- Research Article
- Open Access
Optimized Paraunitary Filter Banks for Time-Frequency Channel Diagonalization
- Ziyang Ju1Email author,
- Thomas Hunziker1 and
- Dirk Dahlhaus1
https://doi.org/10.1155/2010/172751
© Ziyang Ju et al. 2010
- Received: 17 June 2009
- Accepted: 31 December 2009
- Published: 18 January 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.
Keywords
- Filter Bank
- Cyclic Prefix
- Tight Frame
- Gabor Frame
- Frequency Domain Equalization
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


















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






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
- (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.,
).
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




- (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.
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











- (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.
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















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)).
















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.,
).





Model error for different oversampling factors.
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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.








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.
Declarations
Authors’ Affiliations
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