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

Transcription

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

2 Review from Last Time Butterworth Approximations TLP ( jω) Analytical formulation: All pole approximation Magnitude response is maximally flat at ω=0 Goes to 0 at ω= Assumes value at ω= + Assumes value of at ω=0 Characterized by {n,} Emphasis almost entirely on performance at single frequency "On the Theory of Filter Amplifiers", Wireless Engineer (also called Experimental Wireless and the Radio Engineer), Vol. 7, 930, pp

3 Review from Last Time Poles of T BW (s) Butterworth Approximation Im Im for n even for n odd π n n Re π n n Re / n pk+ = sin [ + k] ± jcos [ + k] n n π n k=0,,... π pn [ j0] / n = + π / n pk = sin [ + k] ± jcos [ + k] n n π k=0,... n-3

4 Review from Last Time Butterworth Approximation What is the Q of the highest Q pole for the BW approximation? π π / n p 0 = sin + jcos = α + jβ n n π n n Q MAX = α + β α QMAX π n sin + cos n n = = n π π sin sin n n π Q MAX = π sin n

5 Review from Last Time Butterworth Approximation Order needs to be rather high to get steep transition Figure from Passive and Active Network Analysis and Synthesis, Budak

6 Review from Last Time Butterworth Approximation = π sin n Figure from Passive and Active Network Analysis and Synthesis, Budak Phase is quite linear in passband (benefit unrelated to design requirements)

7 Butterworth Approximation Review from Last Time Summary Widely Used Analytical Approximation Characterized by {,n} Maximally flat at ω=0 Almost all emphasis placed on characteristics at single frequency (ω=0) Transition not very steep (requires large order for steep transition) Pole Q is quite low Pass-band phase is quite linear (no emphasis was placed on phase!) Poles lie on a circle Simple closed-form analytical expressions for poles and T(jω)

8 Approximations Magnitude Squared Approximating Functions H A (ω ) Inverse Transform - H A (ω ) T A (s) Collocation Least Squares Approximations Pade Approximations Other Analytical Optimizations Numerical Optimization Canonical Approximations Butterworth Chebyschev Elliptic Bessel Thompson TLP ( jω)

9 Pafnuty Lvovich Chebyshev Born May 6, 8 Died December 8,894 Nationality Russian FieldsMathematician

10 Type I Chebyshev Approximations Analytical formulation: All pole approximation Magnitude response bounded between and in the pass band Assumes the value of at ω= + Goes to 0 at ω= Assumes extreme values maximum no times in [0 ] Characterized by {n,} Based upon Chebyshev Polynomials Chebyshev polynomials were first presented in: P. L. Chebyshev (854) "Théorie des mécanismes connus sous le nom parallelogrammes," Mémoires des Savants étrangers présentes à l'academie de Saint-Pétersbourg, vol. 7, pages T LP jω +

11 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 ω= + LP ( jω) Value of at ω=0 Assumes extreme values maximum times in [ ] Characterized by {n,} Based upon Chebyshev Polynomials

12 Chebyshev Polynomials The Chebyshev polynomials are characterized by the property that the polynomial assumes the extremum values of 0 and a maximum number of times in the interval [0,] and go to for x large. In polynomial form they can be expressed as C 0 (x)= C (x)=x C n+ (x)=xc n (x) - C n- (x) Or, equivalently, in trigonometric form as ( ) cos n arccos x x [,] Cn ( x) = cosh( n arccosh x ) x n ( ) cosh( n arccosh( x) ) x Figure from Wikipedia

13 Chebyshev Polynomials The first 9 CC polynomials: C ( x) = C x = x C x = x 3 C x = 4x 3x 4 C x = 8x 8x C x = 6x 0x + 5x 6 4 C x = 3x 48x + 8x C x = 64x x + 56x 7x 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)

14 Type H ω = + ω BW n Butterworth H ω = + F ω n A General Form Observation: F n (ω ) close to in the pass band and gets very large in stop-band The square of the Chebyshev polynomials have this property H ω = + C ω CC n This is the magnitude squared approximating function of the Type CC approximation

