- Research Article
- Open Access
- Published:
Optimized Paraunitary Filter Banks for Time-Frequency Channel Diagonalization
EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 172751 (2010)
Abstract
We adopt the concept of channel diagonalization to time-frequency signal expansions obtained by DFT filter banks. As a generalization of the frequency domain channel representation used by conventional orthogonal frequency-division multiplexing receivers, the time-frequency domain channel diagonalization can be applied to time-variant channels and aperiodic signals. An inherent error in the case of doubly dispersive channels can be limited by choosing adequate windows underlying the filter banks. We derive a formula for the mean-squared sample error in the case of wide-sense stationary uncorrelated scattering (WSSUS) channels, which serves as objective function in the window optimization. Furthermore, an enhanced scheme for the parameterization of tight Gabor frames enables us to constrain the window in order to define paraunitary filter banks. We show that the design of windows optimized for WSSUS channels with known statistical properties can be formulated as a convex optimization problem. The performance of the resulting windows is investigated under different channel conditions, for different oversampling factors, and compared against the performance of alternative windows. Finally, a generic matched filter receiver incorporating the proposed channel diagonalization is discussed which may be essential for future reconfigurable radio systems.
1. Introduction
Motivated by the heterogeneity of today's world of wireless communications—which includes cellular mobile radio systems of the second and third generations and beyond, wireless local and personal area networks, broadband wireless access systems, digital audio and video broadcast, emerging peer-to-peer radio, and so forth—particular attention is given to reconfigurable radio architectures. Essential in this context are radio resource management solutions on the higher layers and the ability to comply with a range of different air interfaces on the physical layer. Devices comprising the logic for handling multiple air interfaces in the form of parallel implementations are widely available. However, in view of the still increasing number of standards, monolithic transceiver architectures are desirable which enable a uniform processing of different signals by means of reconfigurable multipurpose signal processing units.
A major challenge in the design of a universal baseband receiver architecture is posed by the dispersive radio channel. For dealing with signal dispersion, fundamentally different approaches are followed in traditional radios depending on the type of modulation. Receivers for single-carrier signals typically model the channel as a tapped delay line. For known coefficients of the delay line, the information in the transmitted signal can be recovered by means of a matched filtering followed by a sequence detector or using instead an equalizer followed by a simple detector. The complexity of the coefficient estimation and detection schemes increases with the delay dispersion and thus with the number of taps. Orthogonal frequency-division multiplexing (OFDM) can evade the need for complex equalizers in high data rate systems. The cyclic extensions in OFDM signals facilitate a frequency domain representation of the multipath channel in the form of parallel single-tap lines. On the basis of the frequency domain signal description resulting from the block-wise Discrete Fourier Transform (DFT), the signal mapping by multipath channels can be represented as diagonal matrices. This channel diagonalization enables straightforward demodulation and coefficient estimation and has, along with the availability of Fast Fourier Transform (FFT) algorithms, led to the popularity of OFDM.
The aforementioned approach for a simple channel inversion based on a frequency domain description is not limited to OFDM receivers. Single-carrier modulation with frequency domain equalization (FDE) can achieve similar performance as OFDM if a proper cyclic prefix is appended to each block of signals [1]. In [2] the computational complexities of time and frequency domain equalizers are compared and it is shown that FDE is simpler when the length of the stationary channel impulse response exceeds the sample time by a factor of 5 or more. Processing signals without cyclic prefix result in errors at the block boundaries. These errors have a limited impact at sufficiently large block sizes, which makes FDE an interesting alternative for code-division multiple access receivers [3, 4].
The limitations of OFDM receivers and FDE to time-invariant channels and certain signal formats can be overcome by resorting to alternative signal representations. A natural choice for the signal transform is the discrete-time Gabor expansion [5] based on a system of time-frequency (TF) shifted versions of a certain window function. Even though a TF domain channel diagonalization based on such a Gabor expansion is approximative in the general case of time-variant channels and aperiodic signals, for the typical underspread channels encountered in mobile radio scenarios the inherent model error can be limited to a usually acceptable level by choosing an adequate window underlying the signal transform [6].
The transform of discrete-time signals into the TF domain can be accomplished by DFT filter banks, for which similarly efficient FFT-based implementations are available as for plain DFTs [7]. There is plenty of literature on filter bank design in the context of generalized multicarrier/multitone modulation in wireless/wired communications. Replacing the block-wise inverse DFT and DFT in the transmitter and receiver, respectively, by more general filter banks is a way to get rid of the rigid framework of rectangular windows and cyclic prefixes in OFDM systems. Interference between adjacent sub-bands or multicarrier symbols can be avoided, or at least limited, by choosing appropriate transmit pulses. Filter banks for transmission over dispersive channels with limited interchannel and intersymbol interference are designed in [8–14].
The optimization of filter banks for specific objective functions and constraints can sometimes be formulated as a convex optimization (CO) problem [12]. In [15], CO methods are employed for the design of a two-channel multirate filter bank, in [16] for the design of pulse shapes which minimize intercarrier interference due to frequency offsets in OFDM systems, in [17] for finding optimized prototype filters for filtered multitone modulation used in digital subscriber line systems, and in [18] for the design of filter banks for sub-band signal processing under minimal aliasing and induced distortion. Semidefinite programming (SDP), a branch of CO for which efficient numerical solution methods are available, was employed in [19] for the design of a linear phase prototype filter with high stopband attenuation for cosine-modulated filter banks. In [20] two-channel filter banks are optimized under similar criteria by SDP.
In this paper we are not concerned with the design of transmit pulses. Rather, we optimize filter banks in the context of channel diagonalization. We are interested exclusively in paraunitary filter banks, which are related to the concept of tight Gabor frames [21]. The signal transform associated with discrete-time tight Gabor frames fulfills Parseval's identity. This property is crucial for flexible receivers as it lets the correlation between two time domain signals be computed based on the respective TF signal representations. A main concern of this paper is the design of tight Gabor frames facilitating TF domain channel diagonalization with minimal model error for given channel conditions. More specifically, we minimize the mean-squared error (MSE) resulting from the diagonalization of random channels with known second-order statistical properties, complying with the wide-sense stationary uncorrelated scattering (WSSUS) model, with respect to the TF window function. As we showed in [6], window functions minimizing the MSE appearing in the TF domain can be computed by SDP. In this paper we directly focus on the more relevant MSE in the time domain signal. We show that for weak assumptions on the channel statistics, the optimization problem can likewise be turned into a tractable form through semidefinite relaxation. In order to be able to constrain the windows to constitute tight frames, we extend the parameterization of tight Gabor frames presented in [22]. Optimized windows can then be computed off-line for different channel conditions encountered by reconfigurable receivers, such as the generic matched filter-based inner receiver discussed in this paper.
1.1. Outline of This Paper
In Section 2, the mathematical concepts for TF representation and processing of signals are introduced. A parameterization of tight Gabor frames, needed for the constrained optimization in Section 5 is presented in Section 3. In Section 4, TF domain channel diagonalization is discussed, resulting in a certain error in the case of doubly dispersive channels. As shown in Section 5, semidefinite relaxation lets the window design problem be formulated as a CO problem. Numerical results are shown in Section 6 for different channel conditions. In Section 7, a generic matched filter architecture incorporating the channel diagonalization is presented. Finally, conclusions are drawn in Section 8.
1.2. Notation
We enclose the arguments of functions defined on a discrete domain 
in square brackets in order to distinguish them from functions defined on 
. The Hilbert space of the square summable functions 
is denoted as 
, and the associated inner product 
and 
-norm 
are given by 
and 
, respectively, where the asterisk in the superscript denotes complex conjugation. Furthermore, we use 
to denote convolution, and 
for the one-by-one multiplication of two compatible functions 
and 
, that is, 
corresponds to 
for all 
. Vectors and matrices are denoted by boldface characters. The transpose and Hermitian transpose of a matrix 
are denoted as 
and 
, respectively, 
stands for the paraconjugate of a polynomial matrix 
(
is obtained from 
by transposing it, conjugating all of the coefficients of the rational functions in 
, and replacing 
by 
[7].), 
for the trace, and 
denotes the identity matrix of size 
. The 
th element of the 
th row of a matrix 
is represented as 
. Also, 
denotes the expected value, 
and 
represent the real and imaginary parts, respectively, of complex arguments, 
the modulo operation, 
, and 
.
