**NON-UNIFORM FILTER BANK IMPLEMENTATION**

George, Sapna (Block 315, #06-220 2 Serangon Avenue Singapore 6, 56050, SG)

Absar, Mohammed Javed (Block 411, #09-1006 10 Hougang Avenue Singapore 1, 53041, SG)

George, Sapna (Block 315, #06-220 2 Serangon Avenue Singapore 6, 56050, SG)

*; (IPC1-7): H03H17/02*

**H03H17/02**IEEE CAS (TC-MSA) ET SP (TC-MSP) SOCIETIES: "THE 2001 IEEE WORKSHOP ON SIGNAL PROCESSING SYSTEMS (SIPS), ANTWERP, BELGIUM, SEPTEMBER 26-28, 2001, ADVANCE PROGRAMME" [Online] 26 September 2001 (2001-09-26) XP002199833 Retrieved from the Internet:

DURAND R. BEGAULT: "3-D Sound for Virtual Reality and Multimedia" [Online] April 2000 (2000-04) , NASA CENTER FOR AEROSPACE INFORMATION , HANOVER, MD 21076-1320 XP002199910 Retrieved from the Internet:

1. | A method of searching for a bestmatch decimation vector of decimation factors for a nonuniform filter bank, the bestmatch decimation vector allowing perfect or near perfect reconstruction of an input signal of the nonuniform filter bank, the method including the steps of : a) selecting a partial decimation vector having a nutuber, 1 2 1, of decimation factors, where 1 does not exceed a maximum number, K, of decimation factors of said best match decimation vector; b) processing said I decimation factors to determine whether said partial decimation vector satisfies a feasibility criterion; c) processing a least common multiplier value of said I decimation factors to determine whether said least common multiplier value is greater than a predetermined value; d) processing a maximum decimation value, Dmax of said partial decimation vector to determine whether Dmax is less than one; e) processing a minimum decimation value, Din : of said partial decimation vector to determine whether Dmin is greater than one; and f) incrementing by one the number of decimation factors in the partial decimation vector and repeating steps b) to e) when said feasibility criterion is satisfied and said least common multiplier value is less than said predetermined value and Dmax is not less than one and Dmin is not greater than one. |

2. | The method of claim 1, further including the step of : g) selecting a new partial decimation vector and repeating steps b) to f) when the feasibility criterion is not satisfied or the least common multiplier is greater than the predetermined value or Dmax is less than one or Dmin is greater than one. |

3. | The method of claim 2, further including the step of : h) repeating steps a) to g) until I = K and a bestmatch decimation vector is located. |

4. | The method of claim 1, wherein said feasibility criterion is satisfied if (, nIL p U=o y/J/, where Lpartial is said least common multiple value and f0, fi,..., fi are said decimation factors of said partial decimation vector. |

5. | The method of claim 1, wherein an ideal set of decimation factors is defined by a vector Vspec= (n0, n1, ... nK1) # NK and a vector D is defined as D = (d0, d1, ..., dK1) # NK, such that a region S around Vspec is defined as S = {n0d0,n0d0+1,...,n0,...n0+d0}#{n1d1,...n1+d1}#####{nK1dK1,...nK1+dK1} and wherein Dmax is determined by. |

6. | The method of claim 5, wherein Dmin is determined by. |

7. | A bestmatch decimation vector generated by a method according to any one of claims 1 to 6. |

8. | A bestmatch decimation vector generated by a method according to any one of claims 1 to 6, wherein Vbestmatch has decimation factors as shown in the following table and where the critical band number is the number of a critical frequency band of the human ear. CriticalBand No. Vspec Vbestrnatch (LCM=2520) 1 220 210 2 220 210 3 220 210 4 220 210 5 200 210 6 183 168 7 158 168 8 148 140 9 138 140 10 116 120 11 105 120 12 92 90 13 79 90 14 70 90 15 59 60 16 49 60 17 40 30 18 32 42 19 25 21 20 20 21 21 17 21 22 12 12 23 9 12 24 6 6 25 3 3. |

9. | A codec adapted to execute the method of any one of claims 1 to 6. |

10. | An audio codec adapted to execute the method of any one of claims 1 to 6. |

BACKGROUND OF THE INVENTION Decomposition of a signal into frequency bands, for analysis or processing, has found useful application in a large number of areas such as audio, speech and video processing.

