Overview of Digital Filters and Analysis of Infinite Duration

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1 Issue 4 –APRIL 2015
ISSN [ONLINE]: 2395-1052
Overview of Digital Filters and Analysis of Infinite
Duration Unit Pulse Response (IIR) Filters
Sarita Srivastava 1, Imran Ullah Khan 2, Saif Ahmad 3
Department of Electronics & Communication Engineering
1, 2, 3
Integral University, India
Abstract- This paper deals with the analysis of Infinite
Duration Unit Pulse Response (IIR) Filters and an overview of
Digital Filters. According to the experimental results, it is find
that IIR filters result in a lower order than the corresponding
FIR filter. Some other results are also discussed.
number of non-zero terms, i.e. its impulse response sequence
is of infinite duration. The response of the FIR filter depends
only on the present and past input samples, whereas for the
IIR filter, the present response is a function of the present and
past N values of the excitation as well as past values of the
response.[4]
Keywords- Analog Filter, Digital filter, FIR Filter, IIR Filter.
I. INTRODUCTION
A filter is essentially a network that selectively
changes the wave shape of a signal in a desired manner. The
objective of filtering is to improve the quality of a signal (for
example, to remove noise) or to extract information from
signals. [1,2]
A digital filter is a mathematical algorithm
implemented in hardware/software that operates on a digital
input to produce a digital output. Digital filter often operates
on digitized analogue signals stored in a computer memory.
Digital filters play very important role in DSP. Compared with
analogue filters, digital filters are preferred in a number of
application like data compression, speech processing, image
processing, etc., because of the following advantages.[3]
1.
2.
3.
4.
5.
Digital filters can have characteristics which are not
possible with
analogue filters such as linear phase
response.
The performance of digital filters does not vary with
environmental changes, for example, thermal variation.
The frequency response of a digital filter can be adjusted
if it is implemented using a programmable processor.
Several input signals can be filtered by one digital filter
without the need to replicate the hardware.
Digital filters can be used at very low frequencies.
II. TECHNICAL OVERVIEW OF DIGITAL FILTER
Digital filters are classified either as finite duration
unit pulse response (FIR) filters or infinite duration unit pulse
response (IIR) filters, depending upon the form of the unit
pulse response of the system. In the FIR system, the impulse
response sequence is of finite duration i.e. it has a finite
number of non-zero terms. The IIR system has an infinite
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There are several techniques available for the design
of digital filter having an infinite duration unit impulse
response .The design of an IIR filter involves design of a
digital filter in the analogue domain and transforming the
design into the digital domain.
There are three alternative methods for transforming
the filter into the digital domain. The design techniques for IIR
filters are presented with the restriction that the filters be
realizable and stable. An analogue filter with system function
H(s) is stable if all its poles lie in the left half of the s plane.
As a result , if the conversion techniques are to be effective,
the techniques should possess the following properties.[5]
1.
2.
The j omega axis in the s-plane should map onto the unity
circle in the z- plane. This gives a direct relationship
between the two frequency variables in the two domains.
The left half plane of the s-plane should map into the
inside of the unit circle in the z-plane to convert a stable
analogue filter into a stable digital filter.
Fig.1 s plane to z plane mapping [5]
The mapping of the s plane to the z plane is
illustrated by the above diagram and the following 2 relations.
Lines of any given colour in the s plane maps to lines of the
same colour in the z plane.
s = σ+ jω………….……………………………………(1)
z = eσT + ejωT …………………………………………..(2)
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The above relations given by equation (1) & (2)
shows the following properties:
1.
2.
3.
4.
5.
6.
7.
The imaginary axis of the s plane between minus half the
sampling and plus half the
sampling frequency maps
onto the unit circle in the z plane.
The portion of the s plane to the left of the red line maps
to the interior of the unit circle in the z plane.
The portion of the s plane to the right of the red line maps
to the exterior of the unit circle in the z plane.
The green line (line of constant sigma) maps to a circle
inside the unit circle in the z plane.
Lines of constant frequency in the s plane maps to radial
lines in the z plane.
The origin of the s plane maps to z = 1 in the z plane.
The negative real axis in the s plane maps to the unit
interval 0 to 1 in the z plane.
IIR filters are digital filters with infinite impulse
response.Unlike FIR filters, they have the feedback and are
known as recursive digital filters therefore [6].
Fig. 2. Representation of FIR and IIR Filter
For this reason IIR filters have much better frequency
response than FIR filters of the same order. Unlike FIR filters,
their phase characteristic is not linear which can cause a
problem to the systems which need phase linearity. For this
reason, it is not preferable to use IIR filters in digital signal
processing when the phase is of the essence. Otherwise, when
the linear phase characteristic is not important, the use of IIR
filters is an excellent solution.
There is one problem known as a potential instability
that is typical of IIR filters only. FIR filters do not have such a
problem as they do not have the feedback. For this reason, it is
always necessary to check after the design process whether the
resulting IIR filter is stable or not.
IIR filters can be designed using different methods.
