MAXIMALLY FLAT FIR FILTERS

Transcription

1 MAXIMALLY FLAT FIR FILTERS This sectio describes a family of maximally flat symmetric FIR filters first itroduced by Herrma [2]. The desig of these filters is particularly simple due to the availability of closed form solutios. I additio, they ca be implemeted efficietly usig certai hardware architectures. The derivatio of the maximally flat filters by Herrma begis with a trasformatio of the problem usig a chage of variables. A CHANGE OF VARIABLES The followig chage of variables is used to covert a Type I liearphase trasfer fuctio H(z) ito a polyomial P (x). The values of P (x) for real values of x betwee ad will give the values of H(z) for complex values of z o the uit circle. To describe the chage of variables we begi with a polyomial P (x) of degree M. P (x) = p + p x + p 2 x p M x M = M m= p m x m The chage of variables, used to obtai a Type I FIR filter, is give by, x = z + 2 z. () I. Selesick EL 73 Lecture Notes

2 Usig this trasformatio, we obtai the followig trasfer fuctio H(z). ( ) z + 2 z H(z) = P (2) M ( ) z + 2 z m = p m (3) The filter will be a Type I FIR filter. The symmetric impulse respose h() will be of legth N = 2M + ; h() will be supported o M M. For example, if the H(z) = 32 ad P (x) = 2 9 x + 3 x 2 6 x 3 () ( 3 z z z z + 8 z z 3). h() = {3, 8, 3, 6, 3, 8, 3} 32 where 6 represets h(). The followig Matlab program x2z coverts the polyomial coefficiets p to filter coefficiets h(). The program is based directly o equatio (3). fuctio h = x2z(p) N = legth(p); h = p(); g = ; for k = 2:N g = cov(g,[- 2 -]/); h = [ h ] + p(k)*g; This program assumes that the coefficiets of the polyomial P (x) are ordered i icreasig powers (this is cotrary to the Matlab covetio for polyomials). I. Selesick EL 73 Lecture Notes 2 (5)

3 >> p = [ ]; >> h = x2z(p); >> h*32 as = The frequecy respose of the filter H(z) ca be directly related to P (x) by otig the followig idetity. z + 2 z = z=e jω 2 (ejω + e jω ) = 2 2 cos(ω) Now the frequecy respose of the filter is give by evaluatig P (x) appropriately. H f (ω) := H(e jω ) ( = P 2 ) 2 cos(ω) Therefore, the frequecy respose of the filter obtaied via (2) is similar to the graph of P (x). The oly differece is a warpig of the idepet variable, give by, x = 2 2 cos(ω). (6) This warpig is show i the followig figure for ω π. I. Selesick EL 73 Lecture Notes 3

4 cos(ω) π.5π.75π π ω Note that x ad ω have the correspodig values: ω x ω = x = ω = π x = For the polyomial P i (), the graph of P (x) ad the frequecy respose H f (ω) are show i the followig figure. 2.5 P(x) x 2.5 H f (ω) ω/π Notice that the frequecy warpig is most proouced ear the poits of the iterval [, ]. I. Selesick EL 73 Lecture Notes

5 We ca also trasform ay Type I FIR filter ito a polyomial, by reversig the procedure. performs the reverse trasformatio. fuctio p = z2x(h) N = legth(h); M = (N-)/2; P = []; g = ; for k = :M g = cov(g,[- 2 -]/); for k = :M [q,r] = decov(h,g); p(m+-k) = q; h = r(2:-); g = decov(g,[- 2 -]/); The followig Matlab program z2x We ca illustrate that the programs z2x ad x2z are iverses of oe aother as i the followig example. >> % check that z2x(x2z(p)) gives p >> p = [ ]; >> z2x(x2z(p)) as = >> % check that x2z(z2x(h)) gives h >> h = [ ]; >> x2z(z2x(h)) as = I. Selesick EL 73 Lecture Notes 5

