1. Introduction

Multirate systems, that is, digital systems with signals of different sampling rates, have wide applications in control [1], signal processing [2], communications, econometrics and numerical mathematics. There are several reasons for this [3]: (1) in large scale multivariable digital systems, often it is unrealistic, or sometimes impossible, to sample all physical signals uniformly at one single rate; in such situations, one is forced to use multirate sampling; (2) multirate systems can often achieve objectives that cannot be achieved by single-rate systems. The study of multirate systems goes back to late 1950s [4]. A renaissance of research in multirate systems has occurred since 1980 in the control, signal processing, and communications communities.

(i)In the control community, two directions of research stand out: first, using multirate control to achieve what single rate control cannot as well as the limitations of doing this, and second, the optimal design of multirate controllers [1].(ii)The driving force for studying multirate systems in signal processing comes from the need of sampling rate conversion, subband coding, and their ability to generate wavelets [2].(iii)In the communications community, multirate sampling is used for blind system identification and equalization.

Intersymbol interference (ISI) is a common problem in telecommunication systems, such as terrestrial television broadcasting, digital data communication systems, and cellular mobile communication systems. Usually the channel distortion results in ISI, which, if left uncompensated, causes high error rates. The solution to the ISI problem is to design a receiver that employs a means for compensating or reducing the ISI in the received signal. The compensator for the ISI is the so-called equalizer. For these problems, the filterbank approach [5–10] has gained practical interest recently.

The multirate filterbank-based transceiver model was proposed in [7] as a unifying framework able to encompass existing modulations and equalization schemes. Further in [9], the multirate filterbank-based transceiver model, as shown in Figure 1, was studied in detail. The filterbank-based transceiver introduces transmitter redundancy using filterbank precoders and generalizes existing modulations including OFDM, DMT, TDMA, and CDMA schemes encountered with single- and multiuser communications [9]. Recently in [8], based on the filterbank approach, the optimal channel equalization is studied: an optimal receiver filterbank, which is a modified Kalman filter, is obtained.

Figure 1: Multirate discrete-time baseband equivalent transmitter/channel/receiver model.

Motivated by the generality and importance of filterbank-based transceivers, this paper focuses on frequency-domain analysis and optimal design issues for filterbank-based transceivers. The contributions of this work are as follows.

(i)Contrary to a time-domain study [9], our work is developed along the line of frequency-domain analysis. Frequency-domain models are obtained using the blocking technique. Based on such models, an composite error criterion is proposed to quantify the degree to perfect reconstruction. This frequency-domain criterion captures the traditional distortions.(ii)In the transceiver design, we apply and incorporate the model matching methodology [11]: instead of designing for perfect reconstruction, we design for close to perfect reconstruction with effective band separation. This idea was explored in filterbanks [11] and transmultiplexers [12]. Under some mild condition, one can always get arbitrarily close to perfect reconstruction by trading off filter complexity and time delay in reconstruction.(iii)Control of filter stopband energy is incorporated in our design. This is important, since narrowband noise could induce serious impairment due to poor stopbands of filters involved [13]: if the receiving filters have poor stopband attenuation, all the neighboring bands would be affected when there is a strong narrowband noise. The resulting ISI can seriously degrade the system performance [14].

The rest of the paper is organized as follows. In Section 2 the blocked model of the filterbank-based transceiver and the perfect reconstruction are briefly reviewed. In Section 3 the transceiver system is analyzed in several aspects based on the blocked model. Section 4 formulates the optimal design problem as a least squares one and develops an iterative design procedure for the filterbank-based transceiver. The proposed design method is illustrated in detail with an example in Section 5. Finally, Section 6 offers some concluding remarks.

We conclude this section by introducing some notation. The signals are denoted by small letters, for example, . (underlining denotes blocking) is the blocked signal. denotes the -transform of . The systems are represented as time-domain operators, denoted by capital letters, for example, . If a system is linear time invariant (LTI), its transfer function (matrix) is written as . The notation stands for the 2-norm for transceiver matrices.

