IRJET- Design and Implementation of Butterworth, Chebyshev-I Filters for Digital Signal Analysis

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 05 Issue: 03 | Mar-2018 www.irjet.net p-ISSN: 2395-0072
© 2018, IRJET | Impact Factor value: 6.171 | ISO 9001:2008 Certified Journal | Page 2707
Design and Implementation of Butterworth, Chebyshev-I Filters for
Digital Signal Analysis
Modi Rishabhkumar N.1, Pramiti Parashar2
1,2 M.E, EC, L. D College of Engineering, Gujarat, India
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - Filters have become a very essential part in
the digital field emerging exponentially in today’s time.
Filters are responsible for removing the undesirable
components from a specific signal and help in extracting
the required frequency range for the respective output.
Filters have many applications in the signal processing
and communication systems, like, noise reduction, channel
equalization, circuit analysis, audio-video signal
processing, radar field, data analysis. Also, the biomedical
fields have the use of filters for filtering the noisy signals
of the ECG, EEG and EMG. This paper explains the basic
functions of filters, their types, i.e. low pass, high pass,
band pass, band stop, and all pass filters. The IIR filters
have been studied theoretically and also designed using
Python software. The filter responses of the Butterworth,
and Chebyshev-I filters have been observed and compared
successfully.
Key Words: Filters, Digital Filters, Impulse Response,
Butterworth Filter, Chebyshev-I Filter
1.INTRODUCTION
The field of signal processing includes filtering as a basic
and very essential process. It is a linear system which is
used for the removal of noise and all other unwanted
components from the signal and gets the desired signal in
the output. Using the filters, the desired amplitude phase
and frequency of a signal can be obtained from the original
signal. Both the digital and analog filters are a part of
filtering. The digital filters are more preferable as
compared to the analog filters in many fields as it is
efficient for detecting and filtering the noise signals.[1]
For the digital filtering the input analog signal is converted
into digital signal using sampling and then it is processed
and converted back to the analog form and received as the
output. The digital filters are classified into many various
types based on several factors and characteristics that
have been discussed in the further section. The IIR filters
and the FIR filters have been explained in detail with the
transfer functions and henceforth the Chebyshev-I and
Butterworth filters are explained in detail. Also, they have
been designed using Python Software for the comparison
of their respective response.
2. TYPES OF FILTERS
The digital filters function differently on the basis of the
requirement of the user. There are different characteristics
exhibited by the digital filters. On the basis of their
different characteristics, they can be classified into various
different types.
The two main classifications of the digital filters based on
their functioning and the response. The very first basic
classification is done based on their magnitude
characteristics. They are:
1) Low pass filter: - As the name suggests, the low pass
filters allow only the low frequencies required up to
the cut-off frequency to pass through and the other
frequencies are attenuated.
2) High pass filter: - The high pass filter attenuates the
frequencies lower than the cut-off frequencies and
allows the higher frequencies to pass through.
3) Band pass filter: - This filter allows the selected band
of frequency lying between the lower and higher cut-
off frequencies to pass through and attenuates the
rest of the frequencies.
4) Band stop filter: - This filter attenuates the frequency
band between the lower and higher cut-off
frequencies and lets all the other frequencies pass
through.
5) All pass filter: - This filter passes all the frequencies of
equal gain. [2]
The other classification of filters is based on the time
domain. They are:
1) Infinite Impulse Response (IIR) Filters
2) Finite Impulse Response (FIR) Filters
2.1 IIR Filters
IIR Filters are the digital filters that have an infinite
impulse response. They are also known as recursive filters
as they have a feedback and hence they produce a better
frequency response. The IIR filters do not possess linear
phase characteristics. This being their limitation, they
cannot be preferred for a linear phase system. The IIR
filters acquire less memory and also include fewer
calculations. The transfer function of the IIR filters is as
shown in equation 1 below.
( )
∑
∑
(1)

