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Filtering process giving the advantage of selecting a particular frequency range or all frequencies after or before selected cut off frequency, the use of filtering can be seen in many digital signal processing applications where the frequency range according to the application selected with the help of such a filtering process. One can find the Fourier transform of the output signal with the help of the multiplication of the Fourier transform of the input signal and the Fourier transform of system frequency response. When someone see the frequency response of the ideal filter to compare it with the frequency response of his practical filter he found that the ideal filter shows the real and non negative response, we can say that it has a zero phase characteristic. We know that due to the zero phases characteristic the time shift is introduced in the pass band of filter, so that the shape of the signal in the pass band is not distorted. We very well know that if we have Fourier transform of a stable impulse response than it will be continuous function of'w, that's why we cannot get a stable ideal filter. We know that frequency response should be a continuous function of the realizable filter; hence the magnitude response of low pass filter with some acceptable tolerance is specified. When we don't know the phase response or it is not mentioned than one can prefer to use the IIR filter. For the designing of IIR filters we convert the specifications of  digital filter to analogue low pass prototype filter, so that we can obtain the Ha(s) which is analogue filter function that can satisfy all the specifications, now this analogue transfer function is converted into digital function H(z). Reasons to use this method:

The approximation techniques for analog filter are very advanced.

This method provides highly efficient solution.

For the analog filter extensive table is available so that one can prefer it.

In many applications there is the need of digital solution of analog filter.

We have following methods for smoothing a sequence of numbers in order to approximate a low-pass filter: the polynomial fit and the moving average. We found that in case of polynomial fit the higher degree of polynomial helps to produce a sharper cut in the magnitude response of the filter and the efficient results also produced by using the higher degree of polynomial approximation. One of the other parts is that to produce efficient response we should use the least-squares quadratic as compared to quadratic on original 'noisy' data.


To design a low pass finite impulse response filter (FIR), and applying NPSO algorithm on it to optimize its result. Novel particle swarm optimization (NPSO) is an advance optimization technique that uses velocity vector and swarm updating concept to optimize result and enhancing search capability, PSO uses the modification of inertia weight to enhance the search capability and hence the solution quality is improved. So that the probability of getting the global optimum solution is very high.

To compare the fitness of each particle vector with respect to the each particle best (pbest). On based on above criteria if the current solution is better than its particle best (pbest), then replace its particle best (pbest) by the current solution.

Comparison of the fitness of each particle vector with each group best (gbest). If the fitness of any particle vector is better than its group best (gbest), then replace its gbest.

We will obtain global optimal solution by velocity vector and swarm updating process in search space for FIR filter.

As well as we will obtain global optimal solution by number of iterations for the filter coefficients like mean, minimum, maximum, standard values with respect to the frequency variation.

On behalf of above objective a low pass filter shows the characteristic that it passes the frequency lower than the cut off frequency and attenuates signals with frequencies higher than the cut-off frequency. The attenuation amount for each frequency depends on the design of filter.


Many times the frequency responses are used to design FIR filter. With the help of the inverse Fourier transform the equivalent sampled impulse response, with the help of which coefficients of the FIR filter can be determined, can then be found. Due to the presence of feedback in the output the filter having Infinite Impulse Response. Transformation technique like bilinear transformation helps to map the poles and zeros of the analog filters into the plan called s-plan, because practical IIR filters are mostly based on analog equivalent filters like Butterworth and Dolph Chebyshev.

When we see the cut off frequency of digital filters then we got that it is tangentially warped in nature as compared to analog filter from which it has designed. Before going to design the analog filter it is important to pre-warp the required cut-off frequencies for compensating the undesirable effect. Hence the cut off frequencies (desired set) of digital filter we will determined first. The filter cut-off frequencies are converted to a new set of analogue cut-off frequencies. With the help of appropriate warped cut-off frequencies the analog filter is finally designed. Now to obtain the digital filter from this analog filter apply the bilinear transformation. In the lower order IIR filter of recursive type we can have desired frequency response as compared to the non recursive filter. According to the choice of designer the poles and zeros can be selected by them in the case of recursive filter because this type of filter having both poles and zeros. So when we compare recursive filter with non recursive of same order we got that in recursive we will have more number of free parameters. The only condition is that the zeros can be variable. When we got the pole location of an IIR filter near to the unit circle these poles need to be specified accurately. That means it should be represent in 3 to 6 places of decimal and this is done to avoid the instability.

FIR has following advantage over IIR Filter:

FIR filter is Finite IR filter and IIR filter is Infinite IR filter.  

FIR filters are non-recursive. That is, there is no feedback involved. Where as an IIR filter is recursive. There is feedback involved.

The impulse response of an FIR filter will eventually reach zero. The impulse response of an IIR filter may very well keep "ringing" ad-infinitum.

IIR filters may be designed to accurately simulate "classical" analog filter responses where as FIR filters, in general, cannot do this.

The controlling of FIR filters can be done easily and it shows the linear phase characteristic and the IIR filter having no particular phase and also typical to control it.

FIR filter is stable and IIR filter is unstable.

FIR filter depend only on I/P where as IIR filter depend upon I/P and O/p.

On the basis of Poles and Zeros we analyze that FIR consisting only zeros and IIR consist both poles and zeros in its transfer function.

1.4 FIR Filter:

The abbreviation of the FIR filter is finite impulse response filter. When the impulse is using as an input in this filter, after the '1' sample, zeroes will come out, through the delay line of the filter its way has made. In signal processing, FIR filter called finite impulse filter because its impulse response is of finite duration, because this response will settle down to the zero position after a certain time. The output of the Kronecker delta function means the impulse response of any N-th order discrete time FIR filter lasts exactly N+ 1 sample that means from nonzero element at start to the nonzero element at last, before it will settle to zero. Commonly the impulse response is finite because of no use of feedback in FIR. The reason behind the guaranty of response to finite is 'no feedback'.  Therefore, the term "finite impulse response" is nearly synonymous with "no feedback". If someone tries to apply feedback in FIR filter then after applying the feedback the filter remains the FIR and it impulse response is also of the finite duration.  The moving average filter can be taken as example for this, when new sample comes in the nth prior sample is subtracted (fed back) every time. The output will always zero for such filters after N samples of an impulse.

In the case of FIR filter of order N, output sequence will be the weighted sum of:

p[n]=a_0 q[n]+a_1 q[n-1]+'+a_N q[n-N]  

p[n]='_(i=0)^N''a_(i )  .q[n-i] '                                                                                         [1.1]  


the q[n] represents the input signal,

the p[n] represents the value of output signal,

The N represents the order of the filter; a N th-order filter has (N+1) terms exist in the right hand side.

a_(i )  Shows the value of the impulse response for 0'i'N in the case of N th-order FIR filter. If the filter is a direct form FIR filter then b_(i  )is also a coefficient of the filter. Another name of this is known as discrete convolution.

Where the term x[n-i]  is commonly referred to as taps, tapped delay line structure that in many implementations or block diagrams provides the delayed inputs to the multiplication operations. Over a finite duration impulse response of the filter is nonzero. Impulse response (including zeros) is the infinite sequence:

h[n]='_(i=0)^N''b_i  .''[n-i]=[b_n '    0'n'N     [0  otherwise'  '                              [1.2]

The range of the nonzero values of non-causal FIR filter of its impulse response can start before n = 0, with the defining formula appropriately generalized. FIR filters having number of properties that can be very useful so that that make it much precious and good one to use. FIR filters:

Feedback is not important. So rounding errors are not compounded or summed every time so that the error will minimize. So the same relative error occurs in each calculation. So that this also makes implementation simpler.

The output is nothing but sum of finite multiples of the input values in the finite number, so it can be no greater than ''b_(i ) times the largest value appearing in the input.

Linear phase designing of this can be done by making the coefficient sequence symmetric. In phase-sensitive applications this required sometime, for example data communications, crossover filters, and mastering.

FIR filters having disadvantage that it requires more power for general purpose processor as compared to an IIR filter and it having sharpness or selectivity similarly to IIR, especially when the lower cut off frequency are needed. In many digital signal processors many hardware features are there to make FIR filter efficient similar to the IIR filter for many applications.

