Time Series Analysis: 4. Digital Linear Filters. P. F. Góra

Similar documents
Time Series Analysis: 4. Linear filters. P. F. Góra

EE482: Digital Signal Processing Applications

Digital Signal Processing

The Approximation Problem

Lecture 3 - Design of Digital Filters

DIGITAL SIGNAL PROCESSING UNIT III INFINITE IMPULSE RESPONSE DIGITAL FILTERS. 3.6 Design of Digital Filter using Digital to Digital

ESS Finite Impulse Response Filters and the Z-transform

Digital Control & Digital Filters. Lectures 21 & 22

Filter Analysis and Design

EE 521: Instrumentation and Measurements

w n = c k v n k (1.226) w n = c k v n k + d k w n k (1.227) Clearly non-recursive filters are a special case of recursive filters where M=0.

The Approximation Problem

The Approximation Problem

UNIT - III PART A. 2. Mention any two techniques for digitizing the transfer function of an analog filter?

Lecture 9 Infinite Impulse Response Filters

Stability Condition in Terms of the Pole Locations

INFINITE-IMPULSE RESPONSE DIGITAL FILTERS Classical analog filters and their conversion to digital filters 4. THE BUTTERWORTH ANALOG FILTER

Chapter 7: Filter Design 7.1 Practical Filter Terminology

Lecture 7 Discrete Systems

Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )

Chapter 7: IIR Filter Design Techniques

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform.

Fourier Series Representation of

UNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.

Filters and Tuned Amplifiers

E : Lecture 1 Introduction

Systems & Signals 315

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches

Design of IIR filters

Optimal Design of Real and Complex Minimum Phase Digital FIR Filters

Digital Signal Processing Lecture 8 - Filter Design - IIR

ECE 410 DIGITAL SIGNAL PROCESSING D. Munson University of Illinois Chapter 12

Analog and Digital Filter Design

Digital Signal Processing IIR Filter Design via Bilinear Transform

Convolution. Define a mathematical operation on discrete-time signals called convolution, represented by *. Given two discrete-time signals x 1, x 2,

Theory and Problems of Signals and Systems

Responses of Digital Filters Chapter Intended Learning Outcomes:

Multimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2

UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z).

Square Root Raised Cosine Filter

Discrete-time signals and systems

All-Pole Recursive Digital Filters Design Based on Ultraspherical Polynomials

Experimental Fourier Transforms

ELEG 305: Digital Signal Processing

Digital Filters Ying Sun

2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.

APPLIED SIGNAL PROCESSING

MITOCW watch?v=jtj3v Rx7E

Lecture 19 IIR Filters

Analysis of Finite Wordlength Effects

Filter Banks II. Prof. Dr.-Ing. G. Schuller. Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany

Filter structures ELEC-E5410

Elec4621 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis

Electronic Circuits EE359A

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.

UNIVERSITY OF OSLO. Faculty of mathematics and natural sciences. Forslag til fasit, versjon-01: Problem 1 Signals and systems.

Digital Signal Processing:

LAB 6: FIR Filter Design Summer 2011

PS403 - Digital Signal processing

DIGITAL SIGNAL PROCESSING. Chapter 6 IIR Filter Design

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRONICS AND TELECOMMUNICATIONS

LINEAR-PHASE FIR FILTERS DESIGN

Butterworth Filter Properties

INF3440/INF4440. Design of digital filters

Research Article Design of One-Dimensional Linear Phase Digital IIR Filters Using Orthogonal Polynomials

-Digital Signal Processing- FIR Filter Design. Lecture May-16

Poles and Zeros in z-plane

EE 225D LECTURE ON DIGITAL FILTERS. University of California Berkeley

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A

Ch. 7: Z-transform Reading

Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp For LTI systems we can write

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR

David Weenink. First semester 2007

1 1.27z z 2. 1 z H 2

Biomedical Signal Processing and Signal Modeling

Transform Representation of Signals

Chapter 12 Variable Phase Interpolation

GATE EE Topic wise Questions SIGNALS & SYSTEMS

Chirp Transform for FFT

EEL3135: Homework #4

EE123 Digital Signal Processing. M. Lustig, EECS UC Berkeley

VALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF INFORMATION TECHNOLOGY. Academic Year

EE Experiment 11 The Laplace Transform and Control System Characteristics

Optimum Ordering and Pole-Zero Pairing of the Cascade Form IIR. Digital Filter

Digital Signal Processing

Recursive Gaussian filters

ELEG 5173L Digital Signal Processing Ch. 5 Digital Filters

A system that is both linear and time-invariant is called linear time-invariant (LTI).