2. DFT Filter Banks and Discrete-Time Gabor Frames
In this section, we introduce signal representation concepts needed subsequently. Some important properties of discrete-time Gabor frames are recapitulated with an emphasis on tight frames and the relationship to DFT filter banks. For more insight into Gabor analysis and filter bank theory the reader is referred to the rich literature, for instance [7, 23–26].
Let 
and 
be two positive integer constants and 
. Given a window function 
, the set
with
is referred to as a Gabor system in 
. The elements of the Gabor system can be associated with the grid points 
of a lattice overlaying the TF plane 
. If there exist two positive constants 
and 
such that
then (1) represents a discrete-time Gabor frame. A necessary condition for (3) is that 
.
For an arbitrary signal 
the inner products of 
with every element of the system (1) form a linear TF representation. In the following, the corresponding transform onto 
is represented by the analysis operator
The mapping (4) can be implemented by a 
-channel DFT (analysis) filter bank with a prototype filter with impulse response 
followed by a down-sampling by a factor 
[21]. Conversely, a synthesis operator 
can be defined based on (1) which maps an arbitrary TF representation 
onto an element of 
according to
The signal synthesis (5) can be implemented by an up-sampling by a factor 
followed by a 
-channel DFT (synthesis) filter bank with a prototype filter with impulse response 
.
If (3) holds with 
then (1) represents a (normalized) tight Gabor frame and 
for all 
. These special Gabor frames obey a generalized Parseval's identity
Furthermore, the inner product 
of any two 
can be computed on the basis of the respective TF representations 
and 
, that is,
Henceforth we assume that (1) represents a tight Gabor frame. We note that the range 
of the operator 
is a subspace of 
, and the mapping 
is an isometry. If 
the operator 
represents the orthogonal projection from 
onto 
. As a direct consequence,
and 
.
Tight Gabor frames are associated with paraunitary DFT filter banks. To enable the design of windows with favorable properties, for instance in regard to TF concentration, it is often necessary to indeed choose 
, resulting in oversampled filter banks. Besides of available efficient implementations of paraunitary DFT filter banks, the properties (6) and (7) are of prime interest for reconfigurable baseband receivers since they allow operations for the signal demodulation, such as signal energy computations and crosscorrelations with reference waveforms, to be performed directly in the TF domain.
3. Parameterization of Tight Gabor Frames
The conditions under which (1) represents a tight Gabor frame can be formulated via the polyphase representation. Let 
denote the least common multiple of 
and 
, and define 
and 
such that 
and 
. The 
-component polyphase representation of the 
-transform 
of the window 
reads
where
Furthermore, the polyphase matrix 
of size 
associated with the DFT filter bank implementing (4) can be expressed as [22]
with 
denoting the DFT matrix of size 
(defined as 
) and
Here, 
is the diagonal matrix with diagonal elements 
. The Gabor system (1) represents a tight frame in 
if and only if the polyphase matrix 
is paraunitary with 
. Or, equivalently, if and only if the polynomial matrix 
is paraunitary with 
, since 
.
We observe that 
if 
, where 
. Consequently, 
is paraunitary if and only if the 
matrices 
of size 
, which comprise the possibly nonzero elements of 
according to 
, are all paraunitary. As follows from (12) the elements of the 
matrices are given as
with 
and 
denoting the Kronecker delta.
Note that if the sequences 
were identical for all column indices 
except for differing offsets, then the factor 
could be omitted in (13) without affecting the condition 
. Replacing some 
by the equivalent 
is a way to align the sequences. Having this in mind, we define 
matrices 
of size 
according to
with the index map
Since the polynomial matrices 
are paraunitary if and only if the modified matrices 
are paraunitary, the Gabor system (1) represents a tight frame in 
if and only if
We note that the size of each polynomial matrix 
, their number 
, and the index map 
are fully determined by 
and 
. Given the latter two constants, any tight Gabor frame is uniquely defined by an instance of 
satisfying (16), where the association of the elements of the 
matrices with the samples of the window 
is defined by (14) and (10). The length of the window is related to the polynomial orders of the matrices 
. We define 
as the maximal polynomial order of the 
matrices plus 
. Thus, in the case 
, all elements of the matrices are scalars, and the support of the representable functions 
is limited to 
. This set is usually not of the form 
for some 
but exhibits "gaps" as illustrated in the example of Figure 1. By increasing 
longer windows can be found.
