EE 508 Lecture 15. Filter Transformations. Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject

Transcription

1 EE 508 Lecture 15 Filter Transformations Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject

2 Review from Last Time Thompson and Bessel pproximations Use of Bessel Filters: X I (s) X OUT (s) T( s) -sh T( s ) = e T( j ) = 1 ( ) T j = -h τ G = h It is challenging to build filters with a constant delay filter with a constant group delay and unity magnitude introduces a constant delay Bessel filters are filters that are used to approximate a constant delay Bessel filters are attractive for introducing constant delays in digital systems Some authors refer to Bessel filters as Delay Filters n ideal delay filter would - introduce a time-domain shift of a step input by the group delay - introduce a time-domain shift each spectral component by the group delay - introduce a time-domain shift of a square wave by the group delay

3 Review from Last Time Filter Transformations ( ) ( ) = Tf( s) T( s ) s f s T ( s) T s ( ) Claim: If the imaginary axis in the s-plane is mapped to the imaginary axis in the s-plane with a variable mapping function, the basic shape of the function T(s) will be preserved in the function F(T(s)) but the frequency axis may be warped and/or folded in the magnitude domain Preserving basic shape, in this context, constitutes maintaining features in the magnitude response of F(T(s)) that are in T(s) including, but not limited to, the peak amplitude, number of ripples, peaks of ripples,

4 Review from Last Time LP to BP Filter Transformations ( ) TLP ( s ) s f s TBP ( s) BP ( ) = LP ( ( )) T s T f s TLP ( j) TBP ( j) Lowpass Bandpass Will consider rational fraction mappings ( ) f s = m T i=0 n T i=0 a s Ti b s Ti i i

5 Review from Last Time Standard LP to BP Transformation apping Strategy: s-domain map s=0 to s= j1 map s=j1 to s=j B map s= j1 to s= j T LP (f(s)) -domain map =0 to =1 map =1 to = B map = 1 to = Consider variable mapping ( ) f s = a s + a s+a b s+b T T1 T0 T1 T0 With this mapping, there are 5 D.O.F and 3 mathematical constraints and the additional constraint that the Im axis maps to the Im axis Will now show that the following mapping will meet these constraints ( ) f s s+1 = or equivalently s BW s s+1 s BW This is the lowest-order mapping that will meet these constrains

6 Review from Last Time Standard LP to BP Transformation TLP ( j) s s+1 s BW TBP ( j)

7 Standard LP to BP Transformation Frequency and s-domain appings - Denormalized (subscript variable in LP approximation for notational convenience) s X s+ s BW s X X s+ s BW - BW ( ) s BW± BW s -4 X X s BW ± BW 4 ( ) + X X Exercise: Resolve the dimensional consistency in the last equation

8 Standard LP to BP Transformation Frequency and s-domain appings - Denormalized (subscript variable in LP approximation for notational convenience) T LP (s x ) s s W s X s+1 s BW s X s+ s BW T BP (s) s s+1 s BW s s W ll three approaches give same approximation Which is most practical to use? Often none of them!

9 Standard LP to BP Transformation Frequency and s-domain appings - Denormalized (subscript variable in LP approximation for notational convenience) s X s+1 s BW BP Often most practical to synthesize directly from the T BP and then do the frequency scaling of components at the circuit level rather than at the approximation level

10 Standard LP to BP Transformation Frequency and s-domain appings (subscript variable in LP approximation for notational convenience) Poles and Zeros of the BP approximations s X f s+ 1 s BW solving for s s ( ) 0 T p = LP x ( ) s BW± BW s -4 1 f X X ( ) = ( ) T s T f s BP LP ( ) TLP ( f( p )) = 0 ( ) ( ) T p T f p BP = LP ( ) = 0 Since this relationship maps the complex plane to the complex plane, it also maps the poles and zeros of the LP approximation to the poles and zeros of the BP approximation

