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Analog Filter Design in Mathematica
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by
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Brian L. Evans and James H. McClellan
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Digital Signal Processing Laboratory
School of Electrical Engineering
Georgia Institute of Technology
Atlanta, GA  30332-0250

EMAIL:  evans@eedsp.gatech.edu

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WARNING:  Under Mathematica 1.2, this notebook may take up to 5 minutes to load the signal processing packages and to initialize the examples.  If it is taking longer than this, then open the Initialization section and check for error messages.
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WARNING:  Designing elliptic filters depends on the Mathematica elliptic functions EllipticK, JacobiSN, etc.   Versions of Mathematica on some computers (like Mathematica 1.2 on the MacIntosh) may not load these automatically.
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Introduction
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		In general, linear filters are designed to alter the frequency characteristics of input signals.  The four common kinds of alterations are:
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(1)  Lowpass --  pass all frequencies below a threshold;
(2)  Highpass --  pass all frequencies above a threshold;
(3)  Bandpass --  pass all frequencies in a given range; and
(4)  Bandstop -- reject all frequencies in a given range.
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The part of the input signal corresponding to frequencies outside a passband will be attenuated (reduced in magnitude).  In the ideal case,  these filters would outright reject these frequency components.   In practice, filters approximately reject unwanted frequency components and approximately pass desired frequencies; frequencies in a transition region between a stopband and a passband are partially rejected and partially passed.
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Characterizing Filters
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		Filters are characterized by their impulse response, which is the output of the filter when the input is an impulsive function (a Dirac delta in continuous time and a Kronecker delta in discrete time).  If the impulse response is zero beyond a certain point in time, then the filter is called a finite impulse response (FIR) filter.  Otherwise, the filter is an infinite impulse response (IIR) filter.		This impulse response represents a solution to a linear constant coefficient differential equation (whose coefficients correspond to those of the filter).   When the filter is a linear operator, the transfer function of the filter H(s) is a ratio of two rational polynomials.  Its FIR section corresponds to the numerator polynomial, and its IIR section corresponds to the denominator polynomial.
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Filter Design
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		Designing linear filters is carried out in the s-domain (a.k.a. the Laplace transform domain).  The goal is to fit the magnitude of a complex-valued function as constrained by the specifications.  For a lowpass filter that passes all frequencies up to wp and rejects all frequencies above ws, the magnitude response is constrained to the shaded regions:
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The stopband attenuation is d2 (or delta2) and the maximum passband ripple is d1 (or delta1).  The function H(s) has a magnitude frequency response of  | H(j w) |, which is the absolute value of the complex variable s evaluated at s = j w (w is the frequency variable).  For linear filters, H(s) is a ratio of two rational polynomials (a rational polynomial is a function of s consisting of a sum of products of rational numbers and integer powers of s).  The polynomial coefficients are called filter coefficients.  In this notebook, the filters that we design only have positive coefficients.
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About this Notebook
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 		In this notebook,  we design passive analog filters so they can be implemented with resistors, capacitors, and inductors.  The focus of this notebook, however, is how to design the filter coefficients in the analog domain, and not on choosing the actual resistor, capacitor, and inductor values.  Once we generate a filter object, we can then use it in other expressions (e.g., modulation), specify an alternate input function to it (in place of the Dirac delta function), transform it, or plot its frequency response.  We can also quantize the filter coefficients (or its poles and zeroes) to get a better feeling of how the implemented filter might behave (for example, it is difficult to get more than three digits of accuracy in the filter coefficients when using resistors, capacitors, and inductors).
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References
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[Daniels] Richard Daniels.  "Approximation Methods for Electronic Filter Design".  McGraw-Hill Book Company:  New York.  1976.
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[Evans & McClellan] Brian Evans and James McClellan.  "Symbolic Transforms with Applications to Signal Processing."  Mathematica Journal.  vol. 1, issue 2.  Fall, 1990.
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[Ludeman] Lonnie Ludeman.  "Fundamentals of Digital Signal Processing".  Harper & Row: New York.  1986.
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[Parks & Burrus] Thomas Parks and Cidney Burrus.  "Digital Filter Design."  John Wiley & Sons: New York.  1987.
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[Rabiner & Gold]  Lawrence Rabiner and Bernard Gold.
"Theory and Application of Digital Signal Processing".
Prentice Hall, Inc.: Englewood Cliffs, NJ.  1975.
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Initialization
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*)
Needs[ "SignalProcessing`Analog`" ]
(*
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Loading version 2.0 of the signal processing support\
 
  packages ...
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Support module has been loaded.
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The knowledge base of signal processing operators has been\
 
  loaded.
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Supporting routines for filter design are loaded.
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Region of convergence routines for the transform rule\
 
  bases are loaded.
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Supporting routines for transform rule bases are loaded.
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Support::loaded: 
   Supporting routines, objects, and rules for the signal
    processing package have been loaded successfully.
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Support::change: 
   The loading process has changed such that the files
    Tree.m and DataType.m are no longer automatically
    loaded (the signal processing packages no longer rely
    on the definitions in these files).
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Loading version 2.0 of the analog signal processing\
 
  packages ...
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Piecewise convolution rules have been loaded.
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The one-dimensional analog filter design objects are\
 
  loaded.
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Supporting routines and objects for the Laplace transform\
 
  are loaded.
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The forward Laplace transform rule base LaPlace has been\
 
  loaded.
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The inverse Laplace transform rule base InvLaPlace has\
 
  been loaded.
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The supporting functions are loaded for symbolic Fourier\
 
  transforms.
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The continuous Fourier transform (CTFT) functions are\
 
  loaded.
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LSolve, a differential equation solver, is loaded.
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The analog signal analyzer is now loaded.
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Analog::loaded: 
   Routines, objects, and rules for the analog signal
    processing packages have been loaded successfully.
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Analog signal processing extensions have been loaded.
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Good starting points are the objects ASPAnalyze and\
 
  LaPlace.
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Active packages are maintained in the variable\
 
