software implementation of digital filters

MEE 0828

SOFTWARE IMPLEMENTATION OF DIGITAL FILTERS

Satish kumar Are

Manoranjan Reddy Thangalla

Saikrishna Gajjala

This thesis is presented as part of Degree of

Master of Science in Electrical Engineering

Blekinge Institute of Technology

August 2008

Blekinge Institute of Technology School of Engineering Department of Signal Processing Supervisors: Nedelko Grbic Mikael Swartling

3

ABSTRACT

This thesis proposes to create a MATLAB GUI (Graphical User Interface) to replace an

existing laboration exercise in signal processing at Blekinge Institute of Technology.

MATLAB is a matrix-based technical computing language widely used throughout the

scientific, engineering and mathematical communities. A GUI provides a graphical interface

between the program and the user, facilitating ease and frequency of use. Development of a

MATLAB GUI for this laboration exercise will benefit the students and increase the

awareness towards designing of digital filters.

The developed software provides an interface between audio recording and playback

hardware and the user when exploring filter design parameters. This software is designed for

analyzing digital filter characteristics such as amplitude, phase and pole/zero locations which

are useful in designing an appropriate filter. This can be achieved by entering arbitrary filter

parameters.

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ACKNOWLEDGMENTS

We would like to express our sincerest thanks and gratitude to Mikael Swartling for being

our mentor on this journey. His guidance, patience and support throughout this project have

been a blessing. We would also like to thank our teacher of signal processing Nedelko Grbic

for providing such a good project and interesting lectures on signal processing. Finally we

would like to thank Mikael Åsman for support and suggestions towards the project.

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Table of Contents

Introduction ............................................................................................................................... 7

Chapter 1: DIGITAL FILTERS ...................................................................................................... 8

1.1 Background ....................................................................................................................... 8

1.2 FIR filters ........................................................................................................................... 9

1.3 Types of windows ........................................................................................................... 10

1.3.1 Rectangular window ................................................................................................ 10

1.3.2 Bartlett window ....................................................................................................... 11

1.3.3 Hanning window ...................................................................................................... 12

1.3.4 Hamming window.................................................................................................... 13

1.3.5 Blackman window ................................................................................................... 14

1.4 Design considerations .................................................................................................... 15

1.5 Observations .................................................................................................................. 16

1.6 IIR filters.......................................................................................................................... 17

1.6.1 Bilinear transformation ........................................................................................... 17

1.6.2 Butterworth approximation .................................................................................... 17

1.6.3 Chebyshev approximation ....................................................................................... 18

1.7 Design considerations .................................................................................................... 19

1.8 Observations .................................................................................................................. 20

Chapter 2 : Developed Software ............................................................................................ 21

2.1 Digital filter software GUI ............................................................................................... 21

2.1.1 Filter selection panel ............................................................................................... 22

2.1.2 Input Selection panel ............................................................................................... 22

2.1.3 Filter specifications panel ........................................................................................ 23

2.2 Digital filter analysis ....................................................................................................... 24

2.2.1 Plot panel ................................................................................................................. 24

2.2.2 Pole/zero plot .......................................................................................................... 25

Conclusion ............................................................................................................................... 26

Appendix .................................................................................................................................. 27

References ............................................................................................................................... 29

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Introduction

Existing problem

For analysis of digital filters, DOS based software was developed years back at Blekinge

Institute of Technology for students of signal processing. However, this software have

drawbacks such as portability, inefficient use of computer resources, less accessibility and

bulk hardware system due to floating point DSP which requires extra hardware for floating

point operations. The developed software has some constraints as it is not user friendly.

Possible solution

With extensive advancements in scientific software, a user friendly graphical user interface

was developed using MATLAB graphical user interface for the analysis of digital filters. This

GUI will overcome the complexities mentioned earlier. The developed software will help the

user to analyze the filters in an efficient manner due to the availability of input signals,

windows, various types of filters such as lowpass, highpass, bandpass, and bandstop filters,

and pole/zero plot with filter coefficients.

