Review Prblems 3 Fur FIR Filter Types Fur types f FIR linear phase digital filters have cefficients h(n fr 0 n M. They are defined as fllws: Type I: h(n = h(m-n and M even. Type II: h(n = h(m-n and M dd. Type III: h(n = -h(m-n and M even. Type IV: h(n = -h(m-n and M dd.. Tw type II filters H (e jw and H (e jw can be expressed as R (w exp(jθ (w and R (w exp(jθ (w respectively. H 3 (e jw is frmed as H (e jw H (e jw. (a Fr H (e jw, give expressins fr R (w and θ (w. (b Express the impulse respnse h 3 (n in terms f h (n and h (n. (c Assciated with H 3 (e jw and h 3 (n are the quantities M 3, R 3 (w, and θ 3 (w. Express θ 3 (w and M 3 in terms f M. Is M 3 even r dd? (d Express R 3 (w in terms f R (w and R (w. Is R 3 (w even r dd? (e What type f filter is h 3 (n?. Linear phase windws fr FIR digital filters can have the same fur types as the impulse respnses described abve prblem. Assume that h d (n is truncated t have (M+ cefficients, and has any f the fur types described abve. (a If h d (n is type II, what type f windw (type I, II, III, r IV shuld be applied, if we want the resulting h(n t als be type II? (b If h d (n is type III, what type f windw (type I, II, III, r IV shuld be applied, if we want the resulting h(n t als be type III? (c If h d (n is a type I lwpass filter and w(n is type III, what is the DC gain f h(n? Des windwing apprximately preserve the shape f the passband f h d (n? 3. Cnsider the type II filter. (a H(e jw can be expressed as R(w exp(jθ(w fr all w, where R(w and θ(w are real functins f w and R(w can be psitive r negative. Give R(w and θ(w. (b Give the zeres f H(z (c If H(e jw = H(e jw exp(jφ(w, give H(e jw in terms f the symbl R(w. (d Cntinuing part (c, give a valid phase φ(w in terms f the symbls R(w and θ(w such that φ(w is an dd functin. (e What types f windws (I, II, III, and/r IV can be applied t the type II filter?
4. Cnsider the type II filter. (a Is h(n a shifted versin f a zer-phase impulse respnse? (b H(e jw can be expressed as R(w exp(jθ(w fr all w, where R(w and θ(w are real functins f w and R(w can be psitive r negative. Give R(w and θ(w. (c Give the zeres f H(z (d If H(e jw = H(e jw exp(jφ(w, give H(e jw in terms f the symbl R(w. (e If H(e jw = H(e jw exp(jφ(w, give a valid phase φ(w in terms f the symbls R(w and θ(w such that φ(w is an dd functin. (f What types f windws (I, II, III, and/r IV can be applied t the type II filter? 5. Given the type II filter f prblem 4, (a Find an efficient expressin fr y(n in terms f x(n, by maing use f the symmetry cnditin: h(n = h(m-n and M is dd. (b In the cnvlutin pseudcde belw fr this filter, allwed values fr n in x(n range frm 0 t x. Allwed values fr n in y(n range frm 0 t y. Give y in terms f x and M. (c Give A and B (d Give expressins fr C and D. Fr 0 n y y(n = A Fr 0 B C = D = If(0 C x y(n = y(n + h(x(c If(0 D x y(n = y(n + h(x(d 6. A highpass type II filter H (e jw with parameter M has cut-ff frequency w and can be expressed as H (e jw = R (w exp(jθ (w. A lwpass type III filter H (e jw with parameter M has cut-ff frequency w can be expressed as H (e jw = j R (w exp(jθ (w. H 3 (e jw is frmed as H (e jw H (e jw. (a Fr H (e jw, give expressins fr R (w and θ (w in terms f M. (b Fr H (e jw, give expressins fr R (w and θ (w in terms f M. (c Assciated with H 3 (e jw and h 3 (n are the quantities M 3, R 3 (w, and θ 3 (w. Express θ 3 (w and M 3 in terms f M and M. Is M 3 even r dd? (d Express R 3 (w in terms f symbls R (w and R (w. Is R 3 (w even r dd? (e What type f filter (LP, BP, HP, r BR is h 3 (n if w < w?
7. Given the type IV filter impulse respnse, h(n, (a Find an efficient expressin fr y(n in terms f x(n, by maing use f the symmetry cnditin: h(n = -h(m-n and M is dd. (b In the cnvlutin pseudcde belw fr this filter, allwed values fr n in x(n range frm 0 t x. Allwed values fr n in y(n range frm 0 t y. Give y in terms f x and M. (c Give A and B (d Give expressins fr C and D. (e Give an expressin fr E Fr 0 n y y(n = A Fr 0 B C = D = If(0 C x y(n = y(n + h(x(c If(0 D x y(n = y(n + E x(d 8. The utput y(n is fund as y(n = h(n*x(n where h(n is a type II filter. (a Find an efficient expressin fr y(n in terms f x(n, by maing use f the symmetry cnditin: h(n = h(m-n and M is dd. (b In the cnvlutin pseudcde belw fr this filter, allwed values fr n in x(n range frm 0 t x. Allwed values fr n in y(n range frm 0 t y. Give y in terms f x and M. (c Give A and B (d Give expressins fr C and D. Fr 0 y y(n = A Fr 0 (M-/ Fr 0 m x Z = h(bx(c y(+m = y(+m + Z y(d = y(d + Z
9. Linear phase windws fr FIR digital filters can have the same fur types as the impulse respnses described abve prblem. Assume that h d (n is truncated t have (M+ cefficients, and has any f the fur types described abve. (a If h d (n is type II, what type f windw (type I, II, III, r IV shuld be applied, if we want the resulting h(n t als be type II? (b If h d (n is type III, what type f windw (type I, II, III, r IV shuld be applied, if we want the resulting h(n t als be type III? (c If h d (n is a type I lwpass filter and w(n is type III, what is the DC gain f h(n? Des windwing apprximately preserve the shape f the passband f h d (n?
