IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 104-114

International Journal of Research in Information Technology (IJRIT) www.ijrit.com

ISSN 2001-5569

Design & Implementation of Adaptive NLMS Filter with Different Target Filters For Optimal Response Mani Nagpal Electronics & Communication Engg. Department Haryana College of Technology & Management, kaithal, India [email protected]

Mr. Munish Verma Electronics & Communication Engg. Department Haryana College of Technology & Management, Kaithal, India [email protected] Abstract—Adaptive filters play an important role in modern day signal processing with applications such as noise cancellation, signal calculation, adaptive response cancellation and echo cancellation. The adaptive filters used in this paper is NLMS (Normalized Least Mean Square) filter, is the most widely used and simplest to implement. NLMS algorithm has low computational complexity, with good convergence speed which makes this algorithm good for echo cancellation. It has minimum steady state error. The noise amplification becomes smaller in size due to the presence of normalized step size. The NLMS algorithm shows far greater stability with unknown signals. Here it uses three different target filters FIR, IIR and multiband Equiripple filter. A detail study of this filter is done by taking into account different cases. The effects of change in parameters are noted within a specific filter and later a comparison between the filters is done. Noise variance is another factor that is considered to learn its consequence. Also parameters of adaptive filter, such as phase size and filter order, are varied to study their outcome on presentation of adaptive filters. The results attained through these test cases are discussed in detail and will help in better understanding of adaptive filters with respect to signal type and filter parameters. The first test led us to conclude the step size increases the convergence speed of a filter from transient to steady state but at the same time increase the error variation in the steady state. This trade-off between the convergence speed and error variation can be set with the help of a suitable step size depending upon signal requirement. The results were implemented in MATLAB R2013a. Keywords- Adaptive NLMS, NLMS using FIR Target, NLMS using IIR target, Equiripple FIR Target.

I. INTRODUCTION Digital filtering is one of the fundamental aspect of Digital Signal Processing. Digital filters are very important portion of DSP. In fact, their performance is one of the key reasons that DSP has become so popular. A Digital filter can be used to pass the signals according to specified pass-band and reject the frequency other than the pass-band specification. DSP (Digital Signal Processing) is one of the field of technology where developments are taking place at a very high speed. The DSP application demands high speed and low power digital filters. A filter is a device used to remove unwanted signal from desired signal. The unwanted signal may be some kind of noise or echo signals, e.g. when a radio tuned to a particular frequency then radio receiver will filter out all other frequencies so that a clear sound can be observed [1]. Digital filters are classified into two categories based on impulse response : 1) Finite Impluse Response Filter(FIR) 2) Infinite Impluse Response Filter(IIR) FIR filters are of non-recursive nature i.e no feedback required. The response of these filters with a given impulse input will decay within finite limit. IIR filters are of recursive type i.e the output of a IIR filter will depend upon the present and past values of input as well as on past output. Further these type of filters have response of such kind which will never dies out. Both these types of filter have some advantages and disadvantages, so no one is suitable for all conditions [2]. The designing of digital filter requires the approved specification with fixed coefficients. If this description is time changing or not accessible then this problem can be manipulated by digital filter with adaptive quantities, which is known as adaptive filter. The adaptive filter is a filter that self-adjusts its transfer function according to an optimization algorithm driven by an error signal. Because of the difficulty of the optimization procedures, most adaptive filters are digital filters. Adaptive

