EE 508 Lecture The Approximation Problem Classical Approximating Functions - Elliptic Approximations - Thompson and Bessel Approximations
Review from Last Time Chebyshev Approximations T Type II Chebyshev Approximations (not so common) Analytical formulation: Magnitude response bounded between 0 and in the stop band Assumes the value of at = LP j Value of at =0 Assumes extreme values maximum times in [ ] Characterized by {n,ε} Based upon Chebyshev Polynomials
Review from Last Time Chebyshev Approximations Chebyshev Polynomials The first 9 CC polynomials: C 0 3 4 5 6 7 8 x C x x C x x 3 C x 4x 3x 4 C x 8x 8x 5 3 C x 6x 0x 5x 6 4 C x 3x 48x 8x 7 5 3 C x 64x x 56x 7x 8 6 4 C x 8x 56x 60x 3x Figure from Wikipedia Even-indexed polynomials are functions of x Odd-indexed polynomials are product of x and function of x Square of all polynomials are function of x (i.e. an even function of x)
Review from Last Time Chebyshev Approximations Type H = + BW n Butterworth H = + F n A General Form Observation: F n ( ) close to in the pass band and gets very large in stop-band The square of the Chebyshev polynomials have this property H = + C CC n This is the magnitude squared approximating function of the Type CC approximation
Review from Last Time Chebyshev Approximations Type H = + C CC n Im Im Inverse Mapping Re Re Poles of T CC (s)
Review from Last Time Chebyshev Approximations Type TCC Even order TCC Odd order T CC (0) alternates between and with index number Substantial pass band ripple Sharp transitions from pass band to stop band
Review from Last Time Comparison of BW and Type CC Responses CC slope at band edge much steeper than that of BW n Slope ( ) n [ Slope ( )] CC BW Corresponding pole Q of CC much higher than that of BW Lower-order CC filter can often meet same band-edge transition as a given BW filter Both are widely used Cost of implementation of BW and CC for same order is about the same
Review from Last Time Transitional BW-Chebyshev Approximations Consider: H= + F n General Form Define F BWk = k F CCk =C n () H = 0 k n + F F BWk CC n-k H= + F F BWk CCn-k 0 Other transitional approximations are possible Transitional approximations have some of the properties of both parents
Review from Last Time Distinguish Between Circuit and Approximation http://www.egr.msu.edu/classes/ece480/capstone/fall/group0/web/documents/how%0to%0design%00%0khz%0filter-vadim.pdf http://www.changpuak.ch/electronics/butterworth_lowpass_active_4db.php Note that what distinguishes between different filter approximations having the same number of cc poles and zeros and the same number of real axis poles and zeros is the component values of a given circuit, not the filter architecture itself
Approximations Magnitude Squared Approximating Functions H A ( ) Inverse Transform - H A ( ) T A (s) Collocation Least Squares Approximations Pade Approximations Other Analytical Optimizations Numerical Optimization Canonical Approximations Butterworth Chebyshev Elliptic Bessel Thompson
Elliptic Filters Can be thought of as an extension of the CC approach by adding complex-conjugate zeros on the imaginary axis to increase the sharpness of the slope at the band edge Im TE j TA Re Ω S Concept Actual effect of adding zeros
Elliptic Filters Basic idea comes from the concept of a Chebyschev Rational Fraction Sometimes termed Cauer filters
Chebyshev Rational Fraction A Chebyshev Rational Fraction is a rational fraction that is equal ripple in [-,] and equal ripple in [-,-] and [, ]
Chebyshev Rational Fractions f(x) f(x) x x Even-order CC rational fraction Odd-order CC rational fraction
Chebyshev Rational Fractions Even-order CC rational fraction n/ k= Rn n/ C x H k= k x -a k x -b Odd-order CC rational fraction n/ k= Rn n/ C x H k= k x x -a k x -b
Elliptic Filters Magnitude-Squared Elliptic Approximating Function H = + C E Rn Inverse mapping to T E (s) exists For n even, n zeros on imaginary axis For n odd, n- zeros on imaginary axis Equal ripple in both pass band and stop band Analytical expression for poles and zeros not available Often choose to have less than n or n- zeros on imaginary axis (No longer based upon CC rational fractions) Termed here full order
Elliptic Filters TE j If of full-order, response completely characterized by {n,ε,a S.Ω S } Any 3 of these paramaters are independent A S Typically ε,ω S, and A S are fixed by specifications (i.e. must determine n) Ω S