Support of the window functions 
representable by matrices 
with maximal polynomial order P − 1 for J = 3, L = 4, P = 1, 2.
4. Time-Frequency Channel Diagonalization
The mapping 
of an input signal 
onto the signal 
at the output of a linear time-variant channel can be expressed as
where 
denotes the time-variant impulse response. We consider random channels where 
represents a two-dimensional zero-mean random process complying with the WSSUS model. The second-order statistics of 
are determined by the time correlation function 
and the delay power spectrum 
according to
The delay power spectrum is related to the frequency correlation function 
through
Of interest in the context of TF signal processing is the time-variant transfer function
reflecting the TF selectivity of a channel realization. In a digital receiver a realization of a doubly dispersive channel can be represented by a sampled version 
of 
, defined by
For compatibility with the TF signal representations introduced in Section 2, the sampling intervals 
and 
are chosen in line with those for the Gabor system (1). The time-variant transfer function represents the complex-valued channel gain over time and frequency. Hence, given the TF representation 
of a signal 
at the channel input, it is straightforward to approximate the signal 
at the channel output as
The approximation of a linear operator by 
, that is, a concatenation of an analysis operation, an element-wise multiplication, and a synthesis operation, appears in the literature under the name Gabor multiplier [27]. Such an approximation is suitable for operators that do not involve TF shifts of large magnitude (i.e., underspread operators). Figure 2 shows an implementation of (22) by filter banks, where 
denotes the 
-transform of 
. The TF channel diagonalization offers several advantages. The flexibility in the choice of the sampling intervals 
and 
can be used for the adaptation to different channel conditions or signal formats, or the limitation of the effort for the coefficient estimation in certain receivers. Furthermore, the channel diagonalization facilitates scalable and efficient receiver processing known from OFDM.
TF domain channel diagonalization with 
, 
.
As a result of the sampling of 
the model (22) is usually only approximative, and 
is an approximation of the channel output. The accuracy of 
depends on the channel characteristics and the underlying Gabor frame. We may expect the model error to be limited if every elementary function 
is concentrated around 
in the TF plane such that 
is essentially constant within the sphere of 
. Window functions fulfilling this can be designed for the typical underspread channels encountered in mobile radio scenarios by CO, as shown in Section 5.
The error from the channel diagonalization is given by
In order to remain general in regard to signal and channel properties, we consider the error signal under the assumptions of
-
(i)
a white random signal at the channel input,
-
(ii)
a random channel
complying with the WSSUS model and unit average channel gain (i.e., 
).
complying with the WSSUS model and unit average channel gain (i.e., 
).To formulate the resulting MSE, we introduce the random signal 
being subject to 
and
with 
an even integer. The error signal corresponding to the truncated white random input signal 
reads
The error signal sample energy relative to the unit average sample energy of the desired signal, in the following termed relative mean-squared sample error (RMSSE), can be expressed as
Making use of the above assumptions, the RMSSE can be written as
as shown in the appendix. Having formulated both conditions for the window 
to define a tight Gabor frame (in Section 3) and the error resulting from the channel diagonalization based on 
, we can now turn to window optimization.
5. Window Design
Let us represent the window to be optimized in vector form 
, choosing 
such that 
comprises the support of 
expressed in Section 3. We consider only real-valued windows. Additionally, in order to eventually arrive at a CO problem, we impose the following restrictions on the channel statistics.
