Digital Speech Processing— Lecture 10 Short-Time Fourier ... ... • Also,based on the...

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  • 1

    Digital Speech Processing— Lecture 10

    Short-Time Fourier Analysis Methods - Filter

    Bank Design

  • 2

    Review of STFT 1

    1 2 3 0

    2

    1

    ˆ ˆ ˆ

    ˆ ˆ

    ˆ ˆ

    ˆ. ( ) [ ] [ ]

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

    ˆ ˆ( ) defined for , , ,...;

    . Interpretations of ( ) ˆ. fix

    ω ω

    ω

    ω

    ω

    ω π

    ∞ −

    =−∞

    = −

    = ≤ ≤

    ∑j j mn m

    j n

    j n

    X e x m w n m e

    n

    X e n

    X e

    n ˆ

    ˆ ˆ

    ˆˆed, variable; ( ) DTFT [ ] [ ]

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

    Linear Filtering view filter bank implementation FBS i

    ω

    ω ω

    ω ω

    ω −

    = = −⎡ ⎤⎣ ⎦ ⇒ ⇒

    = = ∗

    ⇒ ⇒ ⇒

    j n

    j j n n

    X e x m w n m

    n n X e x n e w n

    mplementation

  • 3

    Review of STFT

    3

    2 2

    ˆ ˆ

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

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

    Hamming Window: (Hz) sample

    ω

    ω

    = ⇒

    j n

    j

    S

    x n X e

    W e B B

    FB L

    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

    SF L

    w n L L

  • 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

    L L

    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 L N

    F N

  • 5

    Review of STFT H0(ejω)

    H1(ejω)

    HN-1(ejω)

    +…

    X(ejω) Y(ejω)

    1

    0

    1

    0

    1

    0

    [ ] [ ]

    ( ) ( )

    [ ] [ ] [ ] [ ] [ ]

    [ ] [ ]

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

    need to design digital filters that match criteria for exact reconstr

    ω

    ω ω

    δ

    δ

    δ

    =

    =

    =−∞

    =−∞

    =

    = =

    = = =

    = −

    = − = + +

    %

    %

    %

    kj n k

    N j j

    k k

    N

    k k

    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 lowpass interpolation 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=

    1 1 2

    0

    ( )/( ) ( )

    aliasing addition

    ω ω π −

    =

    = ∑ R

    j j r R d R

    r X e X e

    R↑][nx [ / ] 0, 1, 2,...

    [ ] 0 otherwiseu

    x n R n x n

    = ± ±⎧ = ⎨ ⎩

    ( ) ( ) imaging removal

    ω ω=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 m rR rR

    m W

    N j j n

    rR k N

    j n rR

    k

    j n m

    m W r k

    X k X e w rR m x m e

    y n P k Y e e N

    P k X k f n rR e N

    x m w rR m f n rR P k e N

    ω ω

    ω ω

    ω

    ω

    =

    =

    ∞ −

    ∈ =−∞ =

    = = −

    =

    = −

    ⎡ ⎤ = − −⎢ ⎥

    ⎣ ⎦

    ∑ ∑ 1

    0

    1

    0

    1 1 ( )

    recognizing that when ( ) ,

    ( )

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

    ( ) ( ) ( )

    k

    N

    N j n m

    k q

    r

    P k k

    e n m qN N

    y n x n n

    w rR n qN f n rR q m n

    ω δ

    δ

    − ∞ −

    = =−∞

    =−∞

    • = ∀

    = − −

    • = ∀

    − + − = ⇒ =

    ∑ ∑

    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 n k k

    j j j j k k

    N jj

    k k

    R j j R 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 e N

    H e X e R

    X e

    ω ω

    ω ω ω ω ω ω

    ωω

    π πω ω

    ω

    − −

    =

    − − −

    =

    = =

    = =

    = ⋅

    ⎡ ⎤ ⎢ ⎥ ⎣ ⎦

    =

    ∑ l l

    l

    ( ) 1

    0

    2 21 1

    1 0

    1( ) ( )

    ( ) ( )

    ( ) ( ) ( ) ( )

    ( ) ( ) aliasing terms

    k

    k

    N j j

    k k k

    R Nj jjR R k k

    k

    j j

    P k G e H e RN

    X e P k G e H e RN

    H e X e

    ω ω

    π πω ωω

    ω ω

    =

    − −− −

    = =

    + ⋅

    = ⋅ +

    ∑ ∑ l l

    l

    %

  • 13

    Filter Bank Reconstruction

    1

    0

    21

    0

    1 1

    1 0 1 2 ( )

    ˆ conditions for perfect reconstruction ( ( ) ( ))

    ( ) ( ) ( ) ( )

    and

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

    k

    k

    j j

    N jj j

    k k k

    N jj R k k

    k

    Y e X e

    H e P k G e H e RN

    P k G e H e R RN

    ω ω

    ωω ω

    πωω

    =

    − −

    =

    • =

    = =

    = =

    ∑ l

    %

    l

    Flat Gain

    Alias Cancellation

  • Filter Bank Reconstruction

    14

  • 15

    Decimated Analysis- Synthesis Filter Bank

    2

    1 0 1 1 4

    1 0 4

    ( ) ( )

    ( )

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

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

    and

    ( ) ( ) (

    ω ω ω π ω π

    ω ω π

    − −

    • = = =

    ⎡ ⎤+ =⎣ ⎦ 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 1 1 4

    ( ) ( )

    ( )

    ) ( ) ( ) ( )

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

    ( ) ( )

    ˆ( ) ( )

    ω π ω π

    ω ω

    ω ω π ω

    − −

    − −

    ⎡ ⎤+ =⎣ ⎦

    • = − = =

    ⎡ ⎤− =⎢ ⎥⎣ ⎦ = −

    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

    ω ωω

    ω ω ω

    − −

    =

    =

    =

    ∑%

    %

    k

    N jj

    e k

    j j j e

    H e P k W e RN

    W e W e F e

    h

    1

    0

    [ ] [ ] [ ]