Research Article Optimized Paraunitary Filter Banks for Time-Frequency Channel Diagonalization

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.


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 blockwise 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.

EURASIP Journal on Advances in Signal Processing
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 codedivision multiple access receivers [3,4].
The limitations of OFDM receivers and FDE to timeinvariant 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]basedonasystemoftime-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 timevariant 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][9][10][11][12][13][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] twochannel 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.

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.

Notation.
We enclose the arguments of functions defined on a discrete domain Λ in square brackets in order to distinguish them from functions defined on R n . The Hilbert space of the square summable functions f : Λ → C is denoted as L 2 (Λ), and the associated inner product f , g and L 2 -norm f are given by i∈Λ f [i]g * [i] and f , f , 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 f and g, that is, for all i ∈ Λ. Vectors and matrices are denoted by boldface characters. The transpose and Hermitian transpose of a matrix X are denoted as X T and X H , respectively, X(z) stands for the paraconjugate of a polynomial matrix X(z) ( X(z) is obtained from X(z) by transposing it, conjugating all of the coefficients of the rational functions in X(z), and replacing z by z −1 [7].), tr(·) for the trace, and I N denotes the identity matrix of size N. The nth element of the mth row of a matrix X is represented as [X] m,n . Also, E[·] denotes the expected value, R(·) and I(·) represent the real and imaginary parts, respectively, of complex arguments, mod the modulo operation, j √ −1, and x max{n ∈ Z : n ≤ x}.

DFT Filter Banks and Discrete-Time Gabor Frames
In this section, we introduce signal representation concepts needed subsequently. Some important properties of discretetime 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][24][25][26].
Let N and K be two positive integer constants and Λ Z × {0, . . . , K − 1}. Given a window function g ∈ L 2 (Z), the set is referred to as a Gabor system in L 2 (Z). The elements of the Gabor system can be associated with the grid points {( N, 2πm/K) : ( , m) ∈ Λ} of a lattice overlaying the TF plane Z × [0, 2π). If there exist two positive constants A 0 and B 0 such that then (1) represents a discrete-time Gabor frame. A necessary condition for (3) is that N/K ≤ 1.
For an arbitrary signal x ∈ L 2 (Z) the inner products of x[k] with every element of the system (1) form a linear TF representation. In the following, the corresponding transform onto L 2 (Λ) is represented by the analysis operator The mapping (4) can be implemented by a K-channel DFT (analysis) filter bank with a prototype filter with impulse response g * [−k] followed by a down-sampling by a factor N [21]. Conversely, a synthesis operator G * can be defined based on (1) which maps an arbitrary TF representation Y ∈ L 2 (Λ) onto an element of L 2 (Z) according to The signal synthesis (5) can be implemented by an upsampling by a factor N followed by a K-channel DFT (synthesis) filter bank with a prototype filter with impulse response g [k]. If (3) holds with A 0 = B 0 = 1 then (1) represents a (normalized) tight Gabor frame and G * (Gx) = x for all x ∈ L 2 (Z). These special Gabor frames obey a generalized Parseval's identity Furthermore, the inner product x, y of any two x, y ∈ L 2 (Z) can be computed on the basis of the respective TF representations Gx and Gy, that is, Henceforth we assume that (1) represents a tight Gabor frame. We note that the range F g {(Gx) [k] : x ∈ L 2 (Z)} of the operator G is a subspace of L 2 (Λ), and the mapping G : L 2 (Z) → F g is an isometry. If N/K < 1 the operator GG * represents the orthogonal projection from L 2 (Λ) onto F g . As a direct consequence, and g 2 = N/K. 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 N < K, 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.

