IntroductionSample rate conversion by using multistage implementation of FIR filters is discussed in the previous chapter. FIR filters as decimator work well except in case of narrowband filtering at high rates. The number of taps in an FIR filter is directly proportional to the sampling frequency and inversely proportional to the transition bandwidth. As the ratio of sampling frequency to transition width of decimation filter is usually large for narrowband decimation filters, the number of taps in conventional FIR filters tends to be high. In order to avoid large number of taps in FIR filters, special FIR filter architectures are used as decimation filter. CIC filters are generally use as a special type of FIR filter architecture for decimation purpose for above mentioned cases. CIC filters are linear-phase low pass FIR filters. These filters are also called sinc filters. As the name suggests, CIC filters can be implemented as a cascade of integrator and comb sections. The advantage of CIC decimation filters is the ability of sampling rate conversion without multiplying operations. This is of particular interest when operating at high frequencies. CIC filters of higher order are required to achieve better passband and stopband characteristics. However, higher order CIC filters not only suffer from significant passband droop and limited stopband attenuation but also consume a lot of power and occupy large area. These problems can be corrected to a certain degree by either modifying the conventional structure of CIC filter, or implementing a secondary filter, after the conventional one, to compensate its undesired characteristics. This chapter mainly focuses on selecting the sharpening polynomials using genetic algorithm optimization for passband droop correction and increasing stopband attenuation.The frequency response is improved by optimizing the sharpening polynomial used taking CIC filter as its protype.The chapter is organized as follows. In Section 4.2, conventional CIC decimator filter operation is presented. Starting with the disadvantages related with CIC filter sharpening CIC filters are explained in Section 4.3. Details regarding the selection of sharpening polynomials using GA approach are discussed in Section 4.4. Design examples and results are presented in Section 4.5 and conclusions are drawn in Section 4.6.

II. CASCADED INTEGRATOR-COMB FILTERCascaded integrator-comb (CIC), or Hogenauer filters, are multirate filters used for realizing large sample rate changes in digital systems[1l]. CIC filters are multiplierless structures, consisting of only adders and delay elements which is a great advantage when aiming at low power consumption. They are typically employed in applications that have a large excess sample rate. That is, the system sample rate is much larger than the bandwidth occupied by the signal. CIC filters are frequently used in software defined radio,digital down-converters (DDCs) and digital up-converters[1m]. The CIC filter is a class of hardware-efficient linear phase finite impulse response (FIR) digital filters, consists of an equal number of stages of ideal integrator filters and comb filters. Its frequency response may be tuned by selecting the appropriate number of cascaded integrator and comb filter pairs. The highly symmetric structure of a CIC filter allows efficient implementation in hardware. CIC filters achieve sampling rate decrease (decimation) and sampling rate increase (interpolation) without using multipliers. The CIC filter first performs the averaging operation then follows it with the decimation.

It is clear from the earlier discussion that the CIC filter introduces a droop in the passband of interest and this droop is dependent on the CIC decimation ratio and the order of CIC filter. The limitations of the CIC-based architecture generate a need for a sharpening filter for the aim to improve the frequency response of a CIC-based filter-decimation system.

SHARPENED CIC FILTER

Since in the above described technique the protype filter is chosen to be a CIC filter, the resulting sharpened filter will have significantly reduced passband droop and improved alias rejection.

For illustration purposes, lets assume that the sharpening polynomial g(x) is a Lth order polynomial:

It should be noted the sharpening polynomials in the literature are selected through the mentioned flatness considerations. Hence, these polynomials are not optimized for a particular problem. In the proposed approach, we optimize sharpening polynomial to meet the passband and stopband specifications of particular CIC based decimation systems using a experimental design method called as orthogonal genetic algorithm.