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Chapter 1: Digital Filters

1.1 Background

The term ‘filter’ is frequently used in signal processing. A filter is a frequency selective

device that removes unwanted information from the original message signal. Unwanted

signals can be noise or other undesired information. Digital filters are more versatile when

compared to the analog filters in their characteristics such as programming flexibility, ability

to handle both low as well as high frequency signals accurately. Also the hardware

requirement is relatively simple and compact. In real world signals are analog in nature. A

simple signal flow block diagram that explains how the signal is processed to acquire desired

output signal is shown in figure 1.1.

unfiltered sampled digitally filtered analog digitised filtered analog signal signal signal signal

Figure 1.1: Signal flow block diagram.

Analog to digital conversion is an engineering process that enables digital processor to

interact with real world signals. The input to the processor should be properly sampled and

quantized. Sampling and quantization restrict the amount of information a digital signal

contain. In the figure 1.1 an interface is provided between analog signal and the digital signal

processor called analog to digital converter (ADC). The output from ADC is input to the

processor. In applications output from the processor is to be given to user in analog form such

as speech communications, for this an interface is provided from digital domain to the analog

domain. This interface is called digital to analog converter (DAC). Thus the signal is

provided in analog for to the user as shown in figure 1.1. The processor in figure 1.1 can be

anywhere from a large programmable digital computer to a small microprocessor which

contains digital filters.

ADC

DAC

PROCESSOR

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The digital filters are two types based on their impulse response; finite impulse response

(FIR) and infinite impulse response (IIR) filters. FIR filters have same time delay for all

frequencies (linear phase), relatively insensitive to quantization and are always stable. FIR

filters can be designed in different ways, for example window method, frequency sampling

method, weighted least squares method, minimax method and equiripple method. Out of

these methods, the window technique is most conventional method for designing FIR filters.

1.2 FIR filters

A finite impulse response filter of length with input and output is described by

the difference equation

= + − + ⋯ + − +

= −

where is the set of filter coefficients. The transfer function of this filter in domain can be

represented as

=

A window in filter design provides trade off between resolution that is the width of the peak

and spectral leakage that is the amplitude of the tails of desired impulse response. The desired

frequency response specification for linear phase filter is the Fourier transform of the desired

impulse response, and this can be represented as

=

and the inverse as

=

where is the desired frequency response and is the corresponding impulse

response. As is infinite duration, the sample response must be truncated. Truncation is

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performed by multiplying desired sample response with a window function in time domain

which gives sample response of filter represented as

= where is a window function. Various types of windows were used when designing the

FIR filters.

1.3 Types of Windows

1.3.1 Rectangular window

The rectangular window has excellent resolution characteristics for signals of comparable

strength. The rectangular window is defined as

= , !"# $ = 0,1,2, … ) − 1, *+,*-ℎ*#* / The frequency response of the window function is the Fourier transform which, is defined as

0 =

The amplitude response of the rectangular window function is

|0| = |234/||234/| , − ≤ ≤ and the phase response is

7 = 8 − 9 − :, 234 9 : ≥ − 9 − : + , 234 9 : < 0 /

The actual impulse response can be expressed in frequency domain as convolution which

leads to smoothing of . As increases, 0 becomes narrower, thereby reducing

the smoothing effect. In figure 1.2 it is observed that as increases, the main lobe becomes

narrower. However, the amplitude of the side lobes is unaffected. The frequency response of

a lowpass FIR filter designed using rectangular window is shown in figure 1.3 with cutoff

frequency for different window lengths, where cutoff frequency is the characteristic

frequency which determines the type of the filter.

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Figure 1.2: Frequency response for Rectangular window.

Figure 1.3: Lowpass FIR filter designed with Rectangular Window.

1.3.2 Bartlett window

A Bartlett window is a triangular shaped window function. The Bartlett window has higher

side lobe attenuation than the rectangular window. The Bartlett window is defined as

= − = − − = − The frequency response for Bartlett window is shown in figure 1.4 and figure 1.5 shows the

frequency response of a lowpass FIR filter designed using Bartlett window.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-150

-100

-50

0

Normalized frequency

Magnitude(d

B)

M=9

M=15

M=21

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-140

-120

-100

-80

-60

-40

-20

0

20


Magnitude(d

B)

M=9

M=15

M=21

12

Figure 1.4: Frequency response for Bartlett window.

Figure 1.5: Lowpass FIR filter designed with Bartlett Window.