6 MAXIMALLY FLAT TYPE I FIR FILTERS A maximally flat Type I FIR filter is a filter of miimal legth satisfyig the costraits: H f () = (7) d i H f (ω) dω i =, i =,..., 2L + (8) ω= d i H f (ω) dω i =, i =,..., 2K (9) ω=π Note that the odd derivatives of H f (ω) are automatically zero at ω = ad ω = π for Type I filters. These derivative costraits represet a set of liear equatios for the filter coefficiets h(). By usig the trasformatio x = 2 2 cos(ω) the problem of obtaiig the solutio is simplified. Istead of fidig h(), we ca istead fid a polyomial P (x) which ca be coverted ito a Type I FIR filter. I this case, the problem is to fid a polyomial P (x) of miimal degree satisfyig the costraits: P () = () P (i) () =, i =,..., L () P (i) () =, i =,..., K (2) Equivaletly, P (x) ca be factored as ad P (x) ca be factored as P (x) = A(x) ( x) K (3) P (x) = B(x) x L+. () I. Selesick EL 73 Lecture Notes 6

7 Combiig (3) ad (), we have = A(x) ( x) K + B(x) x L+. (5) Oe method for obtaiig the polyomials A(x) ad B(x) of miimal degree is to use the exted Euclidea algorithm for polyomial GCDs. From the Euclidea algorithm, it is kow that the degrees of A(x) ad B(x) are L ad K respectively. Therefore, the degree of P (x) is L + K. Aother method to obtai A(x) ad B(x) is to rewrite (5) as A(x) = B(x) xl+ ( x) K. ad to the write /( x) K as a Taylor series aroud x = We the have ( x) = K c x. A(x) = ( B(x) x L+) c x. Because the degree of A(x) is L, it must be give by A(x) = L c x. A formula for the costats c is give by the biomial theorem. The biomial theorem states that (a + b) K = where the biomial coefficiet is defied as ( ) K K! = (K )!! ( ) K a K b (6) I. Selesick EL 73 Lecture Notes 7

8 for K, Z. If K is egative, we ca still use the biomial theorem, but we eed to kow how to ext biomial coefficiets to egative values of K. The followig idetity specifies how biomial coefficiets ca be evaluated whe the upper etry is egative []. ( ) ( ) K K + = ( ) Usig this idetity i (6) we get (a + b) = ( ) K a K b K = ( ) ( K + b ( ) a K a Settig a = ad b = x, we get a expressio for /( x) K. ( x) = ( ) K + ( ) ( x) (7) K = ( ) K + x. (8) Therefore, ad A(x) = L ( ) K + x P (x) = ( x) K ) L ( ) K + x. (9) Usig the chage of variables (), we obtai the trasfer fuctio ( ) z z K L ( ) ( ) K + z + 2 z H(z) =. I. Selesick EL 73 Lecture Notes 8

9 Notice that we ever determied B(x). It was ot eeded. With the idetities, z z ( ω ) 2 = cos z=e jω 2 z + 2 z ( ω ) 2, = si z=e jω 2 the frequecy respose H(e jω ) ca be writte as ( ( ω )) 2K L ( ) K + ( ( ω )) 2 H(e jω ) = cos si. 2 2 The followig pseudo program computes the impulse respose h(). Q(z) H(z) for from to L ( ) z + 2 z Q(z) Q(z) ( ) K + H(z) H(z) + Q(z) for k from to K H(z) H(z) ( ) z z If we call c = ( ) K+, we ca ote that the ratio c /c is give by c (K + )! ( )! (K )! = = K +. (2) c! (K )! (K + 2)! I. Selesick EL 73 Lecture Notes 9

10 Therefore, the pseudo program ca be modified to iclude the biomial coefficiet computatio, as follows. Q(z) H(z) c for from to L Q(z) Q(z) c c (K + )/ ( ) z + 2 z H(z) H(z) + c Q(z) for k from to K H(z) H(z) ( ) z z The followig Matlab program maxflati implemets the pseudo program. The vector h is the impulse respose of the filter. fuctio h = maxflati(k,l) h = ; q = ; c = ; for = :L q = cov(q,[- 2 -]/); c = c*(k-+)/; h = [ h ] + c*q; for k = :K h = cov(h,[ 2 ]/); I. Selesick EL 73 Lecture Notes

11 Example Whe K = 3, L = 5, the impulse respose h(), frequecy respose magitude H f (ω), ad zero diagram are as show. Notice that exactly 2L zeros that cotribute to the shape of the passbad, ad exactly 2K zeros are located at z =..7 MAXIMALLY FLAT TYPE I FIR FILTER ZERO DIAGRAM I. Selesick EL 73 Lecture Notes

