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Mathematical Development of the Elliptic Filter

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Mathematical Development of the Elliptic Filter Mathematical Development of the Elliptic Filter Mark Kleehammer Queen’s University August 26, 2013 The elliptic filter is a very powerful tool for signal processing, however it also requires some sophisticated mathematics to properly describe it. Signal process- ing plays a crucial role in a large part in nearly everything electronic today, from phones, to computers to music. We begin with an abstract introduction to signal processing and prove some basic results. Then we will apply the abstract theory to discuss the Butterworth, Chebyshev and elliptic filters, with a primary focus on the elliptic filter. However, before we can discuss the elliptic filter, we transition into an in depth discussion of elliptic functions. 1 Contents 1 Signal Processing 3 1.1 The Ideal Lowpass Filter . .5 1.2 The Causal Filter . .6 1.3 The Transfer Function . .8 2 The Butterworth Filter 10 2.1 Butterworth Polynomials . 10 2.2 TheButterworthFilter .................................. 11 2.3 Conclusions . 13 3 The Chebyshev Filter 15 3.1 Chebyshev Polynomials . 15 3.2 Transfer Function H(s) for the Chebyshev Filter . 16 3.3 The Chebyshev Filter . 17 3.4 Conclusions . 18 4 Elliptic Functions 20 4.1 Elliptic Integrals . 20 4.2 Jacobi’s Elliptic Functions . 22 4.3 The Addition Theorems for the Jacobi Elliptic Functions . 24 4.4 Transformations of Jacobi Elliptic Functions . 27 4.4.1 The First Degree Transformation . 28 4.4.2 The nth Degree Transformation . 29 4.5 The Jacobi Theta Functions . 31 5 Elliptic Rational Function 33 5.1 Statement of the Problems . 33 5.2 Solution to Problem C . 36 5.3 Elliptic Rational Function . 37 5.3.1 Connections Between Texts . 42 5.4 Zeros and Poles of the Elliptic Rational Function . 43 5.5 The Elliptic Rational function for n 1,2,3 . 45 Æ 6 The Elliptic Filter 47 6.1 The Elliptic Rational Function . 47 6.2 The Transfer Function for the Elliptic Filter . 47 6.3 Transfer Function Poles for Order n .......................... 48 6.4 Transfer Function Poles for Orders n 1,2,3 . 49 Æ 6.5 The Elliptic Filter . 52 6.6 Conclusions . 52 7 Appendix 55 References 71 2 1 Signal Processing A signal is a physical quantity or quality that conveys information [LTE01, p. 1]. For exam- ple a person’s voice is a signal, and it may cause the listener to act in a certain manner. This evaluation of the signal is called signal processing. Signals are found in all kinds of places; they could be the voltage produced in an electric circuit, computer graphics, or music. Many electronic signals travel through space, and while traveling they tend to pick up other unde- sirable signals such as white noise, or static. The kinds of signals we will focus on will be sine waves, and we call these signals sinusoidal. The problem which motivates our discussion is given some signal, we must remove the undesirable frequencies from it. A filter takes a signal as its input, performs some operations on the signal which remove these undesirable frequencies, and outputs a "clean" signal. Al- gebraically we think of a filter as a function that maps signals to signals. Definition 1.1. A filter is a function that is used to modify or reshape the frequency spectrum of a signal according to some prescribed requirements [LTE01, p. 241] An everyday example of this problem is used in cell phones all the time. When you make a call (or text, etc...), the phone sends a signal to a satellite in space, and that satellite then redirects the signal to your friend’s phone. When the signal was traveling through space it picks up static, so when your friend gets the signal, his/her phone filters out that static. How does the phone filter out the static? Many electrical engineers are well versed in this problem, and there is a solid amount of literature on the subject from an engineering perspective [Cau58] [Dan74] [LTE01], however there is little if any from mathematicians. As such many of the results are explained with little detail and background, and it’s difficult to see the bigger picture with these details omitted. My goal is to present this material in a straight forward and simple manner, with justifications given for each step. It is important that one can see why we get these results, with less of a "this is how it is" mentality. Many of the functions we will encounter are functions of the complex frequency s σ i! Æ Å (where i p 1) . The variable s comes from the Laplace