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signal processing has been introduced and its growth to the advanced related fields of digital computing, DSP and high speed joined circuit technology has been made rapidly. The least mean square (LMS) adaptive filtering algorithm’s first paper was published in 1959 by Widrow and Hoff. Adaptive filters are extensively used in the variety of application and they had been firstly proposed by Kelly of Bell Telephone Laboratories around 1965, most of the submissions are in telecommunication for the cancellation of noise and echoes in the transmission channel and also used in digital controller for active noise control. In the advanced era of cellular phone, digital television, wireless message and digital multimedia commercial services, progressive adaptive signal processing may give the better solution for the technical problem [3]. The designing of digital filter requires the approved specification with fixed coefficients. If this requirement is time varying or not accessible then this problem can be manipulated by digital filter with adaptive coefficients, which is known as adaptive filter. To design Adaptive filters, LMS, NLMS and RLS algorithm is used. Many other algorithms have been developed based on Linear Programming, Quadratic Programming and Heuristic methods in Artificial Intelligence. Remez Exchange Algorithm(to design equiripple filter) and linear Programming (to design adaptive filter) are optimum in the sense that these methods achieve both a given discrimination and a specified selectivity with a minimum length of the filter impulse response [4]. In the present research work our focus is to design Multiband Equiripple filter, FIR filter using Hanning window and Butterworth IIR filter as target filters with different design parameters. We will also design Adaptive filter using adaptive filtering technique which provide better control of parameter and nearly ideal response. The paper is prepared as follows. In section II, we discuss related work with designing of digital filters. In Section III, it describes the Adaptive NLMS filter response. In Section IV, it describes the design and implementation steps used in designing NLMS filter technique. The results are given in Section V. Finally, conclusion is explained in Section VI.

II. RELATED WORK In literature, authors had discussed FIR filter design method which utilizes the NLMS (Normalized Least Mean Square) adaptive algorithm’s system identification abilities. With such a method, any filters including FIR, IIR or even analog ones can copy its reactions to design the desired FIR filters. The FIR filter generated by this method can have the exact amplitude and phase responses with a target FIR filter which has a smaller or equal length, or attain an amplitude reaction the same with an IIR or analog filter. This also designs a FIR filter with a more excellent roll-off steepness and stop band attenuation than the traditionally designed FIR filter. They had proved that with the NLMS adaptive algorithm, NLMS adaptive filters are able to design any filters including the analog ones whose responses are within the description by the adaptive filters’ length and their FIR structure. [6]. Author had discussed the simulation of Low Pass FIR Adaptive filter using least mean square (LMS) algorithm and least PTH norm algorithm. LMS algorithm is a type of adaptive filter known as stochastic gradient-based algorithms as it utilizes the gradient vector of the filter tap weights to converge on the optimal wiener solution whereas Least PTH does not need to adapt the weighting function involved and no constraints are imposed during the course of optimization. They had shown that least PTH provided better gain as compared to least square. [7]. Some had discussed the comparison between adaptive filtering algorithms that is least mean square (LMS), Normalized least mean square (NLMS), Recursive least square (RLS). Execution aspects of these algorithms, their computational complexity and Signal to Noise ratio are examined. Here, the adaptive behaviour of these algorithms is analysed. Recently, adaptive filtering algorithms have a nice trade-off between the complexity and the convergence speed. The study of all these algorithms cover three performance criteria: the minimum mean square error, the algorithm execution time and the required filter order. Also, comparison is described by SNR improvement table [8]. Authors had described the several parameters of LMS adaptive filter which can affect performance. In most of papers the step size parameter for controlling the performance was discussed. In this paper, three parameters: step-size, filter tap-size and filter form have taken. Here, the regression analysis is used for defining the relation between parameters and performance of LMS adaptive filter with using the system level simulation results. The output presented that all parameters had performance trends in each own particular form, which can be estimated from equations drawn by regression analysis [9]. III.

ADAPTIVE NLMS FILTER

The main drawback of pure LMS algorithm is that it is sensitive to the scaling of its input x(n) which makes it hard to choose a step sizes µ that guarantees stability of the algorithm. The Normalized least mean square filter(NLMS) is a variant of LMS algorithm that solves the above problem by normalizing with the power of input. In other sense, we can say that normalized LMS(NLMS) algorithm is a modified form of the standard LMS algorithm [10]. To reduce the computational complexity and convergence time or increase convergence speed, some alternative LMS based algorithms are used. Adaptive signal processing is more famous due to the property of its digital techniques which is characterized by flexibility and accuracy in the field of communication and control. Adaptive filters are designed to remove the problem of wiener filter. In wiener filters, the processed data will be matched with the prior information for designing. Adaptive filter is totally based on stochastic approach [5]. An adaptive filter is required when either the fixed specifications are unknown or the specifications cannot be satisfied by time-invariant filters. Strictly speaking, an adaptive filter is a nonlinear filter since its characteristics are dependent on the input signal and consequently the homogeneity and additivity conditions are not satisfied. However, if we freeze the filter