Now we can see the effect of the filter on the sequence x[n] and it is described in the frequency domain by the convolution theorem:

f{x*h}=f{x}.f{h}      where y(w)=f{x*h}                                              [1.3]     

and X(w)=f{x}     and H(w)=f{h}

and y[n]=x[n]*h[n]=f^(-1) {X(w).H(w)}                                                   [1.4]                                              

Where operators f and f^(-1)respectively denote the discrete-time Fourier transform (DTFT) and it's inverse. Therefore, the complex-valued, multiplicative function H(w) is the filter's frequency response. It is defined by a Fourier series:

H_(2'' ) (w)''_(n=-')^'''h[n].(e^jw )^(-n )= '_(n=0)^N'b_n '.(e^jw )^(-n )                                   [1.5]

Where 2'' denotes periodicity. Here w represents the frequency in normalized units (radians/sample). The w=2''f substitution is used in many programs, so the unit of frequency f becomes cycles/sample and due to this the periodicity becomes 1.  When the x[n] sequence has a known sampling-rate, f_s  samples/second, the substitution w=2''f/f_s  changes the units of frequency f to cycles/second (hertz) and the periodicity to f_s   the value w=''  corresponds to a frequency of f=f_(s/2)  hz=1/2   cycles/sample, which is the Nyquist frequency.

1.5 Transfer Function:

The frequency responseH_2'' (w)  can also be written as H(e^iw)  where function H is the Z-transform of the impulse response:

H(z)''_(n=-')^'''h[n].z^(-n) '                                                                                   [1.6]

Z is a complex variable, and H (z) is a surface.  One cycle of the periodic frequency response can be found in the region defined by z=e^iw  where -'''w''' which is the unit circle of the z-plane. IIR filter can also be determined by the transfer function. As we have already noted, FIR designs are inherently stable.

1.6 Filter Design:

Filter order and filter coefficients according to specifications are used to determine the design of FIR filter, these can be in the scale of time domain and or this can be from the scale of frequency domain. The cross-correlation between the input signal and a known pulse-shape is performed by 'matched filter'. Terms to describe FIR filters:

Impulse Response - The "impulse response" of a FIR filter is actually just the set of FIR coefficients. (If you put an "impulse" into a FIR filter which consists of a "1" sample followed by many "0" samples, the output of the filter will be the set of coefficients, as the 1 sample moves past each coefficient in turn to form the output.)

Tap - A FIR "tap" is simply a coefficient/delay pair. The number of FIR taps, (often designated as "N") is an indication of 1)what is the amount of memory required to implement a filter, 2) how many number of calculations needed, and 3)in how much amount a filter can do filtering; in effect, more taps means more stop band attenuation, less ripple, narrower filters, etc.

Multiply-Accumulate (MAC) - In a FIR context, a "MAC" is the operation of multiplying a coefficient by the corresponding delayed data sample and accumulating the result. FIRs usually require one MAC per tap. When a single instruction cycle is there the MAC operation can be done by many DSP processors.

Transition Band - The band of frequencies between pass band and stop band edges. There are more number of taps are required to implement the filter when transition band is narrow (A "small" transition band results in a "sharp" filter.).

Delay Line - In the case of the FIR calculation set of memory elements those needed to implement "Z^-1" delay elements.

Circular Buffer - A special buffer which is "circular" because incrementing at the end causes it to wrap around to the beginning, or because decrementing from the beginning causes it to wrap around to the end. Circular buffers are often provided by DSP microprocessors to implement the "movement" of the samples through the FIR delay-line without having to literally move the data in memory. The quality of buffer or process is that it replaces the old value when the new one comes. If we want to find the convolution in the FIR filter then we have to take the cross-correlation between the input signal and a time-reversed copy of the impulse-response. Therefore, the matched-filter's impulse response is "designed" by sampling the known pulse-shape and using those samples in reverse order as the coefficients of the filter. For getting the particular frequency response we have many methods:

With the help of window

Frequency Sampling method

Weighted least squares design

Parks-McClellan method (also known as the Equiripple, Optimal, or Minimax method). The Remez algorithm method is best for the optimal equirriple set. The desired or required frequency response in this is declared by user, from this response a weighting function for errors, and Order N is also defined by the user. This algorithm specifies the set of N+1 coefficients that minimizes the maximum deviation from the ideal. Practically if we see this method seems quite easy because of program used only in one text and it will give the optimal solution.

FFT algorithm can be use to design the equirriple FIR filter. The algorithm is iterative in nature. For the initial filter design the DFT is computed (h[n] =delta[n] from this we can start if we don't have initial estimate). The frequency can be corrected according to our desired specification in the Fourier domain and also the inverse FFT can be compute.

Coefficients retains only N in scale of time domain (force the other coefficients to zero), we will Compute the FFT once again. Correct the frequency response according to specs. The convenient way is provided by the Software packages (software like MATLAB, GNU Octave, Skylab, and Skippy).

1.7 Implementation of FIR Filters

' The implementation of an FIR requires three basic building blocks

' Multiplication

' Addition

' signal delay

1.7.1 Multiplier:

Precision and speed of processing of multiplier in the case of DSP should be very fast (bit width; think logic circuits) to support the desired application

More multiplication is the need of general low quality filter as compared to that of higher, so throughput suffers if the multiplier is not fast.

 1.7.2 Adder:

DSP having a very basic function of addition.

Combination of addition and multiplication is the requirement of the FIR filter, hence DSP microprocessors feature multiply- accumulate (MAC) units Adders generally operate with just two inputs at a time.

  1.7.3 Unit Delay:

The unit delay provides a one sample signal delay

A sample value is stored in a memory slot for one sample clock cycle, and then made available as an input to the next processing stage

An M-unit delay requires M memory cells (note each memory cell must store say B-bits) configured as a shift register.

1.8 Optimal Filter Design Methods

Lots of methods are we have to get the optimal design. The basic concept behind each method is that in each method coefficients perform again and again up till not getting the efficient value. The various methods are as follows:

Frequency domain design of Least Squared Error.

Approximation method of Weighted Chebyshev.

For maximal ripple FIR filters the nonlinear equation solution.



It is a multi programming numerical computing environment and fourth generation programming language. In starting it is used as matrix programming language in which linear algebra programming was simple. A proprietary programming language developed by Math Works, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, Although MATLAB is intended primarily for numerical computing, an optional toolbox uses the Mu-PAD symbolic engine, allowing access to symbolic computing abilities. Packages like, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems are added. In this analysis, an optimization technique coding is done on the M - file of Matlab software. In the case of technical computing, integrates computation, improving visualization, and ease of programming in an easy-to-use environment given by the mat lab where mathematical notation is used to describe these problems. For documenting and sharing our work lots of features are there in mat lab. It also provides us the facility of integrating our code with other languages, and distributes your MATLAB algorithms and applications. Features include:

For technical computing it act as high level language

To manage the code, files, and data it provides safe environment

When someone going to use the iterative exploration, design, and problem solving it provide an interactive environment.

For the purpose of linear algebra, statistics, Fourier analysis, filtering, optimization, and numerical integration provide a brief mathematical environment.

For the visualization of data in 2-D and 3-D inbuilt graphics functions are there.

For building custom graphical user interfaces tools are available.

MATLAB (matrix laboratory) is a fourth-generation high-level programming language and interactive environment for numerical computation, visualization and programming.  MATLAB is developed by Math Works.  Matrix manipulations; plotting of functions and data; implementation of algorithms; creation of user interfaces; interfacing with programs written in other languages, including C, C++, Java, and FORTRAN; analyze data; develop algorithms; and create models and applications these allowed by this.  It has numerous built-in commands and math functions that help you in mathematical calculations, generating plots, and performing numerical methods.

In the case of computational mathematics the mat lab is used everywhere. Following are some commonly used mathematical calculations where it is used most commonly:

When someone has to deal with Matrices and Arrays

When someone tries to Plotting and graphics 2-D and 3-D

In the case of Linear Algebraic problems

When the functions are non linear

Someone needs to analyze in the form of Statistics

In the case of the data Analysis approach  

Typical problems involving the calculus and differential equations

Larger category of numerical calculations



Curve Fitting


When we have problems of numerically computation, visualization and application development for such a typical problems mat lab is work as high level language.

When we are using the iterative exploration, design and problem solving for these purpose the mat lab provides interactive environment.

When we have problems of mathematical functions for linear algebra, statistics, Fourier analysis, filtering, optimization, numerical integration and solving ordinary differential equations then for such cases it provide us a vast library.


