双通道完全重构滤波器设计滤波器
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. (1-9)
Wherecis a nonzero constant andn0is a positive integer.
If a QMF bank is free from aliasing, amplitude distortion,and phase distortion, it is said to have the PR property. For a PR QMF bank we have
Now (1-4) can be written as
. (1-5)
We say thatT(z) is thedistortion transfer functionor “overall” transfer function of the alias-free system andA(z) is thegain of the aliasing term.Here .(1-6)
We will see that the reconstructed signal differs fromx(n)due to three reasons: aliasing, amplitude distortion, phase distortion. It can be shown that the filters can be designed in such a way that all of these distortions are reduced or eliminated.
At the receiver end, the received signals are decoded and then produce the signalsyk(n)by passing through the upsampler. These signals are filtered by the filtersF0(z)andF1(z), producing the output signal .(H0(z) andF0(z) respectively are the analysis and synthesis filters of thek-th channel, where k=0,1 .)
(1-11)
The analysis and synthesis filters are related as .
For the linear-phase filter bank,H0(z)andH1(z)are chosen respectively to be symmetric and anti-symmetric, which is an even number. The system delay should be equal to , whereN0andN1are respectively thelengths of the lowpass and highpass filters.
,k=0,1.(1-3)
The reconstructed signal is
(1-4)
The termX(-z) takes into account aliasing due to the downsamplers and imaging due to the upsamplers. We refer to this just as the aliasing term or alias component.
Thecoefficients of filtersH0(z)andH1(z)are obtained by solving the following constrained optimization
Subject to (1-11), (1-12)
Whereωpandωsare the passband and stopband cut off freque来自百度文库cies ofH0(z)andH1(z), respectively,α is a weighting constant from 0 to 1, andhisthe vector containing the free variables in the impulse response.
1.2Distortions
1.2.1 Aliasing
From (1-4), it is clear that we can cancel aliasing by choosing the filters such that the quantity is zero.
1.2.2Phase and Amplitude Distortions
Inthe time domain,theconstraintcan be as
.(1-13)
Letting (that is, in the time domain), then theconstrained optimizationwill turn as the following form
s.t. . (1-14)
Design ofTwo-ChannelPRFilter Banks
Abstract：In thispaper,we will address the concepts oftwo-channelPRfilterbanksand perfect-reconstruction (PR) property.Thebasic principle and design method oftwo-channel PR filter banks will be discussed in detail.Thenthefilter coefficients,time domainerror,the amplituderesponse andamplitudedistortionwill beshowed in figures.Finally,the designresultswill beanalyzedand summarized.
Fig.1-1The two-channel QMF bank.
From Fig.1-1we have
， .(1-1)
Thez-transforms of the decimated signalsvk(n)are
，k=0,1.(1-2)
The second term above represents aliasing. The z-transforms ofFk(z)isVk(z2), so that
While . (1-7)
Letting ，we have
. (1-8)
Unless isT(z) allpass, we say that suffers from “amplitude distortion”. Similarly unlessT(z)has linear phase (that is, for constantaandb), suffers from phase distortion.
. (1-10)
forall possible inputsx(n). This, of course, ignores the coding/decoding error and filterround offnoise.
3.Design ofTwo-ChannelPRFilter Banks
For a two-channel critically decimated filter bank, the PR condition can be given by
2.The concept of PR filter banks
A filter bank is called perfect-reconstruction (PR) system if the reconstructed signal is a scaled and delayed version of the input , namely,
1.Two-Channel Filter Banks
1.1Basic principleof the Two-Channel Filter Bank
Fig.1-1shows a two-channel filter bank, popularly called the quadrature mirror filter (QMF) bank. The inputsignalx(n)is first filtered by two filtersH0(z)andH1(z),typically lowpass and highpass filters. Each subbandxk(n)signal is therefore bandlimited to a total bandwidth ofπ. The subband signals are decimated by a factor of 2 to producevk(n). Each decimated signalvk(n)is then coded in such a way that the special properties of the subband are exploited.
Solving thisconstrained optimizationisthe most criticalstep inthe design.
4. Results of the design
We get the result as shown inFig.1-2
Fig.1-2Frequency responseof the two-channel PR filter banks for linear-phase property
Wherecis a nonzero constant andn0is a positive integer.