transform; given some function v(t) Æ ¡ (for example a voltage dependent on time) its Laplace transform is defined as [Dan74, p. 2] Z 1 st V (s) L [v(t)] : v(t)e¡ dt (1.1) Æ Æ 0 For our purposes we will view the Laplace transformation as taking a function in the time domain, and translating it into the frequency domain. Typically we will investigate the filter problem in the frequency domain (since the filter requirements given to an engineer will typ- ically be expressed in the frequency domain), and then use the inverse Laplace transform to express our results in the time domain. Let’s begin by analyzing the following example. Suppose we have a sinusoidal input signal x (t) sin!t, and the engineer only needs this signal for a specific range of frequencies, in Æ say [!a,!b]. The engineer will design a filter that preserves those desired frequencies, and 3 eliminates everything else. And after the signal has been filtered, the output signal will also be a sine wave, but with the amplitude and phase likely altered by the filter, i.e. x (t) C sin(!t ') out Æ Å for some real constants C,' [Dan74, p. 3]. Definition 1.2. The transfer function T (s) is defined to be the ratio of the Laplace transform of the output signal (denoted xout(t)), to the Laplace transform of the input signal (denoted xin(t)) [Dan74, p. 3]. Explicitly Xout(s) L [xout(t)] T (s) (1.2) Æ Xin(s) Æ L [xin(t)] Often we will consider the function H(s) 1/T (s), which is referred to as the input/output Æ transfer function. Theorem 1.1. Let x (t) A sin!t represent an input signal with amplitude A 0, and in Æ 1 1 È x (t) A sin(!t ') represent the output signal with amplitude A 0. Then out Æ 2 Å 2 È A2 T (i!) (1.3) A1 Æ j j This means that for a sinusoidal signal, the ratio of the output amplitude to the input am- plitude is equal to the magnitude of the transfer function evaluated at s i!. Æ Proof. Compute the Laplace transforms of xin and xout Z 1 st ! Xin(s) L [xin(t)] A1 sin(!t)e¡ dt A1 2 2 Æ Æ 0 Æ s ! Å Z !cos' s sin' 1 st Å Xout(s) L [xout(t)] A2 sin(!t ')e¡ dt A2 2 2 Æ Æ 0 Å Æ s ! Å Therefore ¯ ¯ A2 ¯!cos' i!sin'¯ T (i!) ¯ Å ¯ j j Æ A1 ¯ ! ¯ A2 Æ A1 We should take note that, in general the frequency of the input sine wave !, is not con- stant, and the amplitude A1 depends on the frequency !. While the theorem above seems to implicitly assume that these are constant. So to be more precise we will denote the ampli- tude as amp(!) and reformulate the theorem we just proved (the proof still works the same). Perhaps we should also denote the signals x(t) as a function of two variables x(!,t), however the frequency ! – while non-constant – is not completely independent of time (we cannot have two different frequencies occurring at the same time). So we think of the signal x as a function of time, the frequency of the signal ! is not constant, and we want to filter out some frequencies of x that may occur. 4 Theorem 1.2. Let x (t) amp (!)sin!t represent an input signal with amplitude in Æ in amp (!) 0, and x (t) amp (!)sin(!t ') represent the output of the filter with am- in È out Æ out Å plitude ampout(!) 0. Then È amp (!) out T (i!) (1.4) ampin(!) Æ j j We say that a sinusoidal signal has been attenuated if the output signal has a smaller am- plitude than the input [Dan74, p. 3]. With this in mind our problem is to create a filter that attenuates certain frequencies of the signal, while leaving others invariant. Regions of low attenuation are called passbands, and we say that the filter passes such frequencies. On the other hand, regions of high attenuation are called stopbands, and we say the filter stops these frequencies. A lowpass filter passes frequencies less than some specific frequency, and attenuates those higher, and a highpass filter does the opposite [Dan74, p. 4]. We will focus our discussion on the lowpass filter. 1.1 The Ideal Lowpass Filter For the lowpass filter, we pass the frequencies ! with ! ! , where ! is some fixed fre- j j · b b quency [Dan74, p. 8]. That is in the passband {! : ! ! }, we want the input amplitude to j j · b equal the output amplitude, so their ratio is 1, hence for ! ! we require j j · b T (i!) 1 (1.5) j j Æ In the stopband, we want very high attenuation, so the output amplitude should be zero. Thus we also require for ! ! , j j È b T (i!) 0 (1.6) j j Æ Equivalently, in terms of H(s) 1/T (s) we desire Æ ½ 1 : ! ! H(i!) j j · b (1.7) j j Æ : ! ! 1 j j È b Such a function is impossible to realize (with the discontinuities), and so we will examine different ways of approximating this ideal.
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