# Overview of Digital Filters and Analysis of Infinite Duration

## Transcription

Overview of Digital Filters and Analysis of Infinite Duration

IJSART - Volume 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 Page | 156 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) www.ijsart.com IJSART - Volume 1 Issue 4 –APRIL 2015 ISSN [ONLINE]: 2395-1052 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]. Page | 157 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]. www.ijsart.com IJSART - Volume 1 Issue 4 –APRIL 2015 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: Page | 158 www.ijsart.com IJSART - Volume 1 Issue 4 –APRIL 2015 ISSN [ONLINE]: 2395-1052 REFERENCES [1] Design OF IIR Filter by Samir V. Ginde and Josrph AN --LaPs.laps.ufpa.br/aldebaro/classes/04pds-manaus/matlab -filtros.pdf. [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 Processing, Prentice-Hall, 1975 [5] Mitra S.K., Digital Signal Processing –A Computer-based Approach, McGraw Hill, 1998 [6] Proakis J.G. and Manolakis D.G., Digital Signal Processing – Principles, Algorithms and Applications, Prentice-Hall, 1996. [7] A book of Digital Signal Processing by S. Salivahanan, A Vallavaraj and C. Gnanapriya Tata McGraw Hill Education Private Limited, NEW DELHI. [8] Rabiner, Lawrence R., and Gold, Bernard, 1975: Theory and Application of Digital Signal Processing (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.) [9] A.Antoniou (1993). Digital Filters: Analysis, Design, and Applications (2 ed.). McGraw-Hill, New York, NY. [10] A. Antoniou (2006). Digital Signal Processing: Signals, Systems, and Filters. McGraw-Hill, New York, NY. “D.Marpe, T.Wiegand and G.J.Sullivan, “The H.264/MPEG -4 Advance video coding standard and its applications”, IEEE Communications Magazine, vol.44, no.8, pp.134-144, Aug.2006. Page | 159 www.ijsart.com