Recent Trends in the Design of Biorthogonal Modulated Filter Banks

Biorthogonal modulated lter banks when compared to paraunitary ones provide the advantage that the overall system delay can be chosen independently of the lter length thus resulting in low delay lter banks They have recently been studied by several authors In the paper we connect di erent design methods quadratic constrained least squares optimization cascade of sparse self inverse matrices and describe advan tages of the a factorization into Zero Delay and Maximum Delay matrices structure inherent perfect reconstruction no DC leakage of the lter bank low implementation cost Introduction Modulated lter banks have been studied extensively in literature within the last years They have shown to provide a very e cient implementation based on a prototype lter and a fast transform The most popular modulation scheme is cosine modulation However other modulation schemes based on Discrete Fourier Transform DFT also exist Historically the rst modulated lter banks with perfect reconstruction were designed such as to be paraunitary PB RT KV Mau NK In this special case the impulse responses of the synthesis lters are ipped versions of the analysis ones and all lters are derived from one common prototype However since the overall system delay of the lter bank is directly related to the lter length the desired features a a high stopband attenuation of the lters and b a short overall system delay are contradictory This problem has partly been overcome with the design of low delay biorthogonal lter banks where the overall system delay can be chosen independently within some fundamental limits of the lter length and the number of subbands In this class of lter banks the synthesis lters are no longer ipped versions of the analysis lters and the analysis and synthesis lters may be derived from di erent prototypes TICSP Workshop on Transforms and Filter Banks Biorthogonal Modulated Filter Banks Two principal approaches can be observed for the design of biorthogonal cosine modulated lter banks with perfect reconstruction The approach by Schuller et al SS Sch SK uses lter bank realizations that structurally guarantee perfect reconstruction for arbitrary system delays It is mainly based on a factorization of the analysis polyphase matrix into a transform and special sparse matrices which are easy to invert The inverse matrices are then used on the synthesis side The prototype lters are derived by nonlinear optimization methods Completeness of the factorization has been shown for all contiguous prototype lters i e prototype lters that do not have any zero taps within their region of support On the other hand Nguyen et al explicitly derive perfect reconstruction PR constraints for the polyphase components of the prototype lters Ngu NH HKN The lter design is carried out by a quadratic constrained least squares optimization QCLS algorithm using the stopband energy of the prototypes frequency responses as a cost function and the PR conditions as constraints In HKN it has also been shown that the same PR constraints also hold true for a DCT II modulation scheme as proposed in LV Both approaches provide certain advantages Since in the latter case the PR constraints are directly formulated it can be easily veri ed whether or not given prototype lters yield perfect reconstruction Necessary relations between the analysis and synthesis prototype lters are also stated HKN Furthermore it can be shown that for certain combinations of lter lengths and overall system delay some polyphase lters can only have one non zero coe cient This case is not treated in the factorization proposed in Sch SK However the factorization approach from Schuller et al o ers many advantages concerning the implementation of the lter bank First of all the structure automatically guarantees PR This also holds true when using integer valued coe cients because the same factorization coe cients are used on the analysis and synthesis side But even when using coe cients with in nite precision it turns out that the implementation cost is nearly halved when compared to the direct realization of the polyphase lters as assumed by Nguyen et al Another advantage of the approach is that it has been extended to the case of time varying lter banks Sch without much e ort Which of the two optimization methods unconstrained and non linear or constrained and quadratic results in better lter designs highly depends on the chosen optimization procedures and the complexity of the problem i e lter length number of subbands etc In this paper we connect both approaches by showing that for the modulation scheme considered by Nguyen et al a factorization being similar the one proposed by Schuller et al exists The factorization is derived directly from the PR constraints on the polyphase components of the prototype lters and also treats the case that some polyphase lters contain coe cients being equal to zero Instead of dealing with size M M matrices as in Sch SK we just have to deal with size matrices and realize bM c of them in parallel Using this factorization the implementation cost can be signi cantly reduced Furthermore we show how to include certain useful lter bank features in the implementation Such features can be same prototype for analysis and synthesis prototype lters with speci ed zeros at certain frequencies in order to yield lter banks without DC leakage integer coe cient prototype lters Biorthogonal Modulated Filter Banks time varying lter banks The outline of the paper is as follows In Section we recall the PR constraints of the cosine modulated lter banks as derived by Nguyen et al In Section we derive how to realize this lter bank using Zero Delay and Maximum Delay matrices starting from the PR constraints Section shows that we can easily design lter banks with identical analysis and synthesis prototype lter by imposing some constraints on the rst Delay matrix Filter banks without DC leakage are important when treating e g images This feature can also be obtained when choosing the rst matrix of the factorization appropriately as will be derived in Section Section compares the implementation cost of the new factorization with a direct implementation of the polyphase lters and in Section we explain how to use the structure inherent PR property of the factorization in order to design VLSI e cient prototype lters In Section we shortly sketch the extension of the framework to time varying lter banks and summarize the results of the paper in Section Cosine Modulated Filter Banks with Perfect Reconstruction In this section we recall the main steps of the derivation of PR constraints from HKN NH for biorthogonal cosine modulated lter banks The lter bank structure is shown in Figure The analysis lter bank consists of M analysis lters of length Nh with impulse responses hk n k M n Nh and subsequent downsampling by M The input signal is denoted as x n The subband signals are yk m k M where m is the time index at the reduced sampling rate The synthesis lter bank consists of upsamplers by M followed by M synthesis lters with impulse responses fk n k M n Nf The outputs of these lters are summed to form the reconstructed signal x n