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United States Patent Application 
20170206911

Kind Code

A1

Ekstrand; Per

July 20, 2017

Complex Exponential Modulated Filter Bank for High Frequency
Reconstruction or Parametric Stereo
Abstract
An apparatus and method are disclosed for filtering and performing high
frequency reconstruction of an audio signal. The apparatus includes an
analysis filter bank, a phase shifter, a high frequency reconstructor,
and a synthesis filter bank. The analysis filterbank receives realvalued
time domain input audio samples and generates complex valued subband
samples. The phase shifter shifts a phase of the complexvalued subband
samples by an arbitrary amount. The high frequency reconstructor modifies
at least some of the complex valued subband samples. A phase shifter
unshifts a phase of the modified complexvalued subband samples by the
arbitrary amount. The synthesis filter bank receives the modified complex
valued subband samples and generates time domain output audio samples.
The analysis filter bank comprises analysis filters that are complex
exponential modulated versions of a prototype filter with an arbitrary
phase shift.
Inventors: 
Ekstrand; Per; (Saltsjobaden, SE)

Applicant:  Name  City  State  Country  Type  Dolby International AB  Amsterdam Zuidoost  
NL   
Assignee: 
Dolby International AB
Amsterdam Zuidoost
NL

Family ID:

1000002564507

Appl. No.:

15/478797

Filed:

April 4, 2017 
Related U.S. Patent Documents
             
 Application Number  Filing Date  Patent Number 

 14810174  Jul 27, 2015  
 15478797   
 14306495  Jun 17, 2014  9449608 
 14810174   
 13201572  Aug 15, 2011  8880572 
 PCT/EP2010/051993  Feb 17, 2010  
 14306495   
 61257105  Nov 2, 2009  