In August 2009 Mehboob R. et al. [5]: 'Fir Filter Design Methodology for Hardware Optimized Implementation'. They proposed novel method of FIR filters design for optimized hardware implementation. They said that FIR filters are extensively employed in a variety of consumer devices like TVs, mobile phones and docking stations. The optimized implementations of FIR filter furnish high quality signal processing at reasonable cost. They first analyzed the effects of quantization on frequency response of a filter by successively reducing the number of bits in each coefficient. Empirical analysis of coefficient quantization reveals a relationship between the number of bits, number of coefficients and the frequency response. Then they presented a methodology for an optimized design of an FIR filter for hardware implementation. The proposed approach first designed a filter with certain parameters and then redesigns the same filter with tighter constraints resulting in higher filter order. The coefficients of the improved or over designed filter had quantized successively with lesser number of bits by an iterative algorithm to a level where its frequency response matched to the original requirements. The synthesis results shown that a significant reduction in hardware resources can be achieved using this methodology. The proposed approach has so far not been reported for reduction in hardware resources.

In 2010 Alam S. M. Shamsul et al. [6]: 'Performance Analysis of FIR Filter Design by Using Optimal, Blackman Window and Frequency Sampling Methods'. Low pass filter designing with the help of Blackman window method, Frequency sampling method and Optimal method have presented. And also the magnitude graphs of these techniques have demonstrated and comparison of these also done with the ideal filter response. Iteration method has proposed by the author to improve the solution quality. They shown that in optimal FIR filter design the transition band defines the pass band and the stop band ripples. Now the author has applied the Remez exchange algorithm on got optimal solution to analyze the response of the FIR filter. They shown that the degree of flatness in pass band and stop band depend on the weighting factor and width of the transition band. Author concluded that the length of filter defines the degree of flatness of filter design by using Blackman window and frequency sampling technique.

In 2011 Mukherjee S. et al. [7]: 'Linear phase low pass FIR filter design using Improved Particle Swarm Optimization'. They have introduced an optimal design of linear phase digital low pass finite impulse response (FIR) filter using Improved Particle Swarm Optimization (IPSO). According to the author in the design process, the length of the filter, band of the frequencies to pass and band of frequencies to stop, pass band and stop band feasible ripple sizes have specified. They said that the FIR filter design is a multi-modal optimization problem. For the digital filters the conventional gradient methods are not efficient. Now the iteration method has applied by the author to optimize the results. According to the author Genetic algorithm (GA), particle swarm optimization (PSO), improved particle swarm optimization (IPSO) has been used here for the design of linear phase low pass FIR filter. IPSO improves the solution quality by applying the new definition of velocity vector and swarm updating. The author concluded that this method given the best performance as compared to the other one in the face of optimized result for the solution of the multimodal, non-differentiable, highly non-linear, and constrained filter design problems.

In 2011 Kar R. et al. [10]: 'Optimization of linear phase FIR band pass filter using Particle Swarm Optimization with Constriction Factor and Inertia Weight Approach'. For the designing of the digital filters they presented the swarm updating and evolutionary algorithm. For the designing of linear phase digital band pass filter they introduce the concept of construction factor and inertia weight in genetic algorithm and novel particle swarm optimization. According to the author they designed a fitness function this function is depending upon squared error which is in between the response of actual and ideal filter. Especially in a dynamic environment FIR filter is designed, in which filter coefficients have to be adapted and fast convergence is of importance. PSOCFIWA seems to be promising optimization tool. According to the author it is very hard to get all the requirements in the case of FIR digital filter. Author has introduced an iterative method to find the optimal solution of optimal FIR filter design. FIR filter design is a multi-modal optimization problem. According to the analysis the gradient based methods are not efficient in case of the designing o the digital filters. PSOCFIWA algorithm generates a set of filter coefficients and also it tries to show ideal frequency characteristic. Here the author presented the band pass filter of different order. The author demonstrated the magnitude responses for different design techniques of digital FIR filters. Simulation results have compared by the author with the well accepted evolutionary algorithm such as genetic algorithm (GA). On the analysis it found by the author that PSOCFIWA gives the best results in the parameters of accuracy convergence speed and solution quality as compared to GA.

In 2012 Mondal S. et al. [12]: 'Novel particle swarm optimization for high pass FIR filter design'. They used the NPSO technique to implement the high pass filter. According to the author NPSO is an improved particle swarm optimization (PSO) and it can help to improve the solution quality by the velocity vector and swarm updating process. The search capability that leads to a higher probability of obtaining the global optimal solution is going to improve in the case of NPSO by modification of the inertia weight. Feature of the applied modified inertia weight mechanism is to monitor the weights of particles, and it is linearly decrease in the general application. In this design process, the length of the filter, band of frequencies to be pass and band of frequencies to be stop, pass band feasible ripple size and also the stop band feasible ripple sizes are specified. According to the author, the FIR filter design is a multi-modal optimization problem. Evolutionary algorithms like real code genetic algorithm (RGA), particle swarm optimization (PSO), differential evolution (DE), and the novel particle swarm optimization (NPSO) have been used in this work for the design of linear phase FIR high pass (HP) filter. Author concluded that the comparison of simulation results reveals the optimization efficacy of the algorithm over the prevailing optimization techniques for the solution of the designing of such filters.

In 2012 Mukhopadhyay A. et al. [11]: 'Optimal design of linear phase FIR band stop filter using particle swarm optimization with improved inertia weight technique'. They proposed multi-objective swarm optimization algorithm called particle swarm optimization with improved inertia weight (PSOIIW). And author applied it for the design of optimal linear phase digital band stop finite impulse response (FIR) filter. PSOIIW in the purpose of solution quality to be improved adopts the new definition of velocity vector and also of swarm updating. In this they have been modified the inertia weight for the PSO to enhance its search capability to obtain the global optimal solution. The weight of particle can be monitor by inertia weight mechanism, and it is linearly decreases in the general applications. In the designing the length of the filter, band of frequency to pass and band to which the frequencies have to stop, pass band ripples those are feasible and stop band ripple sizes have specified. The simulation results obtained prove the superiority of the algorithm compared to the other prevailing optimization algorithms like real code genetic algorithm (RGA), particle swarm optimization (PSO), and differential evolution (DE) for the solution of the multimodal, non-differentiable, highly non-linear, and constrained FIR filter design problems.

In 2013 Kashyap S. et al. [18]: 'Implementation of High Performance Fir Filter Using Low Power Multiplier and Adder'. They presented the FIR filter face with constraints: speed is high, the throughput is high, and at the same time, power consumption is minimal as possible. One of the important parts of digital signal processing is FIR filter. They tried, constructing the FIR filter that can give not only the high speed of response but also less power consumption and less delay in processing. They considered the elementary structure of FIR filter they got that there are multipliers and delays in the circuit, which in turn have the combination of adders. In order to increase the efficiency and less power consumption and author presented efficient implementation and analysis so this also helps to evaluate the performance of filter multipliers and adders. The power comparison of result of adders and multiplier is taken by the author, to choice low power adder and multiplier to implementation of high performance FIR filter.

In 2013 Tian J. et al. [16]: 'Hardware-efficient parallel structures for linear-phase FIR digital filter Based on fast convolution algorithm'. They have proposed improved parallel FIR filter structures for linear-phase FIR filters where the number of taps is a multiple of parallelism. According to the author the proposed parallel FIR structures not only use fast convolution algorithm to reduce the number of sub filter, but also exploit the symmetric (or anti symmetric) coefficients of linear-phase FIR filter to reduce half the number of multiplications in sub filter section at the expense of additional adders in pre processing and post processing blocks. They have concluded that the proposed parallel FIR structures save a large amount of hardware cost for symmetric (or anti symmetric) coefficients from the reported FFA parallel FIR filter structures, especially when the length of the filter is large, e.g., the proposed 4-parallel FIR filter structure has eight sub filter blocks in total and four sub filter blocks contain symmetric coefficients, whereas the improved FFA structure has nine sub filter blocks in total and four contain symmetric coefficients. Specifically, for a 4-parallel 576-tap filter, the proposed design saves 144 multipliers (14.3%), 135 adders (10.2%) and 143 delay elements (11.1%).

