Design of multiplier-less nonuniform filter bank transmultiplexer using genetic algorithm

Abstract

In this paper, the design of multiplier-less nonuniform filter bank transmultiplexer (ML NUFB TMUX) is presented. Coefficient synthesis of the filters in canonic signed digit (CSD) format is modeled as optimization problem and genetic algorithm (GA) is used for the optimization. A new integer chromosome encoding scheme using a look-up table, which is capable of preserving the canonic property of filter coefficients under genetic operations, is introduced. As compared to corresponding binary chromosome encoding scheme, the new encoding scheme has smaller chromosome size, simple encoding and decoding procedure and low computational complexity. New mutation technique and modified crossover technique are introduced to improve the performance of GA. They also enable GA to use the full range of CSD numbers in the look-up table, which is not possible in the binary coded GA. The performance of the proposed algorithm is compared also with simulated annealing (SA) and recently introduced symbol coded GA. Simulation results show that ML NUFB TMUX designed using the proposed algorithm has better performance than that designed using binary coded GA, symbol coded GA and SA. The proposed algorithm is also faster than the binary coded GA, symbol coded GA and SA.

Introduction

One of the major challenges in information technology is the efficient and successful communication of messages through imperfect channels. Since the number of users keeps on increasing, efficient utilization of the communication channel is required. Multicarrier modulation (MCM) can be used for efficient transmission of information over a distorted channel [1]. Discrete multitone (DMT) and orthogonal frequency division multiplexing (OFDM) are considered as efficient multicarrier communication systems. But the frequency responses of their transmitting and receiving filters are very poor. In many applications it is often desired to have filters with better frequency responses. Filter bank technique can be used to design MCM with better transmitting and receiving filters. So the current trend in research is to adopt filter bank based multicarrier system [2], [3], [4], [5], [6].

Most of the studies of filter bank transmultiplexer (FB TMUX) focus on uniform frequency bandwidth in which the incoming data signals are assumed to have the same sampling rate. But a nonuniform filter bank transmultiplexer (NUFB TMUX) is required to multiplex applications having different incoming data rates and different quality of service requirements. A few techniques were introduced for the design of the NUFB TMUX [4], [5], [6]. The design techniques mentioned in [5], [6] do not consider channel effect and hence they give good results only when the channel is ideal. When the transmission channel is frequency selective, an additional equalizer is needed at the receiver. Design of NUFB TMUX considering the channel effect is given in [4]. In this design, signal to interference ratio (SIR) is formulated as a Rayleigh–Ritz ratio whose solution is well known [7]. The channel effect is taken into consideration at the time of the filter bank optimization. Therefore, only a simple one tap equalizer is required at the receiver for the channel equalization. To improve the SIR, an iterative algorithm similar to that for the design of DFT modulated FB TMUX [2] and cosine modulated FB TMUX [3], is also given in [4]. But the iterative algorithm severely effects the frequency responses of transmitting and receiving filters. It also results in nonlinear phase transmitting and receiving filters.

Multiplier circuits are more complex than adder circuits. In a signal processing system, if the filter coefficients are represented as SPT terms, multiplications can be done by using adders and shifters. The number of these building blocks can be reduced by using minimum number of SPT terms for representing the filter coefficients. Canonic signed digit (CSD) representation of the filter coefficient is one of the most commonly used method for minimizing the number of SPT terms [8], [9], [10], [11], [12], [13]. Use of suitable algorithms to design digital filters with CSD coefficients results in more efficient implementations of the filters [8], [9]. A number of algorithms are also reported for the design of QMF lattice filter bank with CSD coefficients [10], [11], [12], [13].

In this paper, our aim is to design multiplier-less nonuniform filter bank transmultiplexer (ML NUFB TMUX). For this, a continuous coefficient NUFB TMUX is designed and the coefficients of the filters are synthesized in CSD format. Unlike in the case of QMF lattice filter bank, we have to synthesize the coefficients of all the filters in CSD format in such a way that SIR is maximum. For this, coefficient synthesis of filters is modeled as constrained optimization problem. Separate objective functions for the design of each receiving filter are formulated by including SIR, stopband energy of the filter and constraint for the number of SPT terms in the filter. Separate objective functions for the design of each transmitting filter are formulated by including signal to leakage ratio (SLR), stopband energy of the filter and constraint for the number of SPT terms in the filter. Filters with CSD coefficients are obtained by optimizing the objective functions using genetic algorithm (GA). If chromosomes are formulated using ternary CSD filter coefficients, the offspring produced by genetic operations like crossover and mutation may not conform to CSD format. In this case, some restoration techniques should be used to keep the chromosomes in the CSD format [9]. This considerably increases the computational load in GA. As a solution, for the formulation of the chromosome, some encoding techniques which preserve the canonical nature of the chromosomes under the genetic operations are introduced [11], [12]. In [12], the chromosomes are encoded using some symbols. It is required to decode the symbol coded chromosomes in each generation. Another limitation of this encoding is that the maximum number of SPT terms that can be used in each filter coefficient should be kept equal. But it is proved that the performance is better when different filter coefficients are allocated with different number of SPT terms [11], [14]. In [11], the chromosomes are encoded using the binary indices of a look-up table. The main disadvantage of this encoding is that when the full range of the CSD numbers in the look-up table is used, the binary indices resulting after the genetic operations may exceed the maximum index of the look-up table. It is also required to decode the binary coded indices of all the filters in the population in each generation to access the look-up table. So in our design, CSD numbers are encoded as integer indices of the SPT look-up table. For genetic operations, new mutation technique and modified crossover technique are introduced. They improve the performance of GA and also enable GA to use the full range of CSD numbers in the look-up table. Advantages of the proposed GA are (1) The canonic property of the filter coefficients is always preserved under genetic operations and hence restoration algorithm is not required after each genetic operation. (2) Unlike in symbol coded GA, different filter coefficients can be allocated with different number of SPT terms. (3) Unlike in binary coded GA, the full range of CSD numbers in the look-up table can be used. (4) Chromosome size is comparatively small. (5) Coding and decoding are simple. (6) Because of smaller chromosome size and simple coding and decoding, it is faster than binary coded GA and symbol coded GA. (7) Because of new mutation and modified crossover, its performance is better than binary coded GA and symbol coded GA.

The paper is organized as follows. NUFB TMUX, the SIR in NUFB TMUX and the design of continuous coefficient NUFB TMUX are introduced in Section 2. Introduction of GA is given in Section 3. The encoding of CSD filter coefficients using the look-up table is explained in Section 4. In Section 5, the objective functions for the synthesis of filter coefficients are formulated. Design of ML NUFB TMUX is explained in Section 6. Section 7 gives simulation results and conclusions are presented in Section 8.

Notations:

• ${\mathbf{A}}^T$ and ${\mathbf{A}}^{†}$ are used for indicating transpose of A and conjugate transpose of A, respectively.

• For any real number x and any integer T, notation $\mathit{round}(x)$ represents rounding of x to the nearest integer and notation $\mathit{round}(x{)}_{{}_T}$ represents rounding of x to the nearest integer in $[-T,T]$.

• $E\{•\}$ represents statistical expectation.