Scholarly article on topic 'Based Second Order ∑Δ Modulator for Digital Hearing Aid Applications'

Based Second Order ∑Δ Modulator for Digital Hearing Aid Applications Academic research paper on "Materials engineering"

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Academic journal
Procedia Computer Science
OECD Field of science
{"Genetic algorithm (GA)" / "Sigma Delta Modulator (∑ΔM)" / "Signal-to-Noise Ratio (SNR)"}

Abstract of research paper on Materials engineering, author of scientific article — Rijo Sebastian, Babita R. Jose, Shahana T.K., Jimson Mathew

Abstract This paper describes the design and simulation of a sigma delta modulator (∑ΔM) architecture that can be used in digital hearing aid applications. A model of a non-ideal second order feed-forward ∑Δ Modulator, adopted for this purpose, is also presented. The loop coefficients of this architecture are optimized using Genetic Algorithm. The architecture together with optimized coefficients achieves 99dB SNDR and a dynamic range of 101dB. A sampling frequency (fs) of 1.28MHz and a second order modulator makes this architecture less complex and power efficient.

Academic research paper on topic "Based Second Order ∑Δ Modulator for Digital Hearing Aid Applications"


4th International Conference on Eco-friendly Computing and Communication Systems, ICECCS

based Second Order LA Modulator for Digital Hearing Aid


Rijo Sebastian^*, Babita R. Josea, Shahana T. K.a,Jimson Mathewb

"Division of Electronics,Cochin University of Science and Technology,Kochi,682022,India bDepartment of Computer Science, University of Bristol,BS8 1TH, U.K


Available online at


Procedía Computer Science 70 (2015) 274 - 281


This paper describes the design and simulation of a sigma delta modulator (EAM) architecture that can be used in digital hearing aid applications. A model of a non-ideal second order feed-forward EA Modulator, adopted for this purpose, is also presented. The loop coefficients of this architecture are optimized using Genetic Algorithm. The architecture together with optimized coefficients achieves 99 dB SNDR and a dynamic range of 101 dB. A sampling frequency (fs) of 1.28MHz and a second order modulator makes this architecture less complex and power efficient.

© 2015 The Authors. Publishedby ElsevierB.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-reviewunderresponsibilityoftheOrganizingCommittee of ICECCS2015 Keywords:Genetic algorithm(GA);Sigma Delta Modulator (SAM); Signal-to-Noise Ratio (SNR)

1. Introduction

Analog to Digital (A/D) Converters are widely used in a variety of applications like digital communication, consumer electronics, industrial measurements and many medical equipments. Sigma delta(ZA) analog-to-digital converters are a popular choice among the different types of analog-to-digital converters especially for applications requiring high resolution and low bandwidth. ZA analog-to-digital converter consists of an oversampling modulator followed by a digital/decimation filter and their combined action produces a digital output1. The oversampling and noise shaping techniques used in ZA converters helps this data converter to achieve high resolution2.

* Corresponding author. Tel.: +91-9847422743 E-mail address: jyrijo @

1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of ICECCS 2015 doi:10.1016/j.procs.2015.10.088

The basic HA modulator consists of a loop filter, a quantizer and a feedback loop. A digital to analog converter is needed in the feedback loop since the output of the quantizer is a digital signal. Multi-order modulators shape the quantization noise to even higher frequencies than do the low order modulators 3. The selection of a particular ÜA ADC architecture for an application is extremely important because it takes it into account many factors such as achievable resolution, complexity of the modulator, and sensitivity of the modulator to non-ideal effects.

The genetic algorithm is a search method that imitates the idea of natural evolution. It works on a population of chromosomes and applies the principle of survival of the fittest to produce better approximations to a solution4. The optimum values of the loop coefficients in ÜA architecture can be found easily using GA. These optimum values provide the maximum signal-to-noise ratio and resolution.

The rapid advancements in the field of digital signal processing and recent trends in large scale integration techniques makes digital hearing aids the better choice compared to the analog counterpart. Fig. 1 shows the typical block diagram of a digital HA system 5. In audio applications, the ADC requires a dynamic range as high as 90dB over a bandwidth of 10 kHz. The power consumption of the ADC should be also minimized to optimize the overall power efficiency 6.

Microphone Pre amplifier SA ADC

2A DAC Receiver Driver Receiver

A i i r t i A

Control parameters

Fig. 1. Block diagram of hearing aid system 5

This paper is organized as follows. Section 2 introduces the details of modulator architecture being used in the ADC section. Section 3 describes the various nonidealities present in a switched capacitor (SC) sigma delta modulator. In section 4 we give an overview of genetic algorithm and why it is used as a proper algorithm for the optimization. Section 5 presents the experimental results and section 6 concludes the paper.