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parameters at a given instant of time, most adaptive filters considered are linear in the sense that their output signals are linear functions of their input signals [9]. Target filter (Equiripple,FIR, IIR)

Desired response Output signal

∑ x(n)

Adaptive filter

Error Figure 1: Block Diagram of Proposed System

An adaptive algorithm is a procedure for adjusting the parameters of an adaptive filter to minimize a cost function chosen for the given task. To discuss each algorithm, we choose FIR structure. Such system are more popular than IIR filters since the input-output stability of the FIR filter structures is guaranteed for any set of fixed coefficients, and the algorithms for adjusting the coefficients of FIR filters are more simple in general than those for adjusting the coefficients of IIR filters. The Adaptive algorithm tries to minimize an appropriate objective or error function that involves the input, reference and filter output signal. This algorithm can be consist of three parts, the definition of minimizing algorithm, the definition of objective function and the definition of error signal [11]. Block diagram of proposed system is given in fig. 1. Here we first take a sinusoidal input signal x(n) that is combined with desired response to give output d(n). This desired response is the output of different target filters used in our designing process. These target filters are equiripple filter, FIR filter, Butterworth IIR filter. The desired response d(n) is combined with the adaptive filter response h(n) to give final output. Then we find the error in the output signal. Here the input signal is filtered for the required output and then passed through further processing. The filter’s output is observed by determines its quality for particular application. After measuring the quality it also examined by a circuit whether it is need to improve the quality of the output signal. This processing loop continues until the filter’s parameters are adjusted properly, so the filter’s output quality should be as good as possible [12]. Adaptive filtering can be classified into three categories: adaptive filter structures, adaptive algorithms, and applications. The performance of the adaptive algorithm is important for all systems; it is also essential how adaptive system is functioning. The choice of algorithm is highly dependent on the signals of interest, the operating environment, as well as the convergence time required and computation power available. For any application the adaptive algorithm provide competent performance evaluations for the structures of various filter and adaptive algorithm. An adaptive digital filter can be built up using an IIR (Infinite impulse response) or FIR (Finite impulse response) filter [13].

IV. DESIGN AND IMPLEMENTATION As the NLMS is an extension of the standard LMS algorithm, its practical implementation is very similar to that of the LMS algorithm except that the NLMS algorithm has a time varying step size µ(n). This step size can improve the convergence speed of adaptive filter. Each iteration of the NLMS algorithm requires these steps in the following order [11]. The only difference with respect to LMS is the coefficient updating step (4). 1) The output of the adaptive filter is calculated. (1) 2) An error signal is calculated as the difference between the desired signal and the filter output. (2) 3) The step size value is calculated from the input vector. (3) 4)

The filter tap weights or the coefficients are updated in preparation for the next iteration. (4) The general form of an adaptive FIR filtering algorithm is given by eq. (5):

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(5) Where G(.) is a particular vector-valued nonlinear function, is a step size parameter, e(n) and X(n) are the error signal and input signal vector, respectively, and is a vector of states that store at previous time instants. In the simplest form, is not used, the information used to adjust the parameters are the error signal, input signal and step size [14]. A. Designing Steps of the System Following are the steps which are used to design a filter to get our desired output. These steps are: • Firstly selecting a sinusoidal input at a desired frequency. • Select target filter for NLMS designing like Equiripple FIR filter, Butterworth IIR filter, FIR filter using Hanning window. • Set the following Filter design parameters: Normalized frequency, Sampling frequency, Modulating frequency, Filter order, Step size. • Designing NLMS adaptive filter in MATLAB for all the three target filters individually and finding their magnitude and phase responses. • Design of Multiband Equiripple filter with filter length 99 and comparison of its performance with NLMS filter designed. • Design of IIR target of 10th order and comparison of its performance with NLMS filter designed. • Design of FIR target of 20th order and comparison of its performance with NLMS filter designed. • Then analysis of magnitude response and phase response of above designed filters individually. • Find the error and convergence rate for above designed filters. B. Proposed Parameters 1. RMSE The root mean square error, RMSE (also called root mean square deviation, RMSD) is a measure of the differences between values predicted by a model or an estimator and the values actually observed from the thing being modelled or estimated. These individual differences are also called residuals, and the RMSE aggregate them into a single measure of predictive power. Since the RMSE is a good measure of accuracy, it is ideal if it is small [15]. 2. Convergence Rate Convergence rate is a performance specification of acoustic echo cancellers. The convergence rate should be fast to estimate the desired filter. The convergence rate is defined as the number of iterations required for the algorithm to converge to its steady state mean square error. 3. Complexity The computational complexity is the measure of the number of arithmetic calculations like multiplications, additions and subtractions for different adaptive algorithms [15].

