Digital Filterbank for Spectral Envelope Adjustment

Abstract

An apparatus and method are disclosed for processing an audio signal. The apparatus includes an input interface, a digital filterbank having an analysis part and a synthesis part, a first phase shifter, a spectral envelope adjuster, a second phase shifter, and an output interface. The first phase shifter and the second phase shifter reduce a complexity of the digital filterbank, which includes both analysis and synthesis filters that are complex-exponential modulated versions of a prototype filter.

Claims

1. A signal processing device for filtering and performing high frequency reconstruction of an audio signal, the signal processing device comprising: an analysis filter bank that receives real valued time domain input audio samples and generates complex valued subband samples; a unit that generates modified complex valued subband samples by modifying gains of the complex valued subband samples according to current spectral envelope adjuster settings; and a synthesis filter bank that receives the modified complex valued subband samples and generates time domain output audio samples, wherein the analysis filter bank comprises analysis filters (h.sub.k(n)) that are complex exponential modulated versions of a prototype filter (p.sub.0(n)) according to: $h_k\left(n\right)=p_0\left(n\right)\exp \left\{i\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}-A\right)\right\},0\le n<N,0\le k<M$ where the analysis filter bank has M channels, the prototype filter (p.sub.0(n)) has a length N, the analysis filter bank and synthesis filter bank have a system delay of D samples with D being smaller than N, and A is a constant chosen to enable efficient implementation of the analysis filter bank.

2. The signal processing device of claim 1 wherein the prototype filter p.sub.0(n) is a symmetric low pass prototype filter or an asymmetric low pass prototype filter.

3. The signal processing device of claim 1 wherein the analysis filter bank is a pseudo QMF bank.

4. The signal processing device of claim 1 wherein an order of the prototype filter p.sub.0(n) equals the system delay D.

5. The signal processing device of claim 1 wherein the number of channels in the analysis filter bank is 32 and the number of channels in the synthesis filter bank is 64.

6. A method performed by a signal processing device for filtering and performing high frequency reconstruction of an audio signal, the method comprising: receiving real-valued time domain input audio samples; filtering the real-valued time domain input audio samples with an analysis filter bank to generate complex valued subband samples; generating modified complex valued subband samples by modifying gains of the complex valued subband samples according to current spectral envelope adjuster settings; receiving the modified complex valued subband samples; filtering the modified complex valued subband samples with a synthesis filter bank to generate time domain output audio samples, wherein the analysis filter bank comprises analysis filters (h.sub.k(n)) that are complex exponential modulated versions of a prototype filter (p.sub.0(n)) according to: $h_k\left(n\right)=p_0\left(n\right)\exp \left\{i\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}-A\right)\right\},0\le n<N,0\le k<M$ where the analysis filter bank has M channels, the prototype filter (p.sub.0(n)) has a length N, the analysis filter bank and synthesis filter bank have a system delay of D samples with D being smaller than N, and A is a constant chosen to enable efficient implementation of the analysis filter bank.

7. A non-transitory computer readable medium containing instructions that when executed by a processor perform the method of claim 6.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0046] FIG. 1 illustrates the analysis and synthesis sections of a digital filter bank;

[0047] FIG. 2 (a)-(c) shows the stylized frequency responses for a set of filters to illustrate the adverse effect when modifying the subband samples in a cosine modulated, i.e. real-valued, filter bank;

[0048] FIG. 3 shows a flow diagram of an example of the optimization procedure;

[0049] FIG. 4(a)-(b) shows a time domain plot and the frequency response of an optimized prototype filter for a low delay modulated filter bank having 64 channels and a total system delay of 319 samples; and

[0050] FIG. 5 (a)-(b) illustrates an example of the analysis and synthesis parts of a low delay complex-exponential modulated filter bank system.

[0051] It should be understood that the present teachings are applicable to a range of implementations that incorporate digital filter banks other than those explicitly mentioned in this patent. In particular, the present teachings may be applicable to other methods for designing a filter bank on the basis of a prototype filter.

[0052] In the following, the overall transfer function of an analysis/synthesis filter bank is determined. In other words, the mathematical representation of a signal passing through such a filter bank system is described. A digital filter bank is a collection of M, M being two or more, parallel digital filters that share a common input or a common output. For details on such filter banks, reference is made to “Multirate Systems and Filter Banks” P. P. Vaidyanathan Prentice Hall: Englewood Cliffs, N J, 1993. When sharing a common input the filter bank may be called an analysis bank. The analysis bank splits the incoming signal into M separate signals called subband signals. The analysis filters are denoted H.sub.k(z), where k=0, . . . , M−1. The filter bank is critically sampled or maximally decimated when the subband signals are decimated by a factor M. Thus, the total number of subband samples per time unit across all subbands is the same as the number of samples per time unit for the input signal. The synthesis bank combines these subband signals into a common output signal. The synthesis filters are denoted F.sub.k(z), for k=0, . . . , M−1.

[0053] A maximally decimated filter bank with M channels or subbands is shown in FIG. 1. The analysis part 101 produces from the input signal X(z) the subband signals V.sub.k (z), which constitute the signals to be transmitted, stored or modified. The synthesis part 102 recombines the signals V.sub.k(z) to the output signal {circumflex over (X)}(z).

[0054] The recombination of V.sub.k (z) to obtain the approximation {circumflex over (X)}(z) of the original signal X(z) is subject to several potential errors. The errors may be due to an approximation of the perfect reconstruction property, and includes non-linear impairments due to aliasing, which may be caused by the decimation and interpolation of the subbands. Other errors resulting from approximations of the perfect reconstruction property may be due to linear impairments such as phase and amplitude distortion.

[0055] Following the notations of FIG. 1, the outputs of the analysis filters H.sub.k(z) 103 are

X.sub.k(z)=H.sub.k(z)X(z),  (1)

where k=0, . . . , M−1. The decimators 104, also referred to as down-sampling units, give the outputs

[00008]$\begin{array}{cc}{V}_k\left(z\right)=\frac{1}{M}\underset{l=0}{\stackrel{M-1}{.Math.}}{X}_k\left(z^{1/M}{W}^l\right)=\frac{1}{M}\stackrel{M-1}{\underset{l=0}{.Math.}}{H}_k\left(z^{1/M}{W}^l\right)X\left(z^{1/M}{W}^l\right),& \left(2\right)\end{array}$

where W=e.sup.−i2π/M. The outputs of the interpolators 105, also referred to as up-sampling units, are given by

[00009]$\begin{array}{cc}{U}_k\left(z\right)=V_k\left(z^M\right)=\frac{1}{M}\underset{l=0}{\stackrel{M-1}{.Math.}}{H}_k\left(z{W}^l\right)X\left(z{W}^l\right),& \left(3\right)\end{array}$

and the sum of the signals obtained from the synthesis filters 106 can be written as

[00010]$\begin{array}{cc}\hat{X}\left(z\right)=\stackrel{M-1}{\underset{k=0}{.Math.}}{F}_k\left(z\right)U_k\left(z\right)=\stackrel{M-1}{\underset{k=0}{.Math.}}{F}_k\left(z\right)\frac{1}{M}\stackrel{M-1}{\underset{l=0}{.Math.}}{H}_k\left(z{W}^l\right)X\left(z{W}^l\right)==\frac{1}{M}\stackrel{M-1}{\underset{l=0}{.Math.}}X\left(z{W}^l\right)\stackrel{M-1}{\underset{k=0}{.Math.}}{H}_k\left(z{W}^l\right)F_k\left(z\right)=\frac{1}{M}\stackrel{M-1}{\underset{k=0}{.Math.}}X\left(z{W}^l\right)A_l\left(z\right)& \left(4\right)\\ \mathrm{where}& \\ A_l\left(z\right)=\stackrel{M-1}{\underset{k=0}{.Math.}}{H}_k\left(z{W}^l\right)F_k\left(z\right)& \left(5\right)\end{array}$

is the gain for the l.sup.th alias term X(zW.sup.l). Eq. (4) shows that {circumflex over (X)}(z) is a sum of M components consisting of the product of the modulated input signal X(zW.sup.l) and the corresponding alias gain term A.sub.l(z). Eq. (4) can be rewritten as

[00011]$\begin{array}{cc}\hat{X}\left(z\right)=\frac{1}{M}\left\{X\left(z\right)A_0\left(z\right)+\underset{l=1}{\overset{M-1}{.Math.}}X\left(z{W}^l\right)A_l\left(z\right)\right\}.& \left(6\right)\end{array}$

[0056] The last sum on the right hand side (RHS) constitutes the sum of all non-wanted alias terms. Canceling all aliasing, that is forcing this sum to zero by means of proper choices of H.sub.k(z) and F.sub.k(z), gives

[00012]$\begin{array}{cc}\hat{X}\left(z\right)=\frac{1}{M}X\left(z\right)A_0\left(z\right)=\frac{1}{M}X\left(z\right)\stackrel{M-1}{\underset{k=0}{.Math.}}{H}_k\left(z\right)F_k\left(z\right)=X\left(z\right)T\left(z\right),\mathrm{where}& \left(7\right)\\ T\left(z\right)=\frac{1}{M}\stackrel{M-1}{\underset{k=0}{.Math.}}{H}_k\left(z\right)F_k\left(z\right)& \left(8\right)\end{array}$

is the overall transfer function or distortion function. Eq. (8) shows that, depending on H.sub.k(z) and F.sub.k(z), T(z) could be free from both phase distortion and amplitude distortion. The overall transfer function would in this case simply be a delay of D samples with a constant scale factor c, i.e.

T(z)=cz.sup.−D,  (9)

which substituted into Eq. (7) gives

{circumflex over (X)}(z)=cz.sup.−DX(z).  (10)

[0057] The type of filters that satisfy Eq. (10) are said to have the perfect reconstruction (PR) property. If Eq. (10) is not perfectly satisfied, albeit satisfied approximately, the filters are of the class of approximate perfect reconstruction filters.

[0058] In the following, a method for designing analysis and synthesis filter banks from a prototype filter is described. The resulting filter banks are referred to as cosine modulated filter banks. In the traditional theory for cosine modulated filter banks, the analysis filters h.sub.k(n) and synthesis filters f.sub.k(n) are cosine modulated versions of a symmetric low-pass prototype filter p.sub.0(n), i.e.

[00013]$\begin{array}{cc}{h}_k\left(n\right)=\sqrt{2}{p}_0\left(n\right)\cos \left\{\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{N}{2}\mp \frac{M}{2}\right)\right\},0\le n\le N,0\le k<M& \left(11\right)\\ f_k\left(n\right)=\sqrt{2}{p}_0\left(n\right)\cos \left\{\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{N}{2}\pm \frac{M}{2}\right)\right\},0\le n\le N,0\le k<M& \left(12\right)\end{array}$

respectively, where M is the number of channels of the filter bank and N is the prototype filter order.

[0059] The above cosine modulated analysis filter bank produces real-valued subband samples for real-valued input signals. The subband samples are down sampled by a factor M, making the system critically sampled. Depending on the choice of the prototype filter, the filter bank may constitute an approximate perfect reconstruction system, i.e. a so called pseudo QMF bank described e.g. in U.S. Pat. No. 5,436,940, or a perfect reconstruction (PR) system. An example of a PR system is the modulated lapped transform (MLT) described in further detail in “Lapped Transforms for Efficient Transform/Subband Coding” H. S. Malvar, IEEE Trans ASSP, vol. 38, no. 6, 1990. The overall delay, or system delay, for a traditional cosine modulated filter bank is N.

[0060] In order to obtain filter bank systems having lower system delays, the present document teaches to replace the symmetric prototype filters used in conventional filter banks by asymmetric prototype filters. In the prior art, the design of asymmetric prototype filters has been restricted to systems having the perfect reconstruction (PR) property. Such a perfect reconstruction system using asymmetric prototype filters is described in EP0874458. However, the perfect reconstruction constraint imposes limitations to a filter bank used in e.g. an equalization system, due to the restricted degrees of freedom when designing the prototype filter. It should be noted that symmetric prototype filters have a linear phase, i.e. they have a constant group delay across all frequencies. On the other hand, asymmetric filters typically have a non-linear phase, i.e. they have a group delay which may change with frequency.

[0061] In filter bank systems using asymmetric prototype filters, the analysis and synthesis filters may be written as

[00014]$\begin{array}{cc}{h}_k\left(n\right)=\sqrt{2}{\hat{h}}_0\left(n\right)\cos \left\{\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}\mp \frac{M}{2}\right)\right\},0\le n<N_h,0\le k<M& \left(13\right)\\ f_k\left(n\right)=\sqrt{2}{\hat{f}}_0\left(n\right)\cos \left\{\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}\pm \frac{M}{2}\right)\right\},0\le n<N_f,0\le k<M& \left(14\right)\end{array}$

respectively, where ĥ.sub.0 (n) and {circumflex over (f)}.sub.0 (n) are the analysis and synthesis prototype filters of lengths N.sub.h and N.sub.f, respectively, and D is the total delay of the filter bank system. Without limiting the scope, the modulated filter banks studied in the following are systems where the analysis and synthesis prototypes are identical, i.e.

{circumflex over (f)}.sub.0(n)=ĥ.sub.0(n)=p.sub.0(n), 0≤n<N.sub.h=N.sub.f=N  (15)

where N is the length of the prototype filter p.sub.0(n).

[0062] It should be noted, however, when using the filter design schemes outlined in the present document, that filter banks using different analysis and synthesis prototype filters may be determined.

[0063] One inherent property of the cosine modulation is that every filter has two pass bands; one in the positive frequency range and one corresponding pass band in the negative frequency range. It can be verified that the so-called main, or significant, alias terms emerge from overlap in frequency between either the filters negative pass bands with frequency modulated versions of the positive pass bands, or reciprocally, the filters positive pass bands with frequency modulated versions of the negative pass bands. The last terms in Eq. (13) and (14), i.e. the terms

[00015]$\frac{\pi }{2}\left(k+\frac{1}{2}\right),$

are selected so as to provide cancellation of the main alias terms in cosine modulated filter banks. Nevertheless, when modifying the subband samples, the cancelation of the main alias terms is impaired, thereby resulting in a strong impact of aliasing from the main alias terms. It is therefore desirable to remove these main alias terms from the subband samples altogether.

[0064] The removal of the main alias terms may be achieved by the use of so-called Complex-Exponential Modulated Filter Banks which are based on an extension of the cosine modulation to complex-exponential modulation. Such extension yields the analysis filters h.sub.k(n) as

[00016]$\begin{array}{cc}{h}_k\left(n\right)=p_0\left(n\right)\exp \left\{i\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}\mp \frac{M}{2}\right)\right\},0\le n<N,0\le k<M& \left(16\right)\end{array}$

using the same notation as before. This can be viewed as adding an imaginary part to the real-valued filter bank, where the imaginary part consists of sine modulated versions of the same prototype filter. Considering a real-valued input signal, the output from the filter bank can be interpreted as a set of subband signals, where the real and the imaginary parts are Hilbert transforms of each other.

[0065] The resulting subbands are thus the analytic signals of the real-valued output obtained from the cosine modulated filter bank. Hence, due to the complex-valued representation, the subband signals are over-sampled by a factor two.

[0066] The synthesis filters are extended in the same way to

[00017]$\begin{array}{cc}{f}_k\left(n\right)=p_0\left(n\right)\exp \left\{i\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}\pm \frac{M}{2}\right)\right\},0\le n<N,0\le k<M.& \left(17\right)\end{array}$

[0067] Eq. (16) and (17) imply that the output from the synthesis bank is complex-valued. Using matrix notation, where C.sub.a is a matrix with the cosine modulated analysis filters from Eq. (13), and S.sub.a is a matrix with the sine modulation of the same argument, the filters of Eq. (16) are obtained as C.sub.a+j S.sub.a. In these matrices, k is the row index and n is the column index. Analogously, the matrix C.sub.s has synthesis filters from Eq. (14), and S.sub.s is the corresponding sine modulated version. Eq. (17) can thus be written C.sub.s+j S.sub.s, where k is the column index and n is the row index. Denoting the input signal x, the output signal y is found from

y=(C.sub.s+jS.sub.s)(C.sub.a+jS.sub.a)x=(C.sub.sC.sub.a−S.sub.sS.sub.a)x+j(C.sub.sS.sub.a+S.sub.sC.sub.a)x  (18)

[0068] As seen from Eq. (18), the real part comprises two terms; the output from the cosine modulated filter bank and an output from a sine modulated filter bank. It is easily verified that if a cosine modulated filter bank has the PR property, then its sine modulated version, with a change of sign, constitutes a PR system as well. Thus, by taking the real part of the output, the complex-exponential modulated system offers the same reconstruction accuracy as the corresponding cosine modulated version. In other words, when using a real-valued input signal, the output signal of the complex-exponential modulated system may be determined by taking the real part of the output signal.

[0069] The complex-exponential modulated system may be extended to handle also complex-valued input signals. By extending the number of channels to 2M, i.e. by adding the filters for negative frequencies, and by keeping the imaginary part of the output signal, a pseudo QMF or a PR system for complex-valued signals is obtained.

[0070] It should be noted that the complex-exponential modulated filter bank has one pass band only for every filter in the positive frequency range. Hence, it is free from the main alias terms. The absence of main alias terms makes the aliasing cancellation constraint from the cosine (or sine) modulated filter bank obsolete in the complex-exponential modulated version. The analysis and synthesis filters can thus be given as

[00018]$\begin{array}{cc}{h}_k\left(n\right)=p_0\left(n\right)\exp \left\{i\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}-A\right)\right\},0\le n<N,0\le k<M\mathrm{and}& \left(19\right)\\ f_k\left(n\right)=p_0\left(n\right)\exp \left\{i\frac{\pi }{M}\left(k+\frac{1}{2}\right)\left(n-\frac{D}{2}+A\right)\right\},0\le n<N,0\le k<M& \left(20\right)\end{array}$

where A is an arbitrary (possibly zero) constant, and as before, M is the number of channels, N is the prototype filter length, and D is the system delay. By using different values of A, more efficient implementations of the analysis and synthesis filter banks, i.e. implementations with reduced complexity, can be obtained.

[0071] Before presenting a method for optimization of prototype filters, the disclosed approaches to the design of filter banks are summarized. Based on symmetric or asymmetric prototype filters, filter banks may be generated e.g. by modulating the prototype filters using a cosine function or a complex-exponential function. The prototype filters for the analysis and synthesis filter banks may either be different or identical. When using complex-exponential modulation, the main alias terms of the filter banks are obsolete and may be removed, thereby reducing the aliasing sensitivity to modifications of the subband signals of the resulting filter banks. Furthermore, when using asymmetric prototype filters the overall system delay of the filter banks may be reduced. It has also been shown that when using complex-exponential modulated filter banks, the output signal from a real valued input signal may be determined by taking the real part of the complex output signal of the filter bank.

[0072] In the following a method for optimization of the prototype filters is described in detail. Depending on the needs, the optimization may be directed at increasing the degree of perfect reconstruction, i.e. at reducing the combination of aliasing and amplitude distortion, at reducing the sensitivity to aliasing, at reducing the system delay, at reducing phase distortion, and/or at reducing amplitude distortion. In order to optimize the prototype filter p.sub.0(n) first expressions for the alias gain terms are determined. In the following, the alias gain terms for a complex exponential modulated filter bank are derived. However, it should be noted that the alias gain terms outlined are also valid for a cosine modulated (real valued) filter bank.

[0073] Referring to Eq. (4), the z-transform of the real part of the output signal {circumflex over (x)}(n) is

[00019]$\begin{array}{cc}Z\left\{\mathrm{Re}\left(\hat{x}\left(n\right)\right)\right\}={\hat{X}}_R\left(z\right)=\frac{\hat{X}\left(z\right)+\hat{X}*\left(z\right)}{2}.& \left(21\right)\end{array}$

[0074] The notation {circumflex over (X)}*(z) is the z-transform of the complex-conjugated sequence {circumflex over (x)}(n). From Eq. (4), it follows that the transform of the real part of the output signal is

[00020]$\begin{array}{cc}{\hat{X}}_R\left(z\right)=\frac{1}{M}\stackrel{M-1}{\underset{l=0}{.Math.}}\frac{1}{2}\left(X\left(z{W}^l\right)A_l\left(z\right)+X\left(z{W}^{-l}\right)A_l*\left(z\right)\right),& \left(22\right)\end{array}$

where it was used that the input signal x(n) is real-valued, i.e. X*(zW.sup.l)=X (zW.sup.−l). Eq. (22) may after rearrangement be written

[00021]$\begin{array}{cc}\begin{array}{c}{\hat{X}}_R\left(z\right)=\frac{1}{M}\left(\begin{array}{c}X\left(z\right)\frac{1}{2}\left(A_0\left(z\right)+A_0*\left(z\right)\right)+\\ \underset{l=1}{\overset{M-1}{.Math.}}\frac{1}{2}\left(X\left(z{W}^l\right)A_l\left(z\right)+X\left(z{W}^{M-l}\right)A_l*\left(z\right)\right)\end{array}\right)=\\ =\frac{1}{M}\left(\begin{array}{c}X\left(z\right)\frac{1}{2}\left(A_0\left(z\right)+A_0*\left(z\right)\right)+\\ \underset{l=1}{\overset{M-1}{.Math.}}X\left(z{W}^l\right)\frac{1}{2}\left(A_l\left(z\right)+A_{M-l}*\left(z\right)\right)\end{array}\right)=\\ =\frac{1}{M}\left(X\left(z\right){\stackrel{\_}{A}}_0\left(z\right)+\underset{l=1}{\overset{M-1}{.Math.}}X\left(z{W}^l\right){\stackrel{\_}{A}}_l\left(z\right)\right)\end{array}& \left(23\right)\\ \mathrm{where}& \\ {\stackrel{\_}{A}}_l\left(z\right)=\frac{1}{2}\left(A_l\left(z\right)+A_{M-l^{*}}\left(z\right)\right),0\le l<M& \left(24\right)\end{array}$

are the alias gain terms used in the optimization. It can be observed from Eq. (24) that

Ã.sub.M-l(z)=½(A.sub.M-l(z)+A.sub.l*(z))=Ã.sub.l*(z).  (25)

[0075] Specifically, for real-valued systems

A.sub.M-l*(z)=A.sub.l(z)  (26)

which simplifies Eq. (24) into

Ã.sub.l(z)=A.sub.l(z), 0≤l<M.  (27)

[0076] By inspecting Eq. (23), and recalling the transform of Eq. (21), it can be seen that the real part of a.sub.0(n) must be a Dirac pulse for a PR system, i.e. Ã.sub.0(z) is on the form Ã.sub.0(z)=c z.sup.−D. Moreover, the real part of a.sub.M/2(n) must be zero, i.e. Ã.sub.M/2(z) must be zero, and the alias gains, for l≠0, M/2 must satisfy

A.sub.M-l(z)=−A.sub.l*(z),  (28)

which for a real-valued system, with Eq. (26) in mind, means that all a.sub.l(n), l=1 . . . M−1 must be zero. In pseudo QMF systems, Eq. (28) holds true only approximately. Moreover, the real part of a.sub.0(n) is not exactly a Dirac-pulse, nor is the real part of a.sub.M/2(n) exactly zero.

[0077] Before going into further details on the optimization of the prototype filters, the impact of modifications of the subband samples on aliasing is investigated. As already mentioned above, changing the gains of the channels in a cosine modulated filter bank, i.e. using the analysis/synthesis system as an equalizer, renders severe distortion due to the main alias terms. In theory, the main alias terms cancel each other out in a pair wise fashion. However, this theory of main alias term cancellation breaks, when different gains are applied to different subband channels. Hence, the aliasing in the output signal may be substantial. To show this, consider a filter bank where channel p and higher channels are set to zero gain, i.e.

[00022]$\begin{array}{cc}{v}_k^{\prime }\left(n\right)=g_k{v}_k\left(n\right),\{\begin{array}{cc}{g}_k=1,& 0\le k<p\\ g_k=0,& p\le k<M-1\end{array}& \left(29\right)\end{array}$

[0078] The stylized frequency responses of the analysis and synthesis filters of interest are shown in FIG. 2. FIG. 2(a) shows the synthesis channel filters F.sub.p-1(z) and F.sub.p(z), highlighted by reference signs 201 and 202, respectively. As already indicated above, the cosine modulation for each channel results in one positive frequency filter and one negative frequency filter. In other words, the positive frequency filters 201 and 202 have corresponding negative frequency filters 203 and 204, respectively.

[0079] The p.sup.th modulation of the analysis filter H.sub.p-1(z), i.e. H.sub.p-1(zW.sup.p) indicated by reference signs 211 and 213, is depicted in FIG. 2(b) together with the synthesis filter F.sub.p-1(z), indicated by reference signs 201 and 203. In this Figure, reference sign 211 indicates the modulated version of the originally positive frequency filter H.sub.p-1(z) and reference sign 213 indicates the modulated version of the originally negative frequency filter H.sub.p-1(z). Due to the modulation of order p, the negative frequency filter 213 is moved to the positive frequency area and therefore overlaps with the positive synthesis filter 201. The shaded overlap 220 of the filters illustrates the energy of a main alias term.

[0080] In FIG. 2(c) the p.sup.th modulation of H.sub.p(z), i.e. H.sub.p(zW.sup.p) indicated by reference signs 212 and 214, is shown together with the corresponding synthesis filter F.sub.p(z), reference signs 202 and 204. Again the negative frequency filter 214 is moved into the positive frequency area due to the modulation of order p. The shaded area 221 again pictorially shows the energy of a main alias term and would un-cancelled typically result in significant aliasing. To cancel the aliasing, the term should be the polarity reversed copy of the aliasing obtained from the intersection of filters H.sub.p-1(zW.sup.p), 213, and F.sub.p-1(z), 201, of FIG. 2(b), i.e. the polarity reversed copy of the shaded area 220. In a cosine modulated filter bank, where the gains are unchanged, these main alias terms will usually cancel each other completely. However, in this example, the gain of the analysis (or synthesis) filter p is zero, so the aliasing induced by filters p−1 will remain un-cancelled in the output signal. An equally strong aliasing residue will also emerge in the negative frequency range.