1.3.3 Hanning window

The Hanning window is a raised cosine window and can be used to reduce the side lobes

while preserving a good frequency resolution compared to the rectangular window. It is

commonly used as general purpose window for the analysis of continuous signals. The

Hanning window is defined as

= . ? 9 − @A2 − :

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-150

-100

-50

0


Magnitude(d

B)

M=9

M=15

M=21

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-140

-120

-100

-80

-60

-40

-20

0

20


Magnitude(d

B)

M=9

M=15

M=21

13

The frequency response for Hanning window is shown in figure 1.6 and figure 1.7 shows the

frequency response of a lowpass FIR filter designed using Hanning window.

Figure 1.6: Frequency response for Hanning window.

Figure 1.7: Lowpass FIR filter designed with Hanning Window.

1.3.4 Hamming window

The Hamming window is, like the Hanning window, also a raised cosine window. The

Hamming window exhibits similar characteristics to the Hanning window but further

suppress the first side lobe. The Hamming window is defined as

= . ?B − . BC @A2 −

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-150

-100

-50

0


Magnitude(d

B)

M=9

M=15

M=21

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-140

-120

-100

-80

-60

-40

-20

0

20


Magnitude(d

B)

M=9

M=15

M=21

14

The frequency response for Hamming window is shown in figure 1.8 and figure 1.9 shows

the frequency response of a lowpass FIR filter designed using Hamming window.

Figure 1.8: Frequency response for Hamming window.

Figure 1.9: Lowpass FIR filter designed with Hamming Window.

1.3.5 Blackman window

The Blackman window is similar to the Hanning and the Hamming windows. An advantage

with the Blackman window over other windows is that it has better stopband attenuation and

with less passband ripple. The Blackman window is defined as

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-150

-100

-50

0


Magnitude(d

B)

M=9

M=15

M=21

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-140

-120

-100

-80

-60

-40

-20

0

20


Magnitude(d

B)

M=9

M=15

M=21

15

= . B − . ? @A2 − + . D @A2 B − The frequency response for Blackman window is shown in figure 1.10 and figure 1.11 shows

the frequency response of a lowpass FIR filter designed using Blackman window.

Figure 1.10: Frequency response for Blackman window.

Figure 1.11: Lowpass FIR filter designed with Blackman Window.

1.4 Design considerations

A design consideration when designing digital FIR filter is selecting a window. This can be

done with the help of frequency specifications of the required filter. In general, the frequency

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-150

-100

-50

0


Magnitude(d

B)

M=9

M=15

M=21

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-140

-120

-100

-80

-60

-40

-20

0

20


Magnitude(d

B)

M=9

M=15

M=21

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specification consists of pass and stopband cutoff frequencies and attenuations. The length of

the filter can be determined by the main lobe width.

Table 1 shows the side lobe attenuation and main lobe width for different windows. Table 2

shows the desired impulse response functions for various filters. Specifically, stopband

attenuation provides for a user to select an appropriate window.

Window Side lobe attenuation Approximate Main lobe width Rectangular -20dB 4F )⁄

Bartlett -27dB 8F )⁄

Hanning -40dB 8F )⁄

Hamming -50dB 8F )⁄

Blackman -70dB 12F )⁄

Table 1: Comparison of main lobe width and side lobe attenuation for different window

types.

Filter Type Desired impulse response I

Lowpass JKF ∙ sin JK P$ − Q) − 12 RS

JK P$ − Q) − 12 RS

Highpass T P$ − 9) − 12 :S − JKF ∙ sin JK P$ − Q) − 12 RSJK P$ − Q) − 12 RS

Bandpass 2 cos WJK P$ − 9) − 12 :SX ∙ JKF ∙ sin JK P$ − Q) − 12 RSJK P$ − Q) − 12 RS

Bandstop T P$ − 9) − 12 :S − 2 cos WJK P$ − 9) − 12 :SX ∙ JKF ∙ sin JK P$ − Q) − 12 RSJK P$ − Q) − 12 RS

Table 2: Desired impulse responses for filter types.