Parameterization of Tight Gabor Frames
The conditions under which (1) represents a tight Gabor frame can be formulated via the polyphase representation. Let M denote the least common multiple of N and K, and define L and J such that where Furthermore, the polyphase matrix G P (z) of size K × N associated with the DFT filter bank implementing (4) can be expressed as [22] 4 EURASIP Journal on Advances in Signal Processing with F K denoting the DFT matrix of size K (defined as Here As follows from (12) the elements of the B matrices are given as (m + jL − 1)δ m+ jL,n+ J and δ i, j denoting the Kronecker delta.
Note that if the sequences ( f (m, n)/J ) m=1,...,L were identical for all column indices n = 1, . . . , J except for differing offsets, then the factor z − f (m,n)/J could be omitted in (13) Replacing some R m (z L ) by the equivalent z −L R M+m (z L ) is a way to align the sequences. Having this in mind, we define B matrices W 0 (z), . . . , with the index map Since the polynomial matrices V 0 (z), . . . , V B−1 (z) are paraunitary if and only if the modified matrices W 0 (z), . . . , W B−1 (z) are paraunitary, the Gabor system (1) represents a tight frame in L 2 (Z) if and only if We note that the size of each polynomial matrix W b (z), their number B, and the index map f (m, n) are fully determined by N and K. Given the latter two constants, any tight Gabor frame is uniquely defined by an instance P = 1: Figure 1: Support of the window functions g[k] representable by (16), where the association of the elements of the B matrices with the samples of the window g[k] is defined by (14) and (10). The length of the window is related to the polynomial orders of the matrices W 0 (z), . . . , W B−1 (z). We define P as the maximal polynomial order of the B matrices plus 1. Thus, in the case P = 1, all elements of the matrices are scalars, and the support of the for some a 0 ≤ b 0 but exhibits "gaps" as illustrated in the example of Figure 1. By increasing P longer windows can be found.

Time-Frequency Channel Diagonalization
The mapping H : at the output of a linear time-variant channel can be expressed as The delay power spectrum is related to the frequency 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 H[ , m] of C H (k, ω), defined by . .
Synthesis filter bank .
. . For compatibility with the TF signal representations introduced in Section 2, the sampling intervals N and 2π/K are chosen in line with those for the Gabor system (1 The approximation of a linear operator by G * (H G(·)), that is, a concatenation of an analysis operation, an elementwise 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 G(z) denotes the z-transform of g [k]. The TF channel diagonalization offers several advantages. The flexibility in the choice of the sampling intervals N and 2π/K 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. As a result of the sampling of C H (k, ω) the model (22) is usually only approximative, and y[k] is an approximation of the channel output. The accuracy of y[k] depends on the channel characteristics and the underlying Gabor frame. We may expect the model error to be limited if every elementary function g ,m [k] is concentrated around ( N, 2πm/K) in the TF plane such that C H (k, ω) is essentially constant within the sphere of g ,m [k]. 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, with Q an even integer. The error signal corresponding to the truncated white random input signal 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 g[k] to define a tight Gabor frame (in Section 3) and the error resulting from the channel diagonalization based on g[k], we can now turn to window optimization.

Window Design
Let us represent the window to be optimized in vector form g [g[a 0 ] · · · g[b 0 ]] T , choosing a 0 , b 0 ∈ Z such that [a 0 , b 0 ] comprises the support of g[k] 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 φ t [ ] ≥ 0 for all ∈ Z, as being the case for two-sided exponentially decaying and many other symmetrical Doppler power spectra. We note that | g, g ,m | 2 can be expressed as (g T R(C ,m )g) 2 + (g T I(C ,m )g) 2 and R( (g * S delay ) φ t , g ) as g T R(D 0 )g with appropriate square matrices C ,m and D 0 . As a consequence, the objective function (27) can be expressed in the form for some F ∈ N depending on the support of g[k], where C 1 , . . . , C F , D are real matrices and the constants c 1 , . . . , c F are positive given the above restrictions. Next, we need to incorporate the constraints under which ε RMSSE (g) will be minimized. In order to formulate the constraints (16) where in (32) where C 1 , . . . , C F , D are the matrices resulting from C 1 , . . . , C F , D by permuting the rows and columns in accordance with (29), and d 0} is a convex subset of S T , we now have a CO problem [28]. Having found a matrix H 0 ∈ S T corresponding to a global minimum of (35), we have two possible cases. If rank(H 0 ) = 1, a solution h 0 of (33) is readily obtainable from h 0 h T 0 = H 0 and the optimal window g CO [k] is found through (29). If rank(H 0 ) > 1, which we observe in most of the cases, rank reduction methods must be employed. We compute a possibly suboptimal window g CO [k] by the following three steps.
(i) In order to reduce the rank to 1, we resort to the matrix H 0 = (N/K)v 0 v H 0 composed by the dominant eigenvector v 0 of H 0 , since H 0 is the matrix nearest to H 0 in terms of the Frobenius norm [29].
(ii) We translate √ N/Kv 0 into a window g [k] taking the sample permutation defined in (29) into account.
(iii) We finally obtain g CO [k] 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., g [k]) in terms of the L 2 -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