Signals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters

DFT & Fast Fourier Transform PART-A. 7. Calculate the number of multiplications needed in the calculation of DFT and FFT with 64 point sequence.

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by

ECSE 512 Digital Signal Processing I Fall 2010 FINAL EXAMINATION

Physics 250 Green s functions for ordinary differential equations

Hilbert Transforms in Signal Processing

EE 508 Lecture 11. The Approximation Problem. Classical Approximations the Chebyschev and Elliptic Approximations

Fourier Methods in Digital Signal Processing Final Exam ME 579, Spring 2015 NAME

DISCRETE-TIME SIGNAL PROCESSING

University Question Paper Solution

Summary: ISI. No ISI condition in time. II Nyquist theorem. Ideal low pass filter. Raised cosine filters. TX filters

(Refer Slide Time: 02:11 to 04:19)

Transcription:

Time Series Analysis: 4. Digital Linear Filters P. F. Góra http://th-www.if.uj.edu.pl/zfs/gora/ 2018

Linear filters Filtering in Fourier domain is very easy: multiply the DFT of the input by a transfer function, which is quivalnet to taking a convolution of the input and the response function of a filter. Even though we now shall try to construct filters in the signal domain, it is convenient to analyse them in Fourier domain. Every possible linear filter is represented by H(f) is the transfer function. Y (f) = H(f)X(f). (1) Copyright c 2009-18 P. F. Góra 4 2

A general linear filter in the signal domain has the form y n = q k= s α k x n k + p k=1 β k y n k (2) x n is a (discretized) input signal, y n is the output signal. If s > 0, future values of the input are needed to construct the output. Such filter is non-causal. It cannot be realized on-line (or in real time). Copyright c 2009-18 P. F. Góra 4 3

Eq. (2): y n = q k= s α k x n k + p k=1 β k y n k If p = 0, the filter is a Finite Impulse Response filter (FIR), or a moving average filter. The output of such a filter dies out in a finite time after the input has died out. If p > 0, the filter is Infinite Impulse Response filter (IIR), or an autoregressive filter. Its output can go on infinitely long after the input has died out (in fact, this is a parasitic behaviour). Copyright c 2009-18 P. F. Góra 4 4

a FIR filter: x(t) r(t) y(t) an IIR filter: x(t) r(t) y(t) Copyright c 2009-18 P. F. Góra 4 5

Causal FIR filters y n = q k=0 α k x n k. (3) If the input is stationary, the output also is. q is called the order of the filter. Copyright c 2009-18 P. F. Góra 4 6

To find the transfer function, we Fourier transform Eq. (3). Y m = 1 N 1 N n=0 e 2πimn/N y n = 1 N 1 N n=0 e 2πimn/N q k=0 α k x n k = q k=0 N 1 1 α k N n=0 e 2πimn/N x n k = q k=0 N 1 k 1 α k N n = k e 2πim(n +k)/n x n. (4) Copyright c 2009-18 P. F. Góra 4 7

By the assumption of periodicity of the input, x l x N l. On the other hand, exp(2πim(n l)/n)= exp(2πim) exp(2πim( l)/n)= exp(2πim( l)/n). Thus Y m = q k=0 α k e 2πimk/N 1 N 1 N e 2πimn /N x n } n =0 {{ } X m = q α k e 2πimk/N } k=0 {{ } H m X m. (5) m/n = m/(n ) = f m, where f m is the m-th discrete Fourier frequency. Therefore... Copyright c 2009-18 P. F. Góra 4 8