[0081] When using complex-exponential modulated filter banks, the complex-valued modulation results in positive frequency filters only. Consequently, the main alias terms are gone, i.e. there is no significant overlap between the modulated analysis filters H.sub.p(zW.sup.p) and their corresponding synthesis filters F.sub.p(z) and aliasing can be reduced significantly when using such filter bank systems as equalizers. The resulting aliasing is dependent only on the degree of suppression of the remaining alias terms.

[0082] Hence, even when using complex-exponential modulated filter banks, it is crucial to design a prototype filter for maximum suppression of the alias gains terms, although the main alias terms have been removed for such filter banks. Even though the remaining alias terms are less significant than the main alias terms, they may still generate aliasing which causes artifacts to the processed signal. Therefore, the design of such a prototype filter can preferably be accomplished by minimizing a composite objective function. For this purpose, various optimization algorithms may be used. Examples are e.g. linear programming methods, Downhill Simplex Method or a non-constrained gradient based method or other nonlinear optimization algorithms. In an exemplary embodiment an initial solution of the prototype filter is selected. Using the composite objective function, a direction for modifying the prototype filter coefficients is determined which provides the highest gradient of the composite objective function. Then the filter coefficients are modified using a certain step length and the iterative procedure is repeated until a minimum of the composite objective function is obtained. For further details on such optimization algorithms, reference is made to “Numerical Recipes in C, The Art of Scientific Computing, Second Edition” W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Cambridge University Press, N Y, 1992, which is incorporated by reference.

[0083] For improved alias term minimization (IATM) of the prototype filter, a preferred objective function may be denoted

e.sub.tot(α)=αe.sub.t+(1−α)e.sub.a,  (30)

where the total error e.sub.tot(α) is a weighted sum of the transfer function error e.sub.t and the aliasing error e.sub.a. The first term on the right hand side (RHS) of Eq. (23) evaluated on the unit circle, i.e. for z=e.sup.jω, can be used to provide a measure of the error energy e.sub.t of the transfer function as

[00023]$\begin{array}{cc}{e}_t=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}{.Math.\frac{1}{2}\left(A_0\left(e^{j\omega }\right)+A_0^{*}\left(e^{-j\omega }\right)\right)-P\left(\omega \right)e^{-j\omega D}.Math.}^{2}d\omega ,& \left(31\right)\end{array}$

where P(ω) is a symmetric real-valued function defining the pass band and stop band ranges, and D is the total system delay. In other words, P(ω) describes the desired transfer function. In the most general case, such transfer function comprises a magnitude which is a function of the frequency ω. For a real-valued system Eq. (31) simplifies to

[00024]$\begin{array}{cc}{e}_t=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}{.Math.A_0\left(e^{j\omega }\right)-P\left(\omega \right)e^{-j\omega D}.Math.}^{2}d\omega & \left(32\right)\end{array}$

[0084] The target function P(ω) and the target delay D may be selected as an input parameter to the optimization procedure. The expression P(ω)e.sup.−jwD may be referred to as the target transfer function.

[0085] A measure of the energy of the total aliasing e.sub.a may be calculated by evaluating the sum of the alias terms on the right hand side (RHS) of Eq. (23), i.e. the second term of Eq. (23), on the unit circle as

[00025]$\begin{array}{cc}{e}_a=\frac{1}{2\pi }\underset{l=1}{\overset{M-1}{.Math.}}{\int }_{-\pi }^{\pi }{.Math.{\stackrel{\_}{A}}_l\left(e^{j\omega }\right).Math.}^{2}d\omega ,& \left(33\right)\end{array}$

[0086] For real-valued systems this translates to

[00026]$\begin{array}{cc}{e}_a=\frac{1}{2\pi }\underset{l=1}{\overset{M-1}{.Math.}}{\int }_{-\pi }^{\pi }{.Math.A_l\left(e^{j\omega }\right).Math.}^{2}d\omega .& \left(34\right)\end{array}$

[0087] Overall, an optimization procedure for determining a prototype filter p.sub.0 (n) may be based on the minimization of the error of Eq. (30). The parameter α may be used to distribute the emphasis between the transfer function and the sensitivity to aliasing of the prototype filter. While increasing the parameter α towards 1 will put more emphasis on the transfer function error e.sub.t, reducing the parameter α towards 0 will put more emphasis on the aliasing error e.sub.a. The parameters P(ω) and D may be used to set a target transfer function of the prototype filter p.sub.0(n), i.e. to define the pass band and stop band behavior and to define the overall system delay.

[0088] According to an example, a number of the filter bank channels k may be set to zero, e.g. the upper half of the filter bank channels are given zero gain. Consequently, the filter bank is triggered to generate a great amount of aliasing. This aliasing will be subsequently minimized by the optimization process. In other words, by setting a certain number of filter bank channels to zero, aliasing will be induced, in order to generate an aliasing error e.sub.a which may be minimized during the optimization procedure. Furthermore, computational complexity of the optimization process may be reduced by setting filter bank channels to zero.

[0089] According to an example, a prototype filter is optimized for a real valued, i.e. a cosine modulated, filter bank which may be more appropriate than directly optimizing the complex-valued version. This is because real-valued processing prioritizes far-off aliasing attenuation to a larger extent than complex-valued processing. However, when triggering aliasing as outlined above, the major part of the induced aliasing in this case will typically origin from the terms carrying the main alias terms. Hence, the optimization algorithm may spend resources on minimizing the main aliasing that is inherently non-present in the resulting complex-exponential modulated system. In order to alleviate this, the optimization may be done on a partially complex system; for the alias terms which are free from main aliasing, the optimization may be done using real-valued filter processing. On the other hand, the alias terms that would carry the main alias terms in a real-valued system would be modified for complex-valued filter processing. By means of such partially complex optimization, the benefits of performing the processing using real-valued processing may be obtained, while still optimizing the prototype filter for usage in a complex modulated filter bank system.

[0090] In an exemplary optimization where exactly the upper half of the filter bank channels are set to zero, the only alias term calculated from complex valued filters is the term l=M/2 of Eq. (33). In this example, the function P(ω) of Eq. (31), may be chosen as a unit magnitude constant ranging from −π/2+ε to π/2−ε, where ε is a fraction of π/2, in order to cover the frequency range constituting the pass band. Outside the pass band the function P(ω) may be defined to be zero or be left undefined. In the latter case, the error energy of the transfer function Eq. (31) is only evaluated between −π/2+ε and π/2−ε. Alternatively and preferably, the pass band error e.sub.t could be calculated over all channels k=0, . . . , M−1, from −π to π with P(ω) being constant, while the aliasing is still calculated with a plurality of the channels set to zero as described above.

[0091] Typically the optimization procedure is an iterative procedure, where given the prototype filter coefficients p.sub.0(n) (n=0, . . . , N−1) at a certain iteration step, the target delay D, the number of channels M, the numbers of low band channels set to zero loCut, the number of high band channels set to zero hiCut, and the weighting factor α, a value for the objective function for this iteration step is calculated. Using semi-complex operations, this comprises the steps: [0092] 1. To obtain the pass band error e.sub.t, evaluate Eq. (32) with P(ω) being a constant, using

[00027]$\begin{array}{cc}{A}_0\left(e^{j\omega }\right)=\underset{k=0}{\overset{M-1}{.Math.}}{H}_k\left(e^{j\omega }\right)F_k\left(e^{j\omega }\right).& \left(35\right)\end{array}$ [0093] where H.sub.k(e.sub.jω) and F.sub.k(e.sup.jω) are the DFT transforms of the analysis and synthesis filters h.sub.k(n) and f.sub.k(n) as generated from the prototype filters coefficients at this iteration step from Eq. (13) to (15), respectively. [0094] 2. To obtain the aliasing error e.sub.a, for aliasing terms not subject to significant aliasing, evaluate

[00028]$\begin{array}{cc}{e}_{\mathrm{aReal}}=\frac{1}{2\pi }\underset{\underset{\underset{\begin{array}{c}M-\mathrm{loCut},\\ M-\mathrm{hiCut}\end{array}}{l\ne \mathrm{loCut},\mathrm{hiCut},}}{l=1}}{\overset{M-1}{.Math.}}{\int }_{-\pi }^{\pi }{.Math.A_l\left(e^{j\omega }\right).Math.}^{2}d\omega ,& \left(36\right)\end{array}$ [0095] where A.sub.l(e.sup.jω) is calculated as

