Filter Banks (779443), страница 7
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Processing ofSignalsFinite-Length191(c)(4Figure 6.34. Periodic extension of the input signal; (a) one-dimensional circularconvolution; (b) one-dimensional symmetric reflection; (c) two-dimensional circularconvolution; (d) two-dimensional symmetric reflection.length.
Figure 6.35(a)shows a scheme suitable for the two-band decompositionof an even-length signal with linear-phase odd-length biorthogonal filters. Theinput signal is denoted as ZO, 2 1 , . . . ,27, and the filter impulse responsesare {A,B , C, B , A } for the lowpass and {-a, b, - a } for the highpass. Theupper row shows the extended input signal, where the given input samplesare showninsolidboxes.The lowpass and highpasssubbandsamples, c,and d,, respectively, are computed by takingthe inner products of theimpulse responsesin the displayed positions with the corresponding part of theextended input signal. We see that only four different lowpass and highpasscoefficients occur and have to be transmitted.
A second scheme for the samefilters which also allows the decomposition of even-length signals into lowpassand highpass components of half the length is depicted in Figure 6.35(b). Inorder to distinguish between both methods we say that the starting positionin Figure 6.35(a) is even and the one in Figure 6.35(b) is odd, as indicatedby the indices of the samples. Combinations of both schemes can be used todecompose odd-length signals. Moreover, these schemes can be used for thedecomposition of 2-D objects with arbitrary shape. We will return to thistopic at the end of this section.Schemesfor the decomposition of even-length signals with even-length192............j............X, j X,BanksI X, I X,I I I I..........
........j X, jX, X, X, X, X, ..........XX,, j X,........IAlB CIB AAIB C B AA BlClBlAlA BlClBlAl..................j X, j X,j.....X, .............X,X,............X, X,X, X,XX,,............X, j X, j............IAlBlClB AIAlBlC B AA B C BlAlA BlClBlAlmm+.l..jd,.lc,Chapter 6. Filterm..................:cI do1 Cl I 4 I c2 I d2 I c3 I 4 I.~,.i.4.:...z.::..............................j.....c2 . i . 4 . ~ .
. l . l ~ o I ~ I I ~ l I ~ 2 I ~ 2 I ~ ; I ~ 3 I ~ 4 l . d ~ . i .(4,.............................j X, j X, X, XI X, X; X, X, X6 X, X, j X, j X, :I.............................I II I I I..................j X; j X, j XI XI X, X;..................X, X,X6 X, X,.............X, j X, j..............IAlBlBlAlAlBBIAA I BB l A lAlBlBlAlI a I b I -bl -al a I b- b l - aa l b -bl-ala Ib I-bl-alFigure 6.35.
Symmetricreflection for even-lengthsignals. (a) odd-length filters,segment starting at an evenposition; (b) odd-lengthfilters,segment starting atan odd position; ( c ) even-length filters, segment starting at an even position; (d)even-length filters, segment starting at an odd position.linear-phase filters are depicted in Figures 6.35(c) and (d). The filter impulseresponses are {A,B , BA}, for the lowpass and {-a, -b, b, a } for the highpass.Note that a different type of reflection isused and that we haveothersymmetries in the subbands. While the scheme in Figure 6.35(c) results inthe same number of lowpass and highpass samples, the one in Figure 6.35(d)yields an extra lowpass value, while the corresponding highpass value is zero.However, the additionallowpass samples can beturned intohighpass values bysubtracting them from the following lowpass value and storing the differencesin the highpass band.In object based image coding, for instance MPEG-4 [log], it is required tocarry out subband decompositions of arbitrarily shaped objects.