1.5 Observations

From the window frequency response plots shown in the Figures 1.2, 1.4, 1.6, 1.8 and 1.10,

one can observe that as M increases, the main lobe becomes narrower, side lobe amplitudes

remain unaffected but width of the sidelobes decreases. The rectangular window provides

less width in mainlobe and higher sidelobes in contrast with other windows. Using the

window function the ringing effects at the band edges vanishes which results in lower

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sidelobes, thereby increase in the width of the transition band of the lowpass FIR filter as

shown in Figures 1.3, 1.5, 1.7, 1.9 and 1.11.

1.6 IIR filters

Impulse response functions of IIR filters are non-zero over an infinite length of time. IIR

filters can be described using a difference equation as

= Y

Y − Y − Z[

− where Y and Z are the filter coefficients. The transfer function of IIR filters can be

expressed as

= ∑ YY Y + ∑ Z[ These filters can be designed by the bilinear transformation method. These filters are

designed using their analog counterparts rather than discrete time analysis.

1.6.1 Bilinear transformation

The bilinear transformation method is commonly used in designing digital IIR filters to

obtain filter coefficients. As mentioned in section 1.6, digital filters are designed with their

analog counterparts, so it must be transformed into discrete time domain. This transformation

can be done with the bilinear transform.

] = P − + S

1.6.2 Butterworth approximation

The amplitude response of a Butterworth filter is given as

|Ω| = _` + 9 Ω Ωa:[

where [ is order of the filter, Ωa is the passband frequency, Ω is the analog frequency of the

filter specifications, and _ is the maximum value of the amplitude function.

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Figure 1.12

Ω] is the stopband frequency shown in the figure 1.12.The filter order [ can be approximated

as

[ = bAc de` f − g /hibAc 9Ω]Ωa:

where h is the filter parameter which determines the band edge value of |kΩ| and f is

the maximum allowed value for the amplitude function at the stopband edge.

f = |Ω]|lmn

The system transfer function in polynomial form is given by

] = _9 ]Ωa:[ + Z[ 9 ]

Ωa:[ + ⋯ + Z 9 ]Ωa: +

The Butterworth filter coefficients Zo are provided in appendix table A.

1.6.3 Chebyshev approximation

The amplitude response of a Chebyshev filter is given by

|Ω| = _` + h^[ 9 Ω

Ωa:

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Figure 1.13

Figure 1.13 gives us the specifications for order of the Chebyshev filter, where the ripple is

defined as pYaaq = ∙ bAc + h dB and _ is the maximum value of the amplitude

function, ^[ 9 ΩΩa: is the Chebyshev polynomial. The required filter order can be

approximated as

[ =tu]

vwwx

yzzzz` f −

h|~

tu] 9Ω]Ωa:

where h is the filter parameter related to ripple in the passband and f is the maximum

allowed stopband ripple. The system transfer function in polynomial form is given by

] = _ ∙ Z ∙ [ u √ + h [ /9 ]Ωa:[ + Z[ 9 ]

Ωa:[ + ⋯ + Z

The Chebyshev filter coefficients Zo are provided in appendix tables B1, B2, B3 and B4.

1.7 Design considerations

Designing of IIR filters are tricky when compared with FIR filters, here we always consider a

lowpass filter and then transforms the desired filter to and from this lowpass filter. A

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technique called “pre-warping” is used to map digital frequencies into analog frequencies.

The warping frequencies are given by

Ωa = m4 Qa R

Ω] = m4 Q] R

where a and ] angular frequencies of pass and stop bands of digital filters. After “pre

warping” frequency transformation is needed for highpass, bandpass and bandelimination

filters shown in table 3.

Filter Frequency Transformation

Highpass = and =

Bandpass = Ω − Ω and = Ω − Ω , Ω ∙ Ω = Ω ∙ Ω = Ω

Bandelimination = and = , Ω ∙ Ω = Ω ∙ Ω = Ω

Table 3: Transform to frequencies of lowpass filter.

After the frequency transformation, the designed lowpass filter is transformed into the desired

filter type according to the table 4.

LP HP BP BS

]

]

] + Ω]

Ω] + Ω]

Table 4: Transform back to original filter (HP, BP, BS) from lowpass filter.

1.8 Observations

Implementing an IIR filter with certain stopband-attenuation and transition-band

requirements typically requires far fewer filter taps than an FIR filter meeting the same

specifications. This leads to a significant reduction in the computational complexity required

to achieve a given frequency response. However, filter requires feedback to implement an IIR

system, which introduces stability issues. In addition, the filter is of nonlinear phase.

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Chapter 2: Developed software

2.1 Digital Filter Software GUI

A digital filter software GUI is shown in figure 2.1 for design and analysis of digital filters.

The developed filter software GUI consists of the following parts.

Filter selection panel

Input selection panel

Filter specifications panel

A panel for different plots for analysis

A button to display the pole/zero

A button to exit the GUI

Figure 2.1: A simple GUI for design and analysis of digital filters.

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2.1.1 Filter Selection Panel

The user can choose to design and analyze a FIR or IIR digital filter.

2.1.2 Input Selection Panel

The input selection panel consists of settings for the input signal to the digital filter. These

specifications are input signal type, sampling rate and frequency of the signal which are

essential to the analysis of the digital filters.

I. Input

In the Input dropdown list, different types of input signal types can be selected. The signal

types are sine wave, square wave, sawtooth wave and white Gaussian noise. The different

signal types are shown in figure 2.2.

Figure 2.2: The different signal types used in the analysis of digital filters.

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II. Sampling Frequency

The sampling rate is the rate at which the input signal for the filter is sampled. For perfect

reconstruction of a signal the sampling rate should be greater than twice the maximum

frequency of the signal being sampled.

III. Frequency

The frequency determines the number of cycles per unit time of a signal. As the frequency

increases the number of cycles of the input wave also increases. The range of the frequency is

from 0 to Q R − 1 which can be adjusted with the help of a slider and edit box where the

user can directly enter the desired frequency.

2.1.3 Filter specifications panel

I. Filter Type

The user can create lowpass, highpass, bandpass and bandstop. These filters types can be

selected from the dropdown list in the Input selection panel.

II. Filter Order

The filter order mainly determines the width of the transition band. The higher the order, the

narrower is the transition between the passband and stopband, giving a sharper cutoff in the

frequency response. This can be clearly observed in figure 2.3 for orders of 9, 15 and 31.The

transition bandwidth can be improved by increasing the order of the filter. The filter order can

be adjusted between 0 to 31. The filter order can be adjusted by setting the slider or entering

the value in edit box.

Figure 2.3: Plot for different filter orders.

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III. Cut off frequencies

Cutoff frequency is the frequency where the filter is designed to cross over from the passband

to the stopband. The cutoff frequency can be adjusted by setting the values of sliders or

entering the values in the edit box. The cutoff frequencies are in the range from 0 to Q,2 R − 1 . A single cutoff frequency is available for lowpass and highpass filters where as for bandpass

and bandstop two cutoff frequencies are available.

2.2 Digital filter analysis

After designing a digital filter, the filters can be analyzed by selecting different input signals

and examining the filter and filter response in different plots.

2.2.1 Plot Panel

The plot panel consists of three dropdown lists. The dropdown lists presents options for

displaying input and output signals in time domain, linear and logarithmic FFT plots, and

linear and logarithmic filter amplitude responses.

Dropdown list

Time x: shows the input signal in time domain

Time y: shows the filtered signal in time domain

Time x+y: shows the input and output on top of each other in time domain

FFT linear x: shows the amplitude of the FFT of the input signal in linear scale.

FFT linear y: shows the amplitude of the FFT of the output signal in linear scale.

FFT linear x+y: shows the amplitude of the FFT of the input and output signal on top

of each other in linear scale.

FFT logarithmic x: shows the amplitude of the FFT of the input signal in logarithmic

scale.

FFT logarithmic y: shows the amplitude of the FFT of the output signal in logarithmic

scale.

FFT logarithmic x+y: shows the amplitude of the FFT of the input and output signal

on top of each other in logarithmic scale.

FFT linear h: shows the FFT of impulse response in linear scale.

FFT logarithmic h: shows the FFT of impulse response in logarithmic scale.

The FFT is a standard method for decomposition of a signal into harmonic components. It is

the faster version of the Discrete Fourier Transform (DFT). FFT generates the frequency

spectrum for a time domain waveform. The FFT of the impulse response, referred to as the

25

frequency response function, completely characterizes the system. Once the system frequency

response is known, one can predict how that system will react to any waveform.