Orthogonal Experimental DesignThe so-called experimental design is a plan for performing experiments to study the effects of factor levels on dependent variables [1g]. The factors of an experimental design are variables that have two or more fixed values or levels. Many design experiments use special techniques for determining suitable combinations of factor levels for implementing the experiments and analyzing the results. The so-called orthogonal design is an efficient way of studying the effects of several factors simultaneously in a design experiment.Let us assume that an experimental design consists of N factors with each factorhaving Q levels. The number of all possible combinations of the factor levels would be QN. However, for most practical situations, it is cost prohibitive and tedious to test all these possible combinations. Orthogonal arrays (OAs) are, therefore, developed to provide a means for optimal sampling of experiments. In orthogonal design, all estimable effects are uncorrelated [1h]. For example, with three factors, each at two levels, there are 23 = 8 possible combinations whereas in an orthogonal design the four best representative combinations would be chosen. The concept of orthogonality for this example is illustrated in Fig. 4.2. An OA is a rectangular matrix designated as LM(QN) whereN = the number of columns representing factorsQ = the number of factor levels,

aij (1,2,...Q)The OA can be represented as LM(QN) =[ aij]M*NIn orthogonal experimental designs ,OA are used as tool to arrange experiments an find out the optimal design with fewer experiments when as orthogonal array LM(QN) is tabulated ,the following relation hold for Q odd and M=QJ Now OGA act as heuristic search method and is used for an efficient searching and fast converging tool for the optimum decimation factor detection in the above problem.

Optimization approach for selection of sharpening polynomial

In this section genetic algorithm based approach is presented for selection of sharpening polynomial. The goal here is to minimize the worst case passbandand stopband ripples. In the original work of Kaiser and Hamming, the sharpening polynomials are designed to improve the response of generic filters, [1i]. Here an optimization approach specific for the CIC filters is presented.

The sharpened filter optimization can be carried out by minimizing an objective function based on one, two, or more criteria. One could, L1-norm for passbands and the L2-norm for stopbands. Minimization of an objective function based on the L1-norm yields a minimax solution. An important merit of minimax solutions is that the optimization error tends to become uniform with respect to the frequency range of interest as the solution is approached [1j]. The maximum errors in the amplitude response in the passband and stopband of sharpend filter can be expressed as

respectively, where !p and !a are the passband and stopband edges, respectively, and!s is the sampling frequency. The approximation errors are sampled uniformly at

For a clearly defined above problem the OGA works as follows: The set of positive integers that represents the individual decimation ratio of stages defines the phenotype structure of a chromosome for the optimization process. The OGA creates new generations of populations by applying an orthogonal crossovers and `sign-inverting' mutations to the individuals of a population. In the genotype representation, the chromosomes are encoded in terms of integer values .The phenotype representation is mapped to the genotype representation with integer values limited in a specified range, which enables the search of the OGA to be confined to within a feasible region. By analogy with the orthogonal experimental design in [1g], if we define decimation ratio of each as the `factor' and the range S of the decimation ratio as the `factor level', the design will have N = 2M factors with each factor having S levels. An OA LK(SN) would be necessary for the crossover operation for such a case.For typical values of N and S for a decimation filter the size of an OAcould be prohibitively large. To keep the size of the OA tractable, a fixed OA, such as, L9(34), can be used as a crossover tool. In the proposed method, the number of columns in the array of the OA is adjusted according to the number of elements in the chromosome. The OA as a whole or part of it is re-used after randomly re-arranging its elements to form an enhanced OA(EOA).The solution evolves from generation to generation through the creation ofoffsprings by applying genetic operators such as crossover and mutation. To perform crossovers, a set of randomly selected chromosomes are used as rows in a matrix. An EOA is then applied as a `mask' to re-distribute the chromosome elements in that group. Repeating the process of grouping and masking, a population-wide crossover can be achieved. An example of the orthogonal crossover operation is illustrated in Fig. 4.4. In the `sign-inverting' mutation operation, the values of particular genes are changed by applying random `sign' inversions. A table is constructed with all the individuals generated by the crossover and a population-wide mutation is performed.

