Time-Varying Analysis-SynthesisFilter Banks Iraj Sodagar DavidSarnoffResearchCenter 37.1 Introduction 37.2 Analysis of Time-Varying Filter Banks 37.3 Direct Switching of Filter Banks 37.
Trang 1Iraj Sodagar “Time-Varying Analysis-Synthesis Filter Banks”
2000 CRC Press LLC <http://www.engnetbase.com>.
Trang 2Time-Varying Analysis-Synthesis
Filter Banks
Iraj Sodagar
DavidSarnoffResearchCenter
37.1 Introduction 37.2 Analysis of Time-Varying Filter Banks 37.3 Direct Switching of Filter Banks 37.4 Time-Varying Filter Bank Design Techniques Approach I: Intermediate Analysis-Synthesis (IAS)•Approach II: Instantaneous Transform Switching (ITS)
37.5 Conclusion References
37.1 Introduction
Time-frequency representations (TFR) combine the time-domain and frequency-domain represen-tations into a single framework to obtain the notion of time-frequency TFR offer the time localization
vs frequency localization tradeoff between two extreme cases of time-domain and frequency-domain representations The short-time Fourier transform (STFT) [1,2,3,4,5] and the Gabor transform [6] are the classical examples of linear time-frequency transforms which use time-shifted and frequency-shifted basis functions
In conventional time-frequency transforms, the underlying basis functions are fixed in time and
define a specific tiling of the time-frequency plane The term time-frequency tile of a particular
basis function is meant to designate the region in the plane that contains most of that function’s energy The short-time Fourier transform and the wavelet transform are just two of many possible tilings of the time-frequency plane These two are illustrated in Fig.37.1(a) and (b), respectively In these figures, the rectangular representation for a tile is purely symbolic, since no function can have compact support in both time and frequency Other arbitrary tilings of the time-frequency plane are possible such as the example shown in Fig.37.1(c) In the discrete domain, linear time-frequency transforms can be implemented in the form of filter bank structures
It is well known that the time-frequency energy distribution of signals often changes with time Thus, in this sense, the conventional linear time-frequency transform paradigm is fundamentally mismatched to many signals of interest A more flexible and accurate approach is obtained if the basis functions of the transform are allowed to adapt to the signal properties An example of such
a time-varying tiling is shown in Figure37.1(d) In this scenario, the time-frequency tiling of the transform can be changed from good frequency localization to good time localization and vice versa Time-varying filter banks provide such flexible and adaptive time-frequency tilings
Trang 3FIGURE 37.1: The time-frequency tiling for different time-frequency transforms: (a) The STFT, (b) the wavelet transform, (c) an example of general tiling, and (d) an example of the time-varying tiling
The concept of time varying (or adaptive) filter banks was originally introduced in [7] by Nayebi
et al The ideas underlying their method were later developed and extended to a more general case
in which it was also shown that the number of frequency bands could also be made adaptive [8,9,
10,11] De Queiroz and Rao [12] reported time-varying extended lapped transforms and Herley et
al [13,14,15] introduced another time-domain approach for designing time-varying lossless filter banks Arrowood and Smith [16] demonstrated a method for switching between filter banks using lattice structures In [17], the authors presented yet another formulation for designing time-varying filter banks using a different factorization of the paraunitary transform Chen and Vaidyanathan [18] reported a noncausal approach to time-varying filter banks by using time-reversed filters Phoong and Vaidyanathan [19] studied time-varying paraunitary filter banks using polyphase approach
In [11,20,21,22], the post filtering technique for designing time-varying filter bank was reported The design of multidimensional time-varying filter bank was addressed in [23,24] In this article,
we introduce the notion of the time-varying filter banks and briefly discuss some design methods
37.2 Analysis of Time-Varying Filter Banks
Time-varying filter banks are analysis-synthesis systems in which the analysis filters, the synthesis filters, the number of bands, the decimation rates, and the frequency coverage of the bands are changed (in part or in total) in time, as is shown in Fig.37.2 By carefully adapting the analysis section to the temporal properties of the input signal, better performance can be achieved in processing the signal In the absence of processing errors, the reconstructed outputˆx(n) should closely approximate
a delayed version of the original signalx(n) When ˆx(n − 1) = x(n) for some integer constant, 1,
then we say that the filter bank is perfectly reconstructing (PR) The intent of the design is to choose the time-varying analysis and synthesis filters along with the time-varying down/up samplers so that the system requirements are met subject to the constraint that the analysis-synthesis filter bank be
PR at all times
Trang 4FIGURE 37.2: The time-varying filter bank structure with time-varying filters and time-dependent down/up samplers
One general method for analysis of time-varying filter banks is the time-domain formulation reported in [10,22] In this method, the time-varying impulse response of the entire filter bank is derived in terms of the analysis and synthesis filter coefficients
Figure (37.3) shows the diagram of a time-varying filter bank In this figure, the filter bank is divided into three stages: the analysis filters, the down/up samplers, and the synthesis filters The signalsx(n) and ˆx(n) are the filter bank input and output at time n, respectively The outputs
of the analysis filters are shown by v(n) = [v0(n), v1(n), , v M(n)−1 (n)] T, wherev i (n) is the
output of theith analysis filter at time n The outputs of the down/up samplers at time n is called
w(n) = [w0(n), w1(n), , w M(n)−1 (n)] T.