-
(i)
The time correlation function is subject to
for all 
, as being the case for two-sided exponentially decaying and many other symmetrical Doppler power spectra. -
(ii)
The frequency correlation function fulfills
for all 
, as, for instance, in the case of exponentially decaying delay power spectra.
for all 
, as being the case for two-sided exponentially decaying and many other symmetrical Doppler power spectra.
for all 
, as, for instance, in the case of exponentially decaying delay power spectra.We note that 
can be expressed as 
and 
as 
with appropriate square matrices 
and 
. As a consequence, the objective function (27) can be expressed in the form
for some 
depending on the support of 
, where 
are real matrices and the constants 
are positive given the above restrictions.
Next, we need to incorporate the constraints under which 
will be minimized. In order to formulate the constraints (16) on the window in the time domain, it is helpful to permute the samples in 
. Let us introduce a window 
of length 
defined as
with 
, 
, 
, and 
. The matrices 
and the samples of the permuted window are related through
With (30) we can now translate the polyphase domain constraints (16) into constraints on the permuted window defined by 
.
(
) Case 
. There are 
constraints of form 
. The 
th diagonal matrix 
of size 
is defined as
with 
. Additionally, there are 
constraints of form 
. The corresponding matrices 
can be defined as the elements of the set resulting from deleting duplicate elements and zero-matrices from
where in (32) we let 
equal zero if either 
or 
.
(
) Case 
. From each of the above-defined matrices 
, 
unique block diagonal matrices of dimension 
are reproduced which contain the original matrix as one of the 
diagonal blocks of dimension 
. Hence, there are 
constraints in total. The constraint matrices are mutually orthogonal in the sense that 
for 
.
We can now formulate the optimization problem in the form
where 
are the matrices resulting from 
by permuting the rows and columns in accordance with (29), and 
. This problem is difficult to tackle for large 
. Let us thus introduce 
and reformulate the optimization problem as
where 
denotes the vector space of symmetric matrices of dimension 
. In (34) we have a convex objective function, however, the set 
is nonconvex. Resorting to semidefinite relaxation, we obtain
with 
denoting that 
is positive semidefinite. Since 
is a convex subset of 
, we now have a CO problem [28]. Having found a matrix 
corresponding to a global minimum of (35), we have two possible cases. If 
, a solution 
of (33) is readily obtainable from 
and the optimal window 
is found through (29). If 
, which we observe in most of the cases, rank reduction methods must be employed. We compute a possibly suboptimal window 
by the following three steps.
-
(i)
In order to reduce the rank to
, we resort to the matrix 
composed by the dominant eigenvector 
of 
, since 
is the matrix nearest to 
in terms of the Frobenius norm [29]. -
(ii)
We translate
into a window 
taking the sample permutation defined in (29) into account. -
(iii)
We finally obtain
by the algorithm [30], which yields a window defining a tight frame and at the same time minimizes the distance to a given window (i.e., 
) in terms of the 
-norm.
, we resort to the matrix 
composed by the dominant eigenvector 
of 
, since 
is the matrix nearest to 
in terms of the Frobenius norm [
into a window 
taking the sample permutation defined in (29) into account.
by the algorithm [
) in terms of the 
-norm.Employing steepest descent methods for solving (35) may result in very slow convergence, whereas alternative methods may not be applicable when the number of dimensions is large. Neglecting the quadratic terms in the objective function leads to the simplified optimization problem
As shown in [6], the linear objective function 
reflects the mean-squared deviation of 
from 
, that is, the model error in the TF domain. Problems of the form (36) are dealt with by SDP, a subfield of CO. For the efficient solution of these optimization problems a number of sophisticated software packages are widely available. However, because generally 
the windows resulting from solving (36) do not minimize the time domain error signal, the magnitude of which determines the performance of the channel diagonalization.
