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Fast Signal Processing Algorithms Week 5. Polyphase Implementation and Filter Banks. Motivation. Up- and down sampling combined with filtering are the usual operations in multirate systems. Polypahse approach will yield simple implementations. Outlines. Two basic multirate operations

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Fast signal processing algorithms week 5 l.jpg

Fast Signal Processing AlgorithmsWeek 5

Polyphase Implementation and Filter Banks


Motivation l.jpg
Motivation

  • Up- and down sampling combined with filtering are the usual operations in multirate systems.

  • Polypahse approach will yield simple implementations


Outlines l.jpg
Outlines

  • Two basic multirate operations

    • Polyphase interpolator – upsampling followed by a filter

    • Polyphase decimatator – a filter followed by a decimator

  • Two-channel filter banks

    • Perfect reconstruction condition

    • Quadrature mirror filter (QMF) filter banks

    • Design of two-channel filter banks with PR

  • Multiple-channel filter banks

    • Tree- structured filter banks

    • Octave-band filter banks


Basic multirate operations l.jpg
Basic Multirate Operations

  • Decimation and interpolation

  • Z-domain and Frequency domain analysis of up-and downsampled version of a signal

  • Polyphase decomposition

  • Noble Identities


Decimation and interpolation l.jpg
Decimation and Interpolation

  • Decimation---down-sampling

N

x(n)


Decimation and interpolation6 l.jpg
Decimation and Interpolation

  • Decimation---down-sampling

N


Decimation and interpolation7 l.jpg
Decimation and Interpolation

  • Decimation---down-sampling

N

y(m)


Decimation and interpolation8 l.jpg
Decimation and Interpolation

  • Interpolation --- up-sampling

N


Decimation and interpolation9 l.jpg
Decimation and Interpolation

  • Interpolation --- up-sampling

N


Decimation and interpolation10 l.jpg
Decimation and Interpolation

  • Interpolation --- up-sampling

N


Decimation and interpolation11 l.jpg

H(z)

N

N

G(z)

Decimation and Interpolation

A typical building block of multirate filter bank

We want to know the relationships between the above signals



Decimation and interpolation13 l.jpg
Decimation and Interpolation

Upsampling

when N=2


Decimation and interpolation14 l.jpg
Decimation and Interpolation

Downsampling followed by upsampling

N

N

as

hence


Decimation and interpolation15 l.jpg
Decimation and Interpolation

Downsampling followed by upsampling

N

N


Decimation and interpolation16 l.jpg
Decimation and Interpolation

Downsampling followed by upsampling

N

N

image spectra

original spectrum


Decimation and interpolation17 l.jpg
Decimation and Interpolation

Downsampling followed by upsampling

image spectra

original spectrum


Decimation and interpolation18 l.jpg
Decimation and Interpolation

Downsampling followed by upsampling

when N=2

Image spectra


Decimation and interpolation19 l.jpg
Decimation and Interpolation

Downsampling

N

We know upsampling

We know downsampling +upsampling

We can get downsampling


Decimation and interpolation20 l.jpg
Decimation and Interpolation

Downsampling

N

Example, N=2


Decimation and interpolation21 l.jpg
Decimation and Interpolation

Downsampling

when N=2

Image spectra


Decimation and interpolation22 l.jpg
Decimation and Interpolation

Downsampling

when N=2

Image spectra


Decimation and interpolation23 l.jpg
Decimation and Interpolation

Downsampling

when N=2

Image spectra


Decimation and interpolation24 l.jpg

H(z)

N

N

G(z)

Decimation and Interpolation

A typical building block of multirate filter bank

We want to know the relationships between the above signals



Polyphase decomposition l.jpg
Polyphase Decomposition

  • Polyphase decomposition is the decomposition of a sequence x(n) into sub-sequences x(mN+i)

  • There are four types of polyphase decomposition.

Type-1

[ 0,1,2,3,4,5,6,7,8,9,10,11]

M=3

[ 0,3,6,9]

[ 1,4,7,10]

[ 2,5,8,11]


Polyphase decomposition27 l.jpg
Polyphase Decomposition

Type-1

M

M

+

T

2T

M


Polyphase decomposition28 l.jpg
Polyphase Decomposition

Type-2

[ 0,1,2,3,4,5,6,7,8,9,10,11]

M=3

[ 2,5,8,11]

[ 1,4,7,10]

[ 0,3,6,9]


Polyphase decomposition29 l.jpg
Polyphase Decomposition

Type-3: we want to have

hence

Type-3 is not very straightforward as there is a casualty problem.


Nobel identities l.jpg
Nobel Identities

Identity I

N

G(z)

G(zN)

N


Nobel identities31 l.jpg
Nobel Identities

Identity II

N

G(zN)

G(z)

N


Decimation and interpolation polyphase implementation l.jpg

H(z)

N

N

G(z)

Decimation and Interpolation—polyphase implementation

A typical building block of multirate filter bank

We want to know if there is an efficient way to implement the above system


Polyphase interpolator l.jpg
Polyphase Interpolator

N

G(z)

x(n)

y(m)

v(n)

Multiplications with zeros are involved


Polyphase intepolator l.jpg
Polyphase Intepolator

  • We decompose the filter into polyphase components (Type-1):

  • In z-domain:



Polyphase intepolator36 l.jpg

N

G(z)

x(n)

y(m)

v(n)

Polyphase Intepolator

N

x(n)

y(m)

v(n)


Polyphase intepolator37 l.jpg

N

G(z)

x(n)

y(m)

v(n)

N

N

N

Polyphase Intepolator

Using the second Noble identity:

x(n)

y(m)


Polyphase intepolator38 l.jpg

N

G(z)

x(n)

y(m)

v(n)

N

x(n)

N

y(m)

N

Polyphase Intepolator

For input signal of length M, and G(z) of length L,

convolution of v(n) (of length NM) and g(n) (of length L) requires NML mutiplications


Polyphase intepolator39 l.jpg

N

x(n)

N

y(m)

N

Polyphase Intepolator

  • Each branch has a convolution of y(m) (of length M) with Gl(z) (of length L/N). ML/N multiplications are required;

  • Hence for N branches, ML multiplications are required in total; Computation is greatly reduced


Polyphase intepolator40 l.jpg
Polyphase Intepolator

  • example: M=1024, N=2,L=64, MNL=128k multiplications are required for convolution

  • when using polyphase approach, only ML=64k multiplications are required.