In 2014 Neha et al. [19]: 'Design of Linear Phase Low Pass FIR Filter using Particle Swarm Optimization Algorithm'. They had presented PSO algorithm which is used with constriction factor approach to solve the multimodal, highly non-linear filter design problem. They said that this method has the property of parameter independence and thus ensuring convergence while fully exploring the solution space. The velocity and position updating rules of original PSO algorithm has used for the design of low pass FIR filter of order 20. The extensive simulation results obtained from the proposed method shows superiority of the algorithm. They said that we conclude that proposed method has best convergence characteristics. Comparison of the resulting performance parameters such as maximum pass band ripple, maximum stop band ripple and maximum stop band attenuation(in dB) with the results reported in past literatures shows significant improvement inferring that the proposed algorithm is best of them all. It is revealed that the proposed algorithm converges to the specified optimum result within very small number of iterations and much less execution time. Thus it has good application prospect in practical filter design problems used in various signal processing systems.

In 2014 Narsale R. M. et al. [17]: 'FPGA Based Design & Implementation of Low Power FIR Filter for ECG Signal Processing'. They have proposed design of low power FIR filter and implementations of the same on FPGA. MATLAB has been used for an Equiripple or Remez Exchange (Parks-McClellan) technique of FIR filter design. A notch FIR filter has been designed for the removal of power line interference in ECG signal which is a very important step in the pre-processing stage of ECG signal. Direct-form approach in realizing a digital filter has considered. This approach gave a better performance than the common filter structures in terms of speed of operation, cost, and power consumption in real-time. The minimum power achieved has been 0.073w in FIR filter based on DA to 626 taps, 8 bits inputs and 9 bits coefficients obtained. The proposed FIR filter has been designed using VHDL, simulated, synthesized using Xilinx.

2.2 Problem Domain:

Design problems for FIR filter:-

Multimodal problems cannot solve by traditional methods.

Genetic algorithm was unable to find global optimal solution efficiently.

In the frequency domain are the problems of simultaneous approximation of prescribed magnitude and phase responses or Phase delay.

Similarly, prescribed magnitude and group delay responses.

Similarly, different magnitude and same phase, respectively.

Similarly, time of iterations calculation.

In the past, these problems only approximated means they treated by taking approximate values. Especially (approximate) in the case of solutions of the frequency response approximation problem have served as substitutes for solutions of the magnitude-phase problem. Here at the starting mathematical formulation of both problems is developed and after this, for these problems, the existence of solutions and results on the convergence of the approximation errors are proved. With these we obtained improvement by use of a direct solution of the magnitude-phase response problem on the place of a solution of the frequency response problem, is quantified by computable bounds. There are many typical problems occur in computer and electronic field so for this we require vast concept of signal and system and the sharper or smarter way is require to apply them on it. We are going to take some examples that can explain the practical use of concept of the modeling and analysis in the domain of time, frequency and also in the domain of s and z.

First we can take the example of system belongs to the category of LTI and it is series resistor shunt capacitor RC low pass filter and the formula of impulse response and system function is:

h(t)=1/RC e^((-t)/RC) u(t)'(''L  1/RC.1/(S+1/RC))                                                                                        [2.1]

Here the term RC is nothing but the time constant of this circuit and it is associated with the series resistor and shunt capacitor and is use to defines the filter circuit. And one can ask you to find the discrete time equivalent of this. Suppose h[n] = h (nT) = h (n/fs). The Laplace transform (LT) of h (t) following sampling is:                 

H_S (s)=Laplace{'_(n=-')^'''h(nT)''(t-nT) e^(-st) dt }= '_(n=-')^'''h(nT) e^(-snT )=H(z) |_(z=e^sT ) ''

                                                                                                                        [2.2] 2.2.1 for the implementation giving the fine tune design:

These filters can be expressed as h[n] = an u[n] for 0 < a < 1, and the scale factor of these filter given by the expression 1/ (RC) and a = e'T/ (RC). In particular f = 0 it ensures that the filter gain at this frequency is preserved. The communication system when taken then it is found that the meaning of this is H(s = j0) = 1/ (RC)/ [j0 + 1/ (RC)] = 1 and to place gain factor in the front of HRC (z) selecting it as G to place. And to force the gain factor to be one at:  z = ej0 = 1:H_RC (z) |_(z=1)=G.(1/RC)/(1-e^((-T)/RC) )=1

 G=RC(1-e^((-T)/RC))                                                                                                          [2.3]

Now at last with the value of G included,

H_RC (z)=  (1-e^((-T)/RC))/(1-e^((-T)/RC) z^(-1) )=  (1-a)/(1-az^(-1) )  ,a= e^((-T)/RC)                                                                     [2.4]                                                                      and it is also clearly known that to actually implement this we need the differential equation of this. And it is also very clearly known thing that the system function can be found from the differential equation and the transfer function is nothing but the ration of the output value to the value of the input, and in the z-domain it can be describe in this way: HRC (z) = Y (z)/X (z). To find the difference equation, you first need to cross-multiply X (z) times the numerator of HRC (z) on the right and cross-multiply the denominator times X (z) on the left:

H_RC (z)=  Y(z)/X(z) =  (1-a)/(1-az^(-1) )    

Y(z)[1-az^(-1) ]=X(z)(1-a)                                                                                    [2.5]

Now applying the inverse Z transform on the right hand side, using the linearity and delay properties of the z-transform:

y[n]-ay[n-1]=(1-a)x[n]or y[n]= ay[n-1]+(1-a)x[n]                        [2.6]           




Role of the filter is very cruise in the field of digital signal processing. Because of its large number of applications it makes famous the DSP in present days. Signal separation and signal restoration are two important uses of filters. When a signal has been contaminated with interference there is a need of signal separation, and also when noise, or other signals effects the original signal then also there is the need of signal separation. For example when a voice signal is transmitted from a transmitter device to a particular receiver but suppose another signal having the frequency near to the frequency of that signal than there is interference occurs in that signal than there is need of signal separation process to get the efficient response. Similarly ECG signal when recorded then there must be separation process other name filtering need to get the correct output on monitor screen if this is not done then noisy signal will be there and there is errors in the calculation or analysis of disease.

When we see the use of signal restoration then it is found that when the signal is distorted in some way, For example, suppose we record an audio signal with the device of poor quality then we get distortion in some portion then we use this process to restore the signal. This type of problems can comes in any of the digital or analog filters then there is question arises that which one is a good? Some qualities of analog filters are cheap, fast, and have a large dynamic range in both amplitude and frequency. Some qualities of Digital filters is that the performance quality is very high as well as programmability is there for these filters. On the basis of the performance analysis it is found that the performance of digital filters is very high as compared to performance of analog filters. The accuracy and stability of the resistors and capacitors are the types of required process of rectification and today the emphasizing is there on to does it. But in the digital filters the performance is ignored frequently. The emphasis shifts to the limitations of the signals. Recursion is another process by which we can implement digital filters. Convolution when used to implement a particular filter then, then calculation of every sample in output is done by weighting the samples in the input, and adding them together. Recursive filters are taken as extensive for this, by using previously calculated values from output, besides points from the input. Recursion coefficients define the recursive filters rather than the filter kernel. We very well know that every linear filter having the impulse response, it doesn't matter that you use it to implement that or not. There is very simple step to find the impulse response of the recursive filter, just put impulse in the input of that filter and see what comes in output of that filer. The impulse response of the recursive filters is in exponentially decayed manner on amplitude scale and sinusoidal in nature. Nature wise it is infinitely long. However, the amplitude eventually drops below the round-off noise of the system, and the remaining samples can be ignored.


Everyone now the characteristics of low pass filter that is the behavior of the low pass filter is that it allow a particular frequency range below the selected cut off frequency and attenuates the frequency above that cut off frequency. Due to this characteristic many times it is also called the high cut filter. The low pass filters characteristic is opposite to that of high pass filters and one can design the band pass filter with the help of taking the combination of the low pass and the high pass filter. One can find many forms of analog filters for example hiss filters and signal conditioners, digital filters for smoothing sets of data, acoustic barriers, blurring of images, and so on.

One can consider moving average operation as a low pass filter; the application area of this is the finance and by the same analyzing techniques those are use to analyze the low pass filter that can be analyze also. The low pass functioning is to remove short term fluctuations and leave the long term trends. When someone wants the filter they can get varies category of filters. Bode plot is one of the important tool to plot the frequency response of the filter, and the cutoff frequency and rate of frequency roll off are two important properties to characterized the filters. The input power when analyzed in any one of the case then we found that the input power become half or 3 db in each case. So one can say the important tool to identify the stop band additional attenuation is the order of the filter.   

A first-order filter, when the signal amplitude is reduced by half (so power reduces by a factor of 4, or 6 dB), the frequency becomes double in every time (goes up one octave); more precisely, the power roll off approaches 20 dB per decade in the limit of high frequency. When we saw the magnitude plot of the bode plot of low pass filter we got that for a first-order filter horizontal line below the cutoff frequency is there, and we got a diagonal line above the cutoff frequency. There is also a curve called "knee curve" at the boundary in between two, which smoothly transitions between the two straight line regions. At the case of attenuation of the high frequency if we have to see that there is the pole and zero both in the transfer function of filter then we got that the Bode plot flattens out again; this effect occurs due to leaking of input around one pole; this one-pole'one-zero filter is still a first-order low-pass.

               A second-order filter response in accordance to high pass frequency attenuation it attenuates more steeply. When the frequency response of such filters is seen with the help of Bode plot we got that it resembles that of a first-order filter, except that it falls off more quickly. For example, the signal amplitude is reduces by second order Butterworth filter to one fourth its original level, we saw that every time the frequency becomes double due to this the power becomes decreasing to 12 db per octave or 40 db per octave. The Q factor is very important factor of analyzing this power decay because depending on the Q factor the decay will occur in different circuits but at final the decay will be of 12 db per octave; In the case of the first order filter the higher frequency asymptotes can be changed by the zeros present in the transfer function. One can also define the higher order filter in the similar way as defined the lower order filters. So finally one can define n order all pole filter by its power roll off is 6n db per octave, means the roll of is 20n db per decay.

If the diagonal line to the upper left and the horizontal line when extended to right in the case of any Butterworth filter then the intersection of these will be exactly at the cut off frequency. Here the upper left diagonal line shows the asymptotes of the function.  We get In the case of the first order filter the cut off frequency is 3 db below the horizontal line when the frequency response analyze. The knee curve in the case of Butterworth filter, Chebyshev filter, Bessel filter, etc are different. Frequency responses at the cut off frequency in the case of lots of second order filters it will be above the horizontal line.

One can predict or analyze the 'peaking' or frequency response in the case of third order filter without the use of calculus. The filter will pass either the low frequency or the high frequency it is totally depends on the type of the characteristics of that filter. Many electronic circuits can be design and also exist to work on GHz range of frequency. Also lots of number of circuits also exist that can work on the low frequency.

3.3 Laplace notation

When we have the impulse response of continuous time filters then by taking the Laplace transform of its responses it can be describe in terms of Laplace transform. On the complex plan now it is very easy to analyze these filters with the help of pole zero locations on this complex plane. In the similar way in case of discrete time Z transform is used to describe it.

For example, a first-order low-pass filter can be described in Laplace notation as:

output/input=K 1/(Ts+1)                                                                             

Where s is the Laplace transforms variable, '' is the filter time constant, and K is the gain of the filter in the pass-band First order RC filter.                                               

Figure 3.1: Passive, first order low-pass RC filter

A low pass circuit can be describing as one resister in series and one capacitor in parallel to the load. Due the reactance property of capacitor it resists the low frequencies. The behavior of the capacitor become as short circuit in the case of high frequency. The time constant of this circuit is given by the product of resistance and the capacitance of this circuit i.e. ''=R.C.

The break frequency, also called the turnover frequency or cutoff frequency (in hertz), is determined by the time constant:

f_c=1/2''T=1/2''RC                                                                                                      [3.1]

Or equivalently (in radians per second):

w_c=1/T=1/RC                                                                          [3.2]

The concept can be clearly explained by the charging of capacitor through the resister: When the frequency is low, the capacitor having sufficiently larger time to charge up to the same voltage as practically of the input voltage.

But in the case of the high frequency it can be clearly seems that the capacitor having smaller time to charge itself up to the voltage as of input. The input goes up and down in the particular amount but the output goes up and down in a small amount of fraction as of input. If someone makes the frequency double then the charging goes only for half of amplitude.

Reactance at a particular frequency can be use to understood the concept, Everyone know that DC current cannot flows through the capacitor so it need to flow from V_out(analogous to removing the capacitor), Since it is very well known thing that AC can flow through the capacitor and solid wires. It is also a very basic understanding concept that capacitor is also not an ON-OFF device. The capacitor variably acts between these two extremes. The variability of the capacitor is clearly shown by Bode plot. One can clearly see that the first order RLC circuit consisting of one resister and one inductor and one capacitor either in the series or in the parallel.

Figure 3.2: RLC circuit as a low-pass filter

One can see the second order RLC circuit then they can found it in two manners one is R and L and C can be connected in the series and or can be connected in parallel form. In the RLC circuit the R is resistance, L is inductance, and C is capacitance of the circuit. Harmonic oscillator produces by this circuit for current and it is resonating similarly to the LC circuit resonates. Due to the presence of the resistance the oscillations will settled down to zero after certain time this is the main difference between these circuits but the condition is that the source should not there after some time. This effect of the resistor is called damping.  

The peak resonant frequency is reduced due to the presence of resistance. If a resistor is not specifically used as a component in the circuit then the minimum value or an ideal, pure LC circuit is an abstraction for the purpose of theory. Lots of applications are there for these circuits. Oscillator circuits of different types are the one of the application area for such circuits. Some other applications for these circuits are also available like tuning section in the T.V. sets or the radio receivers, in these applications these circuits used to select a narrow range of frequencies from the radio waves. Due to this role of these circuits they called technically tuned circuits. We know that the RLC circuits can be used to design not only low pass but also high pass and band pass and band reject filters. The RLC circuits are the second order circuit's means these circuits can be used to generate the higher order circuits and also the characteristic of these circuits is linear.

Means the second order differential equation is used to describe these circuits.

                                        Figure 3.3: Higher order passive filters

 If we have a third order low pass filter in Cauer topology form. Then C2=4/3 farad, R4=1 ohm, L1=3/2 Henry and L3=1/2 Henry (if taken as example) the filter behavior is as Butterworth filter and its cut off frequency is ''c=1. If we want to construct a higher order filter then it is also possible to construct it from this filter.

Figure 3.4: An active low-pass filter

Active low pass filter can be considered as another category of electric circuit. In the operational amplifier circuit shown in the figure, cut off frequency is:

f_c=1/2''R_2 C                                                                         [3.3]

Or equivalently (in radians per second):

w_c=1/R_2 C                                                                            [3.4]

The gain in the pass band is 'R2/R1, and the stop band drops off at '6 dB per octave (that is '20 dB per decade) because it is first order filter.

   Figure 3.5: Slow roll off frequency response of Low pass filter

Figure 3.6: Fast roll off frequency response of Low Pass filter

    Figure 3.7: Ripple in Pass-band frequency response of Low pass filter

Figure 3.8: Poor stop-band attenuation frequency response of Low pass filter

Figure 3.9: Good stop-band attenuation frequency response of Low pass filter




Particle Swarm Optimization (PSO) is an evolutionary computation, optimization technique (a search method based on a natural system) developed by Kennedy and Eberhart. The system initially has a population of random selective solutions. PSO technique can generate high quality solutions within shorter calculation time and have more stable convergence characteristics than other stochastic methods. PSO is a met heuristic as it makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solution. The choice of PSO parameter can have a large impact on optimizing performance.PSO is a population based swarm intelligence algorithm driven by the simulation of a social psychological metaphor instead of the survival of the fittest individual. Kennedy and Eberhart developed PSO in 1995. PSO is an efficient, simple and fertile optimization algorithm. PSO randomly initialized the swarm of particles in the search space associated with randomized velocities. Particle's position and velocities are adjusted and the function evaluated with the new coordinates at each time step (iteration).

     Each particle of PSO keeps track of its coordinates in the search space which are associated with best solution it has achieved so far. This is called pbest. The overall best value obtained so far called gbest. The particle velocities constantly adjusted according to the position of particles as:

'  v'_i^(k+1)=wv_i^k+c_1 r_1 (pbest_i^k-x_i^k )+c_2 r_2 (gbest_i^k-x_i^k )                                 [4.1]

c1, c2 are the cognitive and social components, respectively, which are acceleration constant responsible for varying the particle velocity towards pbest and gbest respectively. Where c2 regulate the maximum step size in the direction of the global best particle, and c1 regulates the step size in the direction of the pbest value. r1 and r2 are random function in the range [0, 1], xik is position of particle at kth iteration. 'w' is inertia weight for velocity of ith particle. A suitable value of w provides better optimal solution. In each iteration the weights 'w' is varied as:

w=w_max-((w_max-w_(min')))/itermax iter                                                                                              [4.2]                          

Where itermax is maximum number of iterations and iter is the current iteration. Wmax and wmin, the upper and lower limit of inertia weights respectively.