Current U.S. Class: 
1/1 
Current CPC Class: 
G10L 19/0208 20130101; H03H 17/0201 20130101; G06F 17/11 20130101; G10L 19/26 20130101; G10L 19/008 20130101 
International Class: 
G10L 19/02 20060101 G10L019/02; G06F 17/11 20060101 G06F017/11; G10L 19/26 20060101 G10L019/26; H03H 17/02 20060101 H03H017/02; G10L 19/008 20060101 G10L019/008 
Foreign Application Data
Date  Code  Application Number 
Feb 18, 2009  SE  09002171 
Claims
1. A signal processing device for filtering and processing 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 high frequency reconstructor or
parametric stereo processor that generates modified complex valued
subband samples; 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 ( n ) = p 0
( n ) exp { i .pi. M ( k + 1 2 ) ( n  D 2 
A ) } , 0 .ltoreq. n < N , 0 .ltoreq. k < M
##EQU00034## where A is a phase shift, the analysis filter bank has M
channels, the prototype filter (p.sub.0(n)) has a length N, and the
analysis filter bank and synthesis filter bank have a system delay of D
samples, wherein one or more of the analysis filter bank, the high
frequency reconstructor, the parametric stereo processor, and the
synthesis filter bank is implemented, at least in part, by one or more
hardware elements of the signal processing device.
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.
5. The signal processing device of claim 1 wherein the high frequency
reconstructor performs spectral band replication (SBR).
6. The signal processing device of claim 1 wherein a value of the
arbitrary phase shift constant is chosen to reduce a complexity of an
implementation of the apparatus.
7. The signal processing device of claim 1 wherein the one or more
hardware elements comprise a digital signal processor, a microprocessor,
or a memory.
8. A method performed by an signal processing device for filtering and
processing an audio signal, the method comprising: receiving realvalued
time domain input audio samples; filtering the realvalued time domain
input audio samples with an analysis filter bank to generate complex
valued subband samples; generating modified complex valued subband
samples through a high frequency reconstruction process or a parametric
stereo process; 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 ( n ) = p 0 ( n ) exp { i
.pi. M ( k + 1 2 ) ( n  D 2  A ) } , 0
.ltoreq. n < N , 0 .ltoreq. k < M ##EQU00035## where A is a
phase shift, the analysis filter bank has M channels, the prototype
filter (p.sub.0(n)) has a length N, and the analysis filter bank and
synthesis filter bank have a system delay of D samples, wherein the
signal processing device comprises one or more hardware elements.
9. A nontransitory computer readable medium containing instructions that
when executed by a processor perform the method of claim 8.
Description
[0001] This application is a divisional application of U.S. patent
application Ser. No. 14/810,174 filed Jul. 27, 2015, which is a
continuation of U.S. patent application Ser. No. 14/306, 495 filed on
Jun. 17, 2014 , now U.S. Pat. No. 9,449,608 issued on Sep. 20, 2016,
which is a continuation of U.S. patent application Ser. No. 13/201,572
filed on Aug. 15, 2011 now U.S. Pat. No. 8,880,572 issued on Nov. 4,
2014, which is a national application of PCT application
PCT/EP2010/051993 filed on Feb. 17, 2010 which claims the benefit of the
filing date of U.S. Provisional Patent Application Ser. No. 61/257,105
filed on Nov. 2, 2009 and Sweden application 09002171 filed on Feb. 18,
2009, all of which are hereby incorporated by reference.
[0002] The present document relates to modulated subsampled digital
filter banks, as well as to methods and systems for the design of such
filter banks. In particular, it provides a new design method and
apparatus for a nearperfect reconstruction low delay cosine or
complexexponential modulated filter bank, optimized for suppression of
aliasing emerging from modifications of the spectral coefficients or
subband signals. Furthermore, a specific design for a 64 channel filter
bank using a prototype filter length of 640 coefficients and a system
delay of 319 samples is given.
[0003] The teachings of this document may be applicable to digital
equalizers, as outlined e.g. in "An Efficient 20 Band Digital Audio
Equalizer" A. J. S. Ferreira, J. M. N. Viera, AES preprint, 98.sup.th
Convention 1995 Feb. 2528 Paris, N.Y., USA; adaptive filters, as
outlined e.g. in Adaptive Filtering in Subbands with Critical Sampling:
Analysis, Experiments, and Application to Acoustic Echo Cancellation" A.
Gilloire, M. Vetterli, IEEE Transactions on Signal Processing, vol. 40,
no. 8, August, 1992; multiband companders; and to audio coding systems
utilizing high frequency reconstruction (HFR) methods; or audio coding
systems employing socalled parametric stereo techniques. In the two
latter examples, a digital filter bank is used for the adaptive
adjustment of the spectral envelope of the audio signal. An exemplary HFR
system is the Spectral Band Replication (SBR) system outlined e.g. in WO
98/57436, and a parametric stereo system is described e.g. in EP1410687.
[0004] Throughout this disclosure including the claims, the expressions
"subband signals" or "subband samples" denote the output signal or output
signals, or output sample or output samples from the analysis part of a
digital filter bank or the output from a forward transform, i.e. the
transform operating on the time domain data, of a transform based system.
Examples for the output of such forward transforms are the frequency
domain coefficients from a windowed digital Fourier transform (DFT) or
the output samples from the analysis stage of a modified discrete cosine
transform (MDCT).
[0005] Throughout this disclosure including the claims, the expression
"aliasing" denotes a nonlinear distortion resulting from decimation and
interpolation, possibly in combination with modification (e.g.
attenuation or quantization) of the subband samples in a subsampled
digital filter bank.
[0006] A digital filter bank is a collection of two or more parallel
digital filters. The analysis filter bank splits the incoming signal into
a number of separate signals named subband signals or spectral
coefficients. The filter bank is critically sampled or maximally
decimated when the total number of subband samples per unit time is the
same as that for the input signal. A so called synthesis filter bank
combines the subband signals into an output signal. A popular type of
critically sampled filter banks is the cosine modulated filter bank,
where the filters are obtained by cosine modulation of a lowpass filter,
a socalled prototype filter. The cosine modulated filter bank offers
effective implementations and is often used in natural audio coding
systems. For further details, reference is made to "Introduction to
Perceptual Coding" K. Brandenburg, AES, Collected Papers on Digital Audio
Bitrate Reduction, 1996.
[0007] A common problem in filter bank design is that any attempt to alter
the subband samples or spectral coefficients, e.g. by applying an
equalizing gain curve or by quantizing the samples, typically renders
aliasing artifacts in the output signal. Therefore, filter bank designs
are desirable which reduce such artifacts even when the subband samples
are subjected to severe modifications.
[0008] A possible approach is the use of oversampled, i.e. not critically
sampled, filter banks. An example of an oversampled filter bank is the
class of complex exponential modulated filter banks, where an imaginary
sine modulated part is added to the real part of a cosine modulated
filter bank. Such a complex exponential modulated filter bank is
described in EP1374399 which is incorporated herewith by reference.
[0009] One of the properties of the complex exponential modulated filter
banks is that they are free from the main alias terms present in the
cosine modulated filter banks. As a result, such filter banks are
typically less prone to artifacts induced by modifications to the subband