If a QMF bank is free from aliasing, amplitude distortion,and phase distortion, it is said to have the PR property. For a PR QMF bank we have
Now (1-4) can be written as
. (1-5)
We say thatT(z) is thedistortion transfer functionor “overall” transfer function of the alias-free system andA(z) is thegain of the aliasing term.Here .(1-6)
We will see that the reconstructed signal differs fromx(n)due to three reasons: aliasing, amplitude distortion, phase distortion. It can be shown that the filters can be designed in such a way that all of these distortions are reduced or eliminated.
At the receiver end, the received signals are decoded and then produce the signalsyk(n)by passing through the upsampler. These signals are filtered by the filtersF0(z)andF1(z), producing the output signal .(H0(z) andF0(z) respectively are the analysis and synthesis filters of thek-th channel, where k=0,1 .)
(1-11)
The analysis and synthesis filters are related as .
For the linear-phase filter bank,H0(z)andH1(z)are chosen respectively to be symmetric and anti-symmetric, which is an even number. The system delay should be equal to , whereN0andN1are respectively thelengths of the lowpass and highpass filters.
,k=0,1.(1-3)
The reconstructed signal is
(1-4)
The termX(-z) takes into account aliasing due to the downsamplers and imaging due to the upsamplers. We refer to this just as the aliasing term or alias component.
Thecoefficients of filtersH0(z)andH1(z)are obtained by solving the following constrained optimization
Subject to (1-11), (1-12)
Whereωpandωsare the passband and stopband cut off freque来自百度文库cies ofH0(z)andH1(z), respectively,α is a weighting constant from 0 to 1, andhisthe vector containing the free variables in the impulse response.
1.2Distortions
1.2.1 Aliasing
From (1-4), it is clear that we can cancel aliasing by choosing the filters such that the quantity is zero.
1.2.2Phase and Amplitude Distortions
Inthe time domain,theconstraintcan be as
.(1-13)
Letting (that is, in the time domain), then theconstrained optimizationwill turn as the following form
s.t. . (1-14)
Design ofTwo-ChannelPRFilter Banks
Abstract：In thispaper,we will address the concepts oftwo-channelPRfilterbanksand perfect-reconstruction (PR) property.Thebasic principle and design method oftwo-channel PR filter banks will be discussed in detail.Thenthefilter coefficients,time domainerror,the amplituderesponse andamplitudedistortionwill beshowed in figures.Finally,the designresultswill beanalyzedand summarized.
Fig.1-1The two-channel QMF bank.
From Fig.1-1we have
， .(1-1)
Thez-transforms of the decimated signalsvk(n)are
，k=0,1.(1-2)
The second term above represents aliasing. The z-transforms ofFk(z)isVk(z2), so that
While . (1-7)
Letting ，we have
. (1-8)
Unless isT(z) allpass, we say that suffers from “amplitude distortion”. Similarly unlessT(z)has linear phase (that is, for constantaandb), suffers from phase distortion.
. (1-10)
forall possible inputsx(n). This, of course, ignores the coding/decoding error and filterround offnoise.
3.Design ofTwo-ChannelPRFilter Banks
For a two-channel critically decimated filter bank, the PR condition can be given by
2.The concept of PR filter banks
A filter bank is called perfect-reconstruction (PR) system if the reconstructed signal is a scaled and delayed version of the input , namely,
1.Two-Channel Filter Banks
1.1Basic principleof the Two-Channel Filter Bank
Fig.1-1shows a two-channel filter bank, popularly called the quadrature mirror filter (QMF) bank. The inputsignalx(n)is first filtered by two filtersH0(z)andH1(z),typically lowpass and highpass filters. Each subbandxk(n)signal is therefore bandlimited to a total bandwidth ofπ. The subband signals are decimated by a factor of 2 to producevk(n). Each decimated signalvk(n)is then coded in such a way that the special properties of the subband are exploited.
Solving thisconstrained optimizationisthe most criticalstep inthe design.
4. Results of the design
We get the result as shown inFig.1-2
Fig.1-2Frequency responseof the two-channel PR filter banks for linear-phase property