Consider audio compression where the audio signal x (n) is decomposed into frequency components by a bank of K-filters (also sometimes termed as frequency transformation or sub-band coding) to yield K signal-components Xkrn2, k ranging over O... K-1. The data rate at each filter output is equal to the input rate of x (n). This implies an K-fold increase in the overall data-rate. Decimation is therefore performed to bring down the data-rate.

Decimation by a factor of ni implies that only one out of ni samples out of the filter is considered. If the input rate of the system is R and the decimation factor at the il filter is ni,, then the output rate from the system is Ideally, this should be equal to one (less than one implies a loss of information). When the system is said to be maximally decimated. When the decimation factor is ni, the bandwidth of the corresponding filter Hi (z) must be aproximately 7C/n ;, to prevent aliasing corruption of the signal. If the decimation factor is the same for all filters, the design is simpler, and it is called a uniform-filter bank. In wavelets, the decimation factors, though not the same, vary only as powers of 2. A truly non-uniform filter bank is one in which each ni can assume any arbitrary value.

Following the decomposition and decimation, the signal is analyzed to find means of reducing the data-rate of the system.

Figure 1 shows encoder and decoder blocks for an MPEG-I Audio Codec. An input pulse- code modulated audio signal (PCM Audio) in the encoder passes through an analysis filter bank for decimating the signal. The PCM Audio input signal is also passed to a psycho- acoustic model, which computes the masking curve. The masking information is used by the bit-allocation module to determine the number of bits to be used by the quantizer for quantizing each frequency component. The larger the number of bits used for quantization, the greater the accuracy of the representation. The trade-off is that the compression rate decreases with an increase in the number of quantization bits. Based on masking effects, the bit-allocation module computes the number of bits to be allocated to each frequency component such that quantization noise is rendered (masked) inaudible. Ancillary data are added to the quantized input signal and the composite bit stream is formatted before transmission. The ancillary data are used to transmit information about the audio signal such as sampling rate, the number of audio channels, encoder settings and all such meta- data which are necessary for correct decoding and reproduction of the audio signal at the decoder. At the receiver side, the decoder performs an inverse quantization followed by upsampling (with the same decimation factors used by the encoder) and filtering to obtain a closely matching version of the input signal xnJ.

In the absence of quantization, it is desirable that the system reproduces the original signal x (n) with almost no distortion. A filter bank which achieves such a low level of distortion is called a Perfect Reconstruction Filter Bank. Compared to the non-uniform case, the theory of perfect reconstruction of a uniform filter bank is well understood and documented, and this explains why most existing codecs (MPEG-I Layer III, MPEG-2 AAC, AC-3) use uniform filter banks. The theory concerning non-uniform perfect reconstruction (PR) filter banks is still in its infancy.

It is noted that analysis by the psychoacoustics model is best when the signal decomposition is performed through a non-uniform filter bank that matches closely the

critical frequency bands of the ear. Absence of a good understanding of non-uniform filter banks has in the past forced designers of codecs to settle for the less desirable uniform filter bank. However, recent developments in digital signal processing have enabled further study of this problem.

Consider a non-uniform filter-bank with decimation factors (no, n/,... n ; ). It is well known that an arbitrary set will not necessarily result in a feasible PR system. For maximal-decimation, the condition Condition I must be satisfied.

Let L be the least common multiplier (lcm) of set {n i}i=0K-1 and let ki = L/ni, i # {0, 1,...K- 1/. Perfect reconstruction is possible only if for each I E {1,... K-1} Condition II The symbol-here stands for congruence (e. g.'1l1 mod 5', since 11-1 equals 10 which is divisible by 5). We shall call condition I the maximal-decimation-condition and condition II the feasibility-condition. A set of numbers satisfying both conditions is called a compatible-set.

Usually, the decimation-vector Vspec= (n0, n1, ... nK-1) # Nk, is defined by the application.

Vspec may not satisfy either condition. The problem is to find a vector Vbest march E NK such that it satisfies both conditions and is the closest match, in terms of a pre-defined measure do, to Vspec. That is, for all V e NK such that V satisfies the maximal-decimation and feasibility conditions, d(V, Vbest_match)#, d(Vspec, Vbest_match).