[00029]$\begin{array}{cc}{A}_l\left(e^{j\omega }\right)=\underset{k=\mathrm{loCut}}{\overset{M-1-\mathrm{hiCut}}{.Math.}}{H}_k\left(e^{j\left(\omega -\frac{2\pi }{M}l\right)}\right)F_k\left(e^{j\omega }\right)& \left(37\right)\end{array}$ [0096] and H.sub.k(e.sup.jω) and F.sub.k(e.sup.jω) are the DFT transforms, i.e. the z-transforms evaluated on the unit circle, of the analysis and synthesis filters h.sub.k(n) and f.sub.k(n) from Eq. (13) to (15). [0097] 3. For the terms subject to significant aliasing, evaluate

[00030]$\begin{array}{cc}{e}_{\mathrm{aCplx}}=\frac{1}{2\pi }\underset{\begin{array}{c}l=\mathrm{lo}\mathrm{Cut},\\ \begin{array}{c}\mathrm{hiCut},\\ \begin{array}{c}M-\mathrm{loCut},\\ M-\mathrm{hiCut}\end{array}\end{array}\end{array}}{.Math.}{\int }_{-\pi }^{\pi }{.Math.{\stackrel{\_}{A}}_l\left(e^{j\omega }\right).Math.}^{2}d\omega & \left(38\right)\end{array}$ [0098] where Ã.sub.l(e.sub.jω) is given by Eq. (24), with Men as Eq. (37), with H.sub.k(e.sup.jω) and F.sub.k(e.sup.jω) being the DFT transforms of h.sub.k(n) and f.sub.k(n) from Eq. (19) and (20). [0099] 4. The error is subsequently weighted with a as

e.sub.tot(α)=αe.sub.t+(1−α)(e.sub.aReal+e.sub.aCplx).  (39)

[0100] Using any of the nonlinear optimization algorithms referred to above, this total error is reduced by modifying the coefficients of the prototype filter, until an optimal set of coefficients is obtained. By way of example, the direction of the greatest gradient of the error function e.sub.tot is determined for the prototype filter coefficients at a given iteration step. Using a certain step size the prototype filter coefficients are modified in the direction of the greatest gradient. The modified prototype filter coefficients are used as a starting point for the subsequent iteration step. This procedure is repeated until the optimization procedure has converged to a minimum value of the error function e.sub.tot.

[0101] An exemplary embodiment of the optimization procedure is illustrated in FIG. 3 as a flow diagram 300. In a parameter determination step 301 the parameters of the optimization procedure, i.e. notably the target transfer function comprising the target delay D, the number of channels M of the target filter bank, the number N of coefficients of the prototype filter, the weighting parameter α of the objective error function, as well as the parameters for aliasing generation, i.e. loCut and/or hiCut, are defined. In an initialization step 302, a first set of coefficients of the prototype filter is selected.

[0102] In the pass band error determination unit 303, the pass band error term e.sub.t is determined using the given set of coefficients of the prototype filter. This may be done by using Eq. (32) in combination with Eqs. (35) and (13) to (15). In the real valued aliasing error determination unit 304, a first part e.sub.aReal of the aliasing error term e.sub.a may be determined using Eqs. (36) and (37) in combination with Eqs. (13) to (15). Furthermore, in the complex valued aliasing error determination unit 305, a second part e.sub.aCplx of the aliasing error term e.sub.a may be determined using Eq. (38) in combination with Eqs. (19) and (20). As a consequence, the objective function e.sub.tot may be determined from the results of the units 303, 304 and 305 using Eq. (39).

[0103] The nonlinear optimization unit 306 uses optimization methods, such as linear programming, in order to reduce the value of the objective function. By way of example, this may be done by determining a possibly maximum gradient of the objective function with regards to modifications of the coefficients of the prototype filter. In other words, those modifications of the coefficients of the prototype filter may be determined which result in a possibly maximum reduction of the objective function.

[0104] If the gradient determined in unit 306 remains within predetermined bounds, the decision unit 307 decides that a minimum of the objective function has been reached and terminates the optimization procedure in step 308. If on the other hand, the gradient exceeds the predetermined value, then the coefficients of the prototype filter are updated in the update unit 309. The update of the coefficients may be performed by modifying the coefficients with a predetermined step into the direction given by the gradient. Eventually, the updated coefficients of the prototype filter are reinserted as an input to the pass band error determination unit 303 for another iteration of the optimization procedure.

[0105] Overall, it can be stated that using the above error function and an appropriate optimization algorithm, prototype filters may be determined that are optimized with respect to their degree of perfect reconstruction, i.e. with respect to low aliasing in combination with low phase and/or amplitude distortion, their resilience to aliasing due to subband modifications, their system delay and/or their transfer function. The design method provides parameters, notably a weighting parameter a, a target delay D, a target transfer function P(ω), a filter length N, a number of filter bank channels M, as well as aliasing trigger parameters hiCut, loCut, which may be selected to obtain an optimal combination of the above mentioned filter properties. Furthermore, the setting to zero of a certain number of subband channels, as well as the partial complex processing may be used to reduce the overall complexity of the optimization procedure. As a result, asymmetric prototype filters with a near perfect reconstruction property, low sensitivity to aliasing and a low system delay may be determined for usage in a complex exponential modulated filter bank. It should be noted that the above determination scheme of a prototype filter has been outlined in the context of a complex exponential modulated filter bank. If other filter bank design methods are used, e.g. cosine modulated or sine modulated filter bank design methods, then the optimization procedure may be adapted by generating the analysis and synthesis filters h.sub.k(n) and f.sub.k(n) using the design equations of the respective filter bank design method. By way of example, Eqs. (13) to (15) may be used in the context of a cosine modulated filter bank.

[0106] In the following, a detailed example of a 64 channel low delay filter bank is described. Using the proposed aforementioned optimization method, a detailed example of an alias gain term optimized, low delay, 64-channel filter bank (M=64) will be outlined. In this example the partially complex optimization method has been used and the uppermost 40 channels have been set to zero during the prototype filter optimization, i.e. hiCut=40, whereas the loCut parameter remained unused. Hence, all alias gain terms, except Ã.sub.l, where l=24, 40, are calculated using real-valued filters. The total system delay is chosen as D=319, and the prototype filter length is N=640. A time domain plot of the resulting prototype filter is given in FIG. 4(a), and the frequency response of the prototype filter is depicted in FIG. 4(b). The filter bank offers a pass band (amplitude and phase) reconstruction error of −72 dB. The phase deviation from a linear phase is smaller than ±0.02°, and the aliasing suppression is 76 dB when no modifications are done to the subband samples. The actual filter coefficients are tabulated in Table 1. Note that the coefficients are scaled by a factor M=64 in respect to other equations in this document that are dependent on an absolute scaling of the prototype filter.

[0107] While the above description of the design of the filter bank is based on a standard filter bank notation, an example for operating the designed filter bank may operate in other filter bank descriptions or notations, e.g. filter bank implementations which allow a more efficient operation on a digital signal processor.

[0108] In an example, the steps for filtering a time domain signal using the optimized prototype filter may be described as follows: [0109] In order to operate the filter bank in an efficient manner, the prototype filter, i.e. p.sub.0(n) from Table 1, is first arranged in the poly-phase representation, where every other of the poly-phase filter coefficients are negated and all coefficient are time-flipped as

p′.sub.0(639−128m−n)=(−1).sup.mp.sub.0(128m+n), 0≤n<128, 0≤m<5  (40) [0110] The analysis stage begins with the poly-phase representation of the filter being applied to the time domain signal x(n) to produce a vector x.sub.l(n) of length 128 as

[00031]$\begin{array}{cc}{x}_{127-l}\left(n\right)=\underset{m-0}{\overset{4}{.Math.}}{p}_0^{\prime }\left(128m+l\right)\times \left(128m+l+64n\right),0\le l<128,n=0,1,\mathrm{.Math.}& \left(41\right)\end{array}$ [0111] x.sub.l(n) is subsequently multiplied with a modulation matrix as

[00032]$\begin{array}{cc}{v}_k\left(n\right)=\underset{l=0}{\overset{127}{.Math.}}{x}_l\left(n\right)\exp \left(j\frac{\pi }{128}\left(k+\frac{1}{2}\right)\left(2l+129\right)\right),0\le k<64,& \left(42\right)\end{array}$ [0112] where v.sub.k(n), k=0 . . . 63, constitute the subband signals. The time index n is consequently given in subband samples. [0113] The complex-valued subband signals can then be modified, e.g. according to some desired, possibly time-varying and complex-valued, equalization curve g.sub.k(n), as

v.sub.k.sup.(m)(n)=g.sub.k(n)v.sub.k(n), 0≤k<64.  (43) [0114] The synthesis stage starts with a demodulation step of the modified subband signals as

[00033]$\begin{array}{cc}{u}_l\left(n\right)=\frac{1}{64}\underset{k=0}{\overset{63}{.Math.}}\mathrm{Re}\left\{v_k^{\left(m\right)}\left(n\right)\exp \left(j\frac{\pi }{128}\left(k+\frac{1}{2}\right)\left(2l-255\right)\right)\right\},0\le l<128.& \left(44\right)\end{array}$ [0115] It should be noted that the modulation steps of Eqs. (42) and (44) may be accomplished in a computationally very efficient manner with fast algorithms using fast Fourier transform (FFT) kernels. [0116] The demodulated samples are filtered with the poly-phase representation of the prototype filter and accumulated to the output time domain signal {circumflex over (x)}(n) according to

{circumflex over (x)}(128m+l+64n)={circumflex over (x)}(128m+l+64n)+p′.sub.0(639−128m−l)u.sub.l(l), 0≤l<128, 0≤m<5, n=0,1, . . .  (45) [0117] where {circumflex over (x)}(n) is set to 0 for all n at start-up time.