samples. Nevertheless, other alias terms remain and sophisticated design
techniques for the prototype filter of such a complex exponential
modulated filter bank should be applied in order to minimize the
impairments, such as aliasing, emerging from modifications of the subband
signals. Typically, the remaining alias terms are less significant than
the main alias terms.
[0010] A further property of filter banks is the amount of delay which a
signal incurs when passing through such filter banks. In particular for
real time applications, such as audio and video streams, the filter or
system delay should be low. A possible approach to obtain a filter bank
having a low total system delay, i.e. a low delay or latency of a signal
passing through an analysis filter bank followed by a synthesis filter
bank, is the use of short symmetric prototype filters. Typically, the use
of short prototype filters leads to relatively poor frequency band
separation characteristics and to large frequency overlap areas between
adjacent subbands. By consequence, short prototype filters usually do not
allow for a filter bank design that suppresses the aliasing adequately
when modifying the subband samples and other approaches to the design of
low delay filter banks are required.
[0011] It is therefore desirable to provide a design method for filter
banks which combine a certain number of desirable properties. Such
properties are a high level of insusceptibility to signal impairments,
such as aliasing, subject to modifications of the subband signals; a low
delay or latency of a signal passing through the analysis and synthesis
filter banks; and a good approximation of the perfect reconstruction
property. In other words, it is desirable to provide a design method for
filter banks which generate a low level of errors. Subsampled filter
banks typically generate two types of errors, linear distortion from the
passband term which further can be divided into amplitude and phase
errors, and nonlinear distortion emerging from the aliasing terms. Even
though a "good approximation" of the PR (perfect reconstruction) property
would keep all of these errors on a low level, it may be beneficial from
a perceptual point of view to put a higher emphasis on the reduction of
distortions caused by aliasing.
[0012] Furthermore, it is desirable to provide a prototype filter which
can be used to design an analysis and/or synthesis filter bank which
exhibits such properties. It is a further desirable property of a filter
bank to exhibit a near constant group delay in order to minimize
artifacts due to phase dispersion of the output signal.
[0013] The present document shows that impairments emerging from
modifications of the subband signals can be significantly reduced by
employing a filter bank design method, referred to as improved alias term
minimization (IATM) method, for optimization of symmetric or asymmetric
prototype filters.
[0014] The present document teaches that the concept of pseudo QMF
(Quadrature Mirror Filter) designs, i.e. near perfect reconstruction
filter bank designs, may be extended to cover low delay filter bank
systems employing asymmetric prototype filters. As a result near perfect
reconstruction filter banks with a low system delay, low susceptibility
to aliasing and/or low level of pass band errors including phase
dispersion can be designed. Depending on the particular needs, the
emphasis put on either one of the filter bank properties may be changed.
Hence, the filter bank design method according to the present document
alleviates the current limitations of PR filter banks used in an
equalization system or other system modifying the spectral coefficients.
[0015] The design of a low delay complexexponential modulated filter bank
according to the present document may comprise the steps: [0016] a
design of an asymmetric lowpass prototype filter with a cutoff frequency
of .pi./2M, optimized for desired aliasing and pass band error
rejections, further optimized for a system delay D; M being the number of
channels of the filter bank; and [0017] a construction of an Mchannel
filter bank by complexexponential modulation of the optimized prototype
filter.
[0018] Furthermore, the operation of such a low delay complexexponential
modulated filter bank according to the present document may comprise the
steps: [0019] a filtering of a realvalued time domain signal through
the analysis part of the filter bank; [0020] a modification of the
complexvalued subband signals, e.g. according to a desired, possibly
timevarying, equalizer setting; [0021] a filtering of the modified
complexvalued subband samples through the synthesis part of the filter
bank; and [0022] a computation of the real part of the complexvalued
time domain output signal obtained from the synthesis part of the filter
bank.
[0023] In addition to presenting a new filter design method, the present
document describes a specific design of a 64 channel filter bank having a
prototype filter length of 640 coefficients and a system delay of 319
samples.
[0024] The teachings of the present document, notably the proposed filter
bank and the filter banks designed according to the proposed design
method may be used in various applications. Such applications are the
improvement of various types of digital equalizers, adaptive filters,
multiband companders and adaptive envelope adjusting filter banks used in
HFR or parametric stereo systems.
[0025] According to a first aspect, a method for determining N
coefficients of an asymmetric prototype filter p.sub.0 for usage for
building an Mchannel, low delay, subsampled analysis/synthesis filter
bank is described. The analysis/synthesis filter bank may comprise M
analysis filters h.sub.k and M synthesis filters f.sub.k, wherein k takes
on values from 0 to M1 and wherein typically M is greater than 1. The
analysis/synthesis filter bank has an overall transfer function, which is
typically associated with the coefficients of the analysis and synthesis
filters, as well as with the decimation and/or interpolation operations.
[0026] The method comprises the step of choosing a target transfer
function of the filter bank comprising a target delay D. Typically a
target delay D which is smaller or equal to Nis selected. The method
comprises further the step of determining a composite objective function
e.sub.tot comprising a pass band error term e.sub.t and an aliasing error
term e.sub.a. The pass band error term is associated with the deviation
between the transfer function of the filter bank and the target transfer
function, and the aliasing error term e.sub.a is associated with errors
incurred due to the subsampling, i.e. decimation and/or interpolation of
the filter bank. In a further method step, the N coefficients of the
asymmetric prototype filter p.sub.o are determined that reduce the
composite objective function e.sub.tot.
[0027] Typically, the step of determining the objective error function
e.sub.tot and the step of determining the N coefficients of the
asymmetric prototype filter p.sub.0 are repeated iteratively, until a
minimum of the objective error function e.sub.tot is reached. In other
words, the objective function e.sub.tot is determined on the basis of a
given set of coefficients of the prototype filter, and an updated set of
coefficients of the prototype filter is generated by reducing the
objective error function. This process is repeated until no further
reductions of the objective function may be achieved through the
modification of the prototype filter coefficients. This means that the
step of determining the objective error function e.sub.tot may comprise
determining a value for the composite objective function e.sub.tot for
given coefficients of the prototype filter p.sub.0 and the step of
determining the N coefficients of the asymmetric prototype filter p.sub.0
may comprise determining updated coefficients of the prototype filter
p.sub.0 based on the gradient of the composite objective function