The position of each particle is updated using velocity vector as:

 x_i^(k+1)=x_i^k+v_i^(k+1)                                                                                                              [4.3]                                             


In the convergence of PSO algorithm inertia weight play the important role.

The effect of the previous velocities on the current velocity is controlled by inertia weight.

The tradeoff between the global and exploration capabilities of the particle regulated by inertia weight.

A large inertia weight helps in good global search, while a smaller value facilitate local exploration.

Constant c1 (cognitive parameter) pulls the particles towards a local best position where as c2 (social parameter) pulls it towards the global best position.

PSO technique is follow optimization of a social psychological metaphor in place of survival of the fittest individual. PSO is provided good solution but tremendous potential solution is finding by chaotic particle swarm optimization (CPSO). One of the major drawbacks of the PSO is its premature convergence, especially while handling problems with more local optima. If we want diversity in population of PSO approaches than chaotic sequence is a good alternative for this purpose.

CPSO is also improve the global convergence in substitution of parameter and can be helpful to escape more easily from local minima than the traditional PSO method.

Local, Best PSO:

The nearby best PSO (or best PSO) strategy just permits every molecule to be affected by the best-fit molecule browsed its neighborhood, and it mirrors a ring social topology. Here this social data traded inside of the area of the molecule, meaning neighborhood learning of the earth.

Comparison of gbest to lbest:

Initially, there are two contrasts between the "gbest" PSO and the "lbest" PSO: One is that in light of the bigger molecule bury network of the best PSO, now and then it unites speedier than the best PSO. Another is because of the bigger differences of the lbest PSO; it is less defenseless to being caught in nearby minima.

PSO Algorithm Parameters:

There are a few parameters in PSO calculation that may influence its execution. For any given enhancement issue, some of these parameter's qualities and decisions have huge effect on the productivity of the PSO system, and different parameters have little or no impact. The essential PSO parameters are swarm estimate or number of particles, number of cycles, speed segments, and increasing speed coefficients outlined howl.

Swarm size:

Swarm size or populace size is the quantity of particles n in the swarm. A major swarm produces bigger parts of the pursuit space to be secured per emphasis. Countless may decrease the quantity of cycles need to acquire a decent enhancement result. Interestingly, gigantic measures of particles build the computational intricacy per cycle, and additional tedious. From various exact studies, it has been demonstrated that the vast majority of the PSO executions utilize an interim of for the swarm size.

Iteration numbers:

The quantity of cycles to get a decent result is likewise issue subordinate. A too low number of emphases may stop the pursuit handle rashly, while too expansive cycles has the outcome of pointless included computational unpredictability and additional time required.

PSO parameter control:

There are very few parameter should be tuned in PSO. Here is a rundown of the parameters and their run of the mill values.

The number of particles: the run of the mill extent is 20 - 40. Really for the majority of the issues 10 particles is sufficiently huge to get great results. For some troublesome or unique issues, one can attempt 100 or 200 particles too.

Dimension of particles: It is dictated by the issue to be advanced.

Range of particles: It is additionally dictated by the issue to be streamlined, you can indicate diverse reaches for distinctive measurement of particles.

Vmax: it decides the greatest change one molecule can take amid one cycle. Generally we set the scope of the molecule as the Vmax for instance, the molecule (x1, x2, x3), X1 has a place [-10, 10], then Vmax = 20.

Learning variables: c1 and c2 normally equivalent to 2. Be that as it may, different settings were additionally utilized as a part of diverse papers.


Parameters Values Description

nvar 20 number of variables

wmax 0.6 Upper limit of Inertia weight

wmin 0.1 Lower limit of Inertia weight

c1 2.05 Cognitive  and Social Parameter (it is constant)

c2 2.05 Cognitive  and Social Parameter (it is constant)

wp 0.45 Pass band edge frequency

ws 0.55 Stop band edge frequency

iteration 500 Number of Iteration

delta_p 0.1 Pass band ripple

delta_s 0.01 Stop band ripple

Transition Width   0.10 Difference of pass and stop band edge frequencies

The setup parameters are:

Parameters Values Description

Minimum Value of Population hmin Denoted by:


Maximum Value of Population hmax hmax=[0.05,0.08,0.005,0,0.005,0.1,0,-0.05,0.005,0.4,0.6]

N Sample 128 Number of Samples

Np 25 Number of Population

Number of Dimensions (Filter Coefficients) 11 D= n/2 +1

n = order of filter

4.3 Comparisons between Genetic Algorithm and PSO:

The greater part of transformative systems has the accompanying technique:

1. Initial value of an introductory population

2. Retribution of wellness esteem for every subject. It will specifically rely on upon the separation to the ideal.

3. Proliferation of the populace in light of wellness qualities.

4. On the off chance that necessities are met, then stop. Generally do a reversal to 2.

From the methodology, we can discover that PSO offers numerous normal focuses with GA. Both calculations begin with a gathering of an arbitrarily produced populace; both have wellness qualities to assess the populace. Both upgrade the populace and hunt down the optimum with arbitrary systems. Both frameworks don't promise achievement.

Notwithstanding, PSO does not have hereditary administrators like hybrid and transformation. Particles upgrade themselves with the interior speed. They likewise have memory, which is critical to the calculation.

Contrasted and hereditary calculations (GAs), the data sharing system in PSO is altogether diverse. In GAs, chromosomes offer data with one another. So the entire populace moves like a one gathering towards an ideal zone. In PSO, just gbest (or lbest) gives out the data to others. It is a one - way data sharing system. The advancement searches for the best arrangement. Contrasted and GA, every one of the particles have a tendency to meet to the best arrangement immediately even in the nearby form by and large.

    4.4 Artificial neural system and PSO:

A simulated neural system (ANN) is an investigation worldview that is a basic model of the mind and the back-spread calculation is the standout amongst the most prevalent strategy to prepare the counterfeit neural system. As of late there have been critical examination endeavors to apply developmental calculation (EC) systems for the reasons of advancing one or more parts of counterfeit neural systems. Developmental calculation systems have been connected to three primary characteristics of neural systems: system association weights, system structural engineering (system topology, exchange work), and organize learning calculations.

The vast majority of the work including the advancement of ANN has concentrated on the system weights and topological structure. Generally the weights and/or topological structure are encoded as a chromosome in GA. The determination of wellness capacity relies on upon the exploration objectives. For an order issue, the rate of mischaracterized examples can be seen as the wellness esteem.

The upside of the EC is that EC can be utilized as a part of cases with non-differentiable PE exchange capacities and no angle data accessible. The burdens are:

The execution is not focused in a few issues.

Representation of the weights is troublesome and the hereditary administrators must be precisely chosen or created.

There are a few papers reported utilizing PSO to supplant the back-engendering learning calculation in ANN in the previous quite a while. It demonstrated PSO is a promising system to prepare ANN. It is speedier and shows signs of improvement results much of the time. It additionally maintains a strategic distance from a percentage of the issues GA met.

Here we demonstrate a straightforward case of developing ANN with PSO.   The issue is a benchmark capacity of grouping issue: iris information set. Estimations of four traits of iris blooms are given in every information set record: sepal length, sepal width, petal length, and petal width. Fifty arrangements of estimations are available for each of three assortments of iris blossoms, for a sum of 150 records, or designs. A 3-layer ANN is utilized to do the order. There are 4 inputs and 3 yields. So the information layer has 4 neurons and the yield layer has 3 neurons. One can advance the quantity of shrouded neurons. Then again, for exhibit just, here we assume the concealed layer has 6 neurons. We can advance different parameters in the food forward system. Here we just advance the system weights. So the molecule will be a gathering of weights, there are 4*6+6*3 = 42 weights, so the molecule comprises of 42 genuine numbers. The scope of weights can be set to [-100, 100] (this is only a sample, in genuine cases, one may attempt diverse reaches). Subsequent to encoding the particles, we have to decide the wellness capacity. For the arrangement issue, we encourage every one of the examples to the system whose weights is dictated by the molecule, get the yields and look at it the standard yields. At that point we record the quantity of misclassified examples as the wellness estimation of that molecule. Presently we can apply PSO to prepare the ANN to get lower number of misclassified examples as could be expected under the circumstances. There are relatively few parameters in PSO should be balanced. We just need to change the quantity of shrouded layers and the scope of the weights to show signs of improvement results in distinctive trials.