C. Flowchart for Proposed Work The flowchart of proposed system is shown in fig 2.

Input Signal Acquisition

Filter Design Parameters Set: Cut off Freq., Order, Fitter Type (LPF, HPF, BPF, SBF), Step size, Window Method

Multiband Equiripple as a targetfilter

FIR as a target filter

Butterworth IIR as a target filter

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Obtain its desired Response

Design Adaptive NLMS Filter With Different Design Parameters

Compare adaptive Output with target Output

Analysis of magnitude & phase response filters

Compute MSE

Comparison & analysis with standardized results

Figure 2: Flowchart of Proposed System

V. RESULTS In this section, it provides the results of Adaptive NLMS technique with different target filters and also with different step size and frequency. The target filters used here are FIR filter, Butterworth IIR and Multiband equiripple Filter. 1. Adaptive NLMS with Multiband Equiripple as a Target In this, sinusoidal input is used to generate a desired signal using Multiband Equiripple as a target filter. After selecting the filter type then it has to give value of the step size (µ) and filter order (N) which are required for the algorithm simulation. The filter length used is 99. The step size determines the updating speed of filter coefficients. Here, step size is 1.4. For designing of equiripple filter as the target filter, fix values of different parameters. Also, Parameters having same values are used for designing of adaptive NLMS filter. Table 1: Design Parameters of Multiband Equiripple Filter

Design Parameters Response type Design method Filter order Freq. vector (Normalized) Magnitude vector Weight vector Sampling frequency Modulating frequency Step size

Value Multiband Equiripple FIR 98 (Length=99) [0,0.28,0.3,0.48,0.5,0.69, 0.7,0.8,0.81,1] [0,0,1,1,0,0,1,1,0,0] [1,1,1,1,1] 8000 8000 1.4

Comparing the responses between the converging result and the target, the amplitude and phase responses of both filters are drawn in the same figure of Fig.4. In this fig, magnitude response of target filter & adaptive NLMS almost overlap with each

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other, while phase response is completely overlapping each other at different normalized frequencies. Their responses are approximated to ideal filter response. The resulting frequency response can be a monotone function or an oscillatory function within a certain frequency range. The waveform of frequency response depends on the method used in design process as well as on its parameters. Magnitude Response (dB) and Phase Response

Adaptive: Magnitude Adaptive: Phase

10

-10

-52.8912

-20

-73.0314

-30

-93.1716

-40

-113.3117

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P h a s e (ra d ia n s )

-32.751

M a g n itu d e (d B )

0

-12.6109

0.9

Normalized Frequency (×π rad/sample)

Figure 3: Response of NLMS Adaptive Filter Magnitude Response (dB) and Phase Response 20

2.8774 -4.4339

0

-11.7452

M a g n it u d e (d B )

10

-10

-19.0564

-20

-26.3677

-30

-33.6789

-40

-40.9902

-50

-48.3014

-60

-55.6127

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P h a s e (r a d ia n s )

Equiripple FIR Adaptive NLMS

0.9

Normalized Frequency (×π rad/sample)