Polyphase decimator polyphase implementation l.jpg
Polyphase Decimator – polyphase implementation

H(z)

N

y(m)

x(n)

v(n)

The number of multiplications is: ML


Polyphase decimator l.jpg
Polyphase Decimator

Let i=jN+k

let


Polyphase decimator43 l.jpg
Polyphase Decimator

N

y(m)

x(n)

N

N

The number of multiplications is: N (M/N)(L/N)

=ML/N. Also reduced a lot.


Two channel filter banks l.jpg
Two-Channel Filter Banks

  • It is known that when the signal bandwidth is p (or half of the sampling frequency), there is no room for decimation operation.

  • In this case filter banks can be used to decompose signals into subband components which has narrower band and decimation can be done for each subband component;

  • A parallel processing method


Two channel filter banks45 l.jpg
Two-channel filter banks

  • Two-channel filter banks

    • Perfect reconstruction condition

    • Quadrature mirror filter (QMF) filter banks

    • Design of two-channel filter banks with PR

  • Multiple-channel filter banks

    • Tree- structured filter banks

    • Octave-band filter banks


Two channel filter banks46 l.jpg
Two-Channel Filter Banks

  • Two-channel filter bank --- the simplest filter bank. Hk(z) are analysis filters and Gk(z) are synthesis filters

2

Processing

2

+

output

2

Processing

2


Two channel filter banks47 l.jpg
Two-Channel Filter Banks

  • In order that the decomposition does not involve loss of information, the following system should meet perfect reconstruction (PR) condition. That is

2

2

+

2

2


Two channel filter banks48 l.jpg
Two-Channel Filter Banks

  • Usually H0(z) is a low-pass filter and H1(z) a high pass filter, by which x(n) are decomposed into low and high frequency components respectively

2

2

+

output

2

2




Two channel filter banks perfect reconstruction pr condition l.jpg
Two-channel filter banks --- Perfect Reconstruction (PR) Condition

According to the results associated with downsampling followed by upsampling we have:



Two channel filter banks quadrature mirror filters qmfs l.jpg
Two-channel filter banks --- Quadrature Mirror Filters (QMFs)

The solutions for the above condition are not unique. A natural selection is

The above solution is called QMF, as there are “mirror” relationships among the filters


Two channel filter banks quadrature mirror filters qmfs54 l.jpg
Two-channel filter banks --- Quadrature Mirror Filters (QMFs)

However the QMFs may not yield PR;

The filters are to be carefully chosen so that the PR condition is met:

Numerical optimization algorithms are usually employed to find the suitable H0(z).


Two channel filter banks general cases with pr l.jpg
Two-channel filter banks --- General cases with PR (QMFs)

There are many solutions for PR, for example

We have the following


Two channel filter banks general cases with pr56 l.jpg
Two-channel filter banks --- General cases with PR (QMFs)

Hence we can find a polynomial T(z) such that

And the decompose T(z) into two factors

Then the filters are determined


Two channel filter banks general cases with pr57 l.jpg
Two-channel filter banks --- General cases with PR (QMFs)

Example l=1

Polynomials with only one term of even power of z satisfy the above relation, for example

The coefficients of odd powers of z can be arbitrary and can be chosen to obtain a nice frequency response


Two channel filter banks general cases with pr58 l.jpg
Two-channel filter banks --- General cases with PR (QMFs)

We can also choose l=0

Polynomials with only one term of odd power of z satisfy the above relation, for example

The coefficients of even powers of z can be arbitrary and can be chosen to obtain a nice frequency response


Two channel filter banks general cases with pr59 l.jpg
Two-channel filter banks --- General cases with PR (QMFs)

  • The approach can be summerized as

    • Choose T(z);

    • Compute the roots of T(z) and group them into two sets

    • Multiply out the factors of each group and call the results H0(z) and G0(z)

  • Example:


Two channel filter banks polyphase implementation l.jpg
Two-Channel Filter Banks– Polyphase Implementation (QMFs)

  • Now let us go back to this two-channel filter banks. How to make it simpler?

2

2

+

2

2




Two channel filter banks polyphase implementation63 l.jpg
Two-Channel Filter Banks– Polyphase Implementation (QMFs)

  • The results can be implemented as

2

2

+

+

2

2

+


Two channel filter banks polyphase implementation64 l.jpg
Two-Channel Filter Banks– Polyphase Implementation (QMFs)

  • The synthesis part can also be implemented using polyphase representation (Type-2)

2

2

+

+

+

2

2

+

+


Two channel qmf banks polyphase implementation l.jpg
Two-Channel QMF banks– Polyphase Implementation (QMFs)

  • As H1(z)=H0(-z), hence H10(z)=H00(z), H11(z)=-H01(-z),

2

2

+

+

+

2

2

-

+

+


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