Elliptic Filters TE j TE j Ω S n odd (n-)/ peaks in pass band (n-)/ peaks in stop band Maximum occurs at =0 T(j ) =0 For full-order elliptic approximations Ω S n even n/ peaks in pass band n/ peaks in stop band T(j0) =/sq(+ε ) T(j ) =A S
Elliptic Filters TE j A S Ω S Simple closed-form expressions for poles, zeros, and T E (j) do not exist Simple closed form expressions for relationship between {n,ε,as, and Ω S } do not exist Simple expressions for max pole Q and slope at band edge do not exist Reduced-order elliptic approximations could be viewed as CC filters with zeros added to stop band General design tables not available though limited tables for specific characterization parameters do exist
Elliptic Filters TE j Observations about Elliptic Filters A S Ω S Elliptic filters have steeper transitions than CC filters Elliptic filters do not roll off as quickly in stop band as CC or even BW Highest Pole-Q of elliptic filters is larger than that of CC filters For a given transition requirement, order of elliptic filter typically less than that of CC filter Cost of implementing elliptic filter is comparable to that of CC filter if orders are the same Cost of implementing a given filter requirement is often less with the elliptic filters Often need computer to obtain elliptic approximating functions though limited tables are available Some authors refer to elliptic filters as Cauer filters
Canonical Approximating Functions Butterworth Chebyschev Transitional BW-CC Elliptic Thompson Bessel Thompson and Bessel Approximating Functions are Two Different Names for the Same Approximation
Thompson and Bessel Approximations - All-pole filters - Maximally linear phase at =0
Consider T(j) Thompson and Bessel Approximations phase R IM Nj NR j +jn j T j = = D j D j +jd j N tan I j D tan I j T j NR j DR j IM Phase expressions are difficult to work with Will first consider group delay and frequency distortion
Linear Phase Consider T(j) R IM Nj NR j +jn j T j = = D j D j +jd j N tan I j D tan I j T j NR j DR j IM Defn: A filter is said to have linear phase if the phase is given by the expression T j θ where θ is a constant that is independent of
Distortion in Filters Types of Distortion Frequency Distortion Amplitude Distortion Phase Distortion Nonlinear Distortion Although the term distortion is used for these two basic classes, there is little in common between these two classes
Distortion in Filters Frequency Distortion Amplitude Distortion A filter is said to have frequency (magnitude) distortion if the magnitude of the transfer function changes with frequency Phase Distortion A filter is said to have phase distortion if the phase of the transfer function is not equal to a constant times Nonlinear Distortion A filter is said to have nonlinear distortion if there is one or more spectral components in the output that are not present in the input
Distortion in Filters Phase and frequency distortion are concepts that apply to linear circuits If frequency distortion is present, the relative magnitude of the spectral components that are present will be different than the spectral components in the output If phase distortion is present, at least for some inputs, waveshape will not be preserved Nonlinear distortion does not exist in linear networks and is often used as a measure of the linearity of a filter. No magnitude distortion will be present in a specific output of a filter if all spectral components that are present in the input are in a flat passband No phase distortion will be present in a specific output of a filter if all spectral components that are present in the input are in a linear phase passband Linear phase can occur even when the magnitude in the passband is not flat Linear phase will still occur if the phase becomes nonlinear in the stopband
Filter Passband and Stopband T j T j T j Flat Passband T j T j T j Non-Flat Passband Frequencies where gain is ideally 0 or very small is termed the stopband Frequencies where the gain is ideally not small is termed the passband Passband is often a continuous region in though could be split
Linear and Nonlinear Phase T j T j T j T j T j T j Linear Passband Phase T j T j T j T j T j T j
Preserving the Waveshape: Example: Consider a signal x(t)=sin( t) + 0.5sin(3 t) sin( t) 0.5 0.5*sin(3 t) 0 0 5 0 5 0 5 30 35 40 45 50-0.5-0.5 x(t) = sin( t)+0.5*sin(3 t) S 0-0.5 0 0 0 30 40 50-3 Note the wave shape and spectral magnitude of x(t)
