0 1 1

    ( ) ( )0 0 1 1

    2; , 0,1

    1( ) ( ) ( ) ( ) ( ) ( )21 ( ) ( ) ( ) ( )2

    For this two-band system, we have: Applying the frequency domain analysis we get:

    = = = =

    = +

    + +

    k

    j j j j j j

    j j j j

    R N k k

    Y e G e H e G e H e X e

    G e H e G e H e ( )

    ( ) ( )0 0 1 1

    ( )

    1/ [0], [1]

    ( ) ( ) ( ) ( ) 0

    not including multiplier or the gain factors Aliasing can be eliminated completely by choosing the filters such that:

    (ali

    + =

    j

    j j j j

    X e

    N P P

    G e H e G e H e

    0 0 1 11 ( ) ( ) ( ) ( ) 1 (2

    asing cancelation)

    Perfect reconstruction requires:

    flat gain condition) + = j j j jG e H e G e H e

  • Quadrature Mirror Filter (QMF) Banks

    23

    ( )1 0 1 0

    0 0 0 0

    ( )1 1 1 0

    [ ] [ ] ( ) ( )

    [ ] 2 [ ] ( ) 2 ( )

    [ ] 2 [ ] ( ) 2 ( )

    Alias cancellation achieved if the filters are chosen to satisfythe conditions:

    Note that the

    j n j j

    j j

    j j

    h n e h n H e H e

    g n h n G e H e

    g n h n G e H e

    = =

    = =

    = =

    0[ ]re is only a single lowpass filter, with impulse response

    on which all other filters are based; fi