  $ContextPath.
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  New signal primitives are in SPsignals.
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  New system primitives are in SPoperators.
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  New Mathematica functions are in SPfunctions.
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Note that all signal processing expressions will
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be automatically simplified.
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Filter Classes
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		The four most common classes of analog filters are Butterworth, Chebyshev type I, Chebyshev type II, and Elliptic.  Each filter has a characteristic frequency response and each filter is optimal but in a different sense.  When choosing a filter type, then, it is important to know in what sense each is optimal.  Parks and Burrus write:
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	"The classical Butterworth filter is optimal in the sense that it is the best Taylor series approximation to the ideal lowpass filter magnitude response at both w=0 and w=infinity.
	The Chebyshev [type I] filter gives the smallest maximum magnitude error over the entire passband of any filter that is also a Taylor series approximation at w=infinity to the ideal magnitude characteristic.
	The inverse Chebyshev filter [type II]  is a Taylor series approximation to the ideal magnitude response at w=0 and minimizes the maximum error in the approximation to zero over the stopband.  Or we can say it maximizes the minimum rejection of the filter over the stopband.
	The elliptic function filter (Cauer filter) considers the four parameters of the filter:  the passband ripple, the transition bandwidth, the stopband ripple, and the order of the filter.  For given values of any three of the four, the fourth is minimized.
	Remember that all four of these filter designs are magnitude approximations and do not address the phase frequency response or the time-domain characteristics.  For most designs the Butterworth filter has the smoothest phase curve, followed by the inverse Chebyshev filer, Chebyshev filter, and elliptic function filter....
	In some cases, especially where the time-domain characteristics are important, ripples in the frequency response cause irregularities, such as echoes in the time response.  For that reason the Butterworth and Chebyshev II filters are more desirable than their frequency response alone might indicate."
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[Parks & Burrus, 199-201]
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Butterworth Filters
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		The magnitude squared frequency response of a Butterworth filter is of the form
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1 / ( 1 + (w/wc)^(2 n) )
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where w is the frequency variable, wc is the cutoff frequency, and n is the filter order. The magnitude response of a lowpass Butterworth filter is smooth (no ripples) and monotonically decreases with respect to frequency since  it is the best Taylor series approximation to the ideal lowpass filter magnitude response at both w=0 and
 w=infinity.  Below is the magnitude and phase response of a fourth-order Butterworth filter with a cutoff frequency of 22 rad/sec:
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Chebyshev type I Filters
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Introduction
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		The Chebyshev type I filter gives the smallest maximum magnitude error over the entire passband of any filter that is also a Taylor series approximation at w=infinity to the ideal magnitude characteristic [Parks & Burrus].  This means that the magnitude response of the stopband monotonically decreases with respect to frequency and that the passband features ripples.
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Magnitude Squared Response
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		The magnitude squared response of a Chebyshev type I filter is of the form
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1 / ( 1 + e^2 ChebyshevT[n, w/wc]^2 )
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where e controls passband ripple, w is the frequency variable, and wc is the cutoff frequency.  The term ChebyshevT[n, s] is the nth-order Chebyshev polynomial.
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Example Magnitude Squared Response
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		The cImsr function below defines the magnitude squared response of a Chebyshev type I filter using the parameters mentioned above:
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*)
cImsr[wc_, n_, e_, w_] :=
	1 / ( 1 + e^2 ChebyshevT[n, w/wc]^2 )
(*
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Now, we can plot the passband response, which ripples:
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Plot[ cImsr[3, 4, 0.1, w], {w, 0, 2} ]
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;[o]
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We can also plot the overall response:
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Plot[ cImsr[3, 4, 0.1, w], {w, 0, 10} ]
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Chebyshev type II Filters
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Introduction
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		This filter class is a Taylor series approximation to the ideal magnitude response at w=0 and minimizes the maximum error in the approximation to zero over the stopband.  This hints at the shape of its magnitude response, which is characterized by a monotonically decreasing function in the passband region and by a rippling function in the stopband region:
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Magnitude Squared Response
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		The magnitude squared response of a Chebyshev type II filter is of the form 
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1 / ( 1 + e^2 Xn(w/wc)^2 ) 
:[font = text; inactive; ]
where e controls stopband ripple, w is the frequency variable, and wc is the cutoff frequency.  Here, Xn(v) = Tn(wr) / Tn(wr / v), where Tn(v) is the nth-order Chebyshev polynomial and wr is the ratio of wc over the transition ratio.  That is, as a formula, the magnitude squared response is
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1 / (1 + e^2 (ChebyshevT[n, wr] / ChebyshevT[n, wr wc / w])^2)
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Example Magnitude Squared Response
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		Here is the general definition of the magnitude squared response of a Chebyshev type II filter following the previous variable conventions (wc is the cutoff frequency and so forth):
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*)
cIImsr[wc_, wr_, n_, e_, w_] :=
  1 / ( 1 + e^2 (ChebyshevT[n, wr] / ChebyshevT[n, wr wc / w])^2 )
(*
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Our example is a fourth-order filter with a cutoff frequency of 3 rad/sec, a transition ratio of 2/3, and ripple control parameter of 1.  The frequency variable is w.  Now, we can plot the magnitude squared response over the passband region (we start at 0.1 because w = 0 causes a division-by-zero error):
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Plot[ cIImsr[3, 3 / (2/3), 4, 1, w], {w, 0.1, 2} ]
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Note that the magnitude response of the passband region monotonically decreases with respect to frequency.  We can also plot the stopband region:
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Plot[ cIImsr[3, 3 / (2/3), 4, 1, w], {w, 10, 100} ]
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Note that the stopband response riples.  Here is the overall frequency response:
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Plot[ cIImsr[3, 3 / (2/3), 4, 1, w], {w, 0.1, 10} ]
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Elliptic Filters
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Introduction
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		The elliptic or Cauer filter considers the four parameters of the filter:  the passband ripple, the transition bandwidth, the stopband ripple, and the order of the filter.  For given values of any three of the four, the fourth is minimized [Parks & Burrus, 199-201].  Elliptic filters exhibit equiripple in the passbands and stopbands, as seen here for a fourth-order elliptic filter:
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Magnitude Squared Response
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		The magnitude squared response of an elliptic filter is of the form
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1 / (1 + e^2 Rn(w,L)^2 )  
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where e controls ripple, w is the frequency variable, and wc is the cutoff frequency.  Here, Rn(w, L) is called a Chebyshev rational function such that L is a parameter describing ripple properties.  As a formula, the magnitude squared response is
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1 / ( 1 + e^2 RationalChebyshev[n, w, xL, L]^2 )
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where xLand L are free parameters (although xL can be expressed in terms of L).  
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The Rational Chebyshev Function
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		The design of elliptic filters is based on the rational Chebyshev function Rn(x, L).  This function is implemented as RationalChebyshev [Daniels, chapter 5]:
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?RationalChebyshev
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RationalChebyshev[n, x, xL, L] returns a rational
   polynomial which is the nth order rational Chebyshev
   function evaluated at x for parameters xL and L.
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In filter design terms, the rational Chebyshev parameters n, xL, and L are dependent upon the filter parameters:  xL = 1 / (transition-ratio) and L = 1 / nu = Sqrt[A^2 - 1] / epsilon.  The order n is a function of the transition ration and nu.  In the plot below, the fifth order rational Chebyshev function is plotted for xL = 1.3 and L = 207.1:
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Use options in the Actions Settings dialog box to control
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Elliptic Functions in Mathematica
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		In filter design, elliptic functions (like the complete elliptic integral) are defined slightly differently than Mathematica defines it.  The modulus squared in filter design literature is really the same as the modulus in Mathematica:
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		K(k) = EllipticK[k^2]
		sn(u, k) = JacobiSN[u, k^2]
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Please keep this is mind when using elliptic functions.  The routines provided for filter design already take this into account.
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Bessel Filters
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		The Bessel filter is an all-pole filter whose transfer function is of the form d0 / Bn(s) where Bn(s) is the nth order Bessel polynomial.  Bessel lowpass filters are useful for two reasons:  
:[font = text; inactive; ]
	(1) their group delay is maximally flat at the origin of the complex s plane
		  (s is the Laplace transform variable s= j w), and
:[font = text; inactive; ]
	(2) their step response exhibits low overshoot, typically less than 1%.
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(Group delay of a filter is defined as the derivative of the phase with respect to frequency.)  Bessel filter differ from the previous four classes because the cutoff frequency varies with filter order [Rabiner & Gold, 228-230].
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		The magnitude squared response of a Bessel  filter is smooth (no ripples) and monotonically decreases with respect to frequency.   Therefore, the magnitude response also monotonically decreases with respect to frequency:
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Design of Lowpass Analog Filters
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		In this section, we discuss how to use the signal processing packages to design analog Butterworth, Chebyshev, Elliptic, and Bessel filters.  These filters can be designed using the Mathematica object DesignFilter and analyzed by transform objects like LaPlace and CTFTransform.
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General Design Algorithm for Analog Lowpass Filters
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		Here is the general design algorithm for the design of IIR analog lowpass filters:
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1.  Determine filter parameters
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Interpret the problem so that you can determine
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		(a)  the passband error tolerance (delta1) or epsilon, and 
:[font = text; inactive; ]
		(b)  the stopband error tolerance (delta2) or A.
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This information, completely determines the values of the filter parameters delta1, delta2, epsilon, A, Eripple, ATT, and nu.  In Mathematica, this is done by calling FilterParameters.
:[font = input; startGroup; ]
?FilterParameters
:[font = info; inactive; endGroup; endGroup; ]
FilterParameters[delta1, delta2, epsilon, A, Eripple,
   ATT, nu] fills in the values of the missing
   parameters.  It assumes that either the passband
   ripple (delta1) and the stopband ripple (delta2) or
   epsilon (controls ripple) and A (1 / delta2) are
   given. The values of Eripple, ATT, and nu will be
   computed and assigned to the symbols passed into those
   slots.  Note that the passband  takes on values
   between 1 and 1 - delta1 and that the stopband  takes
   on values between 0 and delta2.  The function returns
   a list of all seven parameters.
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2.  Determine filter order
:[font = text; inactive; ]
Next, determine the ratio of transition from the passband to the stopband.  The transition ratio and nu will determine the necessary filter order.  In Mathematica, this is done by calling FilterOrder.
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?FilterOrder
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FilterOrder[filter-type, transition-ratio, nu] returns the filter order
   for the lowpass filter that meets the transition ratio (for digital
   filters, see BilinearTransformation) and the filter parameter nu
   (which is a function of the passband and stopband ripple
   parameters-- c.f. FilterParameters).
:[font = subsubsection; inactive; startGroup; Cclosed; ]
3. Determine prototype filter
:[font = text; inactive; ]
Now that all of  the necessary filter parameters are known, the filter coefficients for a lowpass filter with a cutoff frequency of 1 rad/sec can be computed.  We can then transform this filter to have the proper cutoff frequency.  In Mathematica, this is done by calling AnalogFilter.
:[font = input; startGroup; ]
?AnalogFilter
:[font = info; inactive; endGroup; endGroup; ]
AnalogFilter[class, time-variable, order, epsilon, A,
   transition-ratio, cutoff-frequency] will return an
   object with head CIIR that represents a
   one-dimensional lowpass analog filter. Here, class is
   one of Bessel, Butterworth, ChebyshevI, ChebyshevII,
   Elliptic.  The last four arguments are filter
   parameters. The value of the cutoff-frequency
   defaults to 1. See also FilterParameters and
   FilterOrder.
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This general design algorithm does not apply to Bessel filters because the cutoff frequency varies with the filter order but does apply to the other classes of filters [Rabiner & Gold, 228-230].
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		In Mathematica, this whole process is grouped together in a routine called DesignFilter (which does handle Bessel filters as a special case).
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?DesignFilter
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DesignFilter[domain, class, type, time-variable, deltap,
   deltas, w1, w2, w3, w4] returns a filter expression
   (involving IIR/FIR or CIIR/CFIR filter sections). 
   Domain is either Analog or Digital. Possibilities for
   class include Butterworth, ChebyshevI, ChebyshevII,
   and Elliptic.  Possible filter types are Lowpass,
   Highpass, Bandpass, and Bandstop.  For lowpass
   filters, the magnitude response of the passband
   resides on [1, 1 - deltap] and the magnitude response
   of the stopband occupies [0, deltas]. The last two
   arguments, w3 and w4, are only necessary for bandpass
   and bandstop filters.  Frequencies w1, w2, w3, and w4
   are measured in rad/sec and should be given in
   ascending order.
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The remaining sections give one design example per filter class which requires using either the general design procedure or the high-level functionality of DesignFilter.
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Butterworth Example (Using Object AnalogFilter)
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Problem:  Design a lowpass Butterworth analog filter that has a -2 dB or better cutoff at 20 rad/sec and at least 10 db of attenuation at 30 rad/sec [Ludeman, 131].
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Step 1:  Determine the order of the filter
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First, we need to determine the order of the Butterworth filter necessary.  At the cutoff frequency, we want a magnitude response of -2 dB or better:
:[font = input; initialization; startGroup; ]
*)
bdelta1 = 1 - N[ 10^(-2/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.2056717652757185
;[o]
0.205672
:[font = text; inactive; ]
In the stopband, we want an attenuation of at least 10 dB:
:[font = input; initialization; startGroup; ]
*)
bdelta2 = N[ 10^(-10/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.3162277660168379
;[o]
0.316228
:[font = text; inactive; ]
From the values of delta1 and delta2, we can determine the other filter parameters:
:[font = input; initialization; startGroup; ]
*)
FilterParameters[ bdelta1, bdelta2, bepsilon, bA, Eripple, ATT, nu ]
(*
:[font = output; inactive; output; endGroup; ]
{0.2056717652757185, 0.3162277660168379, 
  0.7647831015792084, 3.162277660168379, 
  2.466913365931468, 10., 0.2549277005264028}
;[o]
{0.205672, 0.316228, 0.764783, 3.16228, 2.46691, 10., 
 
  0.254928}
:[font = text; inactive; ]
The parameters Eripple and ATT are useful when using filter design charts and are calculated here for completeness (note that the attenuation ATT should be 10 dB in this case).  The values of delta1 and delta2 determine the values of the other parameters.  The value of nu and the transition width (2/3 = 20 rad/sec divided by 30 rad/sec) determine the filter order:
:[font = input; startGroup; Cclosed; ]
FilterOrder[ Butterworth, 2/3, nu ]
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4
;[o]
4
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Now, we know that we need a Butterworth filter of order 4.
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Step 2:  Find the coefficients of the filter
:[font = text; inactive; ]
We now know that we need a fourth order Butterworth filter to meet specifications.  So, now we can ask Mathematica to create a fourth-order Butterworth filter with a transition band from 20 rad/sec to 30 rad/sec.  The actual cutoff frequency needs to be between 21.39 rad/sec and 22.79 rad/sec (this comes as a consequence of the monotonically decreasing nature of the magnitude response of Butterworth filters).  We have chosen 22 rad/sec as the cutoff frequency:
:[font = input; initialization; startGroup; ]
*)
bfilter = AnalogFilter[ Butterworth, t, 4, bepsilon, bA, 2/3, 22 ]
(*
:[font = output; inactive; output; endGroup; ]
234256*CIIR[t, {234256., 27824.56490000732, 
    1652.479364188578, 57.48877045456058, 1}, 
   Roots -> 
    {22*E^((5*I)/8*Pi), 22*E^((7*I)/8*Pi), 
     22*E^((9*I)/8*Pi), 22*E^((11*I)/8*Pi)}]
;[o]
234256 CIIR[t, {234256., 27824.6, 1652.48, 57.4888, 1}]
:[font = text; inactive; ]
This expression represents a fourth-order filter (because there are four roots yielding five feedback coefficients).  The gain is 234526  and the feedback coefficients are {234256, 27824.6, 1652.48, 57.48888, 1}.   Whenever a CIIR object has three instead of two data slots, the CIIR object contains a list of the exact pole locations.  Here is the third argument of the CIIR object embedded in the fourth-order Butterworth filter that we just designed:
:[font = input; initialization; startGroup; ]
*)
bfilter[[2]][[3]]
(*
:[font = output; inactive; output; endGroup; ]
Roots -> {22*E^((5*I)/8*Pi), 22*E^((7*I)/8*Pi), 
   22*E^((9*I)/8*Pi), 22*E^((11*I)/8*Pi)}
;[o]
              (5 I)/8 Pi      (7 I)/8 Pi
Roots -> {22 E          , 22 E          , 
 