12 Geeral Stopbads The procedure used to obtai maximally flat passbads ca be modified to iclude zeros located at poits other tha z =. For example, suppose it is desired that H f (ω s ) = where the ω s is a specified frequecy i the stopbad. The resultig filter will be maximally flat i the passbad, but ot i the stopbad. We ca agai costruct a polyomial P (x) which ca the be coverted ito a Type I FIR filter. Modifyig the costraits o P (x), we have where P (x) = A(x) ( x) K (x s x) x s = 2 2 cos ω s ad A(x) is a polyomial (of degree L) to be determied. before, the derivative costraits at x = ca be icluded by the coditio that P (x) factors as Combiig, we have or P (x) = B(x) x L+. = A(x) ( x) K (x s x) + B(x) x L+ A(x) = B(x) xl+ ( x) K (x s x). If we kow the Taylor series of /[( x) K (x s x)] aroud x =, ( x) K (x s x) = f x, I. Selesick EL 73 Lecture Notes 2 As

13 the we ca write, as before, A(x) = ( B(x) x L+) = f x B(x) x L+ f x f x. Because we kow that the polyomial A(x) if of degree L, ad because the secod term cotais terms of powers greater tha L oly, A(x) must be give by ad P (x) is give by A(x) = L f x (2) P (x) = ( x) K (x s x) L f x. To fid the Taylor series expasio of /[( x) K (x s x)] we ca use ad to write ( x) = ( ) K + 2 c K x, c = x s x = ( d x, d = ( x) K x s ) + x s x = c i x i d j x j i= j= [ ] = c m d m= } {{ } f x I. Selesick EL 73 Lecture Notes 3

14 Therefore, we ca fid the coefficiets f by covolvig c ad d, f = c d. I (2) we oly eed f for L. To compute f over this rage, we must compute c, d for L also. The covolutio of those two fiite-legth sequeces will be of legth 2L +. The values f are obtaied by trucatio dow to legth L +. Example The followig Matlab program implemets this procedure where we use K = 3, L = 5 as i the previous example. frequecy ω s is take to be ω s =.7 π here. K = 3; L = 5; ws =.7*pi; xs = /2 - cos(ws)/2; = :L; c = biom(k+-2,); d = (/xs).^(+); f = cov(c,d); a = f(:l+); k = :K-; p = cov(a,biom(k-,k).*(-).^k); p = cov(p,[xs -]); h = x2z(p); The The followig figures illustrate the results. It ca be see that there are 2(K ) zeros at z = ad 2L zeros correspodig to the passbad. I. Selesick EL 73 Lecture Notes

15 .6 TYPE I FIR FILTER (MAXIMALLY FLAT PASSBAND) ZERO DIAGRAM This procedure is quite geeral. It calls oly for the Taylor series expasio of a certai fuctio (associated with the stopbad). The program biom, for computig biomial coefficiets, is give at the of these otes. I. Selesick EL 73 Lecture Notes 5

16 Ad Geeral Passbads I additio to specified zeros i the stopbad, it may be desired that a flat filter have poits of uity gai i the passbad, at frequecies other tha ω =. For example, we might wat H f (ω) to have a double root at a frequecy ω p. I this case we ca write the coditio as P (x) = B(x) x L (x p x) 2 where x p = 2 2 cos ω p. I this coditio, we have reduced the degree of flatess at x = correspodigly. Placig a zero at x s as i the previous sectio, we have agai P (x) = A(x) ( x) K (x s ). Combiig, we have = A(x) ( x) K (x s ) + B(x) x L (x p x) 2. The procedure described before will ot work here, so lets compute A(x) ad B(x) together by solvig a liear system of equatios. Defiig, R(x) := ( x) K (x s ) (22) S(x) := x L (x p x) 2, (23) we have: = A(x) R(x) + B(x) S(x). I. Selesick EL 73 Lecture Notes 6

17 We ca write this i terms of matrices as r s.... r a.... s. = r K. a L s L+.... r K... s L+ }{{}}{{}}{{} e R S or [ R S ] [ ] a = e. b b. b K This is a square system of equatios, so we ca solve for A(x) ad B(x) directly. Example The followig Matlab program implemets this procedure, with ω p =.5π ad ω s =.75π, K = 3 ad L = 5. The resultig filter is show i the figure. K = 3; L = 5; wp =.5*pi; ws =.75*pi; xp = /2 - cos(wp)/2; xs = /2 - cos(ws)/2; r = cov(biom(k-,:k).*(-).^(:k),[xs -]); r = r(:k+); s = [zeros(,l-) xp^2-2*xp ]; R = covmtx(r,l+); S = covmtx(s,k); V = [R S]; T = V\[; zeros(l+k,)]; a = T(:L+) ; b = T(L+2:L+K+) ; I. Selesick EL 73 Lecture Notes 7