The frequency of mutations is controlled by probability of occurrence Pm. A random number is generated in the range 0 to 1 and if it is less than Pm, mutation is applied; otherwise, mutation is not applied. For the problem at hand, mutation is treated as a supporting operator for the purpose of restoring lost genetic material. In the selection process, chromosomes are ranked on the basis of their fitness.The total population that competes for survival comprises the parents as well as the offsprings created during crossovers and mutations. The algorithm then records a small number of top-ranked solutions as elite solutions and selects a fixed number of best-fit chromosomes as potential parents for the next generation.

For the initialization of the algorithm, a direct-form FIR filter is designed for sharpening passband as suggested by the previous stated methods. The resulting coeffecients are used to construct a chromosome string which is inserted in the initial population of the GA. The remaining chromosomes are created randomly. In each successive generation, half of the chromosomes are taken from the best fits of the previous generation and the rest are generated randomly. The GA is terminated if it fails to improve the best fitness value in a specified number of successive unproductive generations or if a pre-specified maximum number of generations are performed. Eventually, the best chromosome is selected as the desired solution.

Design Examples and Results

To illustrate the proposed approach, two examples are presented here. In both examples, the cascade of two CIC filters is used as the prototype filter, i.e. P(ej!) = [sin(!M=2)=(M sin(!=2))]2. Insufficient stopband attenuation and passband drop of the single stage CIC structure produces the need of this structure.

Example 1: In this case decimation ratio is =20 also the passband frequency is 0.2and stopband frequency 0.3.Figure 5 shows the suggested CIC based low-pass filtering structure for decimation ratio 10. The sharpening polynomial specific for this problem is g(x) =. The pink line with the label 4th order in Figure 4 shows the frequency response of the suggested system. The other curves show the response of the prototype system and the response of the filters having 6th, 8th order optimal sharpening polynomials similarly found through the described genetic algorithm approach.

As shown in Figure 4, the desired passband is the interval of [0 0.1]. For this system, the target rate after the ten times decimated. For the desired bandwidth, the passband droop of the prototype filter shown by red colour is around 8 dB. The described 4th order implementation has a maximum passband ripple of 1 dB and has a stopband attenuation of at least 34 dB. These values can be acceptable in many applications. It should be noted that the sharpening filters of higher orders have further improved droop and stopband characteristics. For the 6th and 8th order sharpening polynomials, the maximum ripple reduces to 0.5 dB and 0.2 dB respectively and the worst case stopband attenuation increases to 44 dB and 50 dB respectively.

The designs shown in Figure 4 are specific for the given passband and stopband pair. the weighting factor W, trading the passband ripple with the stopband attenuation, is chosen as 7 in this example. By changing W, sharpening polynomials having reduced droop at the expense of worse stopband attenuation.

Example 2: In this example, we examine a system with a target decimation ratio of 20. Figure 6 shows the frequency response of decimation ratio 20 system. Different from the earlier example, the passband of this system is[0 0.05]. As in the first example, the results for the sharpening polynomials having the orders of 4, 6 and 8 (designed for the given passband and stopband pair and W = 7) are presented . Figure 6 shows that the 6thorder design shown in green colour has the passband ripple of 0.25 dB and the stopband attenuation of 40 dB. These values are very much welcomed in many applications.

00.10.20.30.40.50.60.70.80.91-120-100-80-60-40-200Magnitude (dB)Frequency/ (rad/sample)StopbandPassband 4th Order6th Order8th Order

The sharpening polynomial of 6th order specific for this problem is g(x) = The designs shown in Figure 6 are specific for the given passband and stopband pair. the weighting factor W, trading the passband ripple with the stopband attenuation, is chosen as 7 in this example.

CONCLUSIONThe main goal of this chapter is to discuss the utilization of the application specific sharpening filters in CIC decimation filter design in contrast to generic sharpening polynomials using genetic algorithm optimization. It has been observed that the optimally sharpened filters can produce high performance decimators virtually eliminating the need of a secondary compensation filter in certain cases. The present approach has also provide an optimization framework for the sharpening of the CIC filters using new techniques.