       (9 I)/8 Pi      (11 I)/8 Pi
   22 E          , 22 E           }
:[font = text; inactive; startGroup; Cclosed; ]
The location of the roots of this all-pole filter characterizes it as Butterworth; that is, the poles are equally spaced along a semicircle located in the left-hand side of the s-plane.
:[font = text; inactive; ]
		There are many ways to extract the poles (roots) from CIIR objects.  In the above example, we took the easy way out.  By viewing the expression contained in bfilter as a tree, the first level is simply a product of 234256 and the filter object.  So, the code
:[font = input; startGroup; ]
bfilter[[2]]
:[font = output; inactive; output; endGroup; ]
CIIR[t, {234256., 27824.56490000732, 1652.479364188578, 
   57.48877045456058, 1}, 
  Roots -> {22*E^((5*I)/8*Pi), 22*E^((7*I)/8*Pi), 
    22*E^((9*I)/8*Pi), 22*E^((11*I)/8*Pi)}]
;[o]
CIIR[t, {234256., 27824.6, 1652.48, 57.4888, 1}]
:[font = text; inactive; ]
extracts the second argument in the product.  Now, if the roots are contained in the CIIR object, then they are contained in the third data slot:
:[font = input; startGroup; ]
bfilter[[2]][[3]]
:[font = output; inactive; output; endGroup; endGroup; ]
Roots -> {22*E^((5*I)/8*Pi), 22*E^((7*I)/8*Pi), 
   22*E^((9*I)/8*Pi), 22*E^((11*I)/8*Pi)}
;[o]
              (5 I)/8 Pi      (7 I)/8 Pi      (9 I)/8 Pi
Roots -> {22 E          , 22 E          , 22 E          , 
 
       (11 I)/8 Pi
   22 E           }
:[font = text; inactive; ]
		Of course, Mathematica can give us help about the object CIIR:
:[font = input; startGroup; ]
?CIIR
:[font = info; inactive; endGroup; endGroup; ]
CIIR[t, {a0, a1, a2, ...}] represents the infinite impulse
   response of an all-pole analog filter with gain 1, where
   a1, a2, ..., are the feedback coefficients and a0 is the
   scaling factor of the output. The Laplace transform of
   such an expression is 1 / ( a0 + a1 s + a2 s^2 + ... ). By
   default, the input to the filter is a Dirac delta
   function. An alternate input h can be specified by the
   code CIIR[t, {a0, a1, a2, ...}][h]. For the generalized
   N-dimensional IIR structure, the argument t becomes a list
   of indices (like {t1, t2} for N=2) and the feedback
   coefficients are specified by an N x N matrix. Supports
   option Roots, which for 1-D filters is a list of poles.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 3:  Verify that the filter meets the specifications
:[font = text; inactive; ]
In order to verify that the filter meets specifications, we need to examine the frequency response of the filter.  This can be done by simply taking the continuous-time Fourier transform of the filter:
:[font = input; initialization; ]
*)
bfiltertransobject = CTFTransform[ bfilter, t, w ];
(*
:[font = input; initialization; startGroup; ]
*)

bfiltertrans = TheFunction[bfiltertransobject]
(*
:[font = output; inactive; output; endGroup; ]
234256/(234256. + 27824.56490000732*I*w - 
    1652.479364188578*w^2 + -57.48877045456058*I*w^3 + 
    w^4)
;[o]
                                           2
234256 / (234256. + 27824.6 I w - 1652.48 w  + 
 
                3    4
    -57.4888 I w  + w )
:[font = text; inactive; ]
Now, we can plot the magnitude and phase response:
:[font = input; startGroup; ]

MagPhasePlot[ bfiltertrans, {w, 0, 100}, MagRangeScale -> Log ]
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:[font = text; inactive; ]
Graphically, the magnitude response seems to meet the passband and stopband specifications, and the phase response clearly shows that the cutoff frequency is about 22 rad/sec. We will now check the magnitude response at the end of the passband and the beginning of the stopband just to make sure:
:[font = input; startGroup; ]
20 Log[10, Abs[bfiltertrans /. w -> 20]]
:[font = output; inactive; output; endGroup; ]
-1.662842529124097
;[o]
-1.66284
:[font = input; startGroup; ]
20 Log[10, Abs[bfiltertrans /. w -> 30]]
:[font = output; inactive; output; endGroup; ]
-11.12473416925727
;[o]
-11.1247
:[font = text; inactive; ]
Here, the error in the passband is no greater than 1.67 dB (which is less than the specification of 2 dB) and the attenuation of the stopband is greater than 11 dB (which is better than the specification of 10 dB).  Therefore, the following filter meets the specifications:
:[font = text; inactive; endGroup; endGroup; ]
234256 CIIR[t, {234526, 27824.6, 1652.48, 57.4888, 1}]
:[font = subsection; inactive; startGroup; Cclosed; ]
Chebyshev Type I Example (Using Object AnalogFilter)
:[font = text; inactive; ]
Problem:  Design a lowpass Chebyshev type I analog filter that has a -2 dB or better cutoff at 20 rad/sec and at least 40 db of attenuation at 30 rad/sec.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 1:  Determine the order of the filter
:[font = text; inactive; ]
First, we need to determine the order of the Chebyshev filter necessary.  At the cutoff frequency, we want a magnitude response of -2 dB or better:
:[font = input; initialization; startGroup; ]
*)
c1delta1 = 1 - N[ 10^(-2/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.2056717652757185
;[o]
0.205672
:[font = text; inactive; ]
In the stopband, we want an attenuation of at least 40 db:
:[font = input; initialization; startGroup; ]
*)
c1delta2 = N[ 10^(-40/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.01
;[o]
0.01
:[font = text; inactive; ]
From the values of delta1 and delta2, we can determine the other filter parameters:
:[font = input; initialization; startGroup; ]
*)
FilterParameters[ c1delta1, c1delta2, c1epsilon, c1A, Eripple, ATT, nu ]
(*
:[font = output; inactive; output; endGroup; ]
{0.2056717652757185, 0.01, 0.7647831015792084, 100., 
  2.466913365931468, 4.99999999999999, 
  0.00764821343602463}
;[o]
{0.205672, 0.01, 0.764783, 100., 2.46691, 40., 
 
  0.00764821}
:[font = text; inactive; ]
The parameters Eripple and ATT are useful when using filter design charts and are calculated here for completeness (note that the attenuation ATT should be 40 dB in this case).  The values of delta1 and delta2 determine the values of the other parameters.  The value of nu and the transition width (2/3 = 20 rad/sec divided by 30 rad/sec) determine the filter order:
:[font = input; startGroup; ]
FilterOrder[ ChebyshevI, 2/3, nu ]
:[font = output; inactive; output; endGroup; ]
6
;[o]
6
:[font = text; inactive; endGroup; ]
Now, we know that we need a Chebyshev type I filter of order 6.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 2:  Find the coefficients of the filter
:[font = text; inactive; ]
We now know that we need a fourth order Butterworth filter to meet specifications.  So, now we can ask Mathematica to create a fourth-order Butterworth filter with a cutoff frequency between 20 rad/sec and 30 rad/sec (we have chosen a cutoff of 22 rad/sec):
:[font = input; initialization; startGroup; ]
*)
c1filter = AnalogFilter[ ChebyshevI, t, 6, c1epsilon,
						 c1A, 2/3, 20 ]
(*
:[font = output; inactive; output; endGroup; ]
2615120.54315816*CIIR[t, 
   {3292241.706686772, 672865.7798417783, 
    123433.8833675754, 6936.119375271063, 
    698.3434983669848, 14.02451413539769, 1}, 
   Roots -> 
    {20*(1.016335980968083*I*Cos[Pi/12] - 
        0.1814905678275217*Sin[Pi/12]), 
     (-1.814905678275217 + 10.16335980968083*I)*
      2^(1/2), 20*(1.016335980968083*I*Cos[(5*Pi)/12] - 
        0.1814905678275217*Sin[(5*Pi)/12]), 
     20*(1.016335980968083*I*Cos[(7*Pi)/12] - 
        0.1814905678275217*Sin[(7*Pi)/12]), 
     (-1.814905678275217 - 10.16335980968083*I)*
      2^(1/2), 20*(1.016335980968083*I*
         Cos[(11*Pi)/12] - 
        0.1814905678275217*Sin[(11*Pi)/12])}]
;[o]
          6                    6
2.61512 10  CIIR[t, {3.29224 10 , 672866., 123434., 
 
    6936.12, 698.343, 14.0245, 1}]
:[font = text; inactive; ]
Like Butterworth filter, Chebyshev type I filters contain only poles.  Here, the DC gain is about 4/5, which is a less than one.  This is true for Chebyshev type I filters with even order.  The DC gain is scaled to compensate for the location of the ripples in the passband (the maximum value in the passband is still one, but it occurs at locations other than DC).  Note that odd-ordered Chebyshev type I and Chebyshev type II filters have a DC gain of unity.  The ripple characteristics in the passband of this Chebyshev type I filter is shown in the magnitude frequency response plot in Step 3 (the next sub-section).
:[font = text; inactive; startGroup; Cclosed; ]
		In order to learn more about the CIIR object, see Step 2 of the Butterworth Example. Like the Butterworth example, the poles of this filter are also embedded in the CIIR object:
:[font = input; startGroup; ]
c1filter[[2]][[3]]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
Roots -> {20*(1.016335980968083*I*Cos[Pi/12] - 
      0.1814905678275217*Sin[Pi/12]), 
   (-1.814905678275217 + 10.16335980968083*I)*2^(1/2), 
   20*(1.016335980968083*I*Cos[(5*Pi)/12] - 
      0.1814905678275217*Sin[(5*Pi)/12]), 
   20*(1.016335980968083*I*Cos[(7*Pi)/12] - 
      0.1814905678275217*Sin[(7*Pi)/12]), 
   (-1.814905678275217 - 10.16335980968083*I)*2^(1/2), 
   20*(1.016335980968083*I*Cos[(11*Pi)/12] - 
      0.1814905678275217*Sin[(11*Pi)/12])}
;[o]
                            Pi                 Pi
Roots -> {20 (1.01634 I Cos[--] - 0.181491 Sin[--]), 
                            12                 12
 