6. Numerical Results
We consider a WSSUS channel with an exponentially decaying delay power spectrum, the sampled version of which reads
with 
denoting the unit step function and 
the root mean-squared (RMS) delay spread [31]. As for the Doppler power spectrum, a two-sided exponentially decaying shape is assumed, which results in the time correlation function
where 
represents the RMS Doppler spread. Since choosing an oversampling factor 
larger than one increases the degrees of freedom in the window design, we restrict our attention to scenarios with 
, involving oversampled filter banks. Figure 3 shows optimized window functions for different channel conditions and their Fourier transforms. The waveforms were obtained numerically by solving (35) using interior point methods [28] for 
, 
, 
amounting to a window length of 240 samples. An RMS delay spread 
of 
samples and an RMS Doppler spread 
of 0.001 
were assumed in Figure 3(a), while 
, 
in Figure 3(b). The two shown optimized windows achieve RMSSEs (27) of 
dB and 
dB. Figures 3(c) and 3(d) show the Fourier transforms of the optimized pulses in (a) and (b), respectively, versus the normalized frequency 
. Obviously, the optimized waveforms become more concentrated in time domain as the Doppler spread increases (see Figure 3(b) versus Figure 3(a)). For increasing Doppler spreads the coherence time of the channel decreases, and the temporal support of the optimized window is reduced in order to limit the RMSSE.
Examples of optimized window functions in time domain ((a) and (b)) and in frequency domain ((c) and (d)) for different channel statistics: 
, 
(in (a) and (c)), 
, 
(in (b) and (d)).
The RMSSEs (27) achievable by optimized windows are shown in Figure 4 for the same lattice constants and similar types of delay/Doppler power spectra. The RMS delay spread 
ranges between 0.5 and 8 samples while the RMS Doppler spread 
equals 0.01 sample
. For every considered 
a window 
was obtained by numerically solving the CO problem (35), and a window 
by solving (36) through SDP, where both approaches required the above-mentioned additional steps for rank reduction. The global minimum of the objective function in (35) at 
, that is prior to the rank reduction, serves as a lower bound in the figure. The offsets of 
and 
from the lower bound reflect the impact of the rank reduction. Additionally, the figure shows the RMSSEs resulting from choosing a window 
with a root-raised-cosine (RRC) shaped magnitude spectrum with width 
and roll-off factor 
. We choose this window function for comparison because it does constitute a tight Gabor frame while exhibiting superior TF localization properties compared to rectangularly shaped windows for instance. Finally, for the verification of 
the signals 
and 
were also obtained by simulations involving filter banks based on the optimized windows 
and random signal and WSSUS channel generators, and the resulting error signal analyzed.
Model errors by windows 
and 
optimized through CO and SDP, respectively, and by window 
with RRC-shaped magnitude spectrum versus 
at 
.
Obviously, solving (35) leads to better windows than solving (36). The considerable offset of the RMSSEs from the lower bound for smaller 
indicates that here the rank reduction has a significant impact on the windows. We observe that rank reduction generally has a limited effect when the delay and Doppler spreads are of similar extent, that is, when in the TF plane the delay spread relative to the sampling interval in time (i.e., 
) is of the same order of magnitude as the Doppler spread relative to the sampling interval in frequency (i.e., 
).
The relatively high RMSSEs found in Figure 4 are a result of the product 
being in the order of 
, a much larger value than encountered in typical mobile radio scenarios. In environments with such severe dispersion in both time and frequency, the model error performance can actually be improved by increasing the oversampling factor 
. This can be seen in Table 1, showing some 
observed under the same conditions as above except for choosing different lattice constants. An RMS delay spread of 
sample is assumed here. The performance clearly improves with the oversampling factor.
7. Generic Matched Filter Receiver
The considered TF channel diagonalization does not rely on a particular signal format, making it suitable for application in multimode receivers [32]. The burst structures defined in the various standards for wireless communications differ substantially. However, commonly the bursts incorporate preamble signals for the channel estimation along with information-bearing signals which are usually subject to a linear modulation scheme. The transmitted baseband signals generally follow the form 
with 
representing 
elementary waveforms, possibly complex exponentials such as in the case of OFDM, or pseudo-noise sequences as in the case of direct-sequence spread-spectrum systems. For performing channel estimation and information recovery the receiver needs to estimate the signals 
on the basis of the known waveforms 
. To this end, the inner receiver correlates the received signal 
with the elementary signals as appearing at the channel output, resulting in
For example, in the case of signal decoding in the presence of additive white Gaussian noise, 
represent a sufficient statistic. In other situations, such as for the channel parameter estimation (Efficient parameter estimators which are applicable in the context of filter bank-based multicarrier transmission are presented in [33].), 
is unknown.