Numerical Results
We consider a WSSUS channel with an exponentially decaying delay power spectrum, the sampled version of which reads with u(q) denoting the unit step function and τ RMS 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 ν RMS represents the RMS Doppler spread. Since choosing an oversampling factor K/N larger than one increases the degrees of freedom in the window design, we restrict our attention to scenarios with K > N, 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 N = 24, K = 32, P = 2 amounting to a window length of 240 samples. An RMS delay spread τ RMS of 3 samples and an RMS Doppler spread ν RMS of 0.001 samples −1 were assumed in Figure 3(a), while τ RMS = 3, ν RMS = 0.01 in Figure 3 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. 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 τ RMS ranges between 0.5 and 8 samples while the RMS Doppler spread ν RMS equals 0.01 samples −1 . For every considered τ RMS a window g CO [k] was obtained by numerically solving the CO problem (35), and a window g SDP [k] 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 H = H 0 , that is prior to the rank reduction, serves as a lower bound in the figure. The offsets of ε RMSSE (g CO ) and ε RMSSE (g SDP ) from the lower bound reflect the impact of the rank reduction. Additionally, the figure shows the RMSSEs resulting from choosing a window g RRC [k] with a root-raised-cosine (RRC) shaped magnitude spectrum with width 2π/K and roll-off factor K/N − 1. 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 ε RMSSE (g CO ) the signals y = H x and y = G * (H (Gx)) were also obtained by simulations involving filter banks based on the optimized windows g CO [k] and random signal and WSSUS channel generators, and the resulting error signal analyzed.
Obviously, solving (35) leads to better windows than solving (36). The considerable offset of the RMSSEs from the lower bound for smaller τ RMS 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., τ RMS /N) is of the same order of magnitude as the Doppler spread relative to the sampling interval in frequency (i.e., ν RMS /K −1 ).
The relatively high RMSSEs found in Figure 4 are a result of the product τ RMS ν RMS being in the order of 10 −2 , 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 K/N. This can be seen in Table 1, showing some ε RMSSE (g CO ) observed under the same conditions as above except for choosing different lattice constants. An RMS delay spread of 1 sample is assumed here. The performance clearly improves with the oversampling factor.

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 8 EURASIP Journal on Advances in Signal Processing For example, in the case of signal decoding in the presence of additive white Gaussian noise, u 1 , . . . , u I 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].), H is unknown.  (27)) ε RMSSE (g CO ) (simulations) Global minimum prior to rank reduction The impact of the TF channel diagonalization on the ith matched filter output can be formulated as where for obtaining expression (41) we exploit that R ∈ F g while GG * represents the orthogonal projection from L 2 (Λ) onto F g . , the aforementioned assumptions, however, do not hold in general. Nevertheless, ε RMSSE (g) 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].

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 N and K, specifying the downsampling 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.

A. Derivation of RMSSE Formula
The RMSSE can be written as Both the input signal power and the gain of the channel are normalized to unity, and therefore ϕ 2 = 1.