The IIR filters can be realized by Butterworth, Chebyshev-I
and Elliptic filters, of which the Butterworth and
Chebyshev-I filters are explained in the further sections.
[3]
2.1 FIR Filters
The design methods of FIR Filters are based on ideal filter
approximation. Using this approximation, the filter
designed is of a higher order, due to which it becomes
complex to implement. The transfer function of FIR filters
is as shown in equation 2 below.
( )
( )
( )
∑ (2)
The designing process takes into consideration the
required characteristics and specifications. The FIR filters
are designed using various windowing techniques as
shown in the figure below.
Fig -1: Windowing Techniques
3. BUTTERWORTH FILTER
The square magnitude function of a N order Butterworth
filter is given by,
( )
(3)
Unity value is achieved by this function at Ω = 0 that is
considered a flat pass band. Zero value is achieved at
infinity and is considered to be a flat stop band. Figure 2
shows the normalized specifications and the response of
Butterworth filter. The following figure gives the
normalized specifications, where Ωp = 1, ( )
and it is required that ( ) .
Where,
Ap = maximum pass-band variation
As = minimum stop band attenuation
N = Order of filter, that means the no. of stages used in
the design of analog filter. [4]
Fig-2: Response of Butterworth Filter
4. CHEBYSHEV-I FILTER
The square magnitude function of a N order Chebyshev-I
filter is as shown.
( )
( )
(4)
Where,
( ) {
, ( )
, ( )
}
In the normalized pass band 0 ≤ Ω ≤ Ωp = 1, this function
alternatingly achieves the values of 1 and at N + 1
points such that ( ) . For N even,
( ) and for N odd, ( ) (equi-
ripple pass band). At infinity, the value of ( ) is
zero and the first 2N-1 derivatives are zero (maximally
flat stop band), as shown in the figure below [5]. The
Figure below shows the response of the Chebyshev-I
filter.
Name of
Window
function
Transiti
on
Width
(Hz)
(Normal
ize-zed)
Pass
band
ripple
(dB)
Main
Lobe
Relati
ve to
Side
Lobe
(dB)
Stopband
attenuation
(dB)
(Maximum)
Window
Function
w(n),
|n|≤(N-1)/2
Rectangular 0.9/N 0.7416 13 21 1
Hanning 3.1/N 0.0546 31 44
( )
Hamming 3.3/N 0.0194 41 53
( )
Blackman 5.5/N 0.0017 57 75
( )
( )
Kaiser
2.93/N
(β=4.54)
0.0274
-
50
* ( [ ] ) +
( )
4.32/N
(β=6.76)
0.00275 70
5.71/N
(β=8.96)
0.000275 90

Fig-3: Response of Chebyshev-I Filter
5. SIMULATION RESULTS
Considering the following signal for filtering,
( ) ( )
( ) (
)
and the desired signal to be
( )
Cutoff frequency = 1MHz
Sampling frequency = 10MHz
The following graphs demonstrate low pass filter design
using different windows and also compare the different
filter response along with filtered output.
Fig-4: Responses of Blackman Windowing Technique.
Fig-5: Responses of Butterworth Filter
Fig-6: Responses of Chebyshev-I Filter
Fig-7: Desired Output and Filtered Output
6. CONCLUSION
In this paper, we have successfully studied the different
types of digital filters and their functions. Also, we have
understood in detail, the Butterworth and Chebyshev-I
filters, their response, and characteristics. The simulations
have been successfully performed and the comparison has
been concluded that using Butterworth filter, the best
output is obtained in reference to the attenuation and
phase response. It has a flat pass band and the stop band
without any ripples. The results obtained using the FIR
filter is very complex and costly as the order of the filter
required is high. This leads us to conclude that the use of
IIR Filters is better. Among the IIR filters, the Chebyshev-I
Filter gives a sharp response.
REFERENCES
[1] Ranjit Singh and Sandeep K. Arya, “Determining
Optimum coefficients of IIR Digital Filter using Analog
to Digital Mapping,” International Journal of
Advancements in Computer Science and Information
Technology, Vol. 01, No. 01, pp.19-23, September
2011.
[2] Design of Low Pass Filter using Hanning and
Hamming Windowing techniques, Priya Yadav,
Priyanka Sahu.
[3] Proakis, J. G. and Manolakis, D. G. 2007. Digital Signal
Processing: Principles, Algorithms, and Applications.
Pearson Education Ltd.
[4] Design and Implementation of Butterworth,
Chebyshev-I, and Elliptic Filter for Speech Signal
Analysis, PrajoyPodder, Md. Mehedi Hasan
[5] P. Ramesh Babu, “Digital Signal Processing”, Fourth
edition, Scitech Publication(India) Pvt. Ltd, Chennai,
2008

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