FIGURE 37.3: Time-varying filter bank as a cascade of analysis filters, down/up samplers, and synthesis filters
The input/output relation of the analysis filters can be expressed by
P(n) is an M(n) × N(n) matrix whose mth row is comprised of the coefficients of the mth analysis
filter at timen and x N (n) is the input vector of length N(n) at time n:
xN (n) = [x(n), x(n − 1), x(n − 2), , x(n − N(n) + 1)] T (37.2) The input/output function of down/up samplers can be expressed in the form
where3(n) is a diagonal matrix of size M(n) × M(n) The mth diagonal element of 3(n), at time
n, is 1 if the input and output of the mth down/up sampler are identical, otherwise it is zero.
Trang 5To write the input/output relationship of the synthesis filters, Q(n) is defined as
Q(n) =
g0(n, 0) g0(n, 1) g0(n, 2) g0(n, N(n) − 1)
g1(n, 0) g1(n, 1) g1(n, 2) g1(n, N(n) − 1)
g2(n, 0) g2(n, 1) g2(n, 2) g2(n, N(n) − 1)
g M(n)−1 (n, 0) g M(n)−1 (n, 1) g M(n)−1 (n, 2) g M(n)−1 (n, N(n) − 1)
= q0(n) q1(n) q2(n) q N(n)−1 (n) (37.4)
where qi (n) = [g0(n, i), g1(n, i), g2(n, i), , g M(n)−1 (n, i)] T, is a vector of lengthM(n) and
g i (n, j) denotes the jth coefficient of the ith synthesis filter At time n, the mth synthesis filter
is convolved with vector[w m (n), w m (n − 1), , w m (n − N(n) + 1)] T and all outputs are added
together Using Eq (37.4), the output of the filter bank at timen can be written as:
ˆx(n) =
N(n)−1X
i=0
qT
If s(n) and ˆw(n) are defined as
s(n) =hqT
0(n), q T
1(n), q T
2, , q T
N(n)−1 (n)iT (37.6)
ˆw(n) =hwT (n), w T (n − 1), w T (n − 2), , w T (n − N(n) + 1)iT , (37.7) then Eq (37.5) can be written in the form of one inner product,
where s(n) and ˆw(n) are vectors of length N(n)M(n) Using Eqs (37.1), (37.3), (37.7), and (37.8), the input/output function of the filter bank can be written as:
ˆx(n) = s T (n)
3(n) P(n) x N (n)
3(n − 1) P(n − 1) x N (n − 1)
3(n − 2) P(n − 2) x N (n − 2)
3(n − N(n) + 1) P(n − N(n) + 1) x N (n − N(n) + 1)
As the lastN(n) − 1 elements of vector x N (n − i) are identical to the first N(n) − 1 elements of
vector xN (n − i − 1), the latter equation can be expressed by
ˆx(n) = s T (n)
3(n) P(n) O O
3(n − 1) P(n − 1) O O
3(n − 2) P(n − 2) O O
O O 3(n − N(n) + 1) P(n − N(n) + 1)
x(n) x(n − 1) x(n − 2)
x(n − 2N(n) + 1)
(37.10)
Trang 6where O is the zero column vector with lengthM(n) Thus, the input/output function of a
time-varying filter bank can be expressed in the form of
where xI (n) = [x(n), x(n − 1), , x(n − I + 1)] T andI (n) = 2N(n) − 1 and z(n) is the
time-varying impulse response vector of the filter bank at timen:
The matrix A(n) is the [2N(n) − 1] × [N(n) M(n)] matrix
A(n) =
P(n) T 3(n)
OT
OT
.