2. Modulator architecture

Digital hearing aids are battery powered systems and the system needs to be more energy efficient. The ADC block in digital hearing aid system accounts mostly for power consumption. The ADC should also meet the dynamic range (DR) requirement for audio applications. The SNR of SAADC depends on over sampling ratio (OSR), order of the modulator and the number of quantizer bits. A high value for OSR is a good choice for improving SNR but it may increase the power consumption. Therefore, the sampling frequency(fs) is kept low to limit the power dissipation. If the order of modulator increases, stability issues may arise. It seems that high order modulators tend to unstable very easily.

Focusing on the requirements of digital hearing aid ADC, we designed a 2nd order feed-forward £A modulator for 10 KHz bandwidth. Fig. 2 shows a second order feed-forward sigma delta modulator architecture that is designed for 10 KHz. The input signal is provided directly to the input of the quantizer. This makes the signal transfer functionunity.The unity signal transfer function indicates that the distortion due to nonidealities in the modulator's loop filter can be reduced 7. Increasing the number of bits in the quantizer reduces the quantization noise power.The loop coefficients can also be scaled-up by using a multibit quantizer.This architecture uses a 4-bit quantizer.

The performance of the modulator mostly depends on the values of the loop filter coefficients. These coefficients are all related to each other and a small variation in one may cause drastic changes in the operation and performance. The values of the loop coefficients have to be selected in such a way that the modulator operates in

stable condition for the whole input range. The values obtained using GA area^l.74, a2=2.97, ^=3.11, c2=0.99.

Discrete-time Real Integrator

Discrete-time Ideal Integrator H

4-bit DAC

Fig.2. Architecture of the 2nd order feed-forward SAM with 4-bit quantizer The output of the HA modulator is given by:

E (z )(1 - z-1)2

Y ( z) = X ( z ) +

1 + 3.41z+ 0.74 z -2

The input signal, the output signal and the quantization noise are denoted as X(z), Y(z) and E(z). The expression for bandwidth and effective number of bits i.e. resolution is shown in (2) and (3).

B.W ■


SNR(dB) -1.76 6.02

OSR,SNR,fs are respectively the over-sampling ratio, the signal-to-noise ratio and the sampling frequency. The dynamic range (DR) depends on the oversampling ratio (OSR), the order of noise shaping L, and the quantizer resolution N.

DR = ! I n+1 I (2n -1)2OSR2L+1

3. EA Modulator nonidealities

The main nonidealities associated with this modulatorarchitecture are the clock jitter at the input sampler; switch thermal noise (KT/C),operational amplifier noise, operational amplifier finite gain, operational amplifier BW, operational amplifier SR, and operational amplifier saturation voltages8.

The various non-idealities affecting the operation of a modulator are modeled using SIMULINK blocks and simulations were performed. The real integrator block describes the main circuit non-idealities like op-amp finite dc gain, slew rate, gainbandwidth product, and amplifier saturation voltage.

4. Genetic algorithm

Genetic algorithm is a problem solving method and finds its application in all fields of science and engineering for optimization. It uses search technique to find approximate solutions for problems9. In GA abinary population of chromosomes is generated randomly. The algorithm loops over an iteration process to make the population evolve. GA consists of Selection, Reproduction, Evaluation and Replacement operations. The algorithm terminates when the population converges towards the optimal solution. Genetic algorithm does not require any huge mathematical calculation or derivative information. Fig.3. shows the GA cycle of operation.


Genetic Operation


, Objective function


Fig.3. GA cycle

To achieve the best performance of the modulator and to maintain the stability, loop coefficients must be optimized10. GA is one of thebest optimization techniques which find a global optimum solution without taking much of the computational power11. In the case of of sigma-delta modulators maximum signal-to-noise and distortion ratio (SNDR) corresponds to the optimum value of loop coefficients.

In this work the initial population is taken as 20 and the maximum number of generations is limited to 30. The selection method employed here is stochastic universal sampling. In GA crossover operation is used for producing new chromosomes and the new individual have some characteristics of both parent's genetic material. Single-point crossover technique is adopted here. Crossover rate and mutation rate were chosen as 0.7 and 0.7/Lind, where Lind is the length of an individual. This work uses a generation gap, GGAP =0.9 and fitness based reinsertion to implement an elitist strategy. Thus, 18 new individuals are produced at each generation. The objective function to be minimized is the noise power of the modulator. The coefficients responsible for the least value of the noise power are the most fit individuals.