Figure 4: Response Comparison of NLMS and Multiband Equiripple Target

2. Adaptive NLMS with FIR as a Target In this, sinusoidal input is used to generate a desired signal using FIR as a target filter. To get desired response, it uses a NLMS adaptive filter with a length of 19, and the target FIR filter is with a length of 21. Windowing functions are most easily understood in the time domain; however, they are often implemented in the frequency domain instead. FFT windows reduce the effects of leakage but cannot eliminate leakage entirely. In effect, they only change the shape of the leakage. In addition, each type of window affects the spectrum in a slightly different way. Hanning window has good frequency resolution, less spectral leakage and good amplitude accuracy. That’s why it is preferred over other windows. Now, in this, it designs FIR filter as target filter & adaptive NLMS filter as proposed filter with the values as given in Table 2. For better comparison, a comparing FIR filter is designed with lower filter length as specified in Table 2. In this filter, windowing technique is used and hanning window is preferred over others. All other parameters are set according to the requirement of desired response. The other parameters are chosen accordingly to get desired results. Table 2: Design Parameters of FIR Filter

Parameters

Value for Target FIR filter

Value for Comparing FIR filter

Value for NLMS Adaptive filter

Response type

Low pass

Low pass

Low pass

Window Used

Hanning

Hanning

Hanning

Filter length

21

19

19

Normalized frequency

0.5

0.5

0.5

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Sampling frequency

30

30

30

Modulating frequency Step size

15

15

15

1.4

1.4

1.4

Magnitude Response (dB) and Phase Response 0

2.8217

NLMS: Magnitude NLMS: Phase

-10

-30

-16.3698

-40

-22.767

-50

-29.1642

-60

-35.5613

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P h a s e (ra d ia n s )

-9.9726

M a g n it u d e (d B )

-20

-3.5755

0.9

Normalized Frequency (×π rad/sample)

Figure 5: Response of NLMS Adaptive Filter with FIR Target Magnitude Response (dB)

FIR (order=20) FIR (order=18) NLMS (order=18)

0 -10

M a g n itu d e (d B )

-20 -30 -40 -50 -60 -70 -80 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Frequency (×π rad/sample)

Figure 6: Comparisons among Target and Comparing Filter Amplitude Responses

The fig 5 shows the response of proposed NLMS filter of order 18. It has low stop-band attenuation and does not have a flat response. It has response of sine wave. As shown in Fig.6, the proposed converging result has comparable roll-off sharpness and with the larger order FIR filters which are better than the comparing same order FIR filter. The filter response shows that it is trying to extract a single frequency component as it was required and expected. The magnitude response is best given by Higher order FIR as compared to adaptive NLMS of order 18. FIR filters can have a linear phase response and they are very simple to implement. But one disadvantage such ways may result is the missing of the Phase Response 5

FIR (order=20) FIR (order=18) NLMS (order=18)

0 -5

P h a s e (ra d ia n s )

-10 -15 -20 -25 -30 -35

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Frequency (×π rad/sample)

Figure 7: Comparisons among Target and Comparing Filters Phase Responses

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FIR filters’ linear-phase feature, which is shown in Fig.7. The proposed NLMS phase response is also abrupt as shown in fig 7. FIR filters can have linear phase characteristic, which is not like IIR filters. A FIR filter is designed by finding the coefficients and filter order that meet certain specifications, which can be in the time-domain and/or the frequency domain. 3. Adaptive NLMS with IIR as a Target The primary advantage of IIR filters over FIR filters is that they typically meet a given set of specifications with a much lower filter order than a corresponding FIR filter. In this, it designs Butterworth Bandpass IIR filter as the target filter and also design NLMS adaptive filter as in MATLAB. In this, Butterworth IIR filter is chosen as a target filter. Because an IIR filter can achieve a similar amplitude response with a much lower order than its FIR. To test the efficiencies of the NLMS adaptive filter modelling a IIR filter with a smaller length, a 10th order Butterworth band pass IIR filter is chosen as the target filter, the length of the NLMS adaptive filter is 99. For a band pass filter, specified Weight as a two-element vector containing the pass band edge frequencies. This allows for a non-causal, zero-phase filtering approach, which eliminates the nonlinear phase distortion of an IIR filter. The Butterworth filter provides the best Taylor Series approximation to the ideal filter. That’s why it is preferred over others. Following are the design requirements for target filter & adaptive NLMS filter: Table 3: Design Parameters of Target IIR Filter

Parameters

Response type Design method Filter order Normalized frequency(Fc1) Normalized frequency(Fc2) Sampling frequency Modulating frequency Step size