e.sub.tot associated with the coefficients of the prototype filter
p.sub.0.
[0028] According to a further aspect, the composite objective error
function e.sub.tot is given by:
e.sub.tot(.alpha.)=.alpha.e.sub.t+(1.alpha.)e.sub.a,
with e.sub.t being the pass band error term, e.sub.a being the aliasing
error term and a being a weighting constant taking on values between 0
and 1. The pass band error term e.sub.t may be determined by accumulating
the squared deviation between the transfer function of the filter bank
and the target transfer function for a plurality of frequencies. In
particular, the pass band error term e.sub.t may be calculated as
e t = 1 2 .pi. .intg.  .pi. .pi. 1 2 (
A 0 ( e j .omega. ) + A 0 * ( e  j
.omega. ) )  P ( .omega. ) e  j .omega.
D 2 d .omega. , ##EQU00001##
with P(.omega.)e.sup.j.omega.D being the target transfer function, and
A 0 ( z ) = k = 0 M  1 H k ( z ) F
k ( z ) , ##EQU00002##
wherein H.sub.k(z) and F.sub.k(z) are the ztransforms of the analysis
and synthesis filters h.sub.k(n) and f.sub.k(n), respectively.
[0029] The aliasing error term e.sub.a is determined by accumulating the
squared magnitude of alias gain terms for a plurality of frequencies. In
particular, the aliasing error term e.sub.a is calculated as
e a = 1 2 .pi. l = 1 M  1 .intg. 
.pi. .pi. A ~ l ( e j .omega. ) 2 d
.omega. , with A ~ l ( z ) = 1 2 (
A l ( z ) + A M  l * ( z ) ) , l = 1
M  1 , for z = e j .omega. and
with ##EQU00003## A l ( z ) = k = 0 M  1
H k ( zW l ) F k ( z ) ##EQU00003.2##
being the l.sup.th alias gain term evaluated on the unit circle with
W=e.sup.i2.pi./M, wherein H.sub.k(z) and F.sub.k(z) are the ztransforms
of the analysis and synthesis filters h.sub.k(n) and f.sub.k(n),
respectively. The notation A.sub.l*(z) is the ztransform of the
complexconjugated sequence a.sub.l (n) .
[0030] According to a further aspect, the step of determining a value for
the composite objective function e.sub.tot may comprise generating the
analysis filters h.sub.k(n) and the synthesis filters f.sub.k(n) of the
analysis/synthesis filter bank based on the prototype filter p.sub.o(n)
using cosine modulation, sine modulation and/or complexexponential
modulation. In particular, the analysis and synthesis filters may be
determined using cosine modulation as
h k ( n ) = 2 p 0 ( n ) cos { .pi. M (
k + 1 2 ) ( n  D 2 .+. M 2 ) } , ##EQU00004##
with n=0 . . . N1, for the M analysis filters of the analysis filter
bank and;
f k ( n ) = 2 p 0 ( n ) cos { .pi. M (
k + 1 2 ) ( n  D 2 .+. M 2 ) } , ##EQU00005##
with n=0 . . . N1, for the M synthesis filters of the synthesis filter
bank.
[0031] The analysis and synthesis filters may also be determined using
complex exponential modulation as
h k ( n ) = p 0 ( n ) exp { i .pi. M ( k
+ 1 2 ) ( n  D 2  A ) } , ##EQU00006##
with n=0 . . . N1, and A being an arbitrary constant, for the M analysis
filters of the analysis filter bank and;
f k ( n ) = p 0 ( n ) exp { i .pi. M ( k
+ 1 2 ) ( n  D 2 + A ) } , ##EQU00007##
with n=0 . . . N1, for the M synthesis filters of the synthesis filter
bank.
[0032] According to another aspect, the step of determining a value for
the composite objective function e.sub.tot may comprise setting at least
one of the filter bank channels to zero. This may be achieved by applying
zero gain to at least one analysis and/or synthesis filter, i.e. the
filter coefficients h.sub.k and/or f.sub.k may be set to zero for at
least one channel k. In an example a predetermined number of the low
frequency channels and/or a predetermined number of the high frequency
channels may be set to zero. In other words, the low frequency filter
bank channels k=0 up to C.sub.low; with C.sub.low greater than zero may
be set to zero. Alternatively or in addition, the high frequency filter
bank channels k=C.sub.high up to M1, with C.sub.high smaller than M1
may be set to zero.
[0033] In such a case, the step of determining a value for the composite
objective function e.sub.tot may comprise generating the analysis and
synthesis filters for the aliasing terms C.sub.low and MC.sub.low and/or
C.sub.high and MC.sub.high using complex exponential modulation. It may
further comprise generating the analysis and synthesis filters for the
remaining aliasing terms using cosine modulation. In other words, the
optimization procedure may be done in a partially complexvalued manner,
where the aliasing error terms which are free from main aliasing are
calculated using real valued filters, e.g. filters generated using cosine
modulation, and where the aliasing error terms which carry the main
aliasing in a realvalued system are modified for complexvalued
processing, e.g. using complex exponential modulated filters.
[0034] According to a further aspect, the analysis filter bank may
generate M subband signals from an input signal using the M analysis
filters h.sub.k. These M subband signals may be decimated by a factor M,
yielding decimated subband signals. Typically, the decimated subband
signals are modified, e.g. for equalization purposes or for compression
purposes. The possibly modified decimated subband signals may be
upsampled by a factor M and the synthesis filter bank may generate an
output signal from the upsampled decimated subband signals using the M
synthesis filters f.sub.k.
[0035] According to another aspect, an asymmetric prototype filter p0(n)
comprising coefficients derivable from the coefficients of Table 1 by any
of the operations of rounding, truncating, scaling, subsampling or
oversampling is described. Any combination of the operations rounding,
truncating, scaling, subsampling or oversampling are possible.
[0036] The rounding operation of the filter coefficients may comprise any
one of the following: rounding to more than 20 significant digits, more
than 19 significant digits, more than 18 significant digits, more than 17
significant digits, more than 16 significant digits, more than 15
significant digits, more than 14 significant digits, more than 13
significant digits, more than 12 significant digits, more than 11
significant digits, more than 10 significant digits, more than 9
significant digits, more than 8 significant digits, more than 7
significant digits, more than 6 significant digits, more than 5
significant digits, more than 4 significant digits, more than 3
significant digits, more than 2 significant digits, more than 1
significant digits, 1 significant digit.
[0037] The truncating operation of the filter coefficients may comprise
any one of the following: truncating to more than 20 significant digits,
more than 19 significant digits, more than 18 significant digits, more
than 17 significant digits, more than 16 significant digits, more than 15
significant digits, more than 14 significant digits, more than 13
significant digits, more than 12 significant digits, more than 11
significant digits, more than 10 significant digits, more than 9
significant digits, more than 8 significant digits, more than 7
significant digits, more than 6 significant digits, more than 5
significant digits, more than 4 significant digits, more than 3
significant digits, more than 2 significant digits, more than 1
significant digits, 1 significant digit.
[0038] The scaling operation of the filter coefficient may comprise
upscaling or downscaling of the filter coefficients. In particular, it
may comprise up and/or downscaling scaling by the number M of filter
bank channels. Such up and/or downscaling may be used to maintain the
input energy of an input signal to the filter bank at the output of the
filter bank.
[0039] The subsampling operation may comprise subsampling by a factor less
or equal to 2, less or equal to 3, less or equal to 4, less or equal to
8, less or equal to 16, less or equal to 32, less or equal to 64, less or
equal to 128, less or equal to 256. The subsampling operation may further
comprise the determination of the subsampled filter coefficients as the
mean value of adjacent filter coefficient. In particular, the mean value
