Optimization of Sigma-Delta Modulator Based on Artificial Immune Algorithms

Gabriel V. Iana, Gheorghe Serban, Petre Anghelescu, Laurentiu Ionescu

University of Pitesti, ROMANIA

E-mail: gbiana2000@gmail.com, serban@upit.ro, petre.anghelescu@upit.ro, ionescu.laurentiu@upit.ro

Abstract- The artificial immune algorithms are implemented

by the human immune system. They inspired a technique that has the ability to find optimal solutions in a nonlinear search space. This kind of technique increases the computing speed and converges faster than the genetic algorithms. In this paper is presented an application of artificial immune algorithms in order to optimize the transfer function of a sigma delta modulator. It is also presented the base method which can perform the adjustment coefficients, in decimal, of the sigma delta modulator. The results are compared with those obtained by the method of optimization based on both mathematical and genetic algorithms.

Keywords- artificial immune system, clone selection principle, optimization methods, sigma-delta modulation, signal/noise ratio.

I. INTRODUCTION

Various aspects of biology have always been an inspiration to the development of new computer models that are used to solve complex problems. The immune system is the one that has been recently spotted the result being the creation of a new field called Artificial Immune System. It gives very useful information in several types of applications such as: pattern recognition, optimization, learning, etc.

The immune system is – by all means – very complex and presents a new and rich field to be thoroughly investigated by researchers. In the field of artificial intelligence only some of its operating modes are taken into account – the only ones that could be understood and applied in numerical calculation [1]. These methods are: the mechanism of negative selection, immune network theory and the clone selection principle.

The mechanism of negative selection is based on the interpretation of some aspects of positive selection or negative behavior that occurs during maturation of T cell. This class of algorithms is used for classification or pattern recognition problems.

Immune network theory is based on the idea that the immune system makes cell interconnection network for an idiotype B. These cells have been stimulated or removed so

as to maintain a network that made them. Accessed theory is used in data visualization algorithms, control or clustering.

The third method used in this paper is based on cloning and hipermutation processes which can be found together as the clone selection principle [2].

Clone selection principle inspiration needs a great source for finding new improved techniques [3] in problem optimizations for nonlinear functions but at the same time to mitigate the effects of quantization when digital systems are involved in the calculation. For this reason it is proposed to adapt this technique to optimize the coefficient's noise transfer function of a sigma delta modulator in the digital domain for digital signal processing. The significant advantage is that the sigma-delta modulator converting signals at high resolution using oversampling technique. Moreover, sigma-delta converters are successfully implemented in VLSI Technology (Very Large Scale Integration) implementation involving far less hardware.

Sigma delta modulators are nonlinear and, in order to approximate their behavior the white noise model of quantization error is used and its quantizer is replaced with a stochastic random process. Although it delivers relevant results on signal-noise ratio performance, this model fails to explain the emergence of limit cycles that appear in the spectrum of the output signal. Higher order modulators have some stability problems that are not handled by this model [4].

Gray has developed an accurate analytical method for the first sigma-delta modulator with constant input [5]. Using the same approach, sigma-delta modulator with multi-bit quantizer were analyzed using irrational constant inputs [6].

Rangan calculated the exact formulas for quantization noise in second-order modulator with the quantizer on two bits [7]. It is important that this method can be applied only on second order modulators because algebra used in integration of higher order seems to be very difficult to be applied or even impossible. As far as we know, it seems that no one was able to make an extension to higher order modulation. There are used some mathematical methods of autocorrelation found for the quantization error sequence, without overloading, a class of modulators with arbitrary entries Galton [8]. Poorfard conducted a theoretical analysis of sigma-delta modulators only when the entry in a string with zero average [9]. This theory is valid only when entering modulator is a signal and random order modulator is very

1010978-1-4244-7020-4/10/$26.00 '2010 IEEE

2010, 12th International Conference on Optimization of Electrical and Electronic Equipment, OPTIM 2010

high. Methods have been developing on search-based that use the genetic algorithms in the modeling of noise transfer function [10].

II. SIGMA DELTA MODULATOR

The sigma-delta modulation is based on a negative feedback loop in which a low quality quantization is performed at a high sampling frequency, and a big amount of the quantization noise is moved into a superior area of the input signal frequency band [11]. The block diagram of first order one bit quantizer sigma-delta modulator is shown in Fig. 1.