Transfer function of a causal FIR filter has the form H(f m ) = q k=0 where α(z) is the following polynomial of order q α k ( e 2πif m ) k = α ( e 2πif m ), (6) α(z) = q k=0 α k z k (7) (coefficients of the filter become coefficients of the polynomial (7)). Usually, for the sake of simplicity, we assume that = 1, or that the sampling time is the time unit. Remember: With this notation, frequencies become dimensionless and the Nyquist interval equals [ 1/2, 1/2]. Copyright c 2009-18 P. F. Góra 4 9

Transfer function of a general FIR filter The above can be easily generalized to arbitrary (non-causal) FIR filters. The transfer function becomes H(f m ) = q k= s α k ( e 2πif m ) k = α ( e 2πif m ), (8) where α( ) is now an appropriate rational function. Copyright c 2009-18 P. F. Góra 4 10

A simple low-pass filter Consider a filter y n = 1 4 x n 1 + 1 2 x n + 1 4 x n+1. (9) Its transfer function reads H c (f) = 1 4 e2πif + 1 2 + 1 4 e 2πif = 1 2 + 1 2 cos 2πf = cos2 πf. (10) This transfer function is large for small frequencies and approaches zero at the ends of the Nyquist interval. Copyright c 2009-18 P. F. Góra 4 11

A simple high-pass filter Similarly, the transfer function of the filter y n = 1 4 x n 1 + 1 2 x n 1 4 x n+1. (11) reads H s (f) = 1 4 e2πif + 1 2 1 4 e 2πif = 1 2 1 2 cos 2πf = sin2 πf. (12) This transfer function is large near the ends of the Nyquist interval and nearly zero for small frequencies. Copyright c 2009-18 P. F. Góra 4 12

1 (9) is a (poor) low-pass filter. (11) is a (poor) high-pass filter. 1 0.75 0.75 cos 2 πf 0.5 sin 2 πf 0.5 0.25 0.25 0-0.5-0.25 0 0.25 0.5 f 0-0.5-0.25 0 0.25 0.5 f Copyright c 2009-18 P. F. Góra 4 13

The usual moving average The (unweighted) moving average y n = 1 2l + 1 with the transfer function ( xn l + x n l+1 + + x n + + x n+l 1 + x n+l ), (13) H MA (f) = 1 2l + 1 is a poor low-pass filter. (1 + 2 cos 2πf + 2 cos 4πf + + 2 cos 2lπf) (14) Copyright c 2009-18 P. F. Góra 4 14

Transfer functions of unweighted moving averages 1 l=1 l=2 l=3 l=5 0.75 H MA (f) 0.5 0.25 0-0.5-0.25 0 0.25 0.5 f Copyright c 2009-18 P. F. Góra 4 15

An example of the usual moving average (the noise of the order of the signal) Moving Averages l= 0 l= 1 l= 5 l=16 Copyright c 2009-18 P. F. Góra 4 16

Differentiating filters First derivative: dg dx 1 ( g(x) g(x ) x 2 + ) g(x + ) g(x) = 1 1 g(x + ) g(x ) (15a) 2 2 y n = 1 2 x n+1 1 2 x n 1 H(f) = i sin(2πf ) (15b) (15c) Second derivative: y n = 1 4 2x n+1 1 2 2x n + 1 4 2x n 1 H(f) = 1 2 sin(πf ) (16a) (16b) Copyright c 2009-18 P. F. Góra 4 17

A signal 0 1 2 3 4 5 6 7 8 t The derivative 0 1 2 3 4 5 6 7 8 t Copyright c 2009-18 P. F. Góra 4 18

Phase of the transfer function In Fourier domain, Y (f) = H(f)X(f). (17) H(f) = R(f)e iφ(f), R(f) 0. The modulus of the transfer function amplifies/reduces contributions from the corresponding frequencies. What does the phase, φ(f), do? Copyright c 2009-18 P. F. Góra 4 19

The filter (9) is non-causal, but it is easy to find its causal version: we need to introduce a time delay: y n = 1 4 x n 2 + 1 2 x n 1 + 1 4 x n (18) with the transfer function H(f) = e 2πif cos 2 πf. (19) This transfer function differs from that of (9) only by a phase factor. The phase factor in (19) is responsible for the time delay! Copyright c 2009-18 P. F. Góra 4 20