In the design process, the length of the filter, band of frequencies to pass and band of frequencies to stop, feasible pass band and stop band ripple sizes are specified. The category of optimization of FIR filters is multi model. One of the most important advantages of the FIR filter is that its phase response is linear. That is one of the reasons why in literature review all the authors use this filter as compare to IIR. The design procedures are reduced to real-valued approximation problems, where the coefficients have to be optimized with respect to the magnitude response only.

A digital FIR filter is characterized by,

H(Z)= '_(n=0)^N''h (n) Z^(-n) ',n=0,1,'.N                                                                       [5.1]

Where N is the order of the filter which has (N+1) impulse response coefficients, h (n). The values of h (n) responsible to identify the type of that filter, e.g., low pass, high pass, band pass etc. The values of h (n) are to be determined in the design process and N describes the polynomial function order. This thesis presents the even order FIR LP filter design with h (n) as positive even symmetric. Because the h (n) coefficients are symmetrical, the dimension of the problem is halved. Thus, (N/2+1) number of h (n) coefficients is actually optimized, which are finally concatenated to find the required (N+1) number of filter coefficients.

TABLE 5.1: Coefficient of FIR Filter

mean(f) min(f) max(f) std(f)

78.82679 65.22962 93.25139 7.5605512

51.98207 44.19948 58.18716 3.3036941

50.60647 43.46559 56.16859 3.5631587

49.92357 44.4853 56.04444 3.5972527

49.90308 42.63428 58.41779 4.3520635

49.04133 42.96387 57.79743 4.156825

49.76558 43.22149 58.93108 3.929009

49.57641 42.70276 55.65468 3.8306306

49.67879 42.48808 57.26387 3.6186137

48.94372 42.62691 55.58664 3.7534145

48.53934 42.56084 55.05625 3.6318902

49.33073 42.52701 56.44477 4.2146367

49.20308 42.50992 57.28736 4.4284848

49.39037 42.50019 56.42404 3.9241224

49.25064 42.49208 55.54719 3.6733769

48.792 42.49042 55.42809 3.7399311

49.24031 42.48944 55.54811 3.729242

49.1593 42.48887 55.51986 3.7765162

48.393 42.48854 57.10422 3.8290339

47.33343 42.48835 54.61197 3.8552098

48.51627 42.48824 56.36101 4.1065067

47.85509 42.48817 56.46797 4.4272688

47.66552 41.48379 55.79948 4.6331932

47.64854 41.53098 56.01373 4.4683102

47.63875 41.55447 55.64207 4.5353261

47.35009 41.45556 55.23517 4.4198094

46.83876 41.49388 54.48991 4.2754169

47.82453 41.43065 54.44032 4.3692219

46.76473 41.4301 54.20622 4.3228502

47.08357 41.41746 54.72557 4.3280338

47.6454 41.42428 54.56377 4.4733798

47.45047 41.40869 55.2982 4.4759422

47.72325 41.40159 54.78051 4.5255619

47.81575 41.36199 55.42118 4.5119856

47.12588 41.36033 54.92113 4.3760119

44.16113 41.2275 53.59747 3.347323

44.66028 41.13499 55.35216 3.5233983

44.27843 41.14649 55.20761 3.6037494

44.20101 41.14048 55.66255 3.8603332

44.10218 41.13373 54.40147 3.5967134

44.07395 41.13745 54.74982 3.7485758

43.99861 41.12288 54.35571 3.7042962

43.29896 41.07723 50.42942 3.0312446

43.52212 41.02264 50.7686 3.2359149

43.80413 41.00974 51.11751 3.5714766

43.77354 41.00068 51.14403 3.5857401

43.50327 40.99457 50.83701 3.325491

43.37658 40.99694 50.62568 3.1846052

43.84396 40.98439 51.28252 3.6442507

43.84168 40.96187 51.30068 3.6683325

43.88729 40.94913 51.49236 3.741942

43.9602 40.93047 51.7103 3.8523652

43.43295 40.94263 50.72123 3.2903326

43.39649 40.92727 50.67268 3.2710579

43.29353 40.90968 50.50308 3.1775321

43.32151 40.90759 50.55336 3.2208053

43.49441 40.90125 50.79571 3.4110155

43.57576 40.89865 50.94866 3.5000209

43.59111 40.89666 50.96469 3.5179208

43.77856 40.89842 51.22191 3.7083698

43.64474 40.89738 51.08255 3.5715446

43.49522 40.89702 50.74601 3.4078938

43.35651 40.89685 50.55232 3.2616988

43.38876 40.89666 50.5934 3.2947269

43.34558 40.89666 50.53486 3.2491047

43.38434 40.89666 50.58572 3.2891724

43.3819 40.89666 50.58213 3.2863759

43.50751 40.89666 50.793 3.4175924

43.68234 40.89666 51.11792 3.606569

43.60801 40.89666 50.96838 3.5249434

43.31174 40.89666 50.48786 3.213358

43.47976 40.89666 50.71655 3.3880667

43.52171 40.89666 50.8153 3.4322613

43.57747 40.89666 50.91262 3.4922883

43.54039 40.89666 50.84414 3.4517148

43.69865 40.89666 51.13604 3.6232861

43.68976 40.89666 51.12616 3.6141135

43.3972 40.89666 50.60126 3.3022642

43.63114 40.89666 51.06271 3.5521127

43.60795 40.89666 50.96817 3.5248409

43.62706 40.89666 51.05819 3.547825

43.67781 40.89666 51.11285 3.6018672

43.64895 40.89666 51.08216 3.5709736

43.34355 40.89666 50.53053 3.2461977

43.28669 40.89666 50.45581 3.1880086

43.36008 40.89666 50.55232 3.2633818

43.39742 40.89666 50.60154 3.302493

43.31906 40.89666 50.49788 3.2208544

43.43486 40.89666 50.6545 3.3414194

43.34102 40.89666 50.52723 3.2435405

43.57718 40.89666 50.91208 3.4919688

43.42261 40.89666 50.63641 3.3287364

43.30548 40.89666 50.4789 3.2068679

43.52517 40.89666 50.82065 3.4358286

43.51615 40.89666 50.80657 3.4264914

43.43164 40.89666 50.64975 3.3380818

43.26134 40.89666 50.42725 3.1630485

43.25994 40.89666 50.42554 3.1615062

43.45643 40.89666 50.6857 3.3647464

43.52587 40.89666 50.82282 3.4378322

43.68516 40.89666 51.12222 3.6113634

43.64068 40.89666 51.07422 3.5639786

43.30478 40.89666 50.47801 3.2063231

43.21905 40.89666 50.37858 3.1219406

43.36714 40.89666 50.56351 3.2730194

43.21306 40.89666 50.35944 3.1149146

43.31802 40.89666 50.49784 3.2216052

41.23126 40.89666 42.30729 0.45321382

41.32458 40.89663 42.49497 0.55918265

41.30604 40.89658 42.46113 0.53895406

41.26396 40.89655 42.37889 0.49287617

41.29433 40.89651 42.44048 0.52634587

41.27907 40.89647 42.41139 0.50969696

41.29254 40.89648 42.43741 0.52443376

41.29032 40.89648 42.43332 0.52201634

41.30528 40.89647 42.46031 0.53819088

41.31821 40.89647 42.48333 0.55224192

41.30533 40.89647 42.46039 0.53822503

41.26288 40.8964 42.37771 0.49203321

41.25922 40.89641 42.3704 0.48805826

41.2497 40.89636 42.35475 0.47814516

41.28463 40.89634 42.42277 0.51606776

41.27521 40.8963 42.40417 0.50582233

41.29152 40.89629 42.43595 0.52375883

According to this analysis we found that the optimal value for mean is 41.29152 db, minimum value is 40.89629db and maximum value is 42.43595db and standard deviation value is 0.52375883db obtained from 137 iterations and it remains constant until 500 iterations done. The Minimum value of filter function shows that this is the minimum magnitude on which the filter stops the signal. The maximum value of filter shows that this is the maximum magnitude level on which filter to allow pass frequency.