of R adjacent filter coefficients may be determined as the subsampled
filter coefficient, wherein R is the subsampling factor.
[0040] The oversampling operation may comprise oversampling by a factor
less or equal to 2, less or equal to 3, less or equal to 4, less or equal
to 5, less or equal to 6, less or equal to 7, less or equal to 8, less or
equal to 9, less or equal to 10. The oversampling operation may further
comprise the determination of the oversampled filter coefficients as the
interpolation between two adjacent filter coefficients.
[0041] According to a further aspect, a filter bank comprising M filters
is described. The filters of this filter bank are based on the asymmetric
prototype filters described in the present document and/or the asymmetric
prototype filters determined via the methods outlined in the present
document. In particular, the M filters may be modulated version of the
prototype filter and the modulation may be a cosine modulation, sine
modulation and/or complexexponential modulation.
[0042] According to another aspect, a method for generating decimated
subband signals with low sensitivity to aliasing emerging from
modifications of said subband signals is described. The method comprises
the steps of determining analysis filters of an analysis/synthesis filter
bank according to methods outlined in the present document; filtering a
realvalued time domain signal through said analysis filters, to obtain
complexvalued subband signals; and decimating said subband signals.
[0043] Furthermore, a method for generating a real valued output signal
from a plurality of complexvalued subband signals with low sensitivity
to aliasing emerging from modifications of said subband signals is
described. The method comprises the steps of determining synthesis
filters of an analysis/synthesis filter bank according to the methods
outlined in the present document; interpolating said plurality of
complexvalued subband signals; filtering said plurality of interpolated
subband signals through said synthesis filters; generating a
complexvalued time domain output signal as the sum of the signals
obtained from said filtering; and taking the real part of the
complexvalued time domain output signal as the realvalued output
signal.
[0044] According to another aspect, a system operative of generating
subband signals from a time domain input signal are described, wherein
the system comprises an analysis filter bank which has been generated
according to methods outlined in the present document and/or which is
based on the prototype filters outlined in the present document.
[0045] It should be noted that the aspects of the methods and systems
including its preferred embodiments as outlined in the present patent
application may be used standalone or in combination with the other
aspects of the methods and systems disclosed in this document.
Furthermore, all aspects of the methods and systems outlined in the
present patent application may be arbitrarily combined. In particular,
the features of the claims may be combined with one another in an
arbitrary manner.
[0046] The present invention will now be described by way of illustrative
examples, not limiting the scope, with reference to the accompanying
drawings, in which:
BRIEF DESCRIPTION OF DRAWINGS
[0047] FIG. 1 illustrates the analysis and synthesis sections of a digital
filter bank;
[0048] 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. realvalued, filter bank;
[0049] FIG. 3 shows a flow diagram of an example of the optimization
procedure;
[0050] 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
[0051] FIG. 5 (a)(b) illustrates an example of the analysis and synthesis
parts of a low delay complexexponential modulated filter bank system.
[0052] 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.
[0053] 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, . . . , M1. 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, . . . , M1.
[0054] 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).
[0055] 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 nonlinear 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.
[0056] 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, . . . , M1. The decimators 104, also referred to as
downsampling units, give the outputs
V k ( z ) = 1 M l = 0 M  1 X k ( z
1 / M W l ) = 1 M l = 0 M  1 H k
( z 1 / M W l ) X ( z 1 / M W l )
, ( 2 ) ##EQU00008##
where W=e.sup.i2.pi./M. The outputs of the interpolators 105, also
referred to as upsampling units, are given by
U k ( z ) = V k ( z M ) = 1 M l = 0 M
 1 H k ( zW l ) X ( zW l ) , ( 3 )
##EQU00009##
and the sum of the signals obtained from the synthesis filters 106 can be
written as
X ^ ( z ) = k = 0 M  1 F k ( z )
U k ( z ) = k = 0 M  1 F k ( z ) 1 M
l = 0 M  1 H k ( zW l ) X ( zW l ) =
= 1 M l = 0 M  1 X ( zW l ) k =
0 M  1 H k ( zW l ) F k ( z ) = 1 M
l = 0 M  1 X ( zW l ) A l ( z )
( 4 ) where A l ( z ) = k = 0 M  1 H
k ( zW l ) F k ( z ) ( 5 ) ##EQU00010##
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.1(z). Eq.(4) can be rewritten as
X ^ ( z ) = 1 M { X ( z ) A 0 ( z )
+ l = 1 M  1 X ( zW l ) A l ( z ) } .
( 6 ) ##EQU00011##
[0057] The last sum on the right hand side (RHS) constitutes the sum of
all nonwanted 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
X ^ ( z ) = 1 M X ( z ) A 0 ( z ) =
1 M X ( z ) k = 0 M  1 H k ( z ) F k
( z ) = X ( z ) T ( z ) , ( 7 ) where
T ( z ) = 1 M k = 0 M  1 H k ( z )
F k ( z ) ( 8 ) ##EQU00012##
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)
[0058] 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.
[0059] 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 lowpass prototype filter p.sub.0(n), i.e.
h k ( n ) = 2 p 0 ( n ) cos { .pi. M
( k + 1 2 ) ( n  N 2 .+. M 2 ) } , 0 .ltoreq. n
.ltoreq. N , 0 .ltoreq. k .ltoreq. M ( 11 ) f k ( n )
= 2 p 0 ( n ) cos { .pi. M ( k + 1 2 )
( n  N 2 .+. M 2 ) } , 0 .ltoreq. n .ltoreq. N , 0
.ltoreq. k .ltoreq. M ( 12 ) ##EQU00013##
respectively, where M is the number of channels of the filter bank and N
is the prototype filter order.
[0060] The above cosine modulated analysis filter bank produces
realvalued subband samples for realvalued 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.
[0061] 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 nonlinear phase,
i.e. they have a group delay which may change with frequency.
[0062] In filter bank systems using asymmetric prototype filters, the
analysis and synthesis filters may be written as
h k ( n ) = 2 h ^ 0 ( n ) cos { .pi. M
( k + 1 2 ) ( n  D 2 .+. M 2 ) } , 0 .ltoreq.
n < N h , 0 .ltoreq. k < M ( 13 ) f k ( n )
= 2 f ^ 0 ( n ) cos { .pi. M ( k + 1 2 )
( n  D 2 .+. M 2 ) } , 0 .ltoreq. n < N f , 0
.ltoreq. k < M ( 14 ) ##EQU00014##
respectively, where h.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)=h.sub.0(n)=p.sub.0(n),
0.ltoreq.n<N.sub.h=N.sub.f=N (15)
where N is the length of the prototype filter p.sub.0(n).
[0063] 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.
[0064] 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 socalled 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
.pi. 2 ( k + 1 2 ) , ##EQU00015##
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.
[0065] The removal of the main alias terms may be achieved by the use of
socalled ComplexExponential Modulated Filter Banks which are based on
an extension of the cosine modulation to complexexponential modulation.
Such extension yields the analysis filters h.sub.k(n) as
h k ( n ) = p 0 ( n ) exp { i .pi. M (
k + 1 2 ) ( n  D 2 .+. M 2 ) } , 0 .ltoreq. n
< N , 0 .ltoreq. k < M ( 16 ) ##EQU00016##
using the same notation as before. This can be viewed as adding an
imaginary part to the realvalued filter bank, where the imaginary part
consists of sine modulated versions of the same prototype filter.
Considering a realvalued 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. The resulting
subbands are thus the analytic signals of the realvalued output obtained