2. Blocked Models and Perfect Reconstruction

Figure 1 shows the discrete-time multirate filterbank model for the baseband communication system [9]. It consists of the transmitter filterbank (), the receiver filterbank (), and the communication channel modeled by the transfer function , which is assumed to be causal and stable. The input serial data stream and its successively time-advanced versions are downsampled by a factor to get parallel substreams as shown in Figure 1. These substreams are then upsampled by a factor and processed by filters ; the combined output is then transmitted over the channel , which is corrupted at the output by an additive noise , assumed to be stationary and white. At the receiver end, the received signal and its successively shifted versions are then downsampled by a factor , upsampled by a factor , and processed by filters ; the combined output forms the reconstructed signal .

By blocking the transmitter, channel, and receiver, respectively, we can obtain the blocked model [6, 8] for the multirate system in Figure 1:

where , and are the transfer matrices of the blocked transmitter filterbank, the blocked channel, and the receiver filterbank, respectively.

The blocked general multirate system (in the absence of noise) is LTI with . The transceiver achieves perfect reconstruction if in Figure 1 is a delayed version of , that is, if there exists nonnegative integer such that , where is the time-delay system with transfer function . Blocking the same way as we blocked , the perfect reconstruction condition is equivalent to [8] where integers and satisfy and , and and are the and identity matrices, respectively. Moreover, if this condition is satisfied, .

3. Analysis

In this section, we study the distortion analysis, the effect of noise, and the frequency selectivity.

3.1. Distortion Analysis

Perfect reconstruction synthesis filterbanks at the transmitter and analysis filterbanks at the receiver allow perfect recovery of communication symbols, but the challenges arise with ISI-inducing channels and noise, either of which destroying the perfect reconstruction property. Many practical transceivers do not achieve perfect reconstruction but get close to perfect reconstruction. In order to measure the degree of closeness to perfect reconstruction, we will introduce three traditional quantities to measure sources of distortions: aliasing distortion, magnitude and phase distortions [2]. These distortion measures are based on the blocked model .

Lemma 1 (see [15]). An LPTV (linear periodic time variant) system with period can be uniquely decomposed into satisfying the two properties
(i) is the optimal LTI approximation of in the sense that it minimizes over the class of LTI 's ( denotes the blocked system , similarly for . Here is the lifting operator, and the inverse lifting operator [1]). (ii) .

Back to our transceiver problem, the system from to is LPTV with period ; decompose this into , where is the LTI component and the time-varying component. Thus we haveAliasing distortion in the system is defined by

Even if AD is zero, and then the LPTV system reduces to an LTI system , it may still have errors in magnitude and phase compared with the ideal time delay ; define the following quantities:Note that MD and PD are defined across all frequencies: is the energy of the magnitude distortion and PD the energy of sine of the phase distortion . It is worth noting that there are two reasons why we apply sine to characterize the phase distortion PD: (1) if is within , which is usually the case, is a good indicator of the size of ; (2) a connection with the 2-norm-based distortion measure to be introduced, could be conveniently established.

Next we propose a composite distortion measure which captures all the three types of distortions and is relatively easy to use in design. The new distortion measure is the 2-norm of the blocked error transfer matrix:Such a measure is appropriate because in the next theorem we establish connections between and the three types of distortions discussed earlier.

Lemma 2 (see [16]). Let be a stable LTI system. Comparing with the time delay , one has the following inequalities:

Theorem 1. AD and MD relate to via whereas and relate to via

Proof. From (4) and Lemma 1 we getNote (5) to getNow by definition,Invoke Lemma 2 to get
Combining these two equalities with (13) and noting the definitions of MD and PD in (6), one has
The proof is complete by noting (12) and the above two inequalities.

Then it is clear from Theorem 1 that all distortions (AD, MD, and PD) are bounded above by . Therefore, it makes sense to minimize in transceiver design because this suboptimizes the three distortions simultaneously.

3.2. Noise Suppression

In Figure 2, represents the noise effect at the receiver end. If we want to minimize the root-mean-square value of , it is equivalent to minimize the -norm of the blocked transfer matrix from (standard white noise) to . Therefore, the objective function for attenuating the output noise effect can be

Figure 2: A model of the noise effect at the receiver end.

3.3. Frequency Selectivity of Filters

Filters with frequency selectivity are of particular importance in communications systems. In addition, frequency responses of designed filters would be deteriorated in the iterative design procedure, if no constraints are imposed on the filters. In order to obtain better bandlimiting property, we minimize the stopband energy of filters involved. To see this, take the objective function(s) for in the receiver filterbank as an example; we havewhere defines the stopband frequency interval(s) for . Similarly, can be computed.