OT
OT
P(n − 1) T 3(n − 1)
OT
.
OT
OT
.
OT
P(n − N(n) + 1) T 3(n − N(n) + 1)
.
(37.13) For a perfect reconstruction filter bank with a delay of1, it is necessary and sufficient that all elements
but the(1 + 1)th in z(n) be equal to zero at all times The (1 + 1)th entry of z(n) must be equal to
one If the ideal impulse response is b(n), the filter bank is PR if and only if
A(n) s(n) = b(n) for alln. (37.14)
37.3 Direct Switching of Filter Banks
Changing from one arbitrary filter bank to another independently designed filter bank without using
any intermediate filters is called direct switching Direct switching is the simplest switching scheme
and does not require additional steps in switching between two filter banks But such switching will result in a substantial amount of reconstruction distortion during the transition period This is because during the transition, none of the synthesis filters satisfies the exact reconstruction conditions Figure (37.4) shows an example of a direct switching filter bank Figure (37.5) shows the time-varying impulse response of the above system around the transition periods In this figure,z(n, m) is the
response of the system at timen to the unit input at time m For a PR system, z(n, m) has a height
of 1 along the diagonal and 0 everywhere else in the(m, n)-plane As is shown, the time-varying
filter bank is PR before and after but not during the transition periods In this case, each switching operation generates a distortion with an 8-sample duration One way to reduce the distortion is to switch the synthesis filters with an appropriate delay with respect to the analysis switching time This delay may reduce the output distortion, but it can not eliminate it
37.4 Time-Varying Filter Bank Design Techniques
The basic time-varying filter bank design methods are summarized in Table37.1 These techniques can be divided into two major approaches which are briefly described in the following sections
Trang 7FIGURE 37.4: Block diagram of a time-varying analysis/synthesis filter bank that switches between
a two- and three-band decomposition
TABLE 37.1 Comparison of Time-Varying Filter Bank Different Designing Methods
Intermediate Changing freq Filter bank Computational analysis resolution requirement complexity
de Queiroz
Intermediate Gopinath
transform Redesigning
37.4.1 Approach I: Intermediate Analysis-Synthesis (IAS)
In the first approach, both analysis and synthesis filters are allowed to change during the transition period to maintain perfect reconstruction We refer to this approach as the intermediate analysis-synthesis (IAS) approach
In [16], the authors have chosen to start with the lattice implementation of time-invariant two-band filter banks, originally proposed by Vaidyanathan [25] for time-invariant case Consider the lattice structure shown in Fig.37.6 Figure37.6(a) represents a lossless two-band analysis filter bank, consisting ofJ + 1 lattice stages The corresponding synthesis filter bank is shown in Fig.37.6(b) As
is shown, for each stage in the analysis filter bank, there exists a corresponding stage in the synthesis filter bank with similar, but inverse functionality As long as each two corresponding lattice stages in the analysis and synthesis sections are PR, the overall system is PR To switch one filter bank to another, the lattice stages of the analysis section are changed from one set to another If the corresponding lattice stages of the synthesis section are also changed according to the changes of the analysis section, the PR property will hold during transition Due to the existence of delay elements, any change in the analysis section must be followed with the corresponding change in the synthesis section, but with
an appropriate delay For example, the parameterα j of the analysis and synthesis filter banks can
Trang 8FIGURE 37.5: The time-varying impulse response for direct switching between the two- and the three-band system The filter bank is switched from the two-band to the three-band at timen = 0
and switched back at timen = 13 (a) Surface plot, (b) contour plot.