Figure 5.1 Minimum value of Filter function

Figure 5.2 Mean and standard value of Filter function

Figure 5.1 and 5.2 show minimum, mean and standard value of filter function. When we see the magnitude plot it is clearly seems that the gain is near to one and filter passes the lower frequency efficiently and attenuates high frequencies clearly, so one can say it's a good low pass filter. Now when we see the phase plot it is clearly indicated that the phase goes linearly, only when two frequency discontinuities is there that will not be linear. Here the discontinue size is pi, representing a sign reversal. They do not affect the property of linear phase

Figure 5.3 Minimum and maximum value of Filter function

Figure 5.4 Maximum value of Filter function

Figure 5.3 and 5.4 shows minimum and maximum value of filter function. NPSO converges to much lower error fitness value as compared to RGA and PSO which yield suboptimal higher values of error fitness values. Minimum error fitness value in the case of RGA is 3.109 in 38.1096s, minimum error fitness value in the case of PSO is 2.479 in 19.5108s; whereas, minimum error fitness value in the case of NPSO is 1.01 in 9.7218s.

Figure 5.5 Comparative analysis of Fitness function: Case 1

Figure 5.6 Comparative analysis of Fitness function: Case 2

Figure 5.7 Comparative analysis of Fitness function: Case 3

Figure 5.8 Comparative analysis of Fitness function: Case 4

Figure 5.9 Comparative analysis of Fitness function: Case 5

Figure 5.5, 5.6, 5.7, 5.8 and 5.9 shows comparative analysis of fitness function. The simulation results clearly indicate that PSO demonstrates the best performance in terms of magnitude response, the stop band ripples are minimum and also stop band attenuation are maximum with the transition width of narrow size. The efficient results got by number of iterations performed many times. The fitness function graphs clearly indicate that the ripples will settled down after a certain number of iterations performed, in the form of value we can say that 40.89 db magnitude level. The standard deviation also indicates that the variation in the minimum and maximum value will be 40.895706 db



6.1 Advantages and Disadvantages of Filter

6.1.1 FIR filters primary advantages are:

FIR filters shows linear phase characteristic.

FIR filters are always stable.

For designing the FIR filter used design methods are always linear.

FIR filters can be realized in hardware efficiently.

Transients in filter are of finite duration.

6.1.2 FIR filters primary disadvantages are:

To achieve higher performance FIR filters require much higher filter order.  

The comparison of FIR filter when done with IIR filter on the parameter of delay then it is found that FIR having large delay as compare to IIR.

6.2 Advantages and Limitation of PSO

PSO algorithm is one of the most significant methods for solving the non-smooth global optimization problems while there is some limitation of the PSO algorithm. The advantages and limitation of PSO are discussed below:

6.2.1 Advantages of the PSO algorithm:

1) PSO algorithm is a derivative-free algorithm.

2) It is easy to accomplishment, so it can be functional both in scientific research and engineering problems.

3) It has an inadequate number of parameters and the collision of parameters to the solutions is small compared to other optimization techniques.

4) The calculation in PSO algorithm is very simple.

5) There are some techniques which ensure convergence and the optimum value of the problem calculates easily within a short time.

6.2.2 Limitation of the PSO algorithm:

 1) PSO algorithm suffers from the limited optimism, which degrades the parameter of its speed and direction.

2) Problems with non-coordinate system (for instance, in the energy field) exit.



7.1.1 Noise suppression:

Adaptive filtering is one of the important parts of communication. If the signal is continuously affected by noise, then the need for adaptive filtering arises. Adoptive algorithm is the main part of adoptive filtering process.

 7.1.2 Imaging Processing:

In imaging science, image processing is processing of images using mathematical operations by using any form of signal processing for which the input is an image, such as a photograph or video frame; image processing output related to the image may be either an image or a set of characteristics or parameters. In image-processing technique the images as two dimensional signal treated and it apply in standard signal-processing techniques.

7.1.3 Bio-signals:

A bio-signal is any signal in living beings that can be continually measured and monitored. The term bio-signal is often used to refer to bioelectrical signals, but it may refer to both electrical and non-electrical signals. Usually refer time-varying signals, although spatial parameter variations are sometimes subsumed as well that is the nucleotide sequence determining the genetic code)

7.1.4 Bandwidth limiting:

When the Internet service providers slowing the Internet services intentional then this is band limiting. In the field of communication networks for regular network traffic regulation and also for bandwidth congestion reactive measure is applied.  Different locations can be there in the network where the bandwidth limiting is there. To avoid the server crash on the LAN the admin can apply the band limiting. ISP can also limit the BW to reduce the usage of data. To control the process of uploading and downloading like videos file sharing etc the band limiting can be done. In order to spread a load over a wider network to reduce local network congestion bandwidth throttling is also often used in Internet applications or over a number of servers to avoid overloading individual ones, and so :

Crashing risk will reduce.  

If the bandwidth is not throttled then by compelling users, gain additional revenue, to use more expensive pricing schemes.

7.2 Applications of PSO:

Kennedy and Eberhart set up the first handy use of Particle Swarm Optimization in 1995. It was in the field of neural system preparing and was accounted for together with the calculation itself. PSO have been effectively utilized over an extensive variety of uses, for occasion, information transfers, framework control, information mining, power frameworks, plan, combinatorial improvement, signal handling, system preparing, and numerous different zones. These days, PSO calculations have additionally been produced to take care of compelled issues, multi-target advancement issues, issues with powerfully evolving scenes, and to discover various arrangements, while the first PSO calculation was utilized for the most part to explain unconstrained, single-target streamlining issues. Various areas where PSO is applied are listed:

Antennas Design: The optimal control and design of phased arrays, broadband antenna design and modeling, reflector antennas, design of Yagi-Uda arrays, array failure correction, optimization of a reflect array antenna, far-field radiation pattern reconstruction, antenna modeling, design of planar antennas, conformal antenna array design, design of patch antennas, design of a periodic antenna arrays, near-field antenna measurements, optimization of profiled corrugated horn antennas, synthesis of antenna arrays, adaptive array antennas, design of implantable antennas.

Signal Processing: Recognition of flatness signal, design of IIR filters speech coding, tuning of analogue filter, particle filter optimization, nonlinear adaptive filters, Costas arrays, wavelets, blind detection, blind source separation, localization of acoustic sources, distributed odour source localization.

Networking: Micro grids, congestion management, cellular neural networks, design of radial basis function networks, feed forward neural network training, product unit networks, neural gas networks, design of recurrent neural networks, wavelet neural networks, neuron controllers, wireless sensor network design, estimation of target position in wireless sensor networks, wireless video sensor networks optimization.

Biomedical: Inference of gene regulatory networks, human movement biomechanics optimization, RNA secondary structure determination, phylogenetic tree reconstruction, cancer classification, and survival prediction, DNA motif detection, biomarker selection, protein structure prediction and docking, drug design, radiotherapy planning, analysis of brain magneto encephalography data, electroencephalogram analysis, biometrics and so on.




Novel particle swarm optimization algorithm (PSO) is applied to the solution of the constrained, multi-model FIR low pass filter design problem with optimal filter coefficients.

The results comparison of Pbest, Gbest, with the help of PSO algorithm has been made. It is clearly concluded that the PSO can be use to converge best characteristic or best results in terms of filter coefficients  as compared to others.  

The simulation results clearly indicate that PSO demonstrates the best performance in terms of magnitude response, stop band ripple are minimum and also stop band attenuation is minimum with the transition width of the narrower size. One can utilize the NPSO in the field of digital signal processing to optimize their results.   

For the design of optimal digital filters the objective function involves accurate control of various parameters of frequency spectrum and is thus highly non-uniform, non-linear, non-differentiable and multimodal in nature. Classical optimization methods cannot optimize such objective functions and cannot converge to the global minimum solution. The disadvantages such as:

These methods are highly sensitive to starting points when the number of solution variables, so the solution space size increase.    

Convergence to local optimum solution frequently or divergence or revisiting the same suboptimal solution.

Continuous and differentiable objective cost function required. (Gradient search methods).

Requirement of the piecewise linear cost approximation (linear programming).

Its algorithm complexity is high. (Non-linear programming).

Problem of convergence.                        



       CHAPTER 9



One can do the comparison of the characteristics of Low pass, High pass, Band pass, Band stop filters generated by Particle Swarm Optimization and can compare these simultaneously with the characteristics generated by other methods.

In the field of trans-multiplexer and also in the field of filter bank efficient design means negligible error are the important research part. One can optimize the results with the help of inbuilt functions for example the spline function is one of the inbuilt function. Some other types over whom parameters one can control easily and result will be very efficient like Particle Swarm Optimization and genetic algorithm etc. One can say also that the main research field of trans-multiplexer is in 2-D.

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