Fig 1. Block diagram of first order sigma-delta modulator in continuous time

In this paper, the quantization error white noise model is

used, which simplifies the analyses of sigma-delta modulators and gives important information upon the system functionality. The model replaces the quantizer with an independent input stochastic randomize process; it can use the conventional theory of linear systems. In Fig. 2 is shown, in discrete time, a block diagram for the first order sigma-delta modulator that incorporates a single-bit quantizer and the transfer function of the integrator given in relation (1).

/ 1 (1)

Fig 2. Block diagram of first order sigma-delta modulator in discrete time

with quantizer approximated by a white error In discrete time, the relations of the sigma-delta modulator

are as follows: · · 1 (2) Where, the signal transfer function (STF) and noise transfer

function (NTF) are as follows:

and 1 (3)

If we compare the first order sigma-delta modulator with the second order sigma-delta modulator , the second one provides more quantization noise

suppressing over the low frequency signal band, and more amplification of the noise outside the signal band as is shown in Fig. 3.

On the same figure, the plot of noise quantization is

observed, which is generated with a single bit quantizer at rate Nyquist compared with an oversampling quantizer.

When more the order modulator increases, more noise is shifted to higher area. There is not difficult to obtain the high order sigma-delta modulator, but some aspects appear as the limit cycles in the spectrum of output signal or some stability problems that affect the modulator.

III. SELECTIVE CLONING PRINCIPLE

The immune system detects foreign elements and acts to eliminate them. Such elements are prerequisites of the immune system like bacteria, viruses, etc. and are found as the antigen. Detection elements of the immune system in vertebrates are distributed in the body and interact with elements considered intruders by the immune system. For some detectors there are mechanisms to improve their recognition performance. Affinity maturation process consists of an iterative process of generating clones, variation and selection and will still participate in the process of evolution. The best detectors resulting from clone selection are retained in memory cells.

The process of cloning is lymphocytes proliferation that recognizes the antigens. Lymphocytes interaction with the antigens is the result of their activation. When an antigen activates a lymphocyte, it just does not make antibody to bind antigen and clones are moved to have a better affinity with the antigen detected. The latter process is called somatic hyper mutation. The repetitive exposure to antigen of the immune system helps it to learn to adapt itself to various forms of antigen. This process is called clone selection theory [14].

It is well known that the most important steps used in the selective cloning principle are [12]:

First order NTF

Second order NTF

Third order NTF Nyquist rate (1 bit)

Bf2SfFrequency

Fig. 3. Magnitude spectral for NTFs

Mag

nitu

de

oversampling

1

1

1 −

−

− zz∑

∑+

−

( )zX ( )zY( )zE

∑ Integrator

1bit quantizer

+ -

x(t) y(t)

1011

- generation of new random genetic changes subsequently expressed as diverse antibody patterns by a form of accelerated somatic mutation; - phenotypic restriction and retention of one pattern to one differentiated cell (clone); - proliferation and differentiation on contact of cells with antigens.

IV. FOURTH ORDER SIGMA-DELTA MODULATOR MODELING

Only some models have been agreed for higher order sigma-delta modulators in digital structure. There are a lot of different ways to realize a sigma-delta modulator in the digital implementation. In order to spread the position of the zeros, we are tempted to introduce many multiplications coefficients. These structures however would be very difficult to be achieved with the currently used hardware, as a large number of multiplications consume too much space area.

For this reason we are concentrating on a structure with fewer multiplications [4]. The fourth-order low pass sigma delta modulator has the advantage of using a low number of multiply coefficients, where a1, a2, a3 and a4 are on the feed-back and take a ±1 value. If the sigma delta modulator has the quantizer only on single bit, then the multiplication operations of coefficients in the feedback are with one or minus one. In this case, the multiply operation needs a minimal number of logic cells into the hardware structure.

The fourth order low pass sigma-delta modulator is shown in Fig. 4. The transfer functions are low pass filter and identical as in following relation:

(4)

Fig. 4. Block diagram of fourth order sigma-delta modulator

The quantization process will be on a single bit. After the computation in discrete time of the differential equations of the fourth order sigma-delta modulator we can determinate

the signal transfer function (5) and the noise transfer function as (6) given in the following relations:

(5)

(6)

Where the terms u1, u2, u3 and u4 are: 4 6 3 (7) 4 2 3 1

The computation process for the multiplying coefficients is

performed using a transfer function of a digital filter characterized by the following relation:

· · · ·· · · · (8)

After the identification of the coefficients from the

relations (8) with transfer function (NTF) of the sigma delta modulator (6) the following relations result between them:

4 6 3 4 2 3 1 (9)

The numerator coefficients, B0 to B4 are known from (6)

by identification and they are {1, -4, 6, -4, 1}. In this case the Hf4 denominator, A1 to A4 can be computed by a Butterworth filter method.