FIR Filter Design Using Windws. An FIR filter impulse respnse can be fund thrugh windwing as h(n = h d (nw(n. (a Give an expressin fr H(e jw in terms f H d (e jw and W(e jw. (b Find h(n in terms f h d (n if P is even and P/ jw W( e = + cs(π n/p cs(wn (c Hw many cefficients des h(n have? (d What ind f phase is H(e jw liely t have? ( linear r zer n=. A chirp signal has the frm x(n = sin(a n fr 0 n x.fr large n, we can write n = n + i where n is cnstant, i varies, and i << n. (a If we want t rewrite x(n as sin(w (n n, give an expressin fr w (n in terms f a and n. (b A zer-phase FIR lwpass filter with cut-ff frequency w c is applied t x(n, yielding y(n. If y(n 0 fr n > x /, give w c in terms f a and x. (c If h(n is nn-zer fr n M/, give an apprpriate Hamming windw fr the filter f part (b. 3. A type I FIR filter impulse respnse can be fund thrugh windwing as h(n = h d (nw(n. (a Give an expressin fr H(exp(jw in terms f H d (e jw and W(e jw. (b H d (e jw can be expressed as R(w exp(jθ(w and W(e jw can be expressed as S(w exp(jφ(w. If h d (n = h d (M-n and w(n = w(m-n, find expressins fr θ(w and φ(w. R(w and S(w may be psitive r negative. (c. Using the results f part (b, simplify yur expressin frm part (a. 4. A causal, linear-phase bandpass FIR digital filter h(n is t be designed with cut-ff frequencies f.3 radian and.3 radians, and with a phase f -3w. (a Find an expressin fr h d (n using the inverse DTFT. (b Find the filter's time delay in samples. (c If h d (n is windwed t get h(n, find the largest pssible value fr. (d Give the apprpriate Hamming windw fr the length- causal filter f part (c.
5. Fr w π, the desired amplitude and phase respnses, fr a causal FIR digital filter with cefficients, are as fllws: jw 4 π 4 π H ( e = r( w + r( w d φ ( w = 64 w + sin(5 w d π π 4 π Here, r( is the cntinuus ramp functin. (a Ling at the linear part f the desired phase, what is an apprpriate value fr? (b What is the filter s cut-ff frequency? (c The pseudcde belw uses the inverse DFT r FFT t generate h(n. Give the crrect value fr W, in the first line f cde. (d Give the crrect expressin fr X, in terms f, in the secnd line f cde. (e Give the crrect expressin fr Y, in the third line f cde. (f Give the crrect expressin fr Z, in the fifth line f cde. H(0 = H(exp(jW Fr X w( = Y H( = H(exp(jw( H(- = Z h(n = DFT - {H(}
IIR Frequency Selective Filter Design. A prttype Butterwrth lwpass filter, with a cut-ff frequency f radian/sec, has the transfer functin, (s= s H + s + (a Find the impulse respnse f H (s. (b Using impulse invariance and assuming T=, find H(z s that the digital filter cut-ff frequency is radian. (c Find H(z frm H (s as a functin f the sampling perid T, using the bilinear transfrm, assuming n pre-warping. (d Fr the filter H(z f part (c, give the cut-ff frequency in radians as a functin f T.. A prttype Butterwrth lwpass filter H (s with even rder m has n real ples. The ples therefre cme in cmplex cnjugate pairs. The first m/ stable ples are s = e jθ(. (a Give an expressin fr b in terms f θ( if H (s is written as H (s = m/ = ( s + b s + (b Fr the crrespnding Chebyshev versin f H (s, which is m/ = s H ' (s = c ( + d s + e give expressins fr d and e in terms f β and θ(, remembering that β multiplies the real part f e jθ(. (c H (s frm part (b abve can be transfrmed int the lwpass filter H a (s = f ( + g + h m/ = s s Give expressins fr f, g, and h in terms f c, d, e, and the desired H a (s cut-ff frequency Ω c.
3. A 4 th rder prttype Butterwrth lwpass filter H (s has n real ples. The ples therefre cme in cmplex cnjugate pairs. The first tw stable ples are s = e jθ( fr = and. (a Give an expressin fr a in terms f θ( if H (s is written as H (s = ( + a s + = s (b Fr the crrespnding Chebyshev versin f H (s, which is b H ' (s = = ( s + d s + c give expressins fr c and d in terms f β and θ(, remembering that β multiplies the real part f e jθ(. (c H (s frm part (b abve can be transfrmed int the bandpass filter H ( + i + h + g s + f e s a(s = 4 3 = s s s Give expressins fr e f, g, h, and i in terms f the symbls b,c, d, B w, and (Ω. Give expressins fