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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Polyphase Implementation and Filter Banks
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G(z)
Decimation and InterpolationA typical building block of multirate filter bank
We want to know the relationships between the above signals
Downsampling followed by upsampling
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image spectra
original spectrum
Downsampling
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We know upsampling
We know downsampling +upsampling
We can get downsampling
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G(z)
Decimation and InterpolationA typical building block of multirate filter bank
We want to know the relationships between the above signals
Type1
[ 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]
Type3: we want to have
hence
Type3 is not very straightforward as there is a casualty problem.
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G(z)
Decimation and Interpolation—polyphase implementationA typical building block of multirate filter bank
We want to know if there is an efficient way to implement the above system
G(z)
x(n)
y(m)
v(n)
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x(n)
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y(m)
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Polyphase IntepolatorFor 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
x(n)
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y(m)
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Polyphase IntepolatorH(z)
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y(m)
x(n)
v(n)
The number of multiplications is: ML
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y(m)
x(n)
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The number of multiplications is: N (M/N)(L/N)
=ML/N. Also reduced a lot.
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Processing
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Processing
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According to the results associated with downsampling followed by upsampling we have:
PR means that
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
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).
There are many solutions for PR, for example
We have the following
Hence we can find a polynomial T(z) such that
And the decompose T(z) into two factors
Then the filters are determined
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
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
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