be changed instantaneously But any change in parameterα j−1in the analysis filter bank must be followed with the similar change in the synthesis filter bank after one sample delay Because of such delays, switching between two PR filter banks can occur only by going through a transition period in which both analysis and synthesis filter banks are changing in time
In [12,26], the design of time-varying extended lapped transform (ELT) [27,28] was reported The extended lapped transform is a cosine-modulated filter bank with an additional constraint on the filter lengths Here, the design procedure is based on factorization of the time-domain transform matrix into permutation and rotation matrices As the ELT is paraunitary, the inverse transform can be obtained by reversing the order of the matrix multiplication Since any orthogonal transform
is a succession of plane rotations, any changes in these rotation angles result in changing the filter bank without losing the orthogonality property The authors derived a general frame work for
M-band ELT transforms compared to the two-M-band case approach in [16] This method parallels the lattice technique [16] except with the mild modification of imposing the additional ELT constraints
In [17], the authors presented yet another formulation for designing time-varying filter banks In this paper, a different factorization of the paraunitary transform has been shown which is not based
on plane rotations unlike the ones in [12,26] Using this factorization, a paraunitary filter bank can be implemented in the form of some cascade structures Again, to switch one filter bank to
Trang 9FIGURE 37.6: The block diagram of a two-band paraunitary filter bank in lattice form: (a) analysis lattice, (b) synthesis lattice
another, the corresponding structures in the analysis and synthesis filter bank are changed similarly but with an appropriate delay If the orthogonality property in each cascade structure is maintained, the time-varying filter bank remains PR This formulation is very similar to the ones in [12,16,26], but represent a more general form of factorization In fact, all above procedures consider similar frameworks of structures that inherently guarantee the exact reconstruction
Herley et al [13,14,15,29] introduced a time-domain method for designing time-varying pa-raunitary filter banks In this approach, the time-invariant analysis transforms do not overlap As a simple example, consider the case of switching between two paraunitary time-invariant filter banks The analysis transform around the transition period can be written as
T=
PT
The matrices P1and P2represent paraunitray transforms and therefore are unitary matrices Their
nonzero columns also do not overlap with each other The matrix PT represents the analysis filter
bank during the transition period In order to find this filter bank, the matrix PT is initially replaced
with a zero matrix Then, the null space of the transform T is found Any matrix that spans this subspace can be a candidate vector for PT By choosing enough independent vectors of this null space and applying the Gram-Schimidt procedure to them, an orthogonal transform can be selected
for PT This method has also been applied to time-varying modulated lapped transforms [24] and two-dimensional time-varying paraunitary filter banks [30]
The basic property of all above procedures is the use of intermediate analysis transforms in the transition period The characteristics of these analysis transforms are not easy to control and typically the intermediate filters are not well-behaved
Trang 1037.4.2 Approach II: Instantaneous Transform Switching (ITS)
In the second approach, the analysis filters are switched instantaneously and time-varying synthesis filters are used in the transition period We refer to this approach as the instantaneous transform switching (ITS) approach In the ITS approach, the analysis filter bank may be switched to another set of analysis filters arbitrarily This means that the basis vectors and the tiling of the time-frequency plane can be changed instantaneously To achieve PR at each time in the transition period, a new synthesis section is designed to ensure proper reconstruction
In the least squares (LS) method [10], for any given set of analysis filters, a LS solution of Eq (37.14) can be used to obtain the “best” synthesis filters of the corresponding system (inL2 norm):
The advantage of the LS approach is that there is no limitation on the number of analysis filter banks that can be used in the system The disadvantage of the LS method is that it does not achieve PR However, experiments have shown that the reconstruction is significantly improved in this method compared to direct switching [10]
In the LS solution, b(n) is projected onto the column space of A(n) For PR, the projection error
should be zero Thus, to obtain time-varying PR filter banks, the reconstruction error,||A(n)s(n) −
b(n)||2, can be brought to zero with an optimization procedure The optimization operates on the
analysis filter coefficients and modifies the range space of A(n) until b(n) ∈ range(A(n)) Although
the s(n)’s at different states are independent of each other, since the A(n)’s have some common
elements, optimization procedures should be applied to all analysis sections at the same time This method is referred to as “redesigning analysis” [10]
The last ITS method, post filtering, uses conventional filter banks with time-varying coefficients followed by a time-varying post filter The post filter provides exact reconstruction during transition periods, while it operates as a constant delay elsewhere Assume at timen0the time-varying filter bank is switched from the first filter bank to the second If the length of the transition period isL
samples, the output of the filter bank in the interval[n0 , n0+L−1] is distorted because of switching.
The post filter removes this distortion The block diagram of such a system is shown in Fig (37.7)
In this figure, z(n) and y(n) are the analysis/synthesis filter bank and post filter impulse responses,
FIGURE 37.7: The block diagram of time-varying filter bank and post filter
respectively If the delays of the filter bank and the post filter are denoted1 and 2, respectively, we
can write
ˆx(n) =
Distorted ifn0≤ n < n0 + L
The desired output of the post filter is