Preserving the waveshape A filter has no frequency distortion for a given input if the output wave shape is preserved (i.e. the output wave shape is a magnitude scaled and possibly time-shifted version of the input) Mathematically, no frequency distortion for V IN (t) if V OUT (t)=kv IN (t-t shift ) Could have frequency distortion for other inputs
Example of No Frequency Distortion Input 0.5 S 0-0.5 0 0 0 30 40 50 3-4 3 Output No Frequency Distortion S 4 0 - - -3 0 0 0 30 40 50 3-4
Example with Frequency Distortion Input 0.5 S 0-0.5 0 0 0 30 40 50 3-5 4 3 0 - - -3 Output Phase Distortion (No Amplitude Distortion) 0 0 0 30 40 50 S 4-4 -5 3
Example with Frequency Distortion Input 0.5 S 0-0.5 0 0 0 30 40 50 3-5 4 3 Output Amplitude Distortion (No Phase Distortion) S 4 0 - - -3-4 -5 0 0 0 30 40 50 3
Example with Frequency Distortion Input 0.5 S 0-0.5 0 0 0 30 40 50 3-6 4 Output Amplitude and Phase Distortion S 4 0 - -4 0 0 0 30 40 50 3-6
Example with Nonlinear Distortion and Frequency Distortion Input 0.5 S 0-0.5-0 0 0 30 40 50 3 8 6 4 0 - -4-6 Amplitude and Phase and Nonlinear Distortion 0 0 0 30 40 50 S 4 3 Nonlinear distortion evidenced by presence of spectral components in output that are not in the input
Frequency Distortion In most audio applications (and many other signal processing applications) there is little concern about phase distortion but some applications do require low phase distortion In audio applications, any substantive magnitude distortion in the pass band is usually not acceptable Any substantive nonlinear distortion in the pass band is unacceptable in most audio applications
Preserving wave-shape in pass band A filter is said to have linear passband phase if the phase in the passband of the filter is given by the expression T j θ where θ is a constant that is independent of If a filter has linear passband phase in a flat passband, then the waveshape is preserved provided all spectral components of the input are in the passband and the output can be expressed as an amplitude scaled and time shifted version of the input by the expression V OUT (t)=kv IN (t-t shift )
Preserving wave-shape in pass band X IN (s) Ts X OUT (s) Example: Consider a linear network with transfer function T(s) Assume In the steady state Xin t = Asin t+θ + Asint+θ X t = A T j sin t+θ T j + A T j sin t+θ T j OUT Rewrite as: T j T j XOUT t = A T j sin t+ +θ + A T j sin t+ +θ T If and are in a flat passband, j T j = T j Can express as: T j T j XOUT t = T j Asin t+ +θ + Asin t+ +θ
Preserving wave-shape in pass band X IN (s) Ts X OUT (s) Example: Xin t = Asin t+θ + Asint+θ If and are in a flat passband, T j = T j T j T j T j XOUT t = T j Asin t+ +θ + Asin t+ +θ If and are in a linear phase passband, T j = k and T j = k k k XOUT t = T j Asin t+ +θ + Asin t+ +θ X t = T j A sin t+k +θ + A sin t+k +θ OUT XOUT t = Tj xin t+k T j k
Preserving wave-shape in pass band X IN (s) Ts X OUT (s) Example: T j = T j XOUT t = Tj xin t+k Xin t = Asin t+θ + Asint+θ T j T j = k T j = k T j k This is a magnitude scaled and time shifted version ot the input so waveshape is preserved A weaker condition on the phase relationship will also preserve waveshape with two spectral components present T j T j
Amplitude (Magnitude) Distortion, Phase Distortion and Preserving wave-shape in pass band X IN (s) Ts X OUT (s) T j T j k If and are any two spectral components of an input signal in which T j T j then the filter exhibits amplitude distortion for this input. If and are any two spectral components of an input signal in which T j then the filter exhibits phase distortion for this input. T j If and are any two spectral components of an input signal that exhibits either amplitude or phase distortion for these inputs, then the waveshape will X t Hx t+k not be preserved OUT in
Amplitude (Magnitude) Distortion, Phase Distortion and Preserving wave-shape in pass band X IN (s) Ts X OUT (s) Amplitude and phase distortion are often of concern in filter applications requiring a flat passband and a flat zero-magnitude stop band Amplitude distortion is usually of little concern in the stopband of a filter Phase distortion is usually of little concern in the stopband of a filter A filter with no amplitude distortion or phase distortion in the passband and a zero-magnitude stop band will exhibit waveform distortion for any input that has a frequency component in the passband and another frequency component in the stopband It can be shown that the only way to avoid magnitude and phase distortion respectively for signals that have energy components in the interval << is to have constants k and k such that T j k T j k for < <
End of Lecture