   (-1.81491 + 10.1634 I) Sqrt[2], 
 
                     5 Pi                 5 Pi
   20 (1.01634 I Cos[----] - 0.181491 Sin[----]), 
                      12                   12
 
                     7 Pi                 7 Pi
   20 (1.01634 I Cos[----] - 0.181491 Sin[----]), 
                      12                   12
 
   (-1.81491 - 10.1634 I) Sqrt[2], 
 
                     11 Pi                 11 Pi
   20 (1.01634 I Cos[-----] - 0.181491 Sin[-----])}
                      12                    12
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 3:  Verify that the filter meets the specifications
:[font = text; inactive; ]
In order to verify that the filter meets specifications, we need to examine the frequency response of the filter.  This can be done by simply taking the continuous-time Fourier transform of the filter:
:[font = input; initialization; ]
*)
c1filtertransobject = CTFTransform[ c1filter, t, w ];
(*
:[font = input; initialization; startGroup; ]
*)

c1filtertrans = TheFunction[c1filtertransobject]
(*
:[font = output; inactive; output; endGroup; ]
2615120.54315816/
  (3292241.706686772 + 672865.7798417783*I*w - 
    123433.8833675754*w^2 + -6936.119375271063*I*w^3 + 
    698.3434983669848*w^4 + 14.02451413539769*I*w^5 - 
    w^6)
;[o]
           6
(2.61512 10 ) / 
 
             6                          2
  (3.29224 10  + 672866. I w - 123434. w  + 
 
                3            4              5    6
    -6936.12 I w  + 698.343 w  + 14.0245 I w  - w )
:[font = input; startGroup; ]

MagPhasePlot[ c1filtertrans, {w, 0, 40}, MagRangeScale -> Log ]
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;[o]
{-Graphics-, -Graphics-}
:[font = text; inactive; ]
Graphically, the magnitude response seems to meet the passband and stopband specifications, and the phase response shows that the cutoff frequency is about 20 rad/sec. We will now check the magnitude response at the end of the passband and the beginning of the stopband just to make sure:
:[font = input; startGroup; ]
20 Log[10, Abs[c1filtertrans /. w -> 20]]
:[font = output; inactive; output; endGroup; ]
-1.999999999999999
;[o]
-2.
:[font = input; startGroup; ]
20 Log[10, Abs[c1filtertrans /. w -> 30]]
:[font = output; inactive; output; endGroup; ]
-41.80756963255439
;[o]
-41.8076
:[font = text; inactive; ]
Here, the error in the passband is no greater than 2 dB (which is the specification) and the attenuation of the stopband is greater than 41 dB (which is better than the specification of 40 dB).  Therefore, the following filter meets the specifications:
:[font = text; inactive; endGroup; endGroup; ]
2615120.5  CIIR[t, {3292241.7,  672865.78,  123433.88,  6936.1194,  698.3535,  14.024514,  1}]

:[font = subsection; inactive; startGroup; Cclosed; ]
Chebyshev Type II Example (Using Object DesignFilter)
:[font = text; inactive; ]
Problem:  Design a lowpass Chebyshev type II analog filter that has a -2 dB or better cutoff at 20 rad/sec and at least 40 db of attenuation at 30 rad/sec.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 1:  Determine the filter coefficients
:[font = text; inactive; ]
First, we need to determine the order of the Chebyshev filter necessary.  At the cutoff frequency, we want a magnitude response of -2 dB or better:
:[font = input; initialization; startGroup; ]
*)
c2delta1 = 1 - N[ 10^(-2/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.2056717652757185
;[o]
0.205672
:[font = text; inactive; ]
In the stopband, we want an attenuation of at least 40 db:
:[font = input; initialization; startGroup; ]
*)
c2delta2 = N[ 10^(-40/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.01
;[o]
0.01
:[font = text; inactive; ]
From the values of delta1 and delta2, we can determine the entire filter using the DesignFilter command (this abstracts the general design algorithm discussed in the Butterworth and Chebyshev type I examples):
:[font = input; initialization; startGroup; ]
*)
c2filter = DesignFilter[Analog, ChebyshevII, Lowpass,
						t, c2delta1, c2delta2, 20, 30 ]
(*
:[font = output; inactive; output; endGroup; ]
0.01*CIIR[t, {233280000., 30115373.8914797, 
     2332678.053462828, 118311.5674218439, 
     4096.698036875321, 90.4994810689577, 1}, 
    Roots -> 
     {(42.47672063846061*I*Cos[Pi/12])/
        (2.004746440219806*Cos[Pi/12]^2 + 
          1.004746440219806*Sin[Pi/12]^2) - 
       (30.07111232059474*Sin[Pi/12])/
        (2.004746440219806*Cos[Pi/12]^2 + 
          1.004746440219806*Sin[Pi/12]^2), 
      (-9.99208621360889 + 14.11424526522084*I)*
       2^(1/2), (42.47672063846061*I*Cos[(5*Pi)/12])/
        (2.004746440219806*Cos[(5*Pi)/12]^2 + 
          1.004746440219806*Sin[(5*Pi)/12]^2) - 
       (30.07111232059474*Sin[(5*Pi)/12])/
        (2.004746440219806*Cos[(5*Pi)/12]^2 + 
          1.004746440219806*Sin[(5*Pi)/12]^2), 
      (42.47672063846061*I*Cos[(7*Pi)/12])/
        (2.004746440219806*Cos[(7*Pi)/12]^2 + 
          1.004746440219806*Sin[(7*Pi)/12]^2) - 
       (30.07111232059474*Sin[(7*Pi)/12])/
        (2.004746440219806*Cos[(7*Pi)/12]^2 + 
          1.004746440219806*Sin[(7*Pi)/12]^2), 
      (-9.99208621360889 - 14.11424526522084*I)*
       2^(1/2), (42.47672063846061*I*Cos[(11*Pi)/12])/
        (2.004746440219806*Cos[(11*Pi)/12]^2 + 
          1.004746440219806*Sin[(11*Pi)/12]^2) - 
       (30.07111232059474*Sin[(11*Pi)/12])/
        (2.004746440219806*Cos[(11*Pi)/12]^2 + 
          1.004746440219806*Sin[(11*Pi)/12]^2)}][CFIR[
     t, {23328000000., 0, 38879999.99999999, 0., 
     16200., 0., 1}, Roots -> 
     {(30*I)/Cos[Pi/12], (60*I)/2^(1/2), 
      (30*I)/Cos[(5*Pi)/12], (30*I)/Cos[(7*Pi)/12], 
      (-60*I)/2^(1/2), (30*I)/Cos[(11*Pi)/12]}]]
;[o]
                       8            7            6
0.01 CIIR[t, {2.3328 10 , 3.01154 10 , 2.33268 10 , 
 
     118312., 4096.7, 90.4995, 1}][CFIR[t, 
 
              10             7
    {2.3328 10  , 0, 3.888 10 , 0., 16200., 0., 1}]]
:[font = text; inactive; endGroup; ]
We have a Chebyshev type II filter of order 6.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 2:  Verify that the filter meets the specifications
:[font = text; inactive; ]
In order to verify that the filter meets specifications, we need to examine the frequency response of the filter.  This can be done by simply taking the continuous-time Fourier transform of the filter object:
:[font = input; initialization; ]
*)
c2filtertransobject = CTFTransform[ c2filter, t, w ];
(*
:[font = input; initialization; startGroup; ]
*)
c2filtertrans = TheFunction[c2filtertransobject]
(*
:[font = output; inactive; output; endGroup; ]
(0.01*(23328000000. - 38879999.99999999*w^2 + 
      16200.*w^4 - w^6))/
  (233280000. + 30115373.8914797*I*w - 
    2332678.053462828*w^2 + -118311.5674218439*I*w^3 + 
    4096.698036875321*w^4 + 90.4994810689577*I*w^5 - w^6
    )
;[o]
                10           7  2           4    6
(0.01 (2.3328 10   - 3.888 10  w  + 16200. w  - w )) / 
 
            8             7                 6  2
  (2.3328 10  + 3.01154 10  I w - 2.33268 10  w  + 
 
                3           4              5    6
    -118312. I w  + 4096.7 w  + 90.4995 I w  - w )
:[font = text; inactive; ]
Now, we can plot the magnitude and phase response:
:[font = input; startGroup; ]
MagPhasePlot[ c2filtertrans, {w, 0, 100}, MagRangeScale -> Log ]
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;[o]
{-Graphics-, -Graphics-}
:[font = text; inactive; ]
Graphically, the magnitude response seems to meet the passband and stopband specifications, and the phase response shows that the cutoff frequency is about 20 rad/sec. We will now check the magnitude response at the end of the passband and the beginning of the stopband just to make sure:
:[font = input; startGroup; ]
20 Log[10, Abs[c2filtertrans /. w -> 20]]
:[font = output; inactive; output; endGroup; ]
-1.416845759315674
;[o]
-1.41685
:[font = input; startGroup; ]
20 Log[10, Abs[c2filtertrans /. w -> 30]]
:[font = output; inactive; output; endGroup; ]
-39.99999999999995
;[o]
-40.
:[font = text; inactive; ]
Here, the error in the passband is no greater than 1.42 dB (which is the specification) and the attenuation of the stopband is greater than 40 dB (which is the specification).  Therefore, the following filter meets the specifications:
:[font = text; inactive; endGroup; endGroup; ]
0.01 
	CIIR[t, {233280000,  30115374,  2332678.1,  118311.57,  4096.698,  90.499481,  1}] [
 		CFIR[ t, {23328000000, 0, 38880000, 0, 16200, 0, 1} ] ]

