Aliasing reduction using complex-exponential modulated filterbanks
The present invention proposes a new method and apparatus for the improvement of digital filterbanks, by a complex extension of cosine modulated digital filterbanks. The invention employs complex-exponential modulation of a low-pass prototype filter and a new method for optimizing the characteristics of this filter. The invention substantially reduces artifacts due to aliasing emerging from independent modifications of subband signals, for example when using a filterbank as an spectral equalizer. The invention is preferably implemented in software, running on a standard PC or a digital signal processor (DSP), but can also be hardcoded on a custom chip. The invention offers essential improvements for various types of digital equalizers, adaptive filters, multiband companders and spectral envelope adjusting filterbanks used in high frequency reconstruction (HFR) systems.
1. A method for reduction of aliasing emerging from modifications of spectral coefficients or subband signals obtained from a digital filterbank, comprising:
providing a symmetric low-pass prototype filter p 0 (n), having filter order N;
building an M-channel filterbank by complex-exponential modulation of said prototype filter;
where said filter has analysis and synthesis filter coefficients based on
h
k
(
n
)
=
p
0
(
n
)
exp
{
ⅈ
π
2
M
(
2
k
+
1
)
(
n
-
N
2
)
}
,
and n=0 . . . N, and k=0 . . . M−1;
filtering a real-valued time domain signal through an analysis part of said filterbank;
modifying complex-valued subband signals obtained from said filtering;
filtering said modified complex-valued subband signals through the synthesis part of said filterbank; and
taking the real part of a complex-valued time domain output signal, where said output signal is a sum of the signals obtained from said synthesis filtering;
wherein said optimization of said low-pass prototype filter is accomplished by minimizing a composite objective function ε tot (α) as
ε tot (α)=αε t +(1−α)ε a
where α is a weighting constant, ε t is the passband error and ε a is the aliasing error; and wherein said passband error ε t is calculated as
ɛ
t
=
1
4
π
∫
-
π
π
(
|
A
0
(
ⅇ
jω
)
+
A
0
*
(
ⅇ
-
jω
)
|
2
M
-
1
)
2
ⅆ
ω
,
and said aliasing error ε a is calculated as
ɛ
a
=
1
8
π
M
2
∫
-
π
π
|
A
M
2
(
ⅇ
jω
)
+
A
M
2
*
(
ⅇ
-
jω
)
|
2
ⅆ
ω
+
1
4
π
M
2
∑
l
=
1
M
2
-
1
∫
-
π
π
|
A
l
(
ⅇ
jω
)
+
A
M
-
l
*
(
ⅇ
-
jω
)
|
2
ⅆ
ω
where A l (e jω ), l=0 . . . M/2, are the alias gain terms evaluated on the unit circle.
2. A method according to claim 1 , wherein said complex-exponential modulated filterbank is used as the envelope adjusting filterbank in a high frequency reconstruction system.
3. A method according to claim 1 , wherein said complex-exponential modulated filterbank is used as the filters in a digital equalizing system.
4. A method according to claim 1 , wherein said complex-exponential modulated filterbank is used in an adaptive filtering system.
5. A method according to claim 1 , wherein said complex-exponential modulated filterbank is used in a multiband compander system.
6. A method according to claim 1 , wherein said low-pass prototype filter has a filter order N that is higher than 2M−1, where M is the number of channels in said digital filterbank.
7. A method according to claim 1 , wherein said alias gain terms are calculated using gain adjusted versions of said analysis and synthesis filters according to
h k (eq) ( n )= g k h k ( n ) and f k (eq) ( n )= g k f k ( n ),
where g k represents randomized or deterministic real-valued gainfactors, n=0 . . . N, and k=0 . . . M−1.
8. A method according to claim 1 , wherein said low-pass prototype filter has a filter order of at least a number N, where N is chosen so that said composite objective function is smaller than a predefined value.
9. The method of claim 1 , in which the step of modifying the complex-valued subband signals includes a step of changing gains of the subband signals or quantizing the subband signals.