Value for Target IIR Filter Band pass Butterworth IIR 10 0.3

Value for NLMS Adaptive filter Band pass Butterworth IIR 98 0.3

0.7

0.7

8000 4000

8000 4000

1.45

1.45

Magnitude Response (dB) and Phase Response 0

2.9996

NLMS: Magnitude NLMS: Phase

-20

-60

-11.7311

-80

-16.6414

-100

-21.5516

-120

-26.4618

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P h a s e (r a d ia n s )

-6.8209

M a g n it u d e ( d B )

-40

-1.9106

0.9

Normalized Frequency (×π rad/sample)

Figure 8: Response of NLMS Adaptive Filter with IIR Target

The response of proposed NLMS filter of order 98 using IIR target is shown in fig 8. The magnitude shows the better response but does not have a flat response so it is approximated to ideal filter response and the phase response is linear as shown in figure8.

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Magnitude Response (dB) and Phase Response

IIR #1: Magnitude 2.6119 NLMS #2: Magnitude -2.0271 IIR #1: Phase NLMS #2: Phase -6.6662

-20

M a g n itu d e ( d B )

-40

-60

-11.3053

-80

-15.9443

-100

-20.5834

-120

-25.2224

-140

P h a s e (ra d ia n s )

0

-29.8615 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Frequency (×π rad/sample)

Figure 9: Response Comparisons of Adaptive NLMS and IIR Target

The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the pass band. It is also referred to as a maximally flat magnitude filter. Shown in Fig.9, both amplitude responses are almost the same though the phase responses are various. When the seemingly same amplitude responses in Fig.9 enlarged, some slight differences on some remote parts (under 100 dB) can be found. The poles of the converging results are always centred at the centre of the circle. IIR filters can be implemented this way. PoleZero plots is an important tool, which helps us to relate the Frequency domain and Z-domain representation of a system. Understanding this relation will help in interpreting results in either domain. It also helps in determining stability of a system. The pole-zero plots gives us a convenient way of visualizing the relationship between the Frequency domain and Z-domain. For a system to be stable, its impulse response must be absolutely summable i.e. lies within range less than 1. In order that a discrete-time system is stable, all poles of the discrete-time system transfer function must be located inside the unit circle, as shown in Figure 10 &11. Pole/Zero Plot 1 0.8

Imaginary P art

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.5

-1

-0.5

0

0.5

1

1.5

Real Part

Figure 10: Pole-Zero Plots ofatIIR System Butterworth Bandpass Filter Order = 10 1 0.8

Imaginary Part

0.6 0.4 0.2

98

0 -0.2 -0.4 -0.6 -0.8 -1 -1.5

-1

-0.5

0

0.5

1

1.5

Real Part

Figure 11: Pole-Zero Plots of Adaptive NLMS System

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4. Comparisons of NLMS Performance with Target Filters All these filters are compared in terms of their RMSE values. Mean square estimation is the average of the squares of the error. Lower the RMSE values, larger its convergence. The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filters. The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the pass band. It is also referred to as a maximally flat magnitude filter. So, Butterworth IIR has lower RMSE value hence it has high convergence rate as compared to other filters. So, at lower order, IIR filter provides better response than any other filter. It provides faster convergence and also provides stability in system. Smaller the error, response is very close to desired response. Table 4: Comparisons of NLMS Performance with Target Filters

Parameter RMSE

5.

Multiband Equiripple 0.55

FIR

Butterworth IIR 0.20

0.91

Comparison of NLMS and LMS Filter

In this case the signal length is taken as 19, the step size used is 0.01 and the 18th filter order is used. The results are shown by figure 12. Comparing the LMS and NLMS Conversion Performance 0.3 LMS Derived Filter Weights NLMS Derived Filter Weights 0.25

0.2

O utp u t

0.15

0.1

0.05

0

-0.05

-0.1

0

5

10

15

20

25

30

35

Frequency

Figure 12: Error Comparison between Adaptive NLMS and LMS Filter

Fig 12 shows the error plot, the smaller step size results in the slow convergence rate at the start but as soon as it enters the steady state we found that smaller step size gives good result by giving less variation. This shows that the smaller step size approaches steady state late but has a good response in steady state. The error plot is the difference of desired signal d(n) and filter output y(n). This difference tells us how close is the filter in producing the desired signal, lower the absolute value of error closer the output of the filter gets to the desired signal. The algorithm LMS and NLMS are also designed and updated according this error value. The error plot gives us an idea how well the filter is performing.