:[font = subsection; inactive; startGroup; Cclosed; ]
Elliptic Example (Using Object AnalogFilter)
:[font = text; inactive; ]
Problem:  Design a lowpass elliptic analog filter that has a -2 dB or better cutoff at 20 rad/sec and at least 40 db of attenuation at 30 rad/sec.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 1:  Determine the order of the filter
:[font = text; inactive; ]
First, we need to determine the order of the elliptic filter necessary.  At the cutoff frequency, we want a magnitude response of -2 dB or better:
:[font = input; initialization; startGroup; ]
*)
edelta1 = 1 - N[ 10^(-2/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.2056717652757185
;[o]
0.205672
:[font = text; inactive; ]
In the stopband, we want an attenuation of at least 40 db:
:[font = input; initialization; startGroup; ]
*)
edelta2 = N[ 10^(-40/20) ]
(*
:[font = output; inactive; output; endGroup; ]
0.01
;[o]
0.01
:[font = text; inactive; ]
From the values of delta1 and delta2, we can determine the other filter parameters:
:[font = input; initialization; startGroup; ]
*)
FilterParameters[ edelta1, edelta2, eepsilon, eA, Eripple, ATT, nu ]
(*
:[font = output; inactive; output; endGroup; ]
{0.2056717652757185, 0.01, 0.7647831015792084, 100., 
  2.466913365931468, 4.99999999999999, 
  0.00764821343602463}
;[o]
{0.205672, 0.01, 0.764783, 100., 2.46691, 40., 
 
  0.00764821}
:[font = text; inactive; ]
The parameters Eripple and ATT are useful when using filter design charts and are calculated here for completeness (note that the attenuation ATT should be 40 dB in this case).  The values of delta1 and delta2 determine the values of the other parameters.  The value of nu and the transition width (2/3 = 20 rad/sec divided by 30 rad/sec) determine the filter order:
:[font = input; startGroup; ]
FilterOrder[ Elliptic, 2/3, nu ]
:[font = output; inactive; output; endGroup; ]
4
;[o]
4
:[font = text; inactive; endGroup; ]
Now, we know that we need an elliptic filter of order 4 (for the same specifications, we needed a Chebyshev filter of order of 6).
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 2:  Find the coefficients of the filter
:[font = text; inactive; ]
We now know that we need a fourth order elliptic filter to meet specifications.  So, now we can ask Mathematica to create a fourth-order elliptic filter with a transition region of 20 rad/sec to 30 rad/sec (a cutoff frequency of 20 rad/sec and a transition ration of 2/3):
:[font = input; initialization; startGroup; ]
*)
efilter = AnalogFilter[ Elliptic, t, 4, eepsilon,
						eA, 2/3, 20 ]
(*
:[font = output; inactive; output; endGroup; endGroup; ]
0.007033143007862013*CIIR[t, 
    {43461.29998099303, 4532.068401061019, 
     529.812583270024, 14.1568140287867, 1}, 
    Roots -> 
     {-5.474525743379237 + 9.1795634894521*I, 
      -5.474525743379237 - 9.1795634894521*I, 
      -1.603881271014111 + 19.43922253941108*I, 
      -1.603881271014111 - 19.43922253941108*I}][CFIR[
     t, {4908550.509229452, 0., 5853.94501025495, 0, 
     1}, Roots -> 
     {-69.56812348597884*I, 69.56812348597884*I, 
      -31.84683979446859*I, 31.84683979446859*I}]]
;[o]
0.00703314 CIIR[t, {43461.3, 4532.07, 529.813, 14.1568, 
 
                            6
     1}][CFIR[t, {4.90855 10 , 0., 5853.95, 0, 1}]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Step 3:  Verify that the filter meets the specifications
:[font = text; inactive; ]
In order to verify that the filter meets specifications, we need to examine the frequency response of the filter.  This can be done by simply taking the continuous-time Fourier transform of the filter object:
:[font = input; initialization; ]
*)
efiltertransobject = CTFTransform[ efilter, t, w ];
(*
:[font = input; initialization; startGroup; ]
*)

efiltertrans = TheFunction[efiltertransobject]
(*
:[font = output; inactive; output; endGroup; ]
(0.007033143007862013*(4908550.509229452 - 
      5853.94501025495*w^2 + w^4))/
  (43461.29998099303 + 4532.068401061019*I*w - 
    529.812583270024*w^2 + -14.1568140287867*I*w^3 + w^4
    )
;[o]
                       6            2    4
(0.00703314 (4.90855 10  - 5853.95 w  + w )) / 
 
                                    2               3
  (43461.3 + 4532.07 I w - 529.813 w  + -14.1568 I w  + 
 
     4
    w )
:[font = input; initialization; startGroup; ]
*)

MagPhasePlot[ efiltertrans, {w, 0, 100}, MagRangeScale -> Log ]
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;[o]
{-Graphics-, -Graphics-}
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Graphically, the magnitude response seems to meet the passband and stopband specifications, and the phase response shows that the cutoff frequency is about 20 rad/sec. We will now check the magnitude response at the end of the passband and the beginning of the stopband just to make sure:
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20 Log[10, Abs[efiltertrans /. w -> 20]]
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-1.999999999999988
;[o]
-2.
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20 Log[10, Abs[efiltertrans /. w -> 30]]
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-43.05701103608025
;[o]
-43.057
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Here, the error in the passband is no greater than 2 dB (which is the specification) and the attenuation of the stopband is greater than 43 dB (which is better than the specification of 40 dB).  Therefore, the following filter meets the specifications:
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0.00703314 
	CIIR[t, {43461.3, 4532.0684, 529.81258, 14.156814, 1}] [
		CFIR[t, {4908550.5, 0, 5853.945,  0, 1}] ]
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Bessel Example (Using Object AnalogFilter)
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		Design a 10th order Bessel filter with a cutoff of 1 rad/sec. [Rabiner & Gold, 230]
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Step 1:  Design the filter
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*)
besfilter = AnalogFilter[Bessel, t, 10, ignored, ignored, ignored, 1]
(*
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CIIR[t, {1, 7.613882366493523, 27.46004432722153, 
   61.9489028996037, 97.1088710551878, 
   110.9063281435781, 93.8253040205482, 
   58.3163124745262, 25.61616595550315, 
   7.223647195028783, 1.000000000000001}, 
  Roots -> {-0.909134732089576 - 0.1139583137364039*I, 
    -0.909134732089576 + 0.1139583137364039*I, 
    -0.868845964129487 - 0.3430008233748756*I, 
    -0.868845964129487 + 0.3430008233748756*I, 
    -0.783769441309402 - 0.5759147538505069*I, 
    -0.783769441309402 + 0.5759147538505069*I, 
    -0.6417513866990496 - 0.817583616719106*I, 
    -0.6417513866990496 + 0.817583616719106*I, 
    -0.408322073286874 - 1.081274842819102*I, 
    -0.408322073286874 + 1.081274842819102*I}]
;[o]
CIIR[t, {1, 7.61388, 27.46, 61.9489, 97.1089, 110.906, 
 
   93.8253, 58.3163, 25.6162, 7.22365, 1.}]
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		Like the other lowpass prototypes, the filter poles are embedded in the CIIR object but not displayed when printed:
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besfilter[[3]]
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Roots -> {-0.909134732089576 - 0.1139583137364039*I, 
   -0.909134732089576 + 0.1139583137364039*I, 
   -0.868845964129487 - 0.3430008233748756*I, 
   -0.868845964129487 + 0.3430008233748756*I, 
   -0.783769441309402 - 0.5759147538505069*I, 
   -0.783769441309402 + 0.5759147538505069*I, 
   -0.6417513866990496 - 0.817583616719106*I, 
   -0.6417513866990496 + 0.817583616719106*I, 
   -0.408322073286874 - 1.081274842819102*I, 
   -0.408322073286874 + 1.081274842819102*I}
;[o]
Roots -> {-0.909135 - 0.113958 I, -0.909135 + 0.113958 I, 
 
   -0.868846 - 0.343001 I, -0.868846 + 0.343001 I, 
 
   -0.783769 - 0.575915 I, -0.783769 + 0.575915 I, 
 
   -0.641751 - 0.817584 I, -0.641751 + 0.817584 I, 
 
   -0.408322 - 1.08127 I, -0.408322 + 1.08127 I}
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Unfortunately, Bessel polynomials are generated iteratively by the Mathematica object BesselPolynomial and the closed form of the roots are not available.
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Step 2:  Verify that the filter meets the specifications
:[font = text; inactive; ]
In order to verify that the filter meets specifications, we need to examine the frequency response of the filter.  This can be done by simply taking the continuous-time Fourier transform of the filter object:
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*)
besfiltertransobject = CTFTransform[ besfilter, t, w ];
(*
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*)

besfiltertrans = TheFunction[besfiltertransobject]
(*
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(1 + 7.613882366493523*I*w - 27.46004432722153*w^2 + 
    -61.9489028996037*I*w^3 + 97.1088710551878*w^4 + 
    110.9063281435781*I*w^5 - 93.8253040205482*w^6 + 
    -58.3163124745262*I*w^7 + 25.61616595550315*w^8 + 
    7.223647195028783*I*w^9 - 1.000000000000001*w^10)^
  (-1)
;[o]
                              2               3
1 / (1 + 7.61388 I w - 27.46 w  + -61.9489 I w  + 
 
             4              5            6
    97.1089 w  + 110.906 I w  - 93.8253 w  + 
 
                7            8              9       10
    -58.3163 I w  + 25.6162 w  + 7.22365 I w  - 1. w  )
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MagPhasePlot[ besfiltertrans,
			  {w, 0, 4},
			  MagRangeScale -> Log,
			  PlotRange -> { -120, 20 } ]
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Now, we did not specify any stopband or passband error, but let us see what it is anyway:
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20 Log[10, Abs[besfiltertrans /. w -> 1]]
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-15.06345697427096
;[o]
-15.0635
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20 Log[10, Abs[besfiltertrans /. w -> 2]]
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;[o]
-61.5673
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Anyway, the following filter is a 10th-order Bessel filter with a cutoff frequency of 1 rad/sec:
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CIIR[t, {1, 7.6138824, 27.460044, 61.948903, 97.108871, 110.90633, 93.825304, 58.316312, 25.16166, 7.2236472, 1} ]
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Design of Other Analog Filters
:[font = text; inactive; ]
		Highpass, bandpass, and bandstop analog filters can be designed from lowpass filter prototypes.  Such a transformation reduces to a algebraic change of variables in the transfer function of the lowpass prototype.  These analog-to-analog transformations are well understood [Parks and Burrus, 201-206].
:[font = text; inactive; ]
		Design specifications, however, are generally given for the original filter, not the lowpass filter prototype.  Therefore, the design specification must be reinterpreted in terms of the lowpass filter prototype.
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Highpass Filters
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		In this case, we want to design a highpass filter that rejects frequencies below w1 and pass frequencies above w2..  This is the same as lowpass filter prototype with wp = 1 / w2 and ws = 1 / w1 whose transfer function is adjusted by the substitution s -> 1 / s.  This algebraic substitution turns the complex plane inside out:  zero frequency maps to infinity, infinite frequency maps to zero, and unity maps to unity.  The highpass filter will have the same order as the lowpass prototype.
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Implementation in Mathematica
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		A highpass filter can be obtained from a lowpass prototype by inverting the locations of the poles and zeroes [Parks and Burrus, 201-202].  The specifications for a highpass filter include the passband and stopband ripples (dp and ds) as well as the passband and stopband  frequencies (w1 and w2).  In order to convert these specifications into their equivalents for the lowpass filter prototype, wp = 1 / w2 and ws = 1 / w1; that is, the values of the passband and stopband frequencies are swapped then inverted.  This way, the highpass filter obtained by inverting the poles and zeroes will respond properly.  In Mathematica, the code for designhp does this:
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filtertransform::=			(* forward Laplace transform  *)
SignalProcessing`Analog`FilterDesign`Private`FilterTransferFunction;