from the cosine modulated filter bank. Hence, due to the complexvalued
representation, the subband signals are oversampled by a factor two.
[0066] The synthesis filters are extended in the same way to
f k ( n ) = p 0 ( n ) exp { 1 .pi. M (
k + 1 2 ) ( n  D 2 .+. M 2 ) } , 0 .ltoreq. n
< N , 0 .ltoreq. k < M . ( 17 ) ##EQU00017##
Eq.(16) and (17) imply that the output from the synthesis bank is
complexvalued. 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+j S.sub.s)(C.sub.a+j
S.sub.a)x=(C.sub.sC.sub.aS.sub.sS.sub.a)x+j(C.sub.sS.sub.a+S.sub.sC.sub.
a)x (18)
[0067] 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 complexexponential modulated system offers the same
reconstruction accuracy as the corresponding cosine modulated version. In
other words, when using a realvalued input signal, the output signal of
the complexexponential modulated system may be determined by taking the
real part of the output signal.
[0068] The complexexponential modulated system may be extended to handle
also complexvalued 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
complexvalued signals is obtained.
[0069] It should be noted that the complexexponential 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 complexexponential
modulated version. The analysis and synthesis filters can thus be given
as
h k ( n ) = p 0 ( n ) exp { i .pi. M (
k + 1 2 ) ( n  D 2  A ) } , 0 .ltoreq. n < N , 0
.ltoreq. k < M ( 19 ) and f k ( n ) =
p 0 ( n ) exp { i .pi. M ( k + 1 2 ) ( n  D
2 + A ) } , 0 .ltoreq. n < N , 0 .ltoreq. k < M ( 20
) ##EQU00018##
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.
[0070] 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 complexexponential function. The prototype filters for the
analysis and synthesis filter banks may either be different or identical.
When using complexexponential 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 complexexponential 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.
[0071] 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.
[0072] Referring to Eq.(4), the ztransform of the real part of the output
signal {circumflex over (x)}(n) is
Z { Re ( x ^ ( n ) ) } = X ^ R ( z ) =
X ^ ( z ) + X ^ ( z ) 2 . ( 21 )
##EQU00019##
[0073] The notation {circumflex over (X)}*(z) is the ztransform of the
complexconjugated sequence {circumflex over (x)}(n). From Eq.(4), it
follows that the transform of the real part of the output signal is
X R ( z ) = 1 M l = 0 M  1 1 2 ( X
( zW l ) A l ( z ) + X ( zW  l ) A l
( z ) ) , ( 22 ) ##EQU00020##
where it was used that the input signal x(n) is realvalued, i.e.
X*(zW.sup.l)=X(zW.sup.l). Eq.(22) may after rearrangement be written
X ^ R ( z ) = 1 M ( X ( z ) 1 2
( A 0 ( z ) + A 0 ( z ) ) + l = 1
M  1 1 2 ( X ( zW l ) A l ( z ) + X
( zW M  l ) A l ( z ) ) ) = = 1 M (
X ( z ) 1 2 ( A 0 ( z ) + A 0 ( z ) ) +
l = 1 M  1 X ( zW l ) 1 2 ( A
l ( z ) + A M  l ( z ) ) ) = = 1 M (
X ( z ) A ~ 0 ( z ) + l = 1 M  1 X (
zW l ) A ~ l ( z ) ) ( 23 ) where
A ~ l ( z ) = 1 2 ( A l ( z ) + A M  l
( z ) ) , 0 .ltoreq. l < M ( 24 ) ##EQU00021##
are the alias gain terms used in the optimization. It can be observed
from Eq.(24) that
.sub.Ml(z)=1/2(A.sub.Ml(z)+A.sub.l*(z))= .sub.l*(z). (25)
[0074] Specifically, for realvalued systems
A.sub.Ml*(z)=A.sub.l(z) (26)
which simplifies Eq.(24) into
.sub.l(z)=A.sub.l(z), 0.ltoreq.l<M. (27)
[0075] 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. .sub.0(z) is on the form .sub.0(z)=c z.sup.D.
Moreover, the real part of a.sub.M/2(n) must be zero, i.e. .sub.M/2(z)
must be zero, and the alias gains, for l.apprxeq.0, M/2 must satisfy
A.sub.Ml(z)=A.sub.l*(z), (28)
which for a realvalued system, with Eq.(26) in mind, means that all
a.sub.l(n), l=1 . . . M1 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 Diracpulse, nor is the real part of a.sub.M/2(n) exactly
zero.
[0076] 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.
v k ' ( n ) = g k v k ( n ) , { g k =
1 , 0 .ltoreq. k < p g k = 0 , p .ltoreq. k < M
 1 ( 29 ) ##EQU00022##
[0077] 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.p1(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.
[0078] The p.sup.th modulation of the analysis filter H.sub.p1(z), i.e.
H.sub.p1(zW.sup.p) indicated by reference signs 211 and 213, is depicted
in FIG. 2(b) together with the synthesis filter F.sub.p1(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.p1(z) and reference sign 213 indicates the modulated
version of the originally negative frequency filter H.sub.p1(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.
[0079] 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 uncancelled 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.p1(zW.sup.p),
213, and F.sub.p1(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 p1 will
remain uncancelled in the output signal. An equally strong aliasing
residue will also emerge in the negative frequency range.
[0080] When using complexexponential modulated filter banks, the
complexvalued 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.
[0081] Hence, even when using complexexponential 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 nonconstrained 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, NY, 1992, which is incorporated by reference.
[0082] For improved alias term minimization (IATM) of the prototype
filter, a preferred objective function may be denoted
e.sub.tot(.alpha.)=.alpha.e.sub.t+(1.alpha.)e.sub.a, (30)
where the total error e.sub.tot(.alpha.) 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.j.omega., can be used to provide a measure of
the error energy e.sub.t of the transfer function as
e t = 1 2 .pi. .intg.  .pi. .pi. 1 2 (
A 0 ( e j .omega. ) + A 0 ( e  j
.omega. ) )  P ( .omega. ) e  j .omega.
D 2 d .omega. , ( 31 ) ##EQU00023##
where P(.omega.) is a symmetric realvalued function defining the pass
band and stop band ranges, and D is the total system delay. In other
words, P(.omega.) describes the desired transfer function. In the most
general case, such transfer function comprises a magnitude which is a
function of the frequency .omega.. For a realvalued system Eq.(31)
simplifies to
e t = 1 2 .pi. .intg.  .pi. .pi. ( A 0
( e j .omega. )  P ( .omega. ) e  j
.omega. D 2 d .omega. ( 32 )
##EQU00024##
[0083] The target function P(.omega.) and the target delay D may be
selected as an input parameter to the optimization procedure. The
expression P(.omega.)e.sup.jwD may be referred to as the target transfer
function.
[0084] 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
e a = 1 2 .pi. l = 1 M  1 .intg.  .pi.
.pi. A ~ l ( e j .omega. ) 2 d
.omega. , ( 33 ) ##EQU00025##
[0085] For realvalued systems this translates to
e a = 1 2 .pi. l = 1 M  1 .intg.  .pi.
.pi. A l ( e j .omega. ) 2 d
.omega. . ( 34 ) ##EQU00026##
[0086] 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 .alpha. may be used to distribute the emphasis
between the transfer function and the sensitivity to aliasing of the
prototype filter. While increasing the parameter .alpha. towards 1 will
put more emphasis on the transfer function error e.sub.t, reducing the
parameter .alpha. towards 0 will put more emphasis on the aliasing error
e.sub.a. The parameters P(.omega.) 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.
[0087] 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.
[0088] 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 complexvalued version. This is
because realvalued processing prioritizes faroff aliasing attenuation
to a larger extent than complexvalued 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 nonpresent in the
resulting complexexponential 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 realvalued filter processing. On the other hand, the alias
terms that would carry the main alias terms in a realvalued system would
be modified for complexvalued filter processing. By means of such
partially complex optimization, the benefits of performing the processing
using realvalued processing may be obtained, while still optimizing the
