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Perfect Reconstruction Two-Channel FIR Filter Banks. A perfect reconstruction two-channel FIR filter bank with linear-phase FIR filters can be designed if the power-complementary requirement between the two analysis filters and is not imposed.

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perfect reconstruction two channel fir filter banks
Perfect Reconstruction Two-Channel FIR Filter Banks
  • A perfect reconstruction two-channel FIR filter bank with linear-phase FIR filters can be designed if the power-complementary requirement

between the two analysis filters and is not imposed

Copyright © 2001, S. K. Mitra

perfect reconstruction two channel fir filter banks2
Perfect Reconstruction Two-Channel FIR Filter Banks
  • To develop the pertinent design equations we observe that the input-output relation of the 2-channel QMF bank

can be expressed in matrix form as

Copyright © 2001, S. K. Mitra

perfect reconstruction two channel fir filter banks3
Perfect Reconstruction Two-Channel FIR Filter Banks
  • From the previous equation we obtain
  • Combining the two matrix equations we get

Copyright © 2001, S. K. Mitra

perfect reconstruction two channel fir filter banks4
Perfect Reconstruction Two-Channel FIR Filter Banks

where

are called themodulation matrices

Copyright © 2001, S. K. Mitra

perfect reconstruction two channel fir filter banks5
Perfect Reconstruction Two-Channel FIR Filter Banks
  • Now for perfect reconstruction we must have and correspondingly
  • Substituting these relations in the equation

we observe that the PR condition is satisfied if

Copyright © 2001, S. K. Mitra

perfect reconstruction two channel fir filter banks6
Perfect Reconstruction Two-Channel FIR Filter Banks
  • Thus, knowing the analysis filters and , the synthesis filters and are determined from
  • After some algebra we arrive at

Copyright © 2001, S. K. Mitra

perfect reconstruction two channel fir filter banks7
Perfect Reconstruction Two-Channel FIR Filter Banks

where

and is an odd positive integer

Copyright © 2001, S. K. Mitra

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Perfect Reconstruction Two-Channel FIR Filter Banks
  • For FIR analysis filters and , the synthesis filters and will also be FIR filters if

wherecis a real number andkis a positive integer

  • In this case

Copyright © 2001, S. K. Mitra

orthogonal filter banks
Orthogonal Filter Banks
  • Let be an FIR filter of odd orderNsatisfying the power-symmetric condition
  • Choose
  • Then

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • Comparing the last equation with

we observe that andk = N

  • Using in

with we get

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • Note:If is a causal FIR filter, the other three filters are also causal FIR filters
  • Moreover,
  • Thus, for a real coefficient transfer function if is a lowpass filter, then is a highpass filter
  • In addition,

Copyright © 2001, S. K. Mitra

orthogonal filter banks12
Orthogonal Filter Banks
  • A perfect reconstruction power-symmetric filter bank is also called anorthogonal filter bank
  • The filter design problem reduces to the design of a power-symmetric lowpass filter
  • To this end, we can design a an even-order whose spectral factorization yields

Copyright © 2001, S. K. Mitra

orthogonal filter banks13
Orthogonal Filter Banks
  • Now, the power-symmetric condition

implies thatF(z) be a zero-phase half-band lowpass filter with a non-negative frequency response

  • Such a half-band filter can be obtained by adding a constant term K to a zero-phase half-band filter Q(z) such that

Copyright © 2001, S. K. Mitra

orthogonal filter banks14
Orthogonal Filter Banks
  • Summarizing, the steps for the design of a real-coefficient power-symmetric lowpass filter are:
  • (1)Design a zero-phase real-coefficient FIR half-band lowpass filterQ(z) of order 2NwithNan odd positive integer:

Copyright © 2001, S. K. Mitra

orthogonal filter banks15
Orthogonal Filter Banks
  • (2) Letddenote the peak stopband ripple of
  • DefineF(z) = Q(z) + dwhich guaranteesthat for allw
  • Note:Ifq[n] denotes the impulse responseofQ(z),then the impulse responsef [n] ofF(z) is given by
  • (3) Determine the spectral factor ofF(z)

Copyright © 2001, S. K. Mitra

orthogonal filter banks16
Orthogonal Filter Banks
  • Example -Consider the FIR filter

whereR(z) is a polynomial in of degreewithNodd

  • F(z) can be made a half-band filter bychoosingR(z) appropriately
  • This class of half-band filters has been called thebinomialormaxflat filter

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • The filterF(z) has a frequency response that is maximally flat atw = 0 and atw = p
  • ForN = 3, resulting in

which is seen to be a symmetric polynomial with 4 zeros located at , a zero at , and a zero at

Copyright © 2001, S. K. Mitra

orthogonal filter banks18
Orthogonal Filter Banks
  • The minimum-phase spectral factor is therefore the lowpass analysis filter
  • The corresponding highpass filter is given by

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • The two synthesis filters are given by
  • Magnitude responses of the two analysis filters are shown on the next slide

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • Comments:(1) The order ofF(z) is of theform 4K+2,whereKis a positive integer
  • Order of isN = 2K+1,which is odd as required

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • (2)Zeros ofF(z) appear with mirror-image symmetry in thez-plane with the zeros on the unit circle being of evenmultiplicity
  • Any appropriate half of these zeros can be grouped to form the spectral factor
  • For example, a minimum-phase can be formed by grouping all the zeros inside the unit circle along with half of the zeros on the unit circle

Copyright © 2001, S. K. Mitra

orthogonal filter banks22
Orthogonal Filter Banks
  • Likewise, a maximum-phase can be formed by grouping all the zeros outside the unit circle along with half of the zeros on the unit circle
  • However, it is not possible to form a spectral factor with linear phase
  • (3)The stopband edge frequency is the same for F(z) and

Copyright © 2001, S. K. Mitra

orthogonal filter banks23
Orthogonal Filter Banks
  • (4)If the desired minimum stopband attenuation of is dB, then the minimum stopband attenuation ofF(z) is dB
  • Example -Design a lowpass real-coefficient power-symmetric filter with the following specifications: , and dB

Copyright © 2001, S. K. Mitra

orthogonal filter banks24
Orthogonal Filter Banks
  • The specifications of the corresponding zero-phase half-band filterF(z) are therefore: and dB
  • The desired stopband ripple is thus which is also the passband ripple
  • The passband edge is at
  • Using the functionremezordwe firstestimate the order ofF(z) and then using thefunctionremezdesignQ(z)

Copyright © 2001, S. K. Mitra

orthogonal filter banks25
Orthogonal Filter Banks
  • The order ofF(z) is found to be 14 implyingthat the order of is 7 which is odd as desired
  • To determine the coefficients ofF(z) we adderr(the maximum stopband ripple) to thecentral coefficientq[7]
  • Next, using the functionrootswe determinethe roots ofF(z) which should theretically exhibit mirror-image symmetry with double roots on the unit circle

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • However, the algorithm s numerically quite sensitive and it is found that a slightly larger value thanerrshould be added to ensuredouble zeros ofF(z) on the unit circle
  • Choosing the roots inside the unit circle along with one set of unit circle roots we get the minimum-phase spectral factor

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • The zero locations ofF(z) and are shown below

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • The gain responses of the two analysis filters are shown below

Copyright © 2001, S. K. Mitra

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Orthogonal Filter Banks
  • Separate realizations of the two filters and would require 2(N+1) multipliers and 2Ntwo-input adders
  • However, a computationally efficient realization requiringN+1 multipliers and 2Ntwo-input adders can be developed by exploiting the relation

Copyright © 2001, S. K. Mitra