Suppose that the phase of the transfer function depends linearly on frequencies, φ(f) = af. Calculate the inverse transform: y k (t) n H(f n )X(f n )e 2πif nk = n R(f n )e iaf n X(f n )e 2πif nk = n R(f n )X(f n )e 2πif n(k a), (20) which corresponds to a time shift of a units ( channels ). Therefore, filters of a linear phase introduce a uniform time shift. Filters that do not have a linear phase introduce phase differences between various Fourier components. Copyright c 2009-18 P. F. Góra 4 21

A phase shift can make a difference! 1.5 sin(2πx) + 0.5sin(3πx + 0.25π) 1 0.5 0-0.5-1 -1.5 1.5 0 1 2 3 4 5 6 7 8 x sin(2πx) + 0.5sin(3πx) 1 0.5 0-0.5-1 -1.5 0 1 2 3 4 5 6 7 8 x Copyright c 2009-18 P. F. Góra 4 22

FIR filters design If we have a FIR filter y n = q k=0 α k x n k, (21) we know that its transfer function has the form H(f m ) = q k=0 α k ( e 2πif m ) k. (22) Note that (22) equals, up to a constant, to the DFT of the filter. Copyright c 2009-18 P. F. Góra 4 23

An inverse problem In practice, we need to deal with an inverse problem : Given an ideal transfer function H(f), find the order, q, and coefficients α k of the filter (21) such that its transfer function is as close as possible to the ideal one. This is a technical term! Copyright c 2009-18 P. F. Góra 4 24

An intuitive approach Take the ideal transfer function H(f). Calculate the inverse transform h(t) = 1/2 1/2 H(f)e 2πift df. (23) Discretize h(t) in as many points, as the desired order of the filter is. This approach usually does not work, we need to take additional steps, like truncating the filter, and this leads to distortions of the transfer function. Remember that if = 1, f Nyq = 1/2. Copyright c 2009-18 P. F. Góra 4 25

Example: A low-pass filter The ideal transfer function reads H(f) = 1 f f 0 < 1/2, 0 f > f 0. The transfer function in the time domain equals (24) h(t) = 1/2 1/2 H(f)e 2πift df = f 0 f 0 e 2πift df = sin 2πf 0 t πt. (25) The amplitude of h(t) falls off very slowly, there is no natural cut-off. A sharp edge contains all Fourier components. We either introduce an arbitrary cut-off, or do something else, mostly multiplying the ideal transfer function by a window function. Copyright c 2009-18 P. F. Góra 4 26

0.4 0.3 0.2 0.1 Coefficients of a low-pass FIR filter, square window a n 0-0.1-0.2-0.3-0.4 0 8 16 32 64 n Copyright c 2009-18 P. F. Góra 4 27

0.5 0.4 0.3 0.2 Coefficients of a low-pass FIR filter, Hannig window a n 0.1 0-0.1-0.2-0.3-0.4 0 8 16 32 64 n Copyright c 2009-18 P. F. Góra 4 28

Low-pass FIR filters discretized on a different number of points 1.2 1.2 1.2 1.2 1.2 n= 8 n= 16 n= 32 n= 64 n=128 1 1 1 1 1 0.8 0.8 0.8 0.8 0.8 H(f) 0.6 0.6 0.6 0.6 0.6 0.4 square Hanning 0.4 square Hanning 0.4 square Hanning 0.4 square Hanning 0.4 square Hanning 0.2 0.2 0.2 0.2 0.2 0-0.5-0.25 0 0.25 0.5 f 0-0.5-0.25 0 0.25 0.5 f 0-0.5-0.25 0 0.25 0.5 f 0-0.5-0.25 0 0.25 0.5 f 0-0.5-0.25 0 0.25 0.5 f Copyright c 2009-18 P. F. Góra 4 29

Remarks Decent FIR filters require large orders. Window functions reduce the ripple (rapid oscillations near the edge of the band), but extend the roll-off. Filter design is an art. Design involves decisions on the ripple, the roll-off, (non)linearity of the phase, requirements on memory and computational complexity. Usually you can t optimize for all of the above simultaneously. There is vast literature on filter design. Copyright c 2009-18 P. F. Góra 4 30