VI. CONCLUSION In this paper, it covers the designing and implementation of adaptive filter. The adaptive filter used is NLMS. Here it uses three different target filters FIR, IIR and multiband Equiripple filter. Also covers the effects of stationary signals on the performance of adaptive filters. We tested the signal with variation in step size, filter order and sample frequency. The effects of changes in parameters were noted within a specific filter and later a comparison between the filters was done. The first test led us to conclude the step size increases the convergence speed of a filter from transient to steady state but at the same time increase the error variation in the steady state. This trade-off between the convergence speed and error variation can be set with the help of a suitable step size depending upon signal requirement. The second test on the filter order helped us to see how the filter frequency response varies with variation in filter order. The filter order can be set according to the expectation of the final filter. If the error minimization requirement is less strict we are allowed to the keep the filter order low. NLMS performs much better the LMS for the non-stationary signal that is much more difficult to handle. The performance can be accessed from error values.

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 104-114

REFFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

“Digital Signal Processing” S Salivahanan, Tata Mcgraw Hill, 2nd edition L.Litwin, October/November 2000, FIR and IIR digital filters Selesnick , EL 713 Lecture Notes, Digital filtering A. Mishra, K.Pachauri and Zaheeruddin, “Design of 1-Dimentional FIR Filter using Modified Widrow-Hoff Neural Network”, International Journal of Computer Applications (0975-8887) Volume 59, no 20, 2012. J. Dhiman, S. Ahmad and K.Gulia, “Comparison between Adaptive filter Algorithms (LMS, NLMS, RLS)”, International Journal of Science, Engineering and Technology Research (IJSETR), Volume 2, Issue 5, 2013. M. G. Bellanger, Adaptive Digital Filters, Second Edition Revised and expended, Marcel Dekker, Inc. 2001 S. Chaudhary, R.Mehra, “Adaptive Filter Design and Analysis Using Least square and Least PthNorm”,International Journal of Advances in Engineering and Technology, May 2013, Vol. 6, Issue 2, pp. 836-841 H. Kaur, Dr. R. Malhotra, Anjali Patki, “Performance Analysis of Gradient Adaptive LMS Algorithm”, International Journal of Science and Research Publication, Vol. 2, Issue 1, January 2012, ISSN 2250-3153 A. Kabir, khandakerabirRahman and Iahtiaquehussain, “Performance Study of LMS and NLMS Adaptive Algorithms in Interference Cancellation of Speech Signals”. Widrow, B. and Hoff, M.E., 1960, “Adaptive switching circuits”, IRE WESCON Convention Record. F. Boroujeny, B.1999, Adaptive Filters Theory and Applications. John wiley and Sons, Newyork “Matlab Simulator for Adaptive Filters”, By Mr. Muhammad Shahid, Department of Electrical Engineering, School of Engineering (Blekinge, TekniskaHogskola, Sweden) Treichler, J. R. and Johnson Jr., R. C. and Larimore, Michael G. Theory and Design of Adaptive Filters. New York: John Wiley and Sons, 1987 Divya, P. Singh, R.mehra, “Performance Analysis of LMS & NLMS Algorithms for noise cancellation”, International Journal of Scientific Resarch Engineering and Technology (IJSRET), Volume 2, Issue 6, pp 366-369 September 2013 Rafaely, B. Elliot, “A Computationally efficient frequency-domain LMS algorithm with constraints on the adaptive filter”, IEEE Transaction on Signal Processing, vol. 6, Issue 6, pp. 1649-1655, 2002. G. Yecai, H.Longquing, “Design &Implimentation of Adaptive Equalizer Based on FPGA”, 8th International Conference on Electronic Measurement & Instruments, pp. 790-794, 2007

Mani Nagpal, IJRIT- 114