15 Type H ω = + C ω CC n Poles of H CC (ω) lie on an ellipse with none on the real axis

16 Type H ω = + C ω CC n Im Im Inverse Mapping Re Re Poles of T CC (s)

17 Type Im α k β + k = sinh arcsinh cosh arcsinh n n Re Ellipse Intersect Points for select n and

18 Type Im Re Poles of T CC (s) π π pk = sin ( +k) sinh arcsinh ± j cos ( +k) cosh arcsinh n n n n Properties of the ellipse pk = αk ± jβk α k β + k = sinh arcsinh cosh arcsinh n n k=0.n-

19 Type TCC ( ω) + TCC ( ω) + T CC (0) alternates between and with index number Substantial pass band ripple + Sharp transitions from pass band to stop band

20 Type Fig from Allen and Huelsman Sharp transitions from pass band to stop band

21 Type From Budak Text CC transition is much steeper than BW transition

22 Comparison of BW and CC Responses CC slope at band edge much steeper than that of BW n Slope ( ω = ) = n [ Slope ( ω )] 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

23 Type From Budak Text Analytically, it can be shown that, at the band-edge dtbw dω dtcc d ( jω ) ( jω ) ω = n = n ( + ) 3 ( + ) 3 CC slope is n times steeper than that of the BW slope

24 Type From Budak Text CC phase is much more nonlinear than BW phase

25 Im Type π π pk = sin ( +k) sinh arcsinh ± j cos ( +k) cosh arcsinh n n n n Maximum pole Q of CC approximation can be obtained by considering pole with index k=0 Re π π p0 = sin sinh arcsinh ± j cos cosh arcsinh n n n n p 0= Recall Q MAX α + = jβ α + β α π cos n QMAX,CC = + π sin sinh sinh n n arc

26 Type Comparison of maximum pole Q of CC approximation with that of BW approximation Q MAX,BW = π sin n QMAX,CC π cos n = + π sin sinh sinh n n arc QMAX,CC π cos n = QMAX,BW + sinh sinh n arc Example compare the Q s for n=0 and = Q BW =3.9 Q CC =35.9 For large n, the CC filters have a very high pole Q!

27 Type H ω = + ω BW n Butterworth H ω = + F ω n A General Form Another General Form H ω = + HCC ω = + Fn /ω C n ( ) ω

28 Type HCC ω = + C n ( ) ω Equal-ripple in stop band Monotone in pass band Both poles and zeros present Poles of Type II CC are reciprocal of poles of Type I Zeros of Type II are inverse of the zeros of the CC Polynomials p k = π π sin ( +k) sinh arcsinh ± j cos ( +k) cosh arcsinh n n n n zk = j π cos n ( k-)

29 Type HCC ω = + C n ( ) ω TCC ( ω) + TCC ( ω) +

30 Type HCC ω = + TCC ( ω) C n ( ) ω + Transition region not as steep as for Type Considerably less popular

31 Type Im HCC ω = + C n ( ) ω Re Pole Q expressions identical since poles are reciprocals Maximum pole Q is just as high as for Type

32 Transitional BW-Chebyshev Approximations Consider: H( ω ) = + F ω n General Form Define F BWk =ω k F CCk =C n (ω) H( ω ) = 0 k n + F F BWk CC n-k H( ω ) = + θ F θ F BWk + ( ) CC( n-k) 0 θ Other transitional approximations are possible Transitional approximations have some of the properties of both parents

33 Transitional BW-CC filters H ABW ω = H ( ω ) n + ACC ω = + ω ( C ) n H H ω ATRAN k = + ω C ω n k 0 k n = + θω + ( θ ) C ( ω) n ω ATRAN n 0 θ Other transitional BW-CC approximations exist as well

34 Transitional BW-CC filters H H ω ATRAN k = + ω C ω ω ATRAN n n k = + θω + ( θ ) C ( ω) n Transitional filters will exhibit flatness at ω=0, passband ripple, and intermediate slope characteristics at band-edge

35 from Spectrum Software: Note that this is introduced as a Chebyshev filter, the source correctly points out that it implements the CC filter in a specific filter topology It is important to not confuse the approximation from the architecture and this Tow-Thomas Structure can be used to implement either BW or CC functions only differing in the choice of the component values

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