Linear IIR filters... take the form y n = q k= s α k x n k + p k=1 β k y n k, (26) where p 1, x n is a (discretized) input signal, y n is the output signal. For convenience we discuss causal filters (s = 0) only; in fact, allowing for noncausality does not change much. Copyright c 2009-18 P. F. Góra 4 31

A problem An IIR filter has a feedback loop. As a matter of principle, an IIR filter can produce a non-zero output infinitely long after the input has ceased. Can we avoid that? How can we be sure that a stationary input signal produces a stationary output? Copyright c 2009-18 P. F. Góra 4 32

Linear difference equations A homogeneous linear difference equation: z n = β 1 z n 1 + β 2 z n 2 + + β p z n p (27) An inhomogeneous linear difference equation: z n = β 1 z n 1 + β 2 z n 2 + + β p z n p + ϕ (28) Theorem: The general solution to an inhomogeneous difference equation equals a sum of the general solution to the corresponding homogeneous linear difference equation and any particular solution to the inhomogeneous equation. Stability of IIR filters is determined by the homogeneous equations. Copyright c 2009-18 P. F. Góra 4 33

Embedding in higher dimensions Let z n = [z n, z n 1,..., z n p+1 ] T R p. Then the homogeneous equation (27) can be written as z n = β 1 β 2 β 3 β p 1 β p 1 0 0 0 0 0 1 0 0 0..... 0 0 0 1 0 z n 1. (29) Solutions to (29) are stable if moduli of all eigenvalues are smaller that 1. Copyright c 2009-18 P. F. Góra 4 34

The characteristic determinant: W p = det β 1 λ β 2 β 3 β p 1 β p 1 λ 0 0 0 0 1 λ 0 0..... 0 0 0 1 λ = λw p 1 + ( 1) p+1 β p = λ 2 W p 2 + ( 1) p+1 β p 1 λ + ( 1) p+1 β p =... = ( 1) p+1 ( λ p + β 1 λ p 1 + β 2 λ p 2 + + β p ) (30) Copyright c 2009-18 P. F. Góra 4 35

Stability condition of an IIR filter An IIR filter (26) is stable if and only if roots of the equation λ p β 1 λ p 1 β 2 λ p 2 β p = 0 (31) lie inside the unit circle. Copyright c 2009-18 P. F. Góra 4 36

A confusion in terminology Sometimes the above condition is formulated for the reciprocals of λ s: An IIR filter (26) is stable if and only if roots of the equation 1 β 1 u β 2 u 2 β p u p = 0 (32) lie outside the unit circle. The conditions (31), (32) are equivalent, but they should not be confused. Copyright c 2009-18 P. F. Góra 4 37

Example Consider a filter y n = β 1 y n 1 + x n. (33) Its characteristic polynomial in the form (32) reads β(z) = 1 β 1 z. (34) It can be seen that if β 1 > 1, y n explodes, and therefore we need to have β 1 < 1, which means that the only root of (34), 1/β 1, lies outside the unit interval (and of course, outside the unit circle). Copyright c 2009-18 P. F. Góra 4 38

IIR filter transfer function Write (26) in the form y n β 1 y n 1 β p y n p = α 0 x n + α 1 x n 1 + + α q x n q, (35) and Fourier transfer it. After some algebra, H(f m ) = q k=0 1 p j=1 α k ( e 2πif m ) k β j ( e 2πif m ) j. (36) Note: The form of the denominator in (36) suggests that the stability condition (32) is natural. Copyright c 2009-18 P. F. Góra 4 39

IIR filters design The trouble with IIR filters design is that one needs to avoid poles that lead to instability. Usually, it is not easy to verify whether a pole lies inside or outside the unit circle. The following bilinear transform is frequently used: Calculate z 2 = 1 iw 1 + iw 1 + i w 1 i w z = 1 iw 1 + iw or w = i z 1 z + 1. (37) = 1 + i w iw + w 2 1 i w + iw + w 2 = 1 + w 2 + 2 Im w 1 + w 2 2 Im w. (38) We can see that Im w > 0 corresponds to z 2 > 1. It is much easier to identify points on the upper or lower half-plane than outside or inside the unit circle. Copyright c 2009-18 P. F. Góra 4 40

