Optimized Paraunitary Filter Banks for Time-Frequency Channel Diagonalization

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.

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

(1)

with

(2)

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

(3)

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

(4)

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

(5)

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

(6)

Furthermore, the inner product of any two can be computed on the basis of the respective TF representations and , that is,

(7)

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,

(8)

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.

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

(9)

where

(10)

Furthermore, the polyphase matrix of size associated with the DFT filter bank implementing (4) can be expressed as [22]

(11)

with denoting the DFT matrix of size (defined as ) and

(12)

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

(13)

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

(14)

with the index map

(15)

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

(16)

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.

The mapping of an input signal onto the signal at the output of a linear time-variant channel can be expressed as

(17)

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

(18)

The delay power spectrum is related to the frequency correlation function through

(19)

Of interest in the context of TF signal processing is the time-variant transfer function

(20)

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

(21)

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

(22)

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.

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

(23)

To formulate the resulting MSE, we introduce the random signal being subject to and

(24)

with an even integer. The error signal corresponding to the truncated white random input signal reads

(25)

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

(26)

Making use of the above assumptions, the RMSSE can be written as

(27)

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.

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.

1. (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.

    
2. (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

(28)

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

(29)

with , , , and . The matrices and the samples of the permuted window are related through

(30)

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

(31)

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

(32)

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

(33)

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

(34)

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

(35)

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.

1. (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].

    
2. (ii)

   We translate into a window taking the sample permutation defined in (29) into account.

    
3. (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

(36)

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.

We consider a WSSUS channel with an exponentially decaying delay power spectrum, the sampled version of which reads

(37)

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

(38)

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.

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.

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.

		Oversampling factor
			dB
			dB
			dB
			dB
			dB
			dB

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

(39)

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 .

The impact of the TF channel diagonalization on the th matched filter output can be formulated as

(40)

(41)

(42)

(43)

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].

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.