One of the most commonly used is via the reference analogue
prototype filter. This method is the best for designing all
standard types of filters such as low-pass, high-pass, bandpass and band-stop filters[7].
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Fig. 3. Block diagram of design method using
reference analog prototype filter
FIR filters can have linear phase characteristic, which
is not typical of IIR filters. When it is necessary to have linear
phase characteristic, FIR filters are the only available solution.
In other cases when linear phase characteristic is not
necessary, such as speech signal processing, FIR filters are not
good solution. IIR filters should be used instead. The resulting
filter order is considerably lower for the same frequency
response [8].
The filter order determines the number of filter delay
lines, i.e. number of input and output samples that should be
saved in order that the next output sample can be computed.
For instance, if the filter order is 10, it means that it is
necessary to save 10 input samples plus 10 output samples
preceding the current sample. All these 21 samples will affect
the next output sample.
The IIR filter transfer function is a ratio of two
polynomials of complex variable z-1. The numerator defines
location of zeros, whereas the denominator defines location of
poles of the resulting IIR filter transfer function.
III. IMPLEMENTATION AND DESIGN
IIR filters may be implemented as either analogue or
digital filter. In digital IIR filters, the output feedback is
immediately apparent in the equations defining the output.
Note that unlike FIR filters, in designing IIR filters it is
necessary to carefully consider the "time zero" case in which
the outputs of the filter have not yet been clearly defined.
Design of digital IIR filters is heavily dependent on
that of their analogue counterparts because there are plenty of
resources, works and straightforward design methods
concerning analogue feedback filter design while there are
hardly any for digital IIR filters. As a result, usually, when a
digital IIR filter is going to be implemented, an analogue filter
(e.g. Chebyshev filter, Butterworth filter , Elliptic filter) is first
designed and then is converted to a digital filter by applying
discretization techniques such as Bilinear transform or
Impulse
invariance Example IIR filters include the
Chebyshev filter and Butterworth filter [9].
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Fig.4 Simple IIR filter block diagram
ISSN [ONLINE]: 2395-1052
(i) The filter should have a specific frequency response.
(ii) The filter should have a specific phase shift or group
delay.
(iii) The filter should have a specific impulse response.
(iv) The filter should be causal.
(v)
The filter should be stable.
(vi) The filter should be localized.
(vii) The computational complexity of the filter should be
low.
(viii) The filter should be implemented in particular hardware
or software
A typical block diagram of an IIR filter looks like the
above. The block is a unit delay. The coefficients and number
of feedback/feed forward paths are implementation-dependent.
V. RESULT WAVEFORM
Stability
The transfer function allows us to judge whether or
not a system is Bounded Input Bounded Output (BIBO) to be
specific, the BIBO stability criteria requires that the ROC of
the system includes the unit circle. For example, for a causal
system, all poles of the transfer function have to have an
absolute value smaller than one. In other words, all poles must
be located within a unit circle in the z-plane.
The poles are defined as the values of which make the
denominator of H(z) equal to 0:
Q
0   a j z j
j 0
Clearly, if aj ≠ 0then the poles are not located at the
origin of the z-plane. This is in contrast to the FIR filter where
all poles are located at the origin, and is therefore always
stable.
IIR filters are sometimes preferred over FIR filters
because an IIR filter can achieve a much sharper transition
region roll off than FIR filter of the same order [10].
The filter design process can be described as an
optimization problem where each requirement contributes
with a term to an error function which should be minimized.
Certain parts of the design process can be automated, but
normally an experienced electrical engineer is needed to get a
good result.
VI. CONCLUSION
In this project, the design of IIR filters was
considered. Several results from theory were verified in the
design. The bilinear transformation was studied in some depth
through its application to the design of two filters. The
characteristics of a number of important approximations –
Butterworth, Chebyshev, and Elliptic – were affirmed from
the results obtained. The design of the low-pass filter was
particularly insightful in comparing the relative merits and
demerits of FIR and IIR filters in general as well as the
individual IIR filter approximations.
The significant observations made in the design process were:
1.
IIR filters result in a lower order than the
corresponding (designed to meet the same
Specification) FIR filter.
2.
IIR filters exhibit non-linear phase. The bilinear
transformation results in a frequency warping of the
higher frequencies.
IV. TYPICAL DESIGN REQUIREMENTS
Typical requirements which are considered in the design
process are:
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[2] www.mikroe.com/chapter/view/73/chapter 3 IIR Filter.
[3] Infinite impulse response- Wikipedia,
encyclopedia en.wikipedia.org/wiki/Infinite
response
the free
impulse
[4] Oppenheim A.V. and Schafer R.W., Digital Signal
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[5] Mitra S.K., Digital Signal Processing –A Computer-based
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[6] Proakis J.G. and Manolakis D.G., Digital Signal
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[7] A book of Digital Signal Processing by S. Salivahanan, A
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C. Gnanapriya Tata McGraw Hill
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[8] Rabiner, Lawrence R., and Gold, Bernard, 1975: Theory
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[10] A. Antoniou (2006). Digital Signal Processing: Signals,
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