The generic matched filter sketched in Figure 5 aims to compute 
in the TF domain. The TF representation 
of the received signal 
is obtained from an analysis filter bank, while TF representations 
of the elementary waveforms are provided by a local repository [32]. These are mapped to the TF representations 
of 
by means of the channel diagonalization (22) discussed in Section 4. Finally, taking advantage of Parseval's identity, 
is computed for 
.
Generic TF domain matched filter.
The impact of the TF channel diagonalization on the 
th matched filter output can be formulated as
where for obtaining expression (41) we exploit that 
while 
represents the orthogonal projection from 
onto 
. The error signal 
is in line with the error signal definition (25). Under the assumptions that the relation between 
and 
found in Section 4 carries over to the relation between 
and 
, and that 
represents a random signal with 
and 
, the RMSSE 
determines the signal-to-noise ratio 
at the matched filter output. Since the pulse 
is typically a component of 
, the aforementioned assumptions, however, do not hold in general. Nevertheless, 
may in practice serve as a rough characterization of the performance of the matched filter in Figure 5. The performance of the TF domain matched filtering in a reconfigurable receiver architecture configured to the reception of direct-sequence spread-spectrum signals is studied in [32].
8. Conclusions
We have derived paraunitary filter banks facilitating diagonalization of doubly dispersive channels at limited inherent MSE. Making use of a suitable parameterization of tight frames, we have shown that the optimization of paraunitary DFT filter banks for given channel statistics and oversampling factors can be formulated as a CO problem. An investigation of the MSE performance achieved by the optimized windows shows that the windows obtained by CO are more favorable than conventional windows with an RRC spectrum. However, in certain configurations the necessary rank reduction following the CO has a significant impact on the window shapes. The induced potential degradation of the MSE performance may be evaded by choosing appropriate lattice constants 
and 
, specifying the down-sampling factor and the number of sub-bands, respectively, or by alternative rank reduction procedures which are yet to be devised. In general, the MSE performance can be improved at the cost of a higher complexity in terms of numbers of coefficients by increasing the oversampling factor.
In this paper our main concern was mathematical techniques for designing optimized filter banks in the context of channel diagonalization. Reconfigurable radios are clearly a prospective field of application. Since tight frames are natural generalizations of orthonormal bases used for the signal transform in OFDM receivers, the efficient handling of dispersive channels in OFDM can be inherited by receivers not limited to signals with cyclic extensions. Flexible radio architectures which incorporate the channel diagonalization considered in this paper have been investigated within the IST project URANUS (Universal RAdio-link platform for effieNt User-centric accesS) [34]. In this project the performance of such flexible receiver architectures has been studied in the context of different air interfaces and on different levels, from the inner receiver performance with perfect and imperfect channel estimation to the link level performance. While channel diagonalization by means of properly designed filter banks has been shown to have a great potential, there are a number of related issues that need to be addressed on the way to practical solutions, such as adequate channel estimation methods, synchronization, radio resource management, and others. A comparison of the performance of flexible receivers taking advantage of the channel diagonalization as compared to conventional receiver architectures has therefore been out of the scope of this paper.
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Appendix
Appendix
A. Derivation of RMSSE Formula
The RMSSE can be written as
where
and
Both the input signal power and the gain of the channel are normalized to unity, and therefore 
.
Furthermore, 
can be expressed as
where 
. To obtain (A.5) from (A.4) we apply (21), (20), and (24), and to arrive at (A.6) we use (18). Using (2) and (19), 
can now be expressed as
Finally 
can be rewritten as
We use (24) to obtain (A.9) from (A.8), and for the derivation of (A.10), (18) is applied. Thus, the RMSSE is given by
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Ju, Z., Hunziker, T. & Dahlhaus, D. Optimized Paraunitary Filter Banks for Time-Frequency Channel Diagonalization. EURASIP J. Adv. Signal Process. 2010, 172751 (2010). https://doi.org/10.1155/2010/172751
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Keywords
- Filter Bank
- Cyclic Prefix
- Tight Frame
- Gabor Frame
- Frequency Domain Equalization