11 Standard LP to BP Transformation Pole appings Claim: With a variable mapping transform, the variable mapping naturally defines the mapping of the poles of the transformed function p X p+1 p BW s X s+1 s BW p ( ) p BW± BW p -4 X X Exercise: Resolve the dimensional consistency in the last equation

12 Standard LP to BP Transformation Pole appings p ( ) p BW± BW p -4 X X {,Q } 0BPH LBPH {,Q } 0BPL LBPL {,Q } 0LP LP Image of the cc pole pair is the two pairs of poles

13 Standard LP to BP Transformation Pole appings {,Q } 0BPH LBPH {,Q } 0BPL LBPL Im Im {,Q } 0LP LP Re Re Can show that the upper hp pole maps to one upper hp pole and one lower hp pole as shown. Corresponding mapping of the lower hp pole is also shown

14 Standard LP to BP Transformation Pole appings p ( ) p BW± BW p -4 X X Im Im Re Re multipliity 6 ote doubling of poles, addition of zeros, and likely Q enhancement

15 LP to BP Transformation Claim: Other variable mapping transforms exist that satisfy the imaginary axis mapping properties needed to obtain the LP to BP transformation but are seldom, if ever, discussed. The Standard LP to BP transform Is by far the most popular and most authors treat it as if it is unique. s X f(s)

16 LP to BP Transformation Pole Q of BP pproximations T ( j ) T ( j) BP LP BW = - = H H L Consider a pole in the LP approximation characterized by { 0LP,Q LP } It can be shown that the corresponding BP poles have the same Q L {,Q } 0LP LP Im {,Q } 0BPH LBPH {,Q } 0BPL LBPL Im Re Re

17 Pole Q of BP pproximations LP to BP Transformation 1 TLP ( j) TBP ( j) BW {,Q } 0BPH BPH {,Q } 0BPL BPL 1 BW = - = H H L L L H {,Q } 0LP LP Define: BW δ = It can be shown that 0LP Q Q = Q = LP BPL BPH δ δ δ QLP For δ small, It can be shown that 0BP Q BP Q δ LP Q BP Q BP = δ ± δ 4 QLP Q LP ote for δ small, Q BP can get very large

18 LP to BP Transformation Pole Q of BP pproximations BW δ = 0LP Q Q = Q = LP BPL BPH δ δ δ QLP

19 LP to BP Transformation Pole locations vs Q LP and δ BW δ = 0LP

20 LP to BP Transformation 1 TLP ( j) TBP ( j) BW 1 L H Classical BP pproximations Butterworth Chebyschev Elliptic Bessel Obtained by the LP to BP transformation of the corresponding LP approximations

21 Standard LP to BP Transformation s+1 s s BW Standard LP to BP transform is a variable mapping transform aps j axis to j axis aps LP poles to BP poles Preserves basic shape but warps frequency axis Doubles order Pole Q of resultant band-pass functions can be very large for narrow pass-band Sequencing of frequency scaling and transformation does not affect final function

22 Example 1: Obtain an approximation that meets the following specifications R S L B BH BW= - B = B ssume that L, BH and satsify - - L BH = BW BW L BH

23 Example 1: Obtain an approximation that meets the following specifications R S 1 1+ε = R = S = R BW= - B = B - - L BH = BW BW L BH ε S = R = L -1 - L BW (actually is is and L that map to 1 and S respectively but show and L for convenience)

24 Example : Obtain an approximation that meets the following specifications BW= - B = B In this example, - - L BH BW BW L BH

25 Example : Obtain an approximation that meets the following specifications R 1 1+ε = R = SH =min, S R SL BW= - = B B ε = R S1 S -1 - L = BW = L - BH BH BW { } = min, S S1 S

26 Example : Obtain an approximation that meets the following specifications R 1 1+ε = R = SH =min, S R SL { } = min, S S1 S ε = R S1-1 - L = BW L BW= - = B B S = BH BH - BW { } = min, S S1 S

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