tofilter::=
SignalProcessing`Analog`FilterDesign`Private`ToFilter;

designhp[ name_, t_, dp_, ds_, w1_, w2_, rest___ ] :=
Block [	{expdenom, expnumer, fir, hptrans, lowpass, lptrans, n, s},
		lowpass = DesignLPFilter[name, t, dp, ds, 1/w2, 1/w1];
		lptrans = filtertransform[lowpass, t, s];
		n = Exponent[ Denominator[lptrans], s ];
		hptrans = lptrans /. s -> 1/s;
		expnumer = Expand[s^n Numerator[hptrans]];
		expdenom = Expand[s^n Denominator[hptrans]];
		tofilter[ expnumer, expdenom, s, t ] ]
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This is the same code used by "FilterDesign.m".  Note that filtertransform has been aliased to an object that takes a Laplace transform of filter structures.  The more general Laplace transform object LaPlace could have been used instead.
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Design a highpass analog Butterworth filter
:[font = text; inactive; ]
		Design a highpass analog IIR Butterworth filter which rejects all frequencies up to 10 rad/sec and passes all frequencies above 20 rad/sec.  The maximum error in either the passband or the stopband is 0.01.
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*)
hpfilter = DesignFilter[Analog, Butterworth,
						Highpass, t, 0.01, 0.01, 10, 20]
(*
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CFIR[t, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
    8.2935479148246*10^-13}][CIIR[t, 
   {1, 0.3958599764153782, 0.07835256046379188, 
    0.0101645180107133, 0.000954163994378573, 
    0.00006760358118199662, 0.000003659073185437494, 
    1.494799462343597*10^-7, 4.41872941636152*10^-9, 
    8.56118498074177*10^-11, 8.2935479148246*10^-13}]]
;[o]
                                                 -13
CFIR[t, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8.29355 10   }]
 
   [CIIR[t, {1, 0.39586, 0.0783526, 0.0101645, 
 
    0.000954164, 0.0000676036, 0.00000365907, 
 
             -7            -9            -11
    1.4948 10  , 4.41873 10  , 8.56118 10   , 
 
              -13
    8.29355 10   }]]
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In this case, we do want to keep all of the coefficients even though numerically, they are very small.  The Mathematica object Chop is useful for zeroing coefficients smaller than 1.0E-10 (10^-10), but applying Chop to hpfilter would alter its highpass characteristics.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Verify that the filter meets specifications.
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*)
hpfiltertrans = TheFunction[ CTFTransform[ hpfilter, t, w ] ]
(*
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(-(8.2935479148246*10^-13)*w^10)/
  (1 + 0.3958599764153782*I*w - 
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    0.000954163994378573*w^4 + 
    0.00006760358118199662*I*w^5 - 
    0.000003659073185437494*w^6 + 
    -(1.494799462343597*10^-7)*I*w^7 + 
    4.41872941636152*10^-9*w^8 + 
    8.56118498074177*10^-11*I*w^9 - 
    8.2935479148246*10^-13*w^10)
;[o]
            -13  10
(-8.29355 10    w  ) / 
 
                                2                 3
  (1 + 0.39586 I w - 0.0783526 w  + -0.0101645 I w  + 
 
                 4                   5
    0.000954164 w  + 0.0000676036 I w  - 
 
                   6             -7    7
    0.00000365907 w  + -1.4948 10   I w  + 
 
              -9  8             -11    9
    4.41873 10   w  + 8.56118 10    I w  - 
 
              -13  10
    8.29355 10    w  )
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MagPhasePlot[ hpfiltertrans, {w, 0.1, 40}, MagRangeScale -> Log ]
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If one began the graph at w = 0, Mathematica generates a ParametricPlot::notpoint warning because the magnitude response of this highpass filter at DC (w = 0) is negative infinity.
:[font = subsection; inactive; startGroup; Cclosed; ]
Bandpass Filters
:[font = text; inactive; ]
		Four frequency parameters (w1, w2, w3, and w4), a passband error tolerance (deltap), and one stopband error tolerance (deltas) fully specify a bandpass filter.  In this case, the first stopband is (0, w1), the passband is (w2, w3), and the second stopband is (w4, infinity).  The bandpass filter will always have twice the order of the lowpass prototype.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Implementation in Mathematica
:[font = input; ]
filtertransform::=			(* forward Laplace transform  *)
SignalProcessing`Analog`FilterDesign`Private`FilterTransferFunction;

tofilter::=
SignalProcessing`Analog`FilterDesign`Private`ToFilter;

designbp[ name_, t_, dp_, ds_, w1_, w2_, w3_, w4_ ] :=
Block [	{bptrans, expdenom, expnumer, lowpass,
		 lptrans, n, s, w0, wp, ws},
		w0 = Sqrt[ w2 w3 ];
		wp = N[ ( w3^2 - w0^2 ) / w3 ];
		ws = N[ Min[ (w4^2 - w0^2 ) / w4, (w0^2 - w1^2 ) / w1 ] ];
		If [ SameQ[name, Elliptic] && ! SameQ[w1 w4, w2 w3],
		     Message[ DesignBPFilter::narrower ] ];
		lowpass = DesignLPFilter[name, t, dp, ds, wp, ws];
		lptrans = filtertransform[lowpass, t, s];
		n = Exponent[ Denominator[lptrans], s ];
		bptrans = lptrans /. (s -> (s^2 + w0^2) / s); (* 2x order *)
		expnumer = Expand[s^n Numerator[bptrans]];
		expdenom = Expand[s^n Denominator[bptrans]];
		tofilter[ expnumer, expdenom, s, t ] ]
:[font = text; inactive; endGroup; ]
This is the same code used by "FilterDesign.m".  Note that filtertransform has been aliased to an object that takes a Laplace transform of filter structures.  The more general Laplace transform object LaPlace could have been used instead.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Design a bandpass analog Chebyshev type I  filter
:[font = text; inactive; ]
		Design a bandpass analog IIR Chebyshev type I filter which rejects all frequencies up to 5 rad/sec, passes frequencies between 10 and 30 rad/sec, and rejects all frequencies above 35 rad/sec.  The maximum error in either the passband or the stopband is 0.01.
:[font = input; initialization; startGroup; ]
*)
bpfilter = DesignFilter[Analog, ChebyshevI, Bandpass, t,
						0.01, 0.01, 10, 30, 70, 90]
(*
:[font = output; inactive; output; endGroup; ]
CFIR[t, {0, 0, 0, 0, 0, 0, 0, 17965885259.73128}][CIIR[
    t, {180108854100000000000000, 
    5.969009633350858*10^21, 8.13627515332595*10^20, 
    2.116357027199465*10^19, 1.432405352299323*10^18, 
    2.88371457335024*10^16, 1265077976821587., 
    19172986886643.53, 602418084200.756, 
    6539035313.719366, 154670699.9567351, 
    1088207.602387619, 19921.82650068143, 
    69.59635767310566, 1}]]
;[o]
                                        10
CFIR[t, {0, 0, 0, 0, 0, 0, 0, 1.79659 10  }][CIIR[t, 
 
                                        21
   {180108854100000000000000, 5.96901 10  , 
 
              20            19            18
    8.13628 10  , 2.11636 10  , 1.43241 10  , 
 
              16            15           13
    2.88371 10  , 1.26508 10  , 1.9173 10  , 
 
              11            9            8
    6.02418 10  , 6.53904 10 , 1.54671 10 , 
 
              6
    1.08821 10 , 19921.8, 69.5964, 1}]]
:[font = text; inactive; endGroup; ]
In this case, we do want to keep all of the coefficients even though numerically, they are very small.  The Mathematica object Chop is useful for zeroing coefficients smaller than 1.0E-10 (10^-10), but applying Chop to bpfilter would alter its bandpass characteristics.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Verify that the filter meets specifications.
:[font = input; initialization; startGroup; ]
*)
bpfiltertrans = TheFunction[ CTFTransform[ bpfilter, t, w ] ]
(*
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(-17965885259.73128*I*w^7)/
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    8.13627515332595*10^20*w^2 + 
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    1.432405352299323*10^18*w^4 + 
    2.88371457335024*10^16*I*w^5 - 
    1265077976821587.*w^6 + -19172986886643.53*I*w^7 + 
    602418084200.756*w^8 + 6539035313.719366*I*w^9 - 
    154670699.9567351*w^10 + 
    -1088207.602387619*I*w^11 + 
    19921.82650068143*w^12 + 69.59635767310566*I*w^13 - 
    w^14)
;[o]
            10    7
(-1.79659 10   I w ) / 
 
                                        21
  (180108854100000000000000 + 5.96901 10   I w - 
 
              20  2              19    3
    8.13628 10   w  + -2.11636 10   I w  + 
 
              18  4             16    5
    1.43241 10   w  + 2.88371 10   I w  - 
 
              15  6             13    7
    1.26508 10   w  + -1.9173 10   I w  + 
 
              11  8             9    9
    6.02418 10   w  + 6.53904 10  I w  - 
 
              8  10              6    11
    1.54671 10  w   + -1.08821 10  I w   + 
 
             12              13    14
    19921.8 w   + 69.5964 I w   - w  )
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MagPhasePlot[ bpfiltertrans, {w, 1, 100}, MagRangeScale -> Log ]
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;[o]
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Now, if the magnitude and phase response were plotted from 0 rad/sec to 100 rad/sec, then Mathematica would have plotted a ParametricPlot::notpoint warning because the magnitude response of this bandpass filter at DC (w = 0) is negative infinity.
:[font = subsection; inactive; startGroup; Cclosed; ]
Bandstop Filters
:[font = text; inactive; ]
		Four frequency parameters (w1, w2, w3, and w4), a passband error tolerance (deltap), and one stopband error tolerance (deltas) fully specify a bandpass filter.  In this case, the first passband is (0, w1), the stopband is (w2, w3), and the second passband is (w4, infinity).  The filter order of the bandstop filter is always twice that of its lowpass prototype.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Implementation in Mathematica
:[font = input; ]
filtertransform::=			(* forward Laplace transform  *)
SignalProcessing`Analog`FilterDesign`Private`FilterTransferFunction;

tofilter::=
SignalProcessing`Analog`FilterDesign`Private`ToFilter;