2.2.2 Pole/zero Plot

A pole/zero plot is the graphical representation of the transfer function in the complex Z

plane which helps to convey properties of the system such as stability, causality, minimum

phase and region of convergence.

A linear time invariant system is said to be minimum phase if the system and its inverse are

stable and casual. For a system to be minimum phase all its poles and zeros must be inside

the unit circle. The user can determine whether a digital filter is stable that is output of digital

filter is bounded for all possible bounded inputs. It is known that for IIR filters to be stable is

that all its poles lie inside unit circle where as for FIR filters all its poles lie inside unit circle.

Figure 2.4: A simple pole/zero plot.

A pole/zero plot GUI is shown in figure 2.4. It contains buttons to add, remove and move

filter coefficients and a plot of poles and zeros. The pole/zero plot GUI also contains clear,

restore and update filter coefficients in the main GUI window. In the pole/zero GUI the user

can also pan and zoom the pole/zero plot.

26

Conclusion

This Master thesis report presents a software implementation of digital filters using Matlab

GUI in a user friendly environment. The developed software is useful for aspirant students in

designing and analysis of the digital filters. The software consists of radio buttons, pop-up

menus, sliders, edit boxes, axes, push buttons and list boxes all these are placed in different

panels and all are working properly. The evaluated performance using the specific theoretical

filter parameters match the performance with the software. The software is working properly

towards accurate results upon manual testing.

27

Appendix

Table A: Coefficients Zo in Butterworth polynomials.

[ Z Z Z ZB Z? ZC Z √2 2 2 B 2.613 3.141 2.613 ? 3.236 5.236 5.236 3.236 C 3.864 7.464 9.141 7.464 3.864 4.494 10.103 14.606 14.606 10.103 4.494 D 5.126 13.138 21.848 25.691 21.848 13.138 5.126

Table B1: Chebyshev filter coefficients Zo.

0.5 # +*¡ = 0.349, ¡ = 0.122.

[ Z Z Z Z ZB Z? ZC Z 2.863 1.516 1.426 0.716 1.535 1.253 B 0.379 1.025 1.717 1.197 ? 0.179 0.752 1.309 1.937 1.172 C 0.095 0.432 1.172 1.589 2.172 1.159 0.045 0.282 0.756 1.648 1.869 2.413 1.151 D 0.024 0.152 0.573 1.148 2.184 2.149 2.657 1.146


1 # +*¡ = 0.509, ¡ = 0.259.

[ Z Z Z Z ZB Z? ZC Z 1.965 1.102 1.098 0.491 1.238 0.989 B 0.276 0.743 1.454 0.953 ? 0.123 0.580 0.974 1.689 0.937 C 0.069 0.307 0.939 1.202 1.931 0.928 0.031 0.214 0.549 1.357 1.429 2.176 0.923 D 0.017 0.107 0.448 0.447 1.837 1.655 2.423 0.920

28


2 # +*¡ = 0.765, ¡ = 0.585.

[ Z Z Z Z ZB Z? ZC Z 1.307 0.823 0.804 0.327 1.022 0.738 B 0.206 0.517 1.256 0.716 ? 0.082 0.459 0.693 1.499 0.705 C 0.051 0.210 0.771 0.867 1.745 0.701 0.020 0.166 0.383 1.144 1.039 1.994 0.698 D 0.013 0.073 0.359 0.598 1.579 1.212 2.242 0.696


3 # +*¡ = 0.998, ¡ = 0.995.

[ Z Z Z Z ZB Z? ZC Z 1.002 0.708 0.645 0.251 0.928 0.597 B 0.177 0.405 1.169 0.581 ? 0.063 0.408 0.549 1.415 0.575 C 0.044 0.163 0.699 0.691 1.663 0.571 0.016 0.146 0.300 1.052 0.831 1.911 0.568 D 0.011 0.056 0.321 0.472 1.467 0.972 2.161 0.567

29

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Proakis, J.G. and Manolakis, D.G., Digital signal processing: principles, algorithms

and applications, Third Edition,(1996), ISBN -81-203-1129-9.

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M.Dahl, S.Johansson, M.Winberg (Department of Telecommunication and Signal

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(2003),Fourth Reprint, ISBN:981-240-215-2.

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software implementation of digital filters

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