[0118] It should be noted that both floating point and fixed point implementations might change the numerical accuracy of the coefficients given in Table 1 to something more suitable for processing. Without limiting the scope, the values may be quantized to a lower numerical accuracy by rounding, truncating and/or by scaling the coefficients to integer or other representations, in particular representations that are adapted to the available resources of a hardware and/or software platform on which the filter bank is to operate.

[0119] Moreover, the example above outlines the operation where the time domain output signal is of the same sampling frequency as the input signal. Other implementations may resample the time domain signal by using different sizes, i.e. different number of channels, of the analysis and synthesis filter banks, respectively. However, the filter banks should be based on the same prototype filter, and are obtained by resampling of the original prototype filter through either decimation or interpolation. As an example, a prototype filter for a 32 channel filter bank is achieved by resampling the coefficients p.sub.0(n) as)

p.sub.0.sup.(32)(i)=½[p.sub.0(2i+1)+p.sub.0(2i)], 0≤i<320.

[0120] The length of the new prototype filter is hence 320 and the delay is D=└319/2┘=159, where the operator └⋅┘ returns the integer part of its argument.

TABLE-US-00001 TABLE 1 Coefficients of a 64 channel low delay prototype filter n P.sub.0(n) 0 −7.949261005955764e−4 1 −1.232074328145439e−3 2 −1.601053942982895e−3 3 −1.980720409470913e−3 4 −2.397504953865715e−3 5 −2.838709203607079e−3 6 −3.314755401090670e−3 7 −3.825180949035082e−3 8 −4.365307413613105e−3 9 −4.937260935539922e−3 10 −5.537381514710146e−3 11 −6.164241937824271e−3 12 −6.816579194002503e−3 13 −7.490102145765528e−3 14 −8.183711450708110e−3 15 −8.894930051379498e−3 16 −9.620004581607449e−3 17 −1.035696814015217e−2 18 −1.110238617202191e−2 19 −1.185358556146692e−2 20 −1.260769256679562e−2 21 −1.336080675156018e−2 22 −1.411033176541011e−2 23 −1.485316243134798e−2 24 −1.558550942227883e−2 25 −1.630436835497356e−2 26 −1.700613959422392e−2 27 −1.768770555992799e−2 28 −1.834568069395711e−2 29 −1.897612496482356e−2 30 −1.957605813345359e−2 31 −2.014213322475170e−2 32 −2.067061748933033e−2 33 −2.115814831921453e−2 34 −2.160130854695980e−2 35 −2.199696217022438e−2 36 −2.234169110698344e−2 37 −2.263170795250229e−2 38 −2.286416556008894e−2 39 −2.303589449043864e−2 40 −2.314344724218223e−2 41 −2.318352524475873e−2 42 −2.315297727620401e−2 43 −2.304918234544422e−2 44 −2.286864521420490e−2 45 −2.260790764376614e−2 46 −2.226444264459477e−2 47 −2.183518667784246e−2 48 −2.131692017682024e−2 49 −2.070614962636994e−2 50 −1.999981321635736e−2 51 −1.919566223498554e−2 52 −1.828936158524688e−2 53 −1.727711874492186e−2 54 −1.615648494779686e−2 55 −1.492335807272955e−2 56 −1.357419760297910e−2 57 −1.210370330110896e−2 58 −1.050755164953818e−2 59 −8.785746151726750e−3 60 −6.927329556345040e−3 61 −4.929378450735877e−3 62 −2.800333941149626e−3 63 −4.685580749545335e−4 64   2.210315255690887e−3 65   5.183294908090526e−3 66   8.350964449424035e−3 67   1.166118535611788e−2 68   1.513166797475777e−2 69   1.877264877027943e−2 70   2.258899222368603e−2 71   2.659061474958830e−2 72   3.078087745385930e−2 73   3.516391224752870e−2 74   3.974674893613862e−2 75   4.453308211110493e−2 76   4.952626097917320e−2 77   5.473026727738295e−2 78   6.014835645056577e−2 79   6.578414516120631e−2 80   7.163950999489413e−2 81   7.771656494569829e−2 82   8.401794441130064e−2 83   9.054515924487507e−2 84   9.729889691289549e−2 85   1.042804039148369e−1 86   1.114900795290448e−1 87   1.189284254931251e−1 88   1.265947532678997e−1 89   1.344885599112251e−1 90   1.426090972422485e−1 91   1.509550307914161e−1 92   1.595243494708706e−1 93   1.683151598707939e−1 94   1.773250461581686e−1 95   1.865511418631904e−1 96   1.959902227114119e−1 97   2.056386275763479e−1 98   2.154925974105375e−1 99   2.255475564993390e−1 100   2.357989864681126e−1 101   2.462418809459464e−1 102   2.568709554604541e−1 103   2.676805358910440e−1 104   2.786645734207760e−1 105   2.898168394038287e−1 106   3.011307516871287e−1 107   3.125994749246541e−1 108   3.242157192666507e−1 109   3.359722796803192e−1 110   3.478614117031655e−1 111   3.598752336287570e−1 112   3.720056632072922e−1 113   3.842444358173011e−1 114   3.965831241942321e−1 115   4.090129566893579e−1 116   4.215250930838456e−1 117   4.341108982328533e−1 118   4.467608231633283e−1 119   4.594659376709624e−1 120   4.722166595058233e−1 121   4.850038204075748e−1 122   4.978178235802594e−1 123   5.106483456192374e−1 124   5.234865375971977e−1 125   5.363218470709771e−1 126   5.491440356706657e−1 127   5.619439923555571e−1 128   5.746001351404267e−1 129   5.872559277139351e−1 130   5.998618924353250e−1 131   6.123980151490041e−1 132   6.248504862282382e−1 133   6.372102969387355e−1 134   6.494654463921502e−1 135   6.616044277534099e−1 136   6.736174463977084e−1 137   6.854929931488056e−1 138   6.972201618598393e−1 139   7.087881675504216e−1 140   7.201859881692665e−1 141   7.314035334082558e−1 142   7.424295078874311e−1 143   7.532534422335129e−1 144   7.638649113306198e−1 145   7.742538112450130e−1 146   7.844095212375462e−1 147   7.943222347831999e−1 148   8.039818519286321e−1 149   8.133789939828571e−1 150   8.225037151897938e−1 151   8.313468549324594e−1 152   8.398991600556686e−1 153   8.481519810689574e−1 154   8.560963550316389e−1 155   8.637239863984174e−1 156   8.710266607496513e−1 157   8.779965198108476e−1 158   8.846258145496611e−1 159   8.909071890560218e−1 160   8.968337036455653e−1 161   9.023985431182168e−1 162   9.075955881221292e−1 163   9.124187296760565e−1 164   9.168621399784253e−1 165   9.209204531389191e−1 166   9.245886139655739e−1 167   9.278619263447355e−1 168   9.307362242659798e−1 169   9.332075222986479e−1 170   9.352724511271509e−1 171   9.369278287932853e−1 172   9.381709878904797e−1 173   9.389996917291260e−1 174   9.394121230559878e−1 175   9.394068064126931e−1 176   9.389829174860432e−1 177   9.381397976778112e−1 178   9.368773370086998e−1 179   9.351961242404785e−1 180   9.330966718935136e−1 181   9.305803205049067e−1 182   9.276488080866625e−1 183   9.243040558859498e−1 184   9.205488097488350e−1 185   9.163856478189402e−1 186   9.118180055332041e−1 187   9.068503557855540e−1 188   9.014858673099563e−1 189   8.957295448806664e−1 190   8.895882558527375e−1 191   8.830582442418677e−1 192   8.761259906419252e−1 193   8.688044201931157e−1 194   8.611140376567749e−1 195   8.530684188588082e−1 196   8.446723286380624e−1 197   8.359322523144003e−1 198   8.268555005748937e−1 199   8.174491260941859e−1 200   8.077214932837783e−1 201   7.976809997929416e−1 202   7.873360271773119e−1 203   7.766956604639097e−1 204   7.657692341138960e−1 205   7.545663748526984e−1 206   7.430967641354331e−1 207   7.313705248813991e−1 208   7.193979757178656e−1 209   7.071895814695481e−1 210   6.947561322714310e−1 211   6.821083135331770e−1 212   6.692573319585476e−1 213   6.562143182387809e−1 214   6.429904538706975e−1 215   6.295973685335782e−1 216   6.160464554756299e−1 217   6.023493418727370e−1 218   5.885176369189331e−1 219   5.745630487304467e−1 220   5.604973280717471e−1 221   5.463322649085826e−1 222   5.320795532569365e−1 223   5.177509557831821e−1 224   5.033582842235876e−1 225   4.889131973708936e−1 226   4.744274511088447e−1 227   4.599125196114154e−1 228   4.453800290341801e−1 229   4.308413090599260e−1 230   4.163077444128621e−1 231   4.017905891818764e−1 232   3.873008819361793e−1 233   3.728496914938361e−1 234   3.584479879275654e−1 235   3.441060828393923e−1 236   3.298346836739700e−1 237   3.156442070098094e−1 238   3.015447421741344e−1 239   2.875462383794429e−1 240   2.736584401802921e−1 241   2.598909819775319e−1 242   2.462531686198759e−1 243   2.327540108460799e−1 244   2.194025590645563e−1 245   2.062071988727463e−1 246   1.931765200055820e−1 247   1.803186073942884e−1 248   1.676410590306998e−1 249   1.551517472268748e−1 250   1.428578337203540e−1 251   1.307662172525294e−1 252   1.188837988250476e−1 253   1.072167300568495e−1 254   9.577112136322552e−2 255   8.455282024161610e−2 256   7.355793885744523e−2 257   6.280513608528435e−2 258   5.229589453075828e−2 259   4.203381031272017e−2 260   3.202301123728688e−2 261   2.226720136600903e−2 262   1.277000586069404e−2 263   3.534672952747162e−3 264 −5.435672410526313e−3 265 −1.413857081863553e−2 266 −2.257147752062613e−2 267 −3.073254829666290e−2 268 −3.861994968092324e−2 269 −4.623245158508806e−2 270 −5.356875686113461e−2 271 −6.062844791918062e−2 272 −6.741087925238425e−2 273 −7.391592258255635e−2 274 −8.014393008412193e−2 275 −8.609517876186421e−2 276 −9.177059647159572e−2 277 −9.717118785672957e−2 278 −1.022983899423088e−1 279 −1.071535873159799e−1 280 −1.117390940373963e−1 281 −1.160565563647874e−1 282 −1.201089957775325e−1 283 −1.238986104503973e−1 284 −1.274286534385776e−1 285 −1.307022037585206e−1 286 −1.337226598624689e−1 287 −1.364936502000925e−1 288 −1.390190836588895e−1 289 −1.413030335001078e−1 290 −1.433497698594264e−1 291 −1.451636222445455e−1 292 −1.467494079461177e−1 