In this paper, we will consider only FIR filters. The frequency response of a real -tap FIR filter is given bywhere(The superscript indicates the complex conjugate transpose.) The objective function in (17) can then be written aswhere the fixed matrix is defined byThe elements for can be calculated easily if is given; for example, assume the filters have the passband over , thenand thus the elements for are

4. Problem Formulation and Design

In view of the new distortion measure discussed in the preceding section, we wish to design transmitter and receiver subsystems to minimize . Thus our optimal filterbank-based transceiver design problem using FIR subsystems can be stated as follows: given the FIR channel and desired reconstruction time delay , design FIR transmitter and receiver subsystems of some given lengths to minimize subject to some constraint on in (16) and the stopband energy constraints on in (20).

In order to incorporate both the noise attenuation and filter bandlimiting constraints, such an optimal design problem can be recast by including penalties on , , and . Because both and are designable, this optimization problem is in general nonlinear and difficult to solve. Thus we take the following iterative design procedure which turns out to be very effective in the design example to follow.

Step 1. Design transmitter subsystems to satisfy desired frequency limiting properties (without considering reconstruction performance); these are used to initiate the iteration.

Step 2. Fixing the transmitter subsystems, design FIR receiver subsystems by minimizing the following objective function (using the blocked models)

Step 3. Fixing the receiver subsystems just designed, now redesign FIR transmitter subsystems by minimizing the following objective function (using the blocked models)

Step 4. Repeat Steps 2 and 3 until the corresponding objective function is sufficiently small.

We note that the idea of iteratively designing transmitter and receiver filters was used effectively in transmultiplexers design in [12]. The advantage of this procedure is evident: by fixing either or in Steps 2 and 3, the optimization problems become mathematically tractable; in fact, they are finite-dimensional, convex optimization with a quadratic cost function, whose global optimal solution can be always computed. Even analytical solutions can be obtained.

For example, looking at the optimal design problem in Step 2, we defineThis system is FIR and hence can be represented by its finitely many coefficient matrices . By Parseval's equality,Since and are given and thus depends on in an affine manner, it follows that relates to the coefficients of (to be designed) too in an affine manner. It is obvious that both and are of quadratic forms, therefore we can rewrite the quantity in (24) in the following way:Here is a column vector containing all the parameters in to be designed, is a column vector depending on only , is a matrix depending on and the way is formed, depends on and , and finally depends on and . The matrices , , , and can be computed and are independent of the design parameters (). Now the optimal design problem in Step 2 becomes a least squares problem:If is invertible, the optimal solution can be obtained to beFrom here we can recover the optimal receiver subsystems. The optimal design problem in Step 3 can be solved similarly.

5. Design Example

The filterbank-based transceiver with and is designed. The channel to be used in this example is

The transmitter and receiver filters involved in design are all FIR and causal with a fixed order of 12. The magnitude Bode plots of the initial transmitter filters are given in Figure 3, and the reconstruction time delay is taken as . The constants , , and , reflecting relative weightings among multiple objectives, are tuned in the design process. In our design, these are taken to be , , .

Figure 3: The magnitude Bode plots for the initial transmitter filters (solid), (dash-dot), and (dotted): dB versus .

Next we apply the iterative design procedure to the example. In the first iteration, , and the resulting . After some iterations, the value of the objective function gradually decreases as the number of iterations increases, and finally converges to the value , , and finally . The above comparison on clearly illustrates the improvement obtained by using the proposed iterative algorithm. The designed transmitter and receiver filters are shown in Figures 4 and 5, respectively.

Figure 4: The magnitude Bode plots for the designed transmitter filters (solid), (dash-dot), and (dotted): dB versus .

Figure 5: The magnitude Bode plots for the designed receiver filters –: dB versus

6. Conclusion

In this paper, we investigated the problem of optimal design of filterbank-based transceivers. We proposed quantities to measure various distortions. We also introduced a composite distortion () that captures all distortions. Finally, by incorporating two important practical issues, the noise suppression and filter bandlimiting property, we developed an iterative design procedure based on minimizing the objective function and successfully applied this procedure to design of a filterbank-based transceiver. At each iteration, the least squares solution should be found, thus the final transmitter and receiver filters can be obtained with relative ease.

Acknowledgments

The authors wish to thank the associate editor and anonymous reviewers for providing many constructive suggestions which have improved the presentation of the paper. This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Foundation of Innovation.