Figure 6.366.9. Processing of Finite-Length Signals193Figure 6.36. Shape adaptive image decomposition using symmetric reflection forodd-lengthtwo-channelfilterbanks;(a) arbitrarily shaped object and horizontalextension with pixel values as indicated;(b) lowpass filter; (c) highpass filter; (d)and (e) lowpass and highpass subbands after horizontal filtering;(f) and (g) lowpassand highpass decompositions of the signal in (d); (h) and (i) lowpass and highpassdecompositions of the signal in (e).shows a schemewhichissuitable for thistask usingodd-length filters.Thearbitrarilyshapedinputsignal isshown in the marked region, andthe extension for the first horizontal decomposition is found outside thisregion. Figures 6.36(d) and (e) show the shape of the lowpass and highpassband, respectively.
Figures 6.36(f)-(i) finally show the object shapes after thevertical decomposition of the signals in Figures 6.36(d) and (e) based on thesame reflection scheme. Such schemes are often called shape adaptive wavelettransforms. Note that the overall number of subband samples is equal to thenumber of input pixels. Moreover, the scheme yields a decomposition wherethe interiorregion of an object is processed as if the objectwas of infinite size.Thus, the actual object shape only influences the subband samples close tothe boundaries. The 2-D decomposition is carried out in such a way that thehorizontal decomposition introduces minimal distortion for the next verticalone and vice versa.194Chapter 6.
Filter Banksl 0 0 1 2 3 4 5 6 7 8 9 9l 0 0 1 2 3 4 5 6 7 8 8 73 2 1 1 2 3 4 5 6 7 8 8 7Figure 6.37. Shape adaptive image decomposition using symmetric reflection foreven-length two-channel filter banks; see the comments to Figure 6.36 for furtherexplanation.A scheme for the decomposition of arbitrarily shaped 2-D objects witheven-length filters is depicted in Figure 6.37. Note that in this case, thelowpass band grows faster than the highpass band.
The shaded regions inFigures 6.37(d)and (e) show the shapeof the lowpass and highpass band afterhorizontal filtering. The brighter regions within the object in Figure 6.37(d)indicate the extra lowpass samples. The zero-marked fields in Figure 6.37(e)are positions where the highpass samples are exactly zero.If the faster growing of the lowpass band is unwanted the manipulationindicated in Figure 6.35(d) can be applied. Then the subbands obtainedwitheven-length filters will have the same shape as the ones in Figure 6.36.In addition to thedirect use of symmetric reflection, one can optimizetheboundary processing schemes in order to achieve better coding properties.Methods for this task have been proposed in [70, 69, 101, 27, 102, 39, 401.These include the two-band, the more general M-band, and the paraunitarycase with non-linear phase filters.6.10.
Transmultiplexers6.10195Transmult iplexersTransmultiplexers are systems that convert time-division multiplexed (TDM)signals into frequency-division multiplexed (FDM) signals and vice versa [151].Essentially, these systems are filter banks as shown in Figure 6.38. Contraryto the subband coding filter banks considered so far, the synthesis filter bankis applied first and the analysis filter bank is then used to recover the subbandsamples yk(m), which may be understood as components of a TDM signal.At the output of the synthesis filter bank we have an FDM signal where eachdata stream yk(m) covers a different frequency band.The transmission from input i to output k is described by the impulseresponsest i , k (m)= q i , k ( m M ) ,(6.152)(6.153)In the noise-free case, perfect reconstruction of the input datawith a delay ofm0 samples can be obtained when the following condition holds:ti,k(rn)= dik,,,,Si , k = 0,1,...
, M- 1.(6.154)Using the notationof modulation matrices thesePR conditions may be writtenasT ( z M )= H ; ( z ) G , ( z ) = M z-,OM I ,(6.155)where the overall transfer matrix depends onz M . This essentially means thatany P R subband codingfilter bank yields a P R transmultiplexer if the overalldelay is a multiple of M .Practical problems with transmultiplexers mainly occur due to non-idealtransmission channels. Thismeans that intersymbol interference, crosstalkbetween different channels, and additive noise need to be considered in thetransmultiplexer design.
An elaborate discussion of this topic is beyond thescope of this section.Figure 6.38. Transmultiplexer filter bank..
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