V. APPLICATION OF SELECTIVE CLONING ALGORITHM

Clone selection algorithm it’s important for studies on immune system and it is able to generate optimization process. It is always trying to find an optimal solution but it can be stopped after a number of cycles predefined by the user.

Selective cloning algorithm follows these steps [13]: 1. Generates a set (P) of candidate solutions, composed of

the subset of memory cells (M) added to the remaining (Pr) population (P = Pr + M);

2. Determines (Selects) the n best individuals of the population (Pn), based on an affinity measure;

3. Reproduces (Clones) these n best individuals of the population, giving rise to a temporary population of clones (C). The clone size is an increasing function of the affinity with the antigen;

∑[ ]nx

∑−+ −+

1a 2a

∑ 1−z

[ ]nH1

+ +∑ 1−z

[ ]nH 2

+ +

[ ]nu1 [ ]nu2

3a

∑−+

[ ]nyQ∑−+

4a

∑ 1−z

[ ]nH 3

+ +∑ 1−z

[ ]nH 4

+ +

[ ]nu3 [ ]nu4

+

1012

4. Submits the population of clones toscheme, where the hypermutation is paffinity of the antibody with the antantibody population is generated (C*);

5. Re-selects the improved individuals frothe memory set M. Some members ofby other improved members of C*;

6. Replaces d antibodies by novel introduction). The lower affinity cprobabilities of being replaced.

VI. TESTING AND RESULTS

Sigma-delta modulator coefficients wethree methods:

- mathematical computation - genetic algorithms and - selective cloning algorithm.

Evaluation functions were done after theIn order to make a comparison between input signal was chosen, with a 0.9 ampliturate of 32. To make a comparison betwcloning method and the genetic algoritfollowing initial configurations were necess

Genetic Algoritm

- A chromosome consists ogene representing one modulator’s

- Population consisted of 3- 5000 generations are asse

Clonal selection algorithm

- A cell contains four ssubdivision representing one modu

- Populations consisted of 2- For each generation th

population with 10 clone cells - In every evolution the l

less fitted are replaced by anothgenerated.

- 5000 evolutions were ma Mahematical algorithm

- The coefficients were caland (9) using a Butterwoth filtefunctions HF4.

Data obtained by running the three methotable 1.

o a hypermutation proportional to the tigen. A maturated

om C* to compose f P can be replaced

ones (diversity cells have higher

S

ere determined by

e signal-noise ratio. them a sinusoidal

ude and a sampling ween the selective thms method, the sary:

of four genes, each s coefficient. 0 chromosomes.

essed.

subdivisions, each ulator’s coefficient 20 cells here is formed a

last two cells with her two randomly

ade.

lculated by (7), (8) er for the transfer

ods are given in

TABLE I. RESULTS

Methods Coefficients

Mathematical

248.56 96.85 20.72 2.02

Genetic Algorithm

332.36 116.09 21.19 9.09

Clonal Selection Principle

645.17 356.22 82.45 20.17

The coefficients computed af

of the computation are proportitime, of the signal/noise ratio isprinciple computation.

Evolution of the implementa

and the clone selection algorithm

Fig. 5. The evolution of fitness over

Fig. 6. The evolution of fitness overalgori

As we can notice from Fi

selection algorithm is able to ohave been found, that are better

OF THE THREE METHODS

Signal/noise ratio

Time of execution

63.3dB -

62dB 3’22”

75dB 2’43”

fter the three different method ional. The best performance in s obtained after clone selection

ation of the genetic algorithm m are shown in Fig. 5 and 6.

generations for a genetic algorithm

r generations for a clone selection ithm

ig. 5 and Fig. 6, the clone optimize and further solutions than the genetic algorithms.

1013

In Fig. 7, 8 and 9 there is shown the powobtained for the three methods.