An exhaustive brute-force search (ie. by evaluating every possible combination of decimation factors) for V ; mo over a set S c NK can be extremely computationally expensive. An efficient algorithm is required to perform the search efficiently.

SUMMARY OF THE INVENTION The number of vectors satisfying the maximal-decimation and feasibility conditions is rather small. Suppose the first 1 components of decimation vector Vpartial are fixed to f0, fi, ... fi-I, respectively. Rather than trying every different combination on the remaining components and checking each resulting complete vector for compatibility, it would be more efficient to perform some preliminary tests on the partial-vector itself to check if it could ever lead to a compatible set. If tests reveal that no matter what the value of the remaining components, the initial assignment can never lead to a feasible (or acceptable) solution, then it is prudent to abandon that subspace and search in another direction. This would decrease the computational requirements of the search algorithm drastically.

The present invention provides a method of searching for a best-match decimation vector of decimation factors for a non-uniform filter bank, the best-match decimation vector allowing perfect or near-perfect reconstruction of an input signal of the non-uniform filter bank, the method including the steps of : a) selecting a partial decimation vector having a number, l # 1, of decimation factors, where 1 does not exceed a maximum number, K, of decimation factors of said best- match decimation vector; b) processing said l decimation factors to determine whether said partial decimation vector satisfies a feasibility criterion; c) processing a least common multiplier value of said I decimation factors to determine whether said least common multiplier value is greater than a predetermined value; d) processing a maximum decimation value, Dmax, of said partial decimation vector to determine whether Dmax is less than one;

e) processing a minimum decimation value, Dmin, of said partial decimation vector to determine whether Dmin is greater than one; and incrementing by one the number of decimation factors in the partial decimation vector and repeating steps b) to e) when said feasibility criterion is satisfied and said least common multiplier value is less than said predetermined value and DIa,, is not less than one and Dmin is not greater than one.

In the preferred embodiment, a branch-and-bound method is inserted into the structure of the search procedure. For the bounding condition, four tests are used: F-Test : A partial-test on feasibility of the partial decimation vector.

L-Test : A check on the growth of the least-comnlon-multiplier value.

Dmi,-Test & Dm,,,-Test : To check on the possibility of convergence to unity of the summation of the reciprocals of the decimation factors (maximal decimation condition).

The above tests are described further below and an example of the improvement in searching efficiency is provided.

BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 is a block diagram of an encoder and decoder used for audio signal compression; and Figure 2 is a block diagram representation of a non-uniform filter bank.

Preferred embodiments of the invention are described hereinafter, by way of example only, with reference to the drawings mentioned above.

DETAILED DESCRIPTION Figure 2 shows a non-uniform filter bank structure where x(n) is the input signal to the encoder of Figure 1 and x'(n) is the output signal from the decoder. Ho (z),..., HK- ! (z) and Fo (z),..., FK l (z) are the input and output transfer functions, respectively for the K input and output filters. Ho (z),..., HK-I(z), in combination with the corresponding decimation factors no, ni,..., nK-l, form the analysis filter bank on the encoder side, as shown in Figure 1, and Fo (z),..., FK_1 (z), with the same decimation factors, form the analysis filter bank at the decoder for inverse filtering. A method of a preferred embodiment concerns determining an optimal decimation vector V= (no, nl,... njy) to enable perfect reconstruction of the decimated input signal.

Suppose the vector Vspec = (n0, n1, ..., nK-1) # NK denotes the required set of decimation factors for a non-uniform filter-bank. That is, the decimation factor following the first filter in the bank equals no, the decimation factor following the second filter is nl, and so on.

If vector Vspec satisfies the maxinwal-decimation and feasibility conditions, then the filter bank design can proceed according to known methods. If not, a closely matching vector Vbest match E NK that satisfies both conditions must be found.

The term'closely matching'must be clearly defined. Suppose we define a measure that computes the difference between two decimation-vectors. Two vectors could then be considered as'closely matching'if the difference computed is quite small. For example, we may define a measure where A= (ao, a1, .., aJr/) and B= (bo, bz, ..., bK-1) . This measure is the summation of the relative difference between each component of A and each corresponding component of B.