DesignBSFilter[ name_, t_, dp_, ds_, w1_, w2_, w3_, w4_ ] :=
	Block [	{bstrans, denom, lowpass, lptrans, n, numer, s,
		 w0, wp, ws, wsdenom1, wsdenom2},

		w0 = Sqrt[ w1 w4 ];
		wp = N[ w1 / ( w0^2 - w1^2 ) ];

		wsdenom1 = w0^2 - w2^2;			(* ws formula could *)
		wsdenom2 = w3^2 - w0^2;			(* have 0 in denom  *)
		ws = Which [ wsdenom1 == 0,
			       w3 / wsdenom2,
			     wsdenom2 == 0,
			       w2 / wsdenom1,
			     True,
			       Min[ w2 / wsdenom1, w3 / wsdenom2 ] ] // N;

		If [ SameQ[name, Elliptic] && ! SameQ[w1 w4, w2 w3],
		     Message[ DesignBSFilter::narrower ] ];
		lowpass = DesignLPFilter[name, t, dp, ds, wp, ws];
		lptrans = FilterTransferFunction[lowpass, t, s];
		n = Exponent[ Denominator[lptrans], s ];
		bstrans = lptrans /. (s -> (s / (s^2 + w0^2)));
		numer = Distribute[(s^2 + w0^2)^n Numerator[bstrans]];
		denom = Distribute[(s^2 + w0^2)^n Denominator[bstrans]];
		ToFilter[ Expand[numer], Expand[denom], s, t ] ]
:[font = text; inactive; endGroup; ]
This is the same code used by "FilterDesign.m".  Note that filtertransform has been aliased to an object that takes a Laplace transform of filter structures.  The more general Laplace transform object LaPlace could have been used instead.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Design a bandstop analog elliptic IIR filter
:[font = text; inactive; ]
		Design a bandpass analog IIR Chebyshev type I filter which rejects all frequencies up to 5 rad/sec, passes frequencies between 10 and 30 rad/sec, and rejects all frequencies above 35 rad/sec.  The maximum error in either the passband or the stopband is 0.01.
:[font = input; startGroup; Cclosed; ]
bsfilter = DesignFilter[Analog, Elliptic, Bandstop, t,
						0.01, 0.01, 10, 30, 70, 90]
:[font = message; inactive; ]
SignalProcessing`Analog`FilterDesign`Private`DesignBSFil
     ter::narrower: 
   One of the transition bandwidths will be narrower
    than specified.
:[font = output; inactive; output; endGroup; ]
CFIR[t, {732774.8210917324, 0, 9708.90113789096, 0, 
    43.36969065133633, 0, 0.07688734519575741, 0, 
    0.00005354282796461276, 0, 1.479789839641969*10^-8, 
    0, 1.378845104332809*10^-12}][CIIR[t, 
   {740176.5869613457, 221707.696007961, 
    44500.96326619352, 5624.624772642595, 
    586.4537789208246, 36.88432091555759, 
    2.19538236096915, 0.04098257879506399, 
    0.0007240170110133636, 0.000007715534667548142, 
    6.782649484254462*10^-8, 3.754639299699588*10^-10, 
    1.392772832659403*10^-12}]]
;[o]
CFIR[t, {732775., 0, 9708.9, 0, 43.3697, 0, 0.0768873, 
 
                                  -8               -12
    0, 0.0000535428, 0, 1.47979 10  , 0, 1.37885 10   }]
 
   [CIIR[t, {740177., 221708., 44501., 5624.62, 
 
    586.454, 36.8843, 2.19538, 0.0409826, 0.000724017, 
 
                             -8            -10
    0.00000771553, 6.78265 10  , 3.75464 10   , 
 
              -12
    1.39277 10   }]]
:[font = text; inactive; endGroup; ]
The warning message simply informs the user that the one of the transition regions will be narrower than specified (this only occurs for bandstop and bandpass elliptic filters when w1 * w4  does not equal w2 * w3).  The resulting filter has coefficients that are small relative to 1. Sometimes, zero coefficients are calculated to be very small non-zero numbers.  In this case, however, want to keep all of the coefficients.  The Mathematica object Chop is useful for zeroing coefficients smaller than 1.0E-10 (10^-10), but applying Chop to bsfilter would alter its bandpass characteristics.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Verify that the filter meets specifications.
:[font = text; inactive; ]
		Here, we compute the continuous-time Fourier transform of the bandstop filter assigned to bsfilter.
:[font = input; startGroup; ]
bsfiltertrans = TheFunction[ CTFTransform[ bsfilter, t, w ] ]
:[font = output; inactive; output; endGroup; ]
(732774.8210917324 - 9708.90113789096*w^2 + 
    43.36969065133633*w^4 - 0.07688734519575741*w^6 + 
    0.00005354282796461276*w^8 - 
    1.479789839641969*10^-8*w^10 + 
    1.378845104332809*10^-12*w^12)/
  (740176.5869613457 + 221707.696007961*I*w - 
    44500.96326619352*w^2 + -5624.624772642595*I*w^3 + 
    586.4537789208246*w^4 + 36.88432091555759*I*w^5 - 
    2.19538236096915*w^6 + -0.04098257879506399*I*w^7 + 
    0.0007240170110133636*w^8 + 
    0.000007715534667548142*I*w^9 - 
    6.782649484254462*10^-8*w^10 + 
    -(3.754639299699588*10^-10)*I*w^11 + 
    1.392772832659403*10^-12*w^12)
;[o]
                   2            4              6
(732775. - 9708.9 w  + 43.3697 w  - 0.0768873 w  + 
 
                  8             -8  10
    0.0000535428 w  - 1.47979 10   w   + 
 
              -12  12
    1.37885 10    w  ) / 
 
                                   2               3
  (740177. + 221708. I w - 44501. w  + -5624.62 I w  + 
 
             4              5            6
    586.454 w  + 36.8843 I w  - 2.19538 w  + 
 
                  7                8
    -0.0409826 I w  + 0.000724017 w  + 
 
                     9             -8  10
    0.00000771553 I w  - 6.78265 10   w   + 
 
               -10    11             -12  12
    -3.75464 10    I w   + 1.39277 10    w  )
:[font = input; startGroup; ]

MagPhasePlot[ bsfiltertrans, {w, 0, 120}, MagRangeScale -> Log ]
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       {109.375, -175.1335123824176}, 
       {110., -176.9342754718763}, 
       {110.3125, -177.8188918388607}, 
       {110.625, -178.693237459126}, 
       {110.78125, -179.1266133228025}, 
       {110.9375, -179.5574864891757}, 
       {111.09375, -179.9858781008359}, 
       {111.25, 179.5881909382844}, 
       {111.5625, 178.7436285115069}, 
       {111.875, 177.9086635562219}, 
       {112.5, 176.2668925693845}, 
       {115., 170.0533965102983}, 
       {120., 159.1047866204347}}]}}, 
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    PlotColor -> Automatic, Axes -> Automatic, 
    PlotLabel -> "Phase Response (degrees)", 
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Tradeoffs Between Filter Design Parameters
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Filter Length
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		All of the five classical filters require one key parameter---  filter length.  As the filter length increases, the transition between passband and stopband regions in the magnitude response will become sharper (that is, occur over a shorter range of frequencies).   We now demonstrate this through animation of a sequence of magnitude responses of a lowpass Butterworth filter.
:[font = text; inactive; ]
		A filter has a cutoff frequency of 22 rad/sec , -2 dB of attenuation at 20 rad/sec, and at least 10 dB of attenuation at 160 / 7 (about 22.85) rad/sec.  To meet specifications, we will need an eleventh-order Butterworth filter:
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FilterParameters[bdelta1, bdelta2, bepsilon, bA, Eripple, ATT, nu]
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{0.2056717652757185, 0.3162277660168379, 
  0.7647831015792084, 3.162277660168379, 
  2.466913365931468, 10., 0.2549277005264028}
;[o]
{0.205672, 0.316228, 0.764783, 3.16228, 2.46691, 10., 
 
  0.254928}
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FilterOrder[Butterworth, 7/8, nu]
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11
;[o]
11
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Because the specifications here are the same as those in the fourth-order Butterworth filter design example in the "Design of Lowpass Analog Filters" section (except that the cutoff frequency has been changed from 30 rad/sec to 23 rad/sec), the variables bdelta1 and bdelta2 already contain the design passband and stopband tolerance, respectively.
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The animation
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		Animate graphics by selecting the cell below with the magnitude response (choose the cell border with the downward pointing arrow) and hit COMMAND-y (or choose the option Animate Selected Graphics).  Note that the frequencies here are in rad/sec and the magnitude response is in decibels (dB):
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:[font = subsubsection; inactive; startGroup; Cclosed; ]
Generating the animation
:[font = text; inactive; ]
		The following function generates a lowpass Butterworth filter of order i and caches time-domain expressions returned by AnalogFilter:
:[font = input; initialization; ]
*)
blpfilter[i_] :=
	blpfilter[i] =
		AnalogFilter[Butterworth, t, i, bepsilon, bA, 7/8, 22]
(*
:[font = text; inactive; ]
The following function computes the frequency response (as a function of w) of the lowpass Butterworth filters calculated by blpfilter and caches the frequency-domain expressions returned by CTFTransform:
:[font = input; initialization; ]
*)
blpfreqresp[i_] :=
	blpfreqresp[i] =
		TheFunction[ CTFTransform[blpfilter[i], t, w] ]
(*
:[font = text; inactive; ]
The following code generates the nine graphics cells grouped for the animation (remember that MagPhasePlot returns a list of two plots, magnitude and phase) :
:[font = input; endGroup; endGroup; ]
Block[ {order, orderstr, plots},
	For[ order = 2, order <= 10, order++,
		 orderstr = ToString[StringForm["order = ``", order]];
		 plots = MagPhasePlot[ blpfreqresp[order],
						       {w, 1, 50},
						       DisplayFunction -> Identity,
						       MagRangeScale -> Log,
						       PhaseRangeScale -> Null,
						       PlotRange -> { -80, 20 } ];
		 Show [ First[plots],
				Graphics[ Text[orderstr, {10, -60}, {-1, 0}] ],
				Graphics[ Point[{0, -80}] ],
				PlotLabel -> "Magnitude Response (dB)",
				PlotRange -> { -80, 20 },
				DisplayFunction :> $DisplayFunction ] ] ]
:[font = subsection; inactive; startGroup; Cclosed; ]
Ripple Control Parameter
:[font = text; inactive; ]
		All five classical classes of filters have a ripple parameter (called Eripple in the definition of FilterParameters), but only elliptic filters and Chebyshev filters.have magnitude responses that ripple.   The ripple control parameter scales the rippling effect.  In both animations below, we examine the effect of the ripple control parameter on the magnitude response of a bandpass elliptic filter that passes frequencies between 30 and 70 rad/sec and rejects those below 10 rad/sec and above 90 rad/sec.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Animation of passband response with varying ripple and fixed filter order
:[font = text; inactive; ]
		This sequence of frames demonstrates the effect of changing the allowable ripple in a bandpass elliptic analog filter while keeping the filter order constant.  The magnitude responses of the following bandpass filters corresponding to maximum ripples of 0.06 to 0.15 at increments of 0.01:
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		What can you say about the effect of varying the ripple control parameter?  By changing the ripple control parameter, do you always change the filter order?
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Animation of passband response with varying ripple and filter order
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		In this animation, we vary the ripple control parameter so that the filter order increases by two each frame.  That is, the ripple control parameters { 0.45, 0.25, 0.1, 0.035, 0.01 } correspond bandpass elliptic filters of order {2, 4, 6, 8, 10}:
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:[font = text; inactive; endGroup; ]
		Examine the filter orders.  Why are they all even?  That is, are bandpass filter designs based on a lowpass filter prototype always even?
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Generating the animation
:[font = text; inactive; ]
		The function ebpfreqresp takes the desired ripple as the lone argument and a graph of the magnitude response of a bandpass elliptic filter.  The reason for a function is that for the animation to make sense we had to change the title of each plot to include the filter order.
:[font = input; initialization; ]
*)
ebpfreqresp[ripple_, {mindB_, maxdB_}, {ripx_, ripy_}] :=
Block [	{filter, format, label, order = "Error", plot,
		 response, s, step, t, transferfunction, w},
		Off[ DesignFilter::narrower ];
		filter = DesignFilter[Analog, Elliptic, Bandpass, t,
				 			  ripple, ripple, 10, 30, 70, 90];
		On[ DesignFilter::narrower ];
				 			  