Fig.4. shows SNDR value in each generation and the maximum value obtained is 99 dB.The coefficients converge to their optimum value from20th generation onwards resulting in a maximum SNDR. The optimum values obtained for the loop coefficients are a1=1.74, a2=2.97, c1=3.11, c2=0.99.

GA can also utilize to estimate the values for finite GBW and slew rate. The unity gain bandwidth (GBW) requirement of switched-capacitor implementations, were about one order of magnitude higher than the sampling frequency for the past so many years. Now recently, this could be relaxed to a factor of 2- 5 fs 12.If we limit the range of GBW in the range 2-5 times fs in GA, the program itself will estimate the GBW corresponding to maximum SNDR. A similar approach can be adopted for obtaining the slew rate. This makes the design of operational amplifier easy.

102 100

S 96 z

0 5 10 15 20 25 30

Number of generations

Fig.4. SNDR value in each generation corresponds to the value of coefficients

Maximum SNDR[dB]= 99.736

@ a1=1.741


c1 =3.118 -


■ [ [

5. Simulation example

A second order feed forward HA modulator is modeled using SIMULINK and the optimization algorithm is working in a MATLAB environment. Fig. 5 shows the power spectral density of the HA modulator. The input frequency is taken as 234Hz, sampling frequency is 1.28 MHz and an OSR of 64. Table 1 shows the details of parameters used in the simulation.

Frequency [Hz]

Fig.5. Measured output spectrum for -3 dBFS

Simulation results points that the proposed architecture achieves a SNDR of 99 dB and an ENOB of 16 bits for 10 KHz bandwidth. The finite BW and SR result in harmonic distortion and SNDR performance of the modulator is degraded.Table 2. shows the SNDR values obtained with and without optimization of loop coefficients. The results indicate that there is an improvement of 6 dB by using optimization technique.

Table 1. Simulation parameters

Parameter Value

Signal bandwidth BW=10KHz

Sampling frequency Fs=1.28MHz

Oversampling ratio OSR=64

No. of samples considered 65636

Loop filter coefficients (using GA) ai=1.74, a2=2.97

c1=3.11, c2=0.99

Sampling jitter 16nS

Switches (kT/C) noise-Cs 1.25pF

Operational amplifier noise 73^Vrms

Finite dc gain 1e3

Finite GBW 3.28 MHz

Slew-rate 13.77 V/^S

Table.2. Comparison of SNDR with and without coefficient optimization Coefficients SNDRpeak

Without GA ai=4, a2=1.8, ci= 0.47, C2= 0.13 93 dB

With GA ai=1.74, a2=2.97, ci=3.11, C2=0.99 99 dB

The dynamic range of the modulator can be evaluated from SNDR vs input amplitude graph. The modulator achieves a dynamic range of 101 dB. Fig.6. shows the SNDR versus input amplitude.

Fig.6.SNDR vs input amplitude

Fig.7. (a) shows the SNDR vs GBW graph.The SNDR remains same for all values of GBW above 0 .5MHz.The SNDR versus slew rate of this modulator is shown in Fig.7 (b).The SNDR remains constant for values of slew rate above 5V/^S

C.S I (.5 2 2.S 3 3.S 4 4.5 S

О 0.2 0.4 0.6

1.2 1.4 1.6 1.8 2

SR (V/s)


(a) (b)

Fig.7.(a) SNDR vs Gain bandwidth; (b) SNDR vs Slew rate The performance summary ofthis modulator architecture is shown in Table 3 Table 3. Performance summary


Architecture Sampling frequency Signal bandwidth OSR

SNDRpeak(with nonidealities) 99 dB @-3dBFS

DR 101 dB

2nd order feed-forward


— 60

6. Conclusion

In this paper a behavioural model of theHA modulator considering various nonidealities is presented. The optimization of loop coefficients using GA helps to improve the SNDR and DR of this modulator for hearing aid application. The simulations shows that this modulator architecture provides 99dB SNDR and a dynamic range of 101dB over a bandwidth of 10KHz.GA based optimization helps to achieve maximum performance for this second order feed-forward £A modulator.


The author acknowledges support from the Government of India for research under Maulana Azad National Fellowship, award number F1-17.1/2014-15/ MANF-2014-15-CHR-KER-47651.


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