293 −1.481116975400198e−1 294 −1.492556249421260e−1 295 −1.501862836334994e−1 296 −1.509089024309573e−1 297 −1.514289033634045e−1 298 −1.517517580141857e−1 299 −1.518832057448775e−1 300 −1.518289202172233e−1 301 −1.515947694390820e−1 302 −1.511866738705995e−1 303 −1.506105955209982e−1 304 −1.498725980913964e−1 305 −1.489787144055076e−1 306 −1.479352185844335e−1 307 −1.467481851768966e−1 308 −1.454239120021382e−1 309 −1.439685961257477e−1 310 −1.423884130127772e−1 311 −1.406896926563808e−1 312 −1.388785953623746e−1 313 −1.369612022106282e−1 314 −1.349437727408798e−1 315 −1.328323917411932e−1 316 −1.306331212230066e−1 317 −1.283520431992394e−1 318 −1.259952253813674e−1 319 −1.235680807908494e−1 320 −1.210755701624524e−1 321 −1.185237142283346e−1 322 −1.159184450952715e−1 323 −1.132654367461266e−1 324 −1.105698782276963e−1 325 −1.078369135648348e−1 326 −1.050716118804287e−1 327 −1.022789198651472e−1 328 −9.946367410320074e−2 329 −9.663069107327295e−2 330 −9.378454802679648e−2 331 −9.092970207094843e−2 332 −8.807051083640835e−2 333 −8.521107266503664e−2 334 −8.235562752947133e−2 335 −7.950789957683559e−2 336 −7.667177989755110e−2 337 −7.385092587441364e−2 338 −7.104866702770536e−2 339 −6.826847016140082e−2 340 −6.551341011471171e−2 341 −6.278658929544248e−2 342 −6.009091369370080e−2 343 −5.742919825387360e−2 344 −5.480383115198150e−2 345 −5.221738078737957e−2 346 −4.967213638808988e−2 347 −4.717023345307148e−2 348 −4.471364025371278e−2 349 −4.230438144160113e−2 350 −3.994384828552555e−2 351 −3.763371362431132e−2 352 −3.537544041600725e−2 353 −3.317035188016126e−2 354 −3.101971215825843e−2 355 −2.892453070357571e−2 356 −2.688575425197388e−2 357 −2.490421725219031e−2 358 −2.298058501129975e−2 359 −2.111545692324888e−2 360 −1.930927680100128e−2 361 −1.756239270089077e−2 362 −1.587511449869362e−2 363 −1.424750749465213e−2 364 −1.267955527855867e−2 365 −1.117125833414906e−2 366 −9.722405440999532e−3 367 −8.332704660914712e−3 368 −7.001789872901951e−3 369 −5.729226040772489e−3 370 −4.514503359783591e−3 371 −3.356946762357950e−3 372 −2.255849987026407e−3 373 −1.210459261524451e−3 374 −2.199474640570699e−4 375   7.167268627887994e−4 376   1.600440185590357e−3 377   2.432366605744087e−3 378   3.213605482343768e−3 379   3.945301462616821e−3 380   4.628665378925932e−3 381   5.264976586624488e−3 382   5.855653555178131e−3 383   6.401634331453516e−3 384   6.903046246257517e−3 385   7.364537203059431e−3 386   7.785917436812734e−3 387   8.168780818165564e−3 388   8.514510536234886e−3 389   8.824526581578384e−3 390   9.100444687042341e−3 391   9.343819821939981e−3 392   9.556089247587111e−3 393   9.738929904236388e−3 394   9.893728065983530e−3 395   1.002221842309897e−2 396   1.012567516563336e−2 397   1.020575952382967e−2 398   1.026389875785943e−2 399   1.030162959448537e−2 400   1.032037849566083e−2 401   1.032154667898522e−2 402   1.030658039367325e−2 403   1.027682791880806e−2 404   1.023360327572998e−2 405   1.017821017226088e−2 406   1.011195224927225e−2 407   1.003602653649432e−2 408   9.951564927254814e−3 409   9.859735321541087e−3 410   9.761689935477358e−3 411   9.658335268268776e−3 412   9.550506541750015e−3 413   9.439239790180602e−3 414   9.325311662898867e−3 415   9.209571052890813e−3 416   9.092729858436259e−3 417   8.975504153186832e−3 418   8.858564024669505e−3 419   8.742547510216072e−3 420   8.627917215653412e−3 421   8.515236113018675e−3 422   8.404834686887089e−3 423   8.297046056582970e−3 424   8.192181771808344e−3 425   8.090558375952284e−3 426   7.992340268718087e−3 427   7.897787592331651e−3 428   7.806979111626161e−3 429   7.720005213599928e−3 430   7.636899169053526e−3 431   7.557692588413262e−3 432   7.482361735247336e−3 433   7.410882580163479e−3 434   7.343084196594709e−3 435   7.278918614409016e−3 436   7.218206312830178e−3 437   7.160843298305507e−3 438   7.106600272887440e−3 439   7.055249359796239e−3 440   7.006591539682229e−3 441   6.960450953203489e−3 442   6.916554770130135e−3 443   6.874623603448978e−3 444   6.834443173086539e−3 445   6.795786363014294e−3 446   6.758476537306303e−3 447   6.722125942626111e−3 448   6.686140904391229e−3 449   6.650228698006217e−3 450   6.614354298921371e−3 451   6.578320578669048e−3 452   6.541865503698597e−3 453   6.504729306516950e−3 454   6.466690242148724e−3 455   6.427556828582072e−3 456   6.387124476277924e−3 457   6.345262303711465e−3 458   6.301766582696827e−3 459   6.256542736138121e−3 460   6.209372064970386e−3 461   6.160215935384255e−3 462   6.108902434484468e−3 463   6.055355267266873e−3 464   5.999473903317320e−3 465   5.941211676077848e−3 466   5.880495927392625e−3 467   5.817286139372493e−3 468   5.751536864441650e−3 469   5.683230954033062e−3 470   5.612375999953358e−3 471   5.538957988293047e−3 472   5.462963107291498e−3 473   5.384396217909888e−3 474   5.303337109336215e−3 475   5.219739772898678e−3 476   5.133623037830525e−3 477   5.045046346880483e−3 478   4.954008597884707e−3 479   4.860588885693231e−3 480   4.764720830452409e−3 481   4.666469548192818e−3 482   4.565946029127366e−3 483   4.463150894014690e−3 484   4.358150755039186e−3 485   4.250967471708103e−3 486   4.141634861746089e−3 487   4.030165355928349e−3 488   3.916597675997815e−3 489   3.800994685405442e−3 490   3.683451012833619e−3 491   3.563914929838276e−3 492   3.442490007998456e−3 493   3.319256438897666e−3 494   3.194250476422174e−3 495   3.067525877056119e−3 496   2.939139106182801e−3 497   2.809151898728351e−3 498   2.677703006241942e−3 499   2.544830774162231e−3 500   2.410617950987095e−3 501   2.275190768887402e−3 502   2.138586519570023e−3 503   2.000881763033976e−3 504   1.862161137529843e−3 505   1.722850651410707e−3 506   1.583005323492318e−3 507   1.442635273572746e−3 508   1.301735673138880e−3 509   1.160531184883257e−3 510   1.018710154718430e−3 511   8.753658738743612e−4 512   7.250868879948704e−4 513   5.901514303345345e−4 514   4.571251178344833e−4 515   3.254504484897777e−4 516   1.951832637892118e−4 517   6.661818101906931e−5 518 −6.002729636107936e−5 519 −1.845163192347697e−4 520 −3.065712811761140e−4 521 −4.259661821125124e−4 522 −5.424773586381941e−4 523 −6.558084462274315e−4 524 −7.659101269870789e−4 525 −8.724859431432570e−4 526 −9.753531169034512e−4 527 −1.074300123306481e−3 528 −1.169143931350576e−3 529 −1.259725653234229e−3 530 −1.345834916989234e−3 531 −1.427339710937440e−3 532 −1.504079803740054e−3 533 −1.575880973843057e−3 534 −1.642633580824677e−3 535 −1.704200291375062e−3 536 −1.760514312756149e−3 537 −1.811458673156579e−3 538 −1.856981580032126e−3 539 −1.897029046447624e−3 540 −1.931585942699363e−3 541 −1.960627084932276e−3 542 −1.984178530495641e−3 543 −2.002288840866127e−3 544 −2.014916352347506e−3 545 −2.022189226793424e−3 546 −2.024254777335021e−3 547 −2.021156706871573e−3 548 −2.013111787438794e−3 549 −2.000212633130633e−3 550 −1.982687042477966e−3 551 −1.960693892404943e−3 552 −1.934407806173517e−3 553 −1.904123563599214e−3 554 −1.870072199436830e−3 555 −1.832519954023970e−3 556 −1.791756667369466e−3 557 −1.747978720577777e−3 558 −1.701541033746949e−3 559 −1.652689459435072e−3 560 −1.601690868666912e−3 561 −1.548954090992685e−3 562 −1.494709797777335e−3 563 −1.439190571857024e−3 564 −1.382763830841281e−3 565 −1.325642967049430e−3 566 −1.268184236874211e−3 567 −1.210596701555163e−3 568 −1.153025111297160e−3 569 −1.095962010293130e−3 570 −1.039553843860894e−3 571 −9.838346246983619e−4 572 −9.290281181623759e−4 573 −8.749810533387956e−4 574 −8.215803921619577e−4 575 −7.706114369075383e−4 576 −7.240453976226097e−4 577 −6.849432723864428e−4 578 −6.499492788836954e−4 579 −6.169265465797999e−4 580 −5.864023580206857e−4 581 −5.585564628691223e−4 582 −5.332623456777386e−4 583 −5.106711356117643e−4 584 −4.907668696713635e−4 585 −4.734587422398502e−4 586 −4.585871522474066e−4 587 −4.460035977692689e−4 588 −4.356377129231574e−4 589 −4.273247732616044e−4 590 −4.208333621911742e−4 591 −4.159437129295563e−4 592 −4.123958508631197e−4 593 −4.100224176114866e−4 594 −4.085466400930828e−4 595 −4.077080867389932e−4 596 −4.073254606881664e−4 597 −4.070933269997811e−4 598 −4.067607615013048e−4 599 −4.061488056951641e−4 600 −4.050555465493161e−4 601 −4.033838274959328e−4 602 −4.008810861049167e−4 603 −3.973769462134710e−4 604 −3.928186163645286e−4 605 −3.870561868619109e−4 606 −3.799993669990150e−4 607 −3.715971708042990e−4 608 −3.617549303005874e−4 609 −3.505340232816606e−4 610 −3.378810708512397e−4 611 −3.237820254163679e−4 612 −3.083797394566325e−4 613 −2.916580376245428e−4 614 −2.737128656378774e−4 615 −2.546266898474145e−4 616 −2.344785058384558e−4 617 −2.134575242388197e−4 618 −1.916264055195752e−4 619 −1.692851860592005e−4 620 −1.466953561242506e−4 621 −1.236855725370398e−4 622 −1.005737421222391e−4 623 −7.750656629326379e−5 624 −5.466984383016220e−5 625 −3.255925659037227e−5 626 −1.096860208856302e−5 627   9.881411051921578e−6 628   2.951496818998434e−5 629   4.810106298036608e−5 630   6.513783951460106e−5 631   8.051456871678129e−5 632   9.429776656872437e−5 633   1.058298511976110e−4 634   1.155823148740170e−4 635   1.229659417867084e−4 636   1.266886375085138e−4 637   1.279376783418106e−4 638   1.216914974923773e−4 639   9.386301157644215e−5