Therefore, Vb,, Snatch iS considered to be the best-matching vector if it satisfies both the maximal-decimation andfeasibility conditions and for all V E NK such that Vsatisfies both conditions, d(V, Vspec)# d(Vbest_match, Vspec).

Searching the entire space NK is of course impossible. We therefore limit the search to a finite region around Vp, c.

Let us define a vector D = (do, dl,..., dK-1) # NK. Using D we define a region around Vspec as S = {n0 - d0,n0 - d0 +1,...,n0,...n0+d0}#{n1-d1,...n1+d1}#####{nK-1-dK-1,...nK-1+ dK-1}.

The search is now limited to region S. Therefore, rephrasing the earlier statement, d (V, Vspec) # d(Vbest_match, Vspec), #V # S : that maximal-decimation(V) and feasible(V).

A brute-force search over S may not be computationally feasible if it is a large set. For example, there are 25 critical bands in the ear-model. The corresponding decimation-vector is something like (220, 220,220, 220,... 9, 6, 3). Taking D to be (100, 100,..., 5,2, 0), the number of vector-evaluations needed by an exhaustive search would be in the order of 1035. Roughly, a single-vector evh'ion on an Tntel Pentium III processor takes 10-5 sec.

Therefore an exhaustive search within even this restricted space would need 2. 5x1023 years.

Instead of enumerating each vector in S and checking if it satisfies both conditions, a intelligent methodical search of some sort is required. Here we propose a branch-and- bound based approach to the problem.

The branch-and-bound algorithm proceeds as follows: the search space is divided into a number of branches. Each branch is investigated in a sequential order. For each branch, it is estimated whether its exploration would result in a solution better than the existing one.

If a particular branch is predicted not to produce any acceptable solution, or it is predicted that the best solution that it could offer would still be inferior to an existing solution, then further exploration of that branch is abandoned (bounded). The search-space is explored recursively and the algorithm is inherently recursive.

Now, the specific search problem is modeled as a branch and bound solution. Suppose we have a partial vector Vpartial= (f0, f1, ... fl-1, x, ..., x) # (N# {x})K. Here the first I components have been assigned some value. The remaining K-l components are denoted as x, meaning that they have not been assigned any value as yet. If l is equal to K-1, then Vpartial has some value assigned to each component. Vparriaz is then checked to see if it satisfies the maximum-decimation and feasibility conditions. If it does, then its closeness to VspeC is measured to see if it is a solution-vector closer than any previously found vector.

If so, then Vpartial is assigned to the variable Vbest_match- Suppose I is less than K-l. We then conduct a number of tests to check if (fo, f1, ... fl-1, x, ..., x) can lead to a compatible-set. If any of the tests fail, the partial vector (branch) is terminated prematurely, and the search moves onto the next branch.

F-Test : The first test is that of feasibility. Suppose that the unknowns in Vpartial were resolved to some ri, rl+,,... rK-I. Let us check if the feasibility condition would be satisfied.

Let L = lcm(f0, f1, ... fl-1, rl, rl+1, ... rK-1) and Lpartial= lcm(f0, f1, ... fl-1). Now elementary arithmetic tells us that L=Lpartial *Ur, where Ur is some constant. Now, ki = Lyi, therefore ki = = (Lpartial / fi)*Ur. Since Lpartial is a multiplier of fi it follows that fi divides Lpartial.

Consider the compatibility test, That is, t 0mod t i=0 J fm This is equivalent to Lparfial Lpar (ial 7 =mod, m I, G..., Il i=O fi)-f.

If this condition is satisfied for all in, Vpartial is considered to pass the F-test. From this it can be seen that, even though Vpa,,, al is partially fixed, it is possible to check feasibility for the already fixed components of a partial decimation vector.

L-Test : Next we impose a least common multiplier test (L-test) on It is noted above that L = Lpartial*Ur,i. e. L > Lpartial. A very large cm (e. g. if f0, f1, ... fl-1, are all mutually- prime) is not very desirable. To prevent the search from veering into mutually-prime sets, a hard limit is placed on the lcm, Lmax. If Lpartial is greater than Litiax5 then no matter what choice is made forrl, rl+1, ... the composite Icm L would exceed Therefore If Lpartial is greater than the L-test fails and the search branch is terminated.