		(* find filter order *)
		transferfunction = TheFunction[LaPlace[filter, t, s]];
		order = Exponent[Denominator[transferfunction], s];
		
		(* compute frequency response and make magnitude plot *)
		response = transferfunction /. s -> I w;
	 	plot = MagPhasePlot[ response,
							 {w, 0.1, 100},
							 DisplayFunction -> Identity,
							 MagRangeScale -> Log,
							 PhaseRangeScale -> Null,
							 PlotRange -> { mindB, maxdB } ];
							 
		(* adjust the plot title to include filter order *)
		format = If [ order > 9, "``", " ``" ];
		label = "Magnitude Response (dB) order = " ~StringJoin~
				ToString[StringForm[format, order]];
		step = ( maxdB - mindB ) / 5;
		Show [ plot,
				Graphics[ Point[{0, mindB}], Point[{0, maxdB}] ],
				Graphics[ Text["ripple:", {ripx, ripy}, {-1, 0}],
					Text[ripple, {ripx, ripy - step}, {-1, 0}] ],
				DisplayFunction :> $DisplayFunction,
				PlotLabel -> label,
				PlotRange -> { mindB, maxdB } ] ]
(*
:[font = text; inactive; ]
The following code generates the  graphics cells grouped for the animation.  In the first animation, we keep the filter order constant and vary the ripple control parameter:
:[font = input; ]
Map[ ebpfreqresp[#1, {-4, 0}, {80, -1}]&,
	 { .06, .07, .08, .09, .10,
	   .11, .12, .13, .14, .15, .16 } ]
:[font = text; inactive; ]
The second animation varies the filter order by two each frame:
:[font = input; endGroup; endGroup; endGroup; ]
Map[ ebpfreqresp[#1, {-5, 0}, {80, -1}]&,
	 { 0.45, 0.25, 0.1, 0.035, 0.01 } ]
:[font = section; inactive; startGroup; Cclosed; ]
Manipulating Filter Objects
:[font = text; inactive; ]
		At this point, we have generated a variety of filter structures whose magnitude responses meet given specifications.  Once we generate a filter object, we can then use it in other expressions (modulation), specify an alternate input function (in lieu of the Dirac delta function) to it, transform it, or plot its frequency response.  We can also quantize the filter coefficients (or its poles and zeroes) to get a better feeling to how the implemented filter might behave (for example, it is difficult to get more than three digits of accuracy when using resistors, capacitors, and inductors).
:[font = subsection; inactive; startGroup; Cclosed; ]
Transform of a Filter Expression
:[font = text; inactive; ]
		For example, we will take the Laplace transform of an analog filter shifted in the frequency domain by w0 rad.  The filter is the fourth order Butterworth filter which was previously designed and assigned to the Mathematica variable bfilter:
:[font = input; startGroup; Cclosed; ]

bfilter
:[font = output; inactive; output; endGroup; ]
234256*CIIR[t, {234256., 27824.56490000732, 
    1652.479364188578, 57.48877045456058, 1}]
;[o]
234256 CIIR[t, {234256., 27824.6, 1652.48, 57.4888, 1}]
:[font = input; startGroup; Cclosed; ]

LaPlace[ Exp[I w0 t] bfilter, t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[234256/
   (234256. + 27824.56490000732*(s + -I*w0) + 
     1652.479364188578*(s + -I*w0)^2 + 
     57.48877045456058*(s + -I*w0)^3 + (s + -I*w0)^4), 
  Rminus[-Re[-I*w0]], Rplus[DirectedInfinity[1]], 
  LVariables[s]]
;[o]
LTransData[234256 / 
 
   (234256. + 27824.6 (s + -I w0) + 
 
                        2                      3
     1652.48 (s + -I w0)  + 57.4888 (s + -I w0)  + 
 
                4
     (s + -I w0) ), Rminus[-Re[-I w0]], Rplus[Infinity], 
 
  LVariables[s]]
:[font = subsection; inactive; startGroup; Cclosed; ]
Symbolic Filtering (Symbolic Convolution)
:[font = text; inactive; ]
		The output of an analog filter is the convolution of the input function with the impulse response of the filter.  If the input function has a Laplace transform, then the filter output is simply the inverse Laplace transform of the product of the Laplace transform of the filter and the Laplace transform of the input function.  This is purely symbolic convolution. 
:[font = text; inactive; ]
		As an example of this, we will use the fourth-order Butterworth filter design earlier in the notebook and assigned to the variable bfilter:
:[font = input; startGroup; ]
bfilter
:[font = output; inactive; output; endGroup; ]
234256*CIIR[t, {234256., 27824.56490000732, 
    1652.479364188578, 57.48877045456058, 1}, 
   Roots -> {22*E^((5*I)/8*Pi), 22*E^((7*I)/8*Pi), 
     22*E^((9*I)/8*Pi), 22*E^((11*I)/8*Pi)}]
;[o]
234256 CIIR[t, {234256., 27824.6, 1652.48, 57.4888, 1}]
:[font = text; inactive; ]
We will use this to filter a sum of two right-sided sinusoids:
:[font = input; startGroup; ]
input = Cos[w0 t] CStep[t] + Cos[w1 t] CStep[t]
:[font = output; inactive; output; endGroup; ]
CStep[t]*Cos[t*w0] + CStep[t]*Cos[t*w1]
;[o]
CStep[t] Cos[t w0] + CStep[t] Cos[t w1]
:[font = input; startGroup; ]
conv = LaPlace[ Convolve[t][input, bfilter], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(234256*(s/(s^2 + w0^2) + s/(s^2 + w1^2)))/
   (234256. + 27824.56490000732*s + 1652.479364188578*s^2 + 
     57.48877045456058*s^3 + s^4), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
LTransData[
 
                         s          s
              234256 (-------- + --------)
                       2     2    2     2
                      s  + w0    s  + w1
   --------------------------------------------------, 
                                  2            3    4
   234256. + 27824.6 s + 1652.48 s  + 57.4888 s  + s
 
  Rminus[0], Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]
InvLaPlace[ conv ]
:[font = print; inactive; startGroup; ]
Breaking up the denominator of
:[font = print; inactive; ]
                      s          s
           234256 (-------- + --------)
                    2     2    2     2
                   s  + w0    s  + w1
--------------------------------------------------
                               2            3    4
234256. + 27824.6 s + 1652.48 s  + 57.4888 s  + s
:[font = print; inactive; ]
into its approximate roots
:[font = print; inactive; ]
            3       2       2
(234256 (2 s  + s w0  + s w1 )) / 
 
  ((8.41904 - 20.3253 I + s) (8.41904 + 20.3253 I + s) 
 
    (20.3253 - 8.41904 I + s) (20.3253 + 8.41904 I + s) 
 
      2     2    2     2
    (s  + w0 ) (s  + w1 ))
:[font = print; inactive; ]
Option Apart is not All so partial
:[font = print; inactive; ]
fractions decomposition on real-valued
:[font = print; inactive; ]
poles was not performed
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
(234256*(2*s^3 + s*w0^2 + s*w1^2))/
  ((8.41903551201782 - 20.32534971523457*I + s)*
    (8.41903551201782 + 20.32534971523457*I + s)*
    (20.32534971528209 - 8.41903551199771*I + s)*
    (20.32534971528209 + 8.41903551199771*I + s)*(s^2 + w0^2)*
    (s^2 + w1^2))
;[o]
            3       2       2
(234256 (2 s  + s w0  + s w1 )) / 
 
  ((8.41904 - 20.3253 I + s) (8.41904 + 20.3253 I + s) 
 
    (20.3253 - 8.41904 I + s) (20.3253 + 8.41904 I + s) 
 
      2     2    2     2
    (s  + w0 ) (s  + w1 ))
:[font = subsection; inactive; startGroup; Cclosed; ]
Coefficient Quantization
:[font = text; inactive; ]
		For example, let's look at the frequency response of the Chebyshev type I bandpass filter previously design and assigned to the Mathematica variable bpfilter.  It has a frequency response of bpfiltertrans.  Without coefficient quantization, it is a bandpass filter that rejects frequencies below 10 rad/sec and above 90 rad/sec and that passes frequencies between 30 rad/sec and 70 rad/sec, as shown below.
:[font = input; startGroup; Cclosed; ]

MagPhasePlot[ bpfiltertrans, {w, 1, 100}, MagRangeScale -> Log ]
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;[o]
{-Graphics-, -Graphics-}
:[font = text; inactive; ]

Now, we will quantize the filter coefficients to 3 digits:
:[font = input; startGroup; Cclosed; ]

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This work was supported in part by the Joint Services Electronics Program
contract #DAAL-03-90-C-0004.  
^*)