18 p = cov(a,r); h = x2z(p);.8 TYPE I FIR FILTER ZERO DIAGRAM I. Selesick EL 73 Lecture Notes 8

19 MAXIMALLY FLAT TYPE II FIR FILTERS As we have see, if H (z) is a Type I trasfer fuctio, the H f (ω) ca be writte as H f (ω) = P where P (x) is a polyomial, ( 2 ) 2 cos(ω) P (x) = p + p x + + p M x M. A Type II trasfer fuctio H 2 (z) ca always be writte as H 2 (z) = 2 ( + z ) H (z) where H (z) is a Type I trasfer fuctio. Note that H f 2 (ω) = 2 ( + e jω ) H f (ω) = 2 e jω/2 ( e jω/2 + e jω/2 ) H f (ω) = e jω/2 cos(ω/2) H f (ω) = e jω/ cos(ω) Hf (ω). (Note double agle formula.) Now if we defie P 2 (x) as P 2 (x) := x P (x) = x ( p + p x + + p M x M) the ( P 2 2 ) 2 cos(ω) = 2 + ( 2 cos(ω) P 2 ) 2 cos(ω) = cos(ω) Hf (ω) = e jω/2 H f 2 (ω) I. Selesick EL 73 Lecture Notes 9

20 Therefore, where H f 2 (ω) = e jω/2 x P (x) (2) x = 2 2 cos(ω). Therefore, to obtai a maximally flat Type II FIR filter H 2 (z) we ca first obtai a polyomial P (x) such that P 2 (x) = x P (x) is maximally flat, ad the covert P (x) to H (z) ad set H 2 (z) = 2 ( + z ) H (z). I this case, the problem is to fid a polyomial P (x) of degree L + K satisfyig the costraits, P 2 () = (25) P (i) 2 () =, i =,..., L (26) P (i) 2 () =, i =,..., K (27) where P 2 (x) = x P (x). Equivaletly, P 2 (x) ca be factored as P 2 (x) = A(x) x ( x) K ad P 2 (x) ca be factored as P 2 (x) = B(x) x L+, where A(x) is a polyomial of degree L. Combiig the coditios, we have = A(x) x ( x) K + B(x) x L+ or A(x) = B(x) xl+. ( x) K+ 2 I. Selesick EL 73 Lecture Notes 2

21 If we write /( x) K+ 2 as a Taylor series aroud x =, the we have ( x) K+ 2 = c x A(x) = ( B(x) x L+) c x = L c x + higher terms. Sice A(x) is of degree L, we have ad A(x) = L c x P (x) = ( x) K L c x. It oly remais to fid the values c i the Taylor expasio of /( x) K+ 2. We ca use the equatio (8) agai, with K + 2 i place of K, ( x) K+ 2 = ( K ) x. The we have P 2 (x) = x ( x) K ( K ) x ad H 2 (z) = 2 (+z ) ( ) z z K L ( K ) ( ) z + 2 z. I. Selesick EL 73 Lecture Notes 2

22 To evaluate the biomial coefficiet for fractioal values of the upper etry, we ca use the Gamma fuctio Γ. The Gamma fuctio iterpolates the factorial fuctio o the itegers, Γ( + ) =! for Z, so we have ( K ) = Γ(K ) Γ(K + 2 ) Γ( + ). The Gamma fuctio also satisfies Γ(α + ) = α Γ(α), so if we defie c() := ( K+ 2 + ) the we ca write the ratio c()/c( ) as c() c( ) = Γ(K ) Γ(K + 2 Γ(K + 2 ) Γ( + ) ) Γ() Γ(K ) = K Also ote that c() = (usig the Taylor series rule for c()). With the rule c() = K + 2 c( ) (28) for > we ca the compute the values c() which are eeded. The followig pseudo program computes H 2 (z). Note the similarity with the pseudo program for the desig of maximally flat Type I I. Selesick EL 73 Lecture Notes 22