prototype filter for usage in a complex modulated filter bank system.
[0089] 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(.omega.) of Eq.(31), may be chosen as a unit magnitude
constant ranging from .pi./2+.epsilon. to .pi./2.epsilon., where
.epsilon. is a fraction of .pi./2, in order to cover the frequency range
constituting the pass band. Outside the pass band the function P(.omega.)
may be defined to be zero or be left undefined.
[0090] In the latter case, the error energy of the transfer function
Eq.(31) is only evaluated between .pi./2+.epsilon. and .pi./2.epsilon..
Alternatively and preferably, the pass band error e.sub.t could be
calculated over all channels k=0, . . . , M1, from .pi. to .pi. with
P(.omega.) 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, . . . ,
N1) 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
.alpha., a value for the objective function for this iteration step is
calculated. Using semicomplex operations, this comprises the steps:
[0092] 1. To obtain the pass band error e.sub.t, evaluate Eq.(32) with
P(.omega.) being a constant, using
A 0 ( e j .omega. ) = k = 0 M  1 H
k ( e j .omega. ) F k ( e j .omega. )
, ( 35 ) ##EQU00027##
where H.sub.k(e.sup.j.omega.) and F.sub.k(e.sup.j.omega.) 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.
[0093] 2. To obtain the aliasing error e.sub.a, for aliasing terms not
subject to significant aliasing, evaluate
e aReal = 1 2 l = 1 l .noteq. loCut , hiCut
, M  loCut , M  hiCut M  1 .intg.  .pi. .pi.
A l ( e j .omega. ) 2 d .omega.
, ( 36 ) ##EQU00028##
where A.sub.l(e.sup.j.omega.) is calculated as
A l ( e j .omega. ) = k = loCut M  1 
hiCut H k ( e j ( .omega.  2 .pi. M l ) ) F
k ( e j .omega. ) ( 37 ) ##EQU00029##
and H.sub.k(e.sup.j.omega.) and F.sub.k(e.sup.j.omega.) are the DFT
transforms, i.e. the ztransforms 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).
[0094] 3. For the terms subject to significant aliasing, evaluate
e aCplx = 1 2 .pi. l = loCut , hiCut , M
 loCut , M  hiCut .intg.  .pi. .pi. A ~ l
( e j .omega. ) 2 d .omega. ( 38 )
##EQU00030##
where .sub.l(e.sup.j.omega.) is given by Eq.(24), with
A.sub.l(e.sup.j.omega.) as Eq.(37), with H.sub.k(e.sup.j.omega.) and
F.sub.k(e.sup.j.omega.) being the DFT transforms of h.sub.k(n) and
f.sub.k(n) from Eq.(19) and (20).
[0095] 4. The error is subsequently weighted with .alpha. as
e.sub.tot(.alpha.)=.alpha.e.sub.t+(1.alpha.)(e.sub.aReal+e.sub.aCplx).
(39)
[0096] 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.
[0097] 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 .alpha. 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.
[0098] 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).
[0099] 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.
[0100] 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.
[0101] 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
.alpha., a target delay D, a target transfer function P(.omega.), 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.
[0102] 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,
64channel 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 .sub.l, where l=24, 40, are calculated using
realvalued 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.degree., 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.
[0103] 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.
[0104] In an example, the steps for filtering a time domain signal using
the optimized prototype filter may be described as follows: [0105] 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 polyphase
representation, where every other of the polyphase filter coefficients
are negated and all coefficient are timeflipped as
[0105] {acute over (p)}.sub.0(639128mn)=(1).sup.mp.sub.0(128m+n),
0.ltoreq.n<128, 0.ltoreq.m<5 (40) [0106] The analysis stage
begins with the polyphase 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
[0106] x 127  l ( n ) = m = 0 4 p 0 ' (
128 m + l ) x ( 128 m + l + 64 n ) , 0
.ltoreq. l < 128 , n = 0 , 1 , ( 41 ) ##EQU00031##
[0107] x.sub.l(n) is subsequently multiplied with a modulation matrix as
[0107] v k ( n ) = l = 0 127 x l ( n )
exp ( j .pi. 128 ( k + 1 2 ) ( 2 l + 129 ) )
, 0 .ltoreq. k < 64 , ( 42 ) ##EQU00032##
where v.sub.k(n), k=0 . . . 63, constitute the subband signals. The time
index n is consequently given in subband samples. [0108] The
complexvalued subband signals can then be modified, e.g. according to
some desired, possibly timevarying and complexvalued, equalization
curve g.sub.k(n), as
[0108] v.sub.k.sup.(m)(n)=g.sub.k(n)v.sub.k(n), 0.ltoreq.k<64. (43)
[0109] The synthesis stage starts with a demodulation step of the
modified subband signals as
[0109] u l ( n ) 1 64 k = 0 63 Re { v k
( m ) ( n ) exp ( j .pi. 128 ( k + 1 2 ) ( 2
l  255 ) ) } , 0 .ltoreq. l < 128. ( 44 )
##EQU00033##
[0110] 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. [0111] The
demodulated samples are filtered with the polyphase representation of
the prototype filter and accumulated to the output time domain signal
{circumflex over (x)}(n) according to
[0111] {circumflex over (x)}(128m+l+64n)={circumflex over
(x)}(128m+l+64n)+{acute over (p)}.sub.0(639128ml)u.sub.l(l),
0.ltoreq.l<128, 0.ltoreq.m<5,n=0,1, . . . (45)
where {acute over (x)}(n) is set to 0 for all n at startup time.
[0112] 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.
[0113] 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)=1/2[p.sub.0(2i+1)+p.sub.0(2i)], 0.ltoreq.i<230.
[0114] The length of the new prototype filter is hence 320 and the delay
is D=.left brktbot.319/2.right brktbot.=159, where the operator .left
brktbot..cndot..right brktbot. returns the integer part of its
argument.
TABLEUS00001
TABLE 1
Coefficients of a 64 channel low delay prototype filter
n P.sub.0(n)
0 7.949261005955764e4
1 1.232074328145439e3
2 1.601053942982895e3
3 1.980720409470913e3
4 2.397504953865715e3
5 2.838709203607079e3
6 3.314755401090670e3
7 3.825180949035082e3
8 4.365307413613105e3
9 4.937260935539922e3
10 5.537381514710146e3
11 6.164241937824271e3
12 6.816579194002503e3
13 7.490102145765528e3
14 8.183711450708110e3
15 8.894930051379498e3
16 9.620004581607449e3
17 1.035696814015217e2
18 1.110238617202191e2
19 1.185358556146692e2
20 1.260769256679562e2
21 1.336080675156018e2
22 1.411033176541011e2
23 1.485316243134798e2
24 1.558550942227883e2
25 1.630436835497356e2
26 1.700613959422392e2
27 1.768770555992799e2
28 1.834568069395711e2
29 1.897612496482356e2
30 1.957605813345359e2
31 2.014213322475170e2
32 2.067061748933033e2
33 2.115814831921453e2
34 2.160130854695980e2
35 2.199696217022438e2
36 2.234169110698344e2
37 2.263170795250229e2
38 2.286416556008894e2
39 2.303589449043864e2
40 2.314344724218223e2
41 2.318352524475873e2
42 2.315297727620401e2
43 2.304918234544422e2
44 2.286864521420490e2
45 2.260790764376614e2
46 2.226444264459477e2
47 2.183518667784246e2
48 2.131692017682024e2
49 2.070614962636994e2
50 1.999981321635736e2
51 1.919566223498554e2
52 1.828936158524688e2
53 1.727711874492186e2
54 1.615648494779686e2
55 1.492335807272955e2
56 1.357419760297910e2
57 1.210370330110896e2
58 1.050755164953818e2
59 8.785746151726750e3
60 6.927329556345040e3
61 4.929378450735877e3
62 2.800333941149626e3
63 4.685580749545335e4
64 2.210315255690887e3
65 5.183294908090526e3
66 8.350964449424035e3
67 1.166118535611788e2
68 1.513166797475777e2
69 1.877264877027943e2
70 2.258899222368603e2
71 2.659061474958830e2
72 3.078087745385930e2
73 3.516391224752870e2
74 3.974674893613862e2
75 4.453308211110493e2
76 4.952626097917320e2
77 5.473026727738295e2
78 6.014835645056577e2
79 6.578414516120631e2
80 7.163950999489413e2
81 7.771656494569829e2
82 8.401794441130064e2
83 9.054515924487507e2
84 9.729889691289549e2
85 1.042804039148369e1
86 1.114900795290448e1
87 1.189284254931251e1
88 1.265947532678997e1
89 1.344885599112251e1
90 1.426090972422485e1
91 1.509550307914161e1
92 1.595243494708706e1
93 1.683151598707939e1
94 1.773250461581686e1
95 1.865511418631904e1
96 1.959902227114119e1
97 2.056386275763479e1
98 2.154925974105375e1
99 2.255475564993390e1
100 2.357989864681126e1
101 2.462418809459464e1
102 2.568709554604541e1
103 2.676805358910440e1
104 2.786645734207760e1
105 2.898168394038287e1
106 3.011307516871287e1
107 3.125994749246541e1
108 3.242157192666507e1
109 3.359722796803192e1
110 3.478614117031655e1
111 3.598752336287570e1
112 3.720056632072922e1
113 3.842444358173011e1
114 3.965831241942321e1
115 4.090129566893579e1
116 4.215250930838456e1
117 4.341108982328533e1
118 4.467608231633283e1
119 4.594659376709624e1
120 4.722166595058233e1
121 4.850038204075748e1
122 4.978178235802594e1
123 5.106483456192374e1
124 5.234865375971977e1
125 5.363218470709771e1
126 5.491440356706657e1
127 5.619439923555571e1
128 5.746001351404267e1
129 5.872559277139351e1
130 5.998618924353250e1
131 6.123980151490041e1
132 6.248504862282382e1
133 6.372102969387355e1
134 6.494654463921502e1
135 6.616044277534099e1
136 6.736174463977084e1
137 6.854929931488056e1
138 6.972201618598393e1
139 7.087881675504216e1
140 7.201859881692665e1
141 7.314035334082558e1
142 7.424295078874311e1
143 7.532534422335129e1
144 7.638649113306198e1
145 7.742538112450130e1
146 7.844095212375462e1
147 7.943222347831999e1
148 8.039818519286321e1
149 8.133789939828571e1
150 8.225037151897938e1
151 8.313468549324594e1
152 8.398991600556686e1
153 8.481519810689574e1
154 8.560963550316389e1
155 8.637239863984174e1
156 8.710266607496513e1
157 8.779965198108476e1
158 8.846258145496611e1
159 8.909071890560218e1
160 8.968337036455653e1
161 9.023985431182168e1
162 9.075955881221292e1
163 9.124187296760565e1
164 9.168621399784253e1
165 9.209204531389191e1
166 9.245886139655739e1
167 9.278619263447355e1
168 9.307362242659798e1
169 9.332075222986479e1
170 9.352724511271509e1
171 9.369278287932853e1
172 9.381709878904797e1
173 9.389996917291260e1
174 9.394121230559878e1
175 9.394068064126931e1
176 9.389829174860432e1
177 9.381397976778112e1
178 9.368773370086998e1
179 9.351961242404785e1
180 9.330966718935136e1
181 9.305803205049067e1
182 9.276488080866625e1
183 9.243040558859498e1
184 9.205488097488350e1
185 9.163856478189402e1
186 9.118180055332041e1
187 9.068503557855540e1
188 9.014858673099563e1
189 8.957295448806664e1
190 8.895882558527375e1
191 8.830582442418677e1
192 8.761259906419252e1
193 8.688044201931157e1
194 8.611140376567749e1
195 8.530684188588082e1
196 8.446723286380624e1
197 8.359322523144003e1
198 8.268555005748937e1
199 8.174491260941859e1
200 8.077214932837783e1
201 7.976809997929416e1
202 7.873360271773119e1
203 7.766956604639097e1
204 7.657692341138960e1
205 7.545663748526984e1
206 7.430967641354331e1
207 7.313705248813991e1
208 7.193979757178656e1
209 7.071895814695481e1
210 6.947561322714310e1
211 6.821083135331770e1
212 6.692573319585476e1
213 6.562143182387809e1
214 6.429904538706975e1
215 6.295973685335782e1
216 6.160464554756299e1
217 6.023493418727370e1
218 5.885176369189331e1
219 5.745630487304467e1
220 5.604973280717471e1
221 5.463322649085826e1
222 5.320795532569365e1
223 5.177509557831821e1
224 5.033582842235876e1
225 4.889131973708936e1
226 4.744274511088447e1
227 4.599125196114154e1
228 4.453800290341801e1
229 4.308413090599260e1
230 4.163077444128621e1
231 4.017905891818764e1
232 3.873008819361793e1
233 3.728496914938361e1
234 3.584479879275654e1
235 3.441060828393923e1
236 3.298346836739700e1
237 3.156442070098094e1
238 3.015447421741344e1
239 2.875462383794429e1
240 2.736584401802921e1
241 2.598909819775319e1
242 2.462531686198759e1
243 2.327540108460799e1
244 2.194025590645563e1
245 2.062071988727463e1
246 1.931765200055820e1
247 1.803186073942884e1
248 1.676410590306998e1
249 1.551517472268748e1
250 1.428578337203540e1
251 1.307662172525294e1
252 1.188837988250476e1
253 1.072167300568495e1
254 9.577112136322552e2
255 8.455282024161610e2
256 7.355793885744523e2
257 6.280513608528435e2
258 5.229589453075828e2
259 4.203381031272017e2
260 3.202301123728688e2
261 2.226720136600903e2
262 1.277000586069404e2
263 3.534672952747162e3
264 5.435672410526313e3
265 1.413857081863553e2
266 2.257147752062613e2
267 3.073254829666290e2
268 3.861994968092324e2
269 4.623245158508806e2
270 5.356875686113461e2
271 6.062844791918062e2
272 6.741087925238425e2
273 7.391592258255635e2
274 8.014393008412193e2
275 8.609517876186421e2
276 9.177059647159572e2
277 9.717118785672957e2
278 1.022983899423088e1
279 1.071535873159799e1
280 1.117390940373963e1
281 1.160565563647874e1
282 1.201089957775325e1
283 1.238986104503973e1
284 1.274286534385776e1
285 1.307022037585206e1
286 1.337226598624689e1
287 1.364936502000925e1
288 1.390190836588895e1
289 1.413030335001078e1
290 1.433497698594264e1
291 1.451636222445455e1
292 1.467494079461177e1
293 1.481116975400198e1
294 1.492556249421260e1
295 1.501862836334994e1
296 1.509089024309573e1
297 1.514289033634045e1
298 1.517517580141857e1
299 1.518832057448775e1
300 1.518289202172233e1
301 1.515947694390820e1
302 1.511866738705995e1
303 1.506105955209982e1
304 1.498725980913964e1
305 1.489787144055076e1
306 1.479352185844335e1
307 1.467481851768966e1
308 1.454239120021382e1
309 1.439685961257477e1
310 1.423884130127772e1
311 1.406896926563808e1
312 1.388785953623746e1
313 1.369612022106282e1
314 1.349437727408798e1
315 1.328323917411932e1
316 1.306331212230066e1
317 1.283520431992394e1
318 1.259952253813674e1
319 1.235680807908494e1
320 1.210755701624524e1
321 1.185237142283346e1
322 1.159184450952715e1
323 1.132654367461266e1
324 1.105698782276963e1
325 1.078369135648348e1
326 1.050716118804287e1
327 1.022789198651472e1
328 9.946367410320074e2
329 9.663069107327295e2
330 9.378454802679648e2
331 9.092970207094843e2
332 8.807051083640835e2
333 8.521107266503664e2
334 8.235562752947133e2
335 7.950789957683559e2
336 7.667177989755110e2
337 7.385092587441364e2
338 7.104866702770536e2
339 6.826847016140082e2
340 6.551341011471171e2
341 6.278658929544248e2
342 6.009091369370080e2
343 5.742919825387360e2
344 5.480383115198150e2
345 5.221738078737957e2
346 4.967213638808988e2
347 4.717023345307148e2
348 4.471364025371278e2
349 4.230438144160113e2
350 3.994384828552555e2
351 3.763371362431132e2
352 3.537544041600725e2
353 3.317035188016126e2
354 3.101971215825843e2
355 2.892453070357571e2
356 2.688575425197388e2
357 2.490421725219031e2
358 2.298058501129975e2
359 2.111545692324888e2
360 1.930927680100128e2
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[0115] In the following, different aspects of practical implementations
are outlined. Using a standard PC or DSP, realtime operation of a low
delay complexexponential modulated filter bank is possible. The filter
bank may also be hardcoded on a custom chip. FIG. 5(a) shows the
structure for an effective implementation of the analysis part of a
complexexponential 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 polyphase coefficients of the prototype filter 503. The
filtered signals are subsequently combined 504 and in parallel
transformed with a DCTIV 505 and a DSTIV 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.
[0116] An effective implementation of the synthesis part of a low delay
complexexponential modulated system is shown in FIG. 5(b). The subband
samples are first multiplied with complexvalued twiddlefactors, i.e.
complexvalued channel dependent constants, 511, and the real part is
modulated with a DCTIV 512 and the imaginary part with a DSTIV 513
transform. The outputs from the transforms are combined 514 and fed
through the polyphase 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.
[0117] 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
complexexponential modulated banks use pure FFT kernels. Implementations
using a direct matrixvector multiplication are also possible but are
inferior in efficiency.
[0118] 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.
[0119] 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. 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.
[0120] 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 settop 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.
* * * * *