The next two tests are designed to test whether converges to one. The maximum value for is l/(ni-di) within the defined area around Vspec. Similarly, the minimum value is l/(ni+di). With (fo, f/,... fl-1) fixed, and (rb rl+l,... rK s) being some arbitrary numbers yet to be fixed, the Dmax-Test (Maximum-Decimation Rate) and Dmin-Test (Minimum- Decimation Rate) compute whether the current fixed assignments f0, f1, ... fl-1 could ever lead to a convergence value of equal to unity.

Dmax-Test (Maximum-Decimation Rate): Let Vpartial= (f0, f1, .. fl-1, x, ..., x) # (N# {x})K be a partially assigned decimation vector. Define : can never be equal to one, and consequently the test fails.

Dmin-Test (Minimum-Decimation Rate): Define:

If D", ;" > 1, then Vparxjal would always lead to a vector whose sum of reciprocals would exceed unity, can never be equal to one.

An algorithm for effecting the four tests (F-test, L-test, Dmsx-test and Dmjn-test) is illustrated in pseudo-code in Appendix 1 hereto.

In table 1 below, the efficiency of the presently described Branch-and-Bound algorithm is compared against the exhaustive Brute-Force method.

Table 1-Efficiency Comparison Number of Evaluations K (# proportional to processor execution time) (number of Brute-Force Branch and Branch and Branch and Filters) Bound Bound Bound (Tests: Dmjn + (Tests: (Tests: D,, in+ Dmax) Dmin+Dmax+L) Dmax+L + F) 3 2,744 211 211 45 4 384,16 2, 866 2866 115 5 537, 824 36,119 35,807 293 6 7, 529,536 401, 716 376, 924 666 7 105, 413, 504 6, 121,843 3, 185,673 1311

In table 2 below, a compatible set of decimation factors which closely match the critical decimation factors (computed from the critical band frequencies) is presented. The critical band frequencies of the human ear are empirically determined and are spread across about 25 distinctly identifiable bands. The bandwidth of each band varies from slightly less than 100Hz at low frequencies to about one third of an octave at higher audible frequencies (eg.

1. 2kHz for a center frequency of 10kHz). The critical bands of the ear are generally continuous such that a tone of an audible frequency has a critical band centered on it.

Table 2-Best match decimation vector Critical-Band No. Vspec Vers, match (LCM=2520) 1 220 210 2 220 210 3 220 210 4 220 210 5 200 210 6 183 168 7 158 168 8 148 140 9 138 140 10 116 120 11 105 120 12 92 90 13 79 90 14 70 90 15 59 60 16 49 60 17 40 30 18 32 42 19 25 21 20 20 21 21 17 21 22 12 12 23 9 12 24 6 6 25 3 3 APPENDIX 1 Initialization : VJpec = (nnZ nl,, nK], D- (do, dJ,, dJc) E Nt, Ls = Limit ou LClII, error/eaS, = any lar, ge number Algorithm branch and bound input : Vparzjal= Cfo, fs, --f, x x) E x, l) ; begin foreach E i-dj,-d+l,... 0, 1, 2... dl) FpM/= (,/7,... -7, 77, X..., ) ;//= + if I. equals. K-I llorie complete solution ? if y}// : =}. and. feasible (Fparr ; at, 1+1) llcheck feasibility and t-o maximal dec. if "'f < enror llis this feasible solution laO rio /yj9/'e ! OUJ' 0/ ! g ? better than //previous one ? erroy « _ I 'f I , Fbesl match-r''porfial a zozo endif endif else if l. is greater than. 0 if (. not. feasible (FpHr, Q, 1+I)) llF Test . or. (Lp-lial > L) IIL-Test . or. 0'1'rS . nr f-'' or- (I + E I) <i/IDmax-Test i=o f i. + di ? ! ex &roy ! c/ ? end end continue ;//si7 ? ce conditions fail, move to rzext branch end end brarzclz and bound (Fpo, r, at, 1+I) ; llbranch to next child end end Function feasible input : Fpa"jaZ os fs, fi l, r,, x) EaVutxv begin ZpnrrrQ ! = jcZ''parrinl l) lllcnt of the first 1 nurnbens foreach i e 1'1, 2,... l-l lliterate if Lna, at k = 0 modCLa. r ; or f J t k) (i) if-0 inod f E."ilfk return False end return True End

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