Fig. 7. Output density power spectrum for a fourthmodulator with mathematical algorithm coefficie

Fig. 8. Output density power spectrum for a fourthmodulator with genetic algorithm coefficients

Fig. 9. Output density power spectrum for a fourth ordmodulator with clonal selection algorithm coefficients

In Fig. 10, 11 and 12 there are represtransfer functions (STF and NTF) for the th

wer spectral density

h order sigma delta ents optimizations.

h order sigma delta s optimizations.

der sigma delta s optimizations.

ented the obtained hree methods.

Fig. 10. STF and NTF obtaine

Fig. 11. STF and NTF obtain

Note that the transfer selection algorithm were modesignal frequency.

Fig. 12. STF and NTF obtained

VII. CONCLUSIO

The algorithm based on thwas applied to optimize a foumodulator with a structure easiwhere a minimum number Coefficients calculation was function in three ways to hdisadvantages of clone selectithat the clone selection algorithless time and it is able to optimto the genetic algorithms. The ssupposed to start from the relatdata system, according to the pthis conclusion, the result of thwith the clone selection after abcame to the outcome of previoabove. To obtain the best possiba lot of clones have been geprocess has been performed fothe less fitness cells.

ed with mathematical method

ned with genetic algorithm

functions obtained by cloning led to find the optimum input

with clonal selection principle

ONS

he principle of clone selection urth degree order sigma-delta ily applied to digital systems, of multipliers are desired. performed for its transfer

highlight the advantages and on algorithm. It was noticed

hm reaches an optimal level in mize a better solution compared

selective cloning principle it’s tively known solutions inmate profile literature. To highlight he genetic algorithm was run bout 500 generations and there ous selective clone algorithm ble optimization performance,

enerated and a hipermutation ollowed by the replacement of

1014

REFERENCES

[1] D. Floreano, C. Claudio Mattiussi, „Bio-Inspired Artificial

Intelligence – Theories, Methods, and Technologies”, The MIT Press Cambrige, pp. 335-398, London, 2008.

[2] G. Ada, G. V. Nossal, “The clonal selection theory”, Scientific American, vol. 257, no. 2, pp. 50-57, 1987.

[3] C. Coello, N. Cortes, „Solving Multiobjective Optimization Problems using an Artificial Immune System”, Kluwer Academic Publishers, Netherlands, 2002.

[4] S. Hein and A. Zakhor, "On the stability of sigma delta modulators," IEEE Trans. Signal Proc, July 1993.

[5] R. M. Gray, "Spectral analysis of quantization noise in a single-loop sigma-delta modulator with dc input," IEEE Trans. Commun., vol. COM-37, June 1989.

[6] N. He, F. Kuhlmann, and A. Buzo, "Multiloop sigma-delta quantization," IEEE Trans. Inform. Theory, vol. 38, pp. 1015-1028, May 1992.

[7] S. Rangan R. and B. Leung, "Quantization noise spectrum of double-loop sigma-delta converter with sinusoidal input," IEEE Trans. Circuit and Systems-II: Analog and Digital Signal Processing, vol. 41, Feb. 1994

[8] Galton, "Granular quantization noise in a class of delta-sigma modulators," IEEE Trans. Inform. Theory, vol. 40, pp. 848-859, May 1994.

[9] R. K. Poorfard and D. A. Johns, "Analysis of delta-sigma modulators with zero mean stochastic input," IEEE Trans. Circuit and Systems-H: Analog and Digital Signal Processing, vol. 42, pp. 164-175, Mar. 1995.

[10] Wang, Y., Muhammad, K., Roy, K “Design of Sigma-Delta Modulators With Arbitrary Transfer Function,” IEEE Trans. on Signal Processing, V55, N2, pp. 677-683, 2007.

[11] M. Kozal, I Kale, Oversampled Delta-Sigma Modulators – Analysis, Applications and Novel Topologies, Kluwer Academic Publishers, 2003.

[12] F. Burnet, ”Clonal Selection and After. in Theoretical Immunology, (Eds.) G. I. Bell, A.S. Perelson & G. H. Pimbley Jr., Marcel Dekker Inc., 1978, pp. 63-85

[13] L. Castro, F. Zuben, „The Clonal Selection Algorithm with Engineering Applications”, Proc. of GECCO’00, Workshop on Artificial Immune Systems and Their Applications, pp. 36-37, 2000

[14] X. Z. Gao, S. J. Ovaska, X. Wang, M. Y. Chow, “Clonal optimization-based negative selection algorithm with applications in motor fault detection”, Neural Comput & Aplic, Springer-Verlag London, 2009

1015