IIR filter design procedure An ideal transfer function H(f) is given. Find a rational function that approximates H(f) sufficiently well. Let H(f) be this function. It must be real and nonnegative. Find poles of H(f). Half of them lie in the upper half-plane, half in the lower half-plane. Take a product of terms with poles from the upper half-plane only, substitute f = i(z 1)/(z + 1), simplify and identify the coefficients. Copyright c 2009-18 P. F. Góra 4 41

Example: Butterworth filter of order N As before, we are trying to design a low-pass filter. The step function is approximated by H(f) = 1 1 + ( f f 0 ) 2N, (39) where f 0 is the cut-off frequency. A filter based on (39) is called Butterworth filter of order N. Its poles lie on a circle with a radius f 0, symmetrically with respect to the real axis. ( f f 0 ) 2N ( 2k 1 = 1 f = f 0 exp 2N ) iπ, k = 0,..., 2N 1. (40) Copyright c 2009-18 P. F. Góra 4 42

Poles of a Butterworth filter of order N = 16 Im z good ploes f 0 Re z bad ploes Copyright c 2009-18 P. F. Góra 4 43

N = 1 is the simplest case. We thus take H(f) = if 0 f if 0 = if 0 i z 1 z+1 if 0 1 i 1 + ( ) f 2 = f f 0 f + i } 0 {{} lower = f 0 f 0 z z 1 f 0 z f 0 = = f 0 1+f 0 + f 0 1+f 0 z 1 1 f 0 1+f 0 z i f f 0 i }{{} upper. (41) f 0 f 0 z (1 + f 0 ) + (1 f 0 )z (42) α 0 = α 1 = f 0 /(1 + f 0 ), β 1 = (1 f 0 )/(1 + f 0 ) are the coefficients of the filter. Copyright c 2009-18 P. F. Góra 4 44

Left: Transfer function of Butterworth filter of order 1. Middle: Transfer function of Butterworth filter of order 4. Right: Phase of Butterworth filter of order 4. 1 1 4π 0.8 0.8 3π 0.6 0.6 H(f) H(f) φ(f) 2π 0.4 0.4 π 0.2 0.2 0-0.4-0.2 0 0.2 0.4 0-0.4-0.2 0 0.2 0.4 0-0.4-0.2 0 0.2 0.4 f f f Copyright c 2009-18 P. F. Góra 4 45

Most popular filters 1. Butterworth filters See above (39). The first 2N 1 derivatives of H(f) vanish at f = 0 the filter is maximally flat. Poles lie on a circle. Used in audio processing. Copyright c 2009-18 P. F. Góra 4 46

2. Chebyshev filters A steeper rolloff, but ripples appear. There are two kinds of Chebyshev filters, with ripples in thepassband: H(f) = and with ripples in the stopband: H(f) = 1 1 + [T N (f/f p )] 2, (43) 1 + 1 [ TN (f s /f p ) T N (f s /f) ] 2. (44) T N in the above stands for a Chebyshev polynomial of order N, f p is the upper bound of the passband, f s > f p is the lower bound of the stopband. Tschebyshef, Tchebycheff, Czebyszew Copyright c 2009-18 P. F. Góra 4 47

Because of ripples, they are not used in audio processing, but it is most excellent if the passband contains a single interesting frequency (for example, if higher harmonics are to be eliminated). 3. Elliptic filters H(f) = 1 1 + [R N (f/f p )] 2, (45) where R N is a rational function, with roots of the numerator within [ 1/2, 1/2], and roots of the denominator outside, with R N (1/z) = 1/R N (z). Poles of such a filter lie on an ellipsis. 4. Bessel filters, with a constant delay in the passband. Copyright c 2009-18 P. F. Góra 4 48

Because a rational approximation is much better than a polynomial approximation, IIR filters require significantly lower orders than FIR filters of similar performance. All above filters have analog realizations, i.e. electric circuits that can filter analog signals. Design of analog and digital filters is very important in electronics, telecommunication etc. Design of good filters involves both science and art Copyright c 2009-18 P. F. Góra 4 49