[0121] In the following, different aspects of practical implementations are outlined. Using a standard PC or DSP, real-time operation of a low delay complex-exponential modulated filter bank is possible. The filter bank may also be hard-coded on a custom chip. FIG. 5(a) shows the structure for an effective implementation of the analysis part of a complex-exponential modulated filter bank system. The analogue input signal is first fed to an A/D converter 501. The digital time domain signal is fed to a shift register holding 2M samples shifting M samples at a time 502. The signals from the shift register are then filtered through the poly-phase coefficients of the prototype filter 503. The filtered signals are subsequently combined 504 and in parallel transformed with a DCT-IV 505 and a DST-IV 506 transform. The outputs from the cosine and sine transforms constitute the real and the imaginary parts of the subband samples respectively. The gains of the subband samples are modified according to the current spectral envelope adjuster setting 507.

[0122] An effective implementation of the synthesis part of a low delay complex-exponential modulated system is shown in FIG. 5(b). The subband samples are first multiplied with complex-valued twiddle-factors, i.e. complex-valued channel dependent constants, 511, and the real part is modulated with a DCT-IV 512 and the imaginary part with a DST-IV 513 transform. The outputs from the transforms are combined 514 and fed through the poly-phase components of the prototype filter 515. The time domain output signal is obtained from the shift register 516. Finally, the digital output signal is converted back to an analogue waveform 517.

[0123] While the above outlined implementations use DCT and DST type IV transforms, implementations using DCT type II and III kernels are equally possible (and also DST type II and III based implementations). However, the most computationally efficient implementations for complex-exponential modulated banks use pure FFT kernels. Implementations using a direct matrix-vector multiplication are also possible but are inferior in efficiency.

[0124] In summary, the present document describes a design method for prototype filters used in analysis/synthesis filter banks. Desired properties of the prototype filters and the resulting analysis/synthesis filter banks are near perfect reconstruction, low delay, low sensitivity to aliasing and minimal amplitude/phase distortion. An error function is proposed which may be used in an optimization algorithm to determine appropriate coefficients of the prototype filters. The error function comprises a set of parameters that may be tuned to modify the emphasis between the desired filter properties. Preferably, asymmetric prototype filters are used. Furthermore, a prototype filter is described which provides a good compromise of desired filter properties, i.e. near perfect reconstruction, low delay, high resilience to aliasing and minimal phase/amplitude distortion.

[0125] While specific embodiments and applications have been described herein, it will be apparent to those of ordinary skill in the art that many variations on the embodiments and applications described herein are possible without departing from the scope of the invention described and claimed herein.

[0126] It should be understood that while certain forms of the invention have been shown and described, the invention is not to be limited to the specific embodiments described and shown or the specific methods described.

[0127] The filter design method and system as well as the filter bank described in the present document may be implemented as software, firmware and/or hardware. Certain components may e.g. be implemented as software running on a digital signal processor or microprocessor. Other component may e.g. be implemented as hardware and or as application specific integrated circuits. The signals encountered in the described methods and systems may be stored on media such as random access memory or optical storage media. They may be transferred via networks, such as radio networks, satellite networks, wireless networks or wireline networks, e.g. the Internet. Typical devices making use of the filter banks described in the present document are set-top boxes or other customer premises equipment which decode audio signals. On the encoding side, the filter banks may be used in broadcasting stations, e.g. in video headend systems.