Abstract
Digital IIR filters have an infinitely long impulse response and therefore can be associated with analog filters that have the same characteristics. For this reason, the classical method for digital IIR filter design is based on the design of an analog filter, which is later transformed into an equivalent digital filter through a mapping in the complex plane. The advantage of such a technique lies in the fact that analog filter design and mapping methods for analog-to-digital (A/D) transformation are well known and have sound theoretical foundations. The process is based on a lowpass filter, and frequency transformation methods are later applied if a different type of frequency selectivity is desired. These frequency transformations are again well-known mapping procedures in the complex plane. In this chapter, we will first discuss the design of the main types of lowpass analog filters, namely Butterworth, Chebyshev and elliptic lowpass filters, all of which represent different ways of approximating the desired ideal frequency response. Then we will learn how to transform the analog lowpass filter into an equivalent IIR digital lowpass filter via bilinear transformation, and finally how to transform the IIR lowpass filter into a highpass, bandpass or bandstop filter, if needed. The appendix to this chapter provides deeper insight into the mathematical facets of elliptic design, discussing elliptic integrals of the first kind, Jacobi elliptic functions, and the elliptic rational function on which the transfer function of the analog elliptic filter is based. It must be explicitly noted that all classically-designed lowpass IIR filters are inadequate for specifications with cutoff frequency close to zero and narrow transition band. In such cases, it may be advisable to preliminarily downsample the signal and then design a filter with a less extreme cutoff frequency, allowing for a wider transition band.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Some particular IIR filters, such as polyphase filters, can have quasi-linear phase response, but they will not be described in this book.
- 2.
The monolateral Laplace transform is defined over \(t\in [0,\infty )\).
- 3.
The Chebyshev polynomial of the first kind has already been defined in Chap. 7.
- 4.
A monic polynomial is such that the coefficient \(a_0\) of the highest-degree term is unitary.
- 5.
The poles are N, but a relation exists among them, so that there are only \(N-1\) degrees of freedom (DOF); the monic polynomial with degree N has N coefficients and therefore one of the coefficients must actually be fixed.
- 6.
The last name is ambiguous, because other functions exist that are called in the same way.
- 7.
This name derives from the fact that these integrals were originally studied in the frame of the problem of calculating the length of an arc of ellipse.
- 8.
Plotting the square instead of the plain function avoids many sources of confusion. Of course, we could use \(|R_N(w)|\) as well. Plots of \(R_N(w)\) can be found in the appendix to this chapter.
- 9.
For a description of the Jacobi elliptic functions cd and sn and of their inverse functions the reader is referred to the appendix.
- 10.
This transformation, and other analogous transformations discussed later, are often expressed in literature in terms of \(z^{-1}\) (unit delay), a form that is useful when the H(z) to be obtained is more conveniently written as the ratio of polynomials in \(z^{-1}\), or even in the corresponding factorized form. We will also, however, write the form in terms of z, considering that in this book, in most cases ratios of polynomials in z or corresponding factorized forms are used.
- 11.
Usually, in literature, these transformation are given in terms of \(z^{-1}\) rather than in terms of z, because the form of H(z) as a rational function in \(z^{-1}\) is considered. Here we prefer to use the variable z directly.
- 12.
The new positions of poles and zeros can be found considering a generic factor \(\left( Z-q_i\right) \) contributing to the lowpass transfer function in the zero-pole gain-form. Here \(q_i\) represents either a zero, or a pole, and of course the factor can be in the numerator or in the denominator, respectively. We can then substitute Z with R(z) and set the factor equal to zero, to find \(q_i\).
- 13.
The final lowpass is obviously required to have a different cutoff frequency.
- 14.
DC stands for direct current: here it means “zero frequency”.
- 15.
This method also requires writing the derivative of the transcendental equation with respect to \(\theta _s\).
- 16.
Obviously we assume that \(\omega _{c1}<\omega _{c2}\): \(\alpha \) can then be positive, zero or negative, but its absolute value never exceeds 1.
- 17.
Actually it is possible to use frequency transformations to pass from a digital lowpass filter, derived from an analog lowpass filter, to a digital multiband filter, but these are high-order transformations that push the order of the final filter up considerably. Moreover, direct design techniques avoid constraints and compromises that are unnecessary.
- 18.
This tolerance level can be taken equal to the machine epsilon \(\varepsilon \), which is defined as the absolute value of the difference between a positive floating point number and the closest higher floating point number. The machine epsilon depends on the particular software used: for example, in Matlab \(\varepsilon = 2^{-52}=2.2204\times 10^{-16}\).
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)
Antoniou, A.: Digital Filters: Analysis, Design, and Applications. McGraw-Hill, Blacklick (1993)
Bochner, S., Chandrasekharan, K.: Fourier Transforms. Princeton University Press, Princeton (1949)
Bracewell, R.N.: The Fourier Transform and Its Applications. McGraw-Hill, Boston (2000)
Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, New York (1971)
Constantinides, A.G.: Spectral transformations for digital filters. Proc. IEEE 117, 1585–1590 (1970)
Kreyszig, E.: Advanced Engineering Mathematics. Wiley, Hoboken (2011)
Lutovac, M.D., Tosic, D.V., Evans, B.L.: Filter Design for Signal Processing Using MATLAB and Mathematica. Prentice-Hall, Upper Saddle River (2001)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)
Orchard, H.J., Willson, A.N.: Elliptic functions for filter design. IEEE Trans. Circuits Syst. I 44, 273–287 (1997)
Orfanidis, S.J.: High-order digital parametric equalizer design. J. Audio Eng. Soc. 53, 1026–1046 (2005)
Orfanidis, S.J.: Lecture Notes on Elliptic Filter Design. http://eceweb1.rutgers.edu/~orfanidi/ece521/notes.pdf (2006)
Oppenheim, A.V., Willsky, A.S., Hamid, S.: Signals and Systems. Prentice Hall, Englewood Cliffs (1996)
Pao, Y.C.: Engeneering Analysis. CRC Press LLC, Boca Raton (1999)
Porat, B.: A Course in Digital Signal Processing. Wiley, New York (1996)
Porat, B., Friedlander, B.: The modified Yule-Walker method of ARMA spectral estimation. IEEE Trans. Aerosp. Electron. Syst. AES-20(2), 158–173 (1984)
Saff, E.B., Snider, A.D.: Fundamentals of Complex Analysis with Applications to Engineering and Science. Prentice Hall, Upper Saddle River (2002)
Shenoi, B.A.: Introduction to Digital Signal Processing and Filter Design. Wiley, New York (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Alessio, S.M. (2016). IIR Filter Design. In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-25468-5_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25466-1
Online ISBN: 978-3-319-25468-5
eBook Packages: EngineeringEngineering (R0)