23 filters. Q(z) H 2 (z) c for from to L Q(z) Q(z) ( ) z + 2 z c c (K + /2)/ H 2 (z) H 2 (z) + c Q(z) for k from to K H 2 (z) H 2 (z) H 2 (z) 2 ( + z ) H 2 (z) ( ) z z The followig Matlab program maxflatii implemets the pseudo program. The vector h is the impulse respose of the filter. fuctio h = maxflatii(k,l) h = ; q = ; c = ; for = :L q = cov(q,[- 2 -]/); c = c*(k-/2+)/; h = [ h ] + c*q; for k = :K h = cov(h,[ 2 ]/); h = cov(h,[ ]/2) I. Selesick EL 73 Lecture Notes 23

24 Example Whe K = 3, L = 5, the impulse respose h(), frequecy respose magitude H f 2 (ω), ad zero diagram are as show. Notice that exactly 2L zeros that cotribute to the shape of the passbad, ad exactly 2K + zeros are located at z =..6 MAXIMALLY FLAT TYPE II FIR FILTER ZERO DIAGRAM I. Selesick EL 73 Lecture Notes 2

25 A PROGRAM FOR BOTH TYPE I AND TYPE II FILTERS Notice that the programs maxflati ad maxflatii are very similar. With a slight modificatio, a sigle program ca be writte to hadle both Type I ad Type II filters. Let us deote the umber of zeros at z = by Z π. The maximally flat Type I FIR filter described above has Z π = 2K zeros at z =. That is, K = Z π /2. Therefore, the rule (2) for computig the values c() ca be writte as c() = K + c() = Z π/2 + c( ) c( ). (29) O the other had, the maximally flat Type II FIR filter has Z π = 2K + zeros at z =. That is, K = Z π /2 /2. Therefore, the rule (28) for computig the values c() ca be writte as c() = K + /2 c() = Z π/2 + c( ) c( ). (3) Because equatios (29) ad (3) are the same whe the parameter Z π is used, the two programs maxflati ad maxflatii ca be combied ito a sigle program. The followig program symflat hadles both Type-I ad Type-II maximally flat FIR filters. I. Selesick EL 73 Lecture Notes 25

26 fuctio h = symflat(zpi,l) % Maximally flat FIR filter (Type-I ad II) % ) Zpi zeros at z=- % 2) 2L zeros away from z=- % 3) legth(h) = Zpi+2L+ % ) If Zp = 2L+2, the h is halfbad h = ; q = ; c = ; for = :L q = cov(q,[- 2 -]/); c = c*(zpi/2-+)/; h = [ h ] + c*q; for k = :Zpi h = cov(h,[ ]/2); I. Selesick EL 73 Lecture Notes 26

27 Appix: A Program for Biomial Coefficiets The followig program for computig the biomial coefficiets ( ) k allows ad k to be scalars or vectors. fuctio a = biom(,k) % % a = biom(,k) % BINOMIAL COEFFICIENTS % % allowable iputs: % : iteger, k : iteger % : iteger vector, k : iteger % : iteger, k : iteger vector % : iteger vector, k : iteger vector (of equal dimesio) % v = ; kv = k; if (legth(v) == ) & (legth(kv) > ) v = v * oes(size(kv)); elseif (legth(v) > ) & (legth(kv) == ) kv = kv * oes(size(v)); a = v; for i = :legth(v) = v(i); k = kv(i); if >= if k >= if >= k c = prod(:)/(prod(:k)*prod(:-k)); else c = ; else c = ; else if k >= c = (-)^k * prod(:k--)/(prod(:k)*prod(:--)); else if >= k I. Selesick EL 73 Lecture Notes 27

28 c = (-)^(-k)*prod(:-k-)/(prod(:-k)*prod(:--)); else c = ; a(i) = c; I. Selesick EL 73 Lecture Notes 28

29 REFERENCES REFERENCES Refereces [] R. L. Graham, D. E. Kuth, ad O. Patashik. Cocrete Mathematics: A Foudatio for Computer Sciece. Addiso-Wesley Publishig Co., 99. [2] O. Herrma. O the approximatio problem i orecursive digital filter desig. IEEE Tras. o Circuit Theory, 8(3): 3, May 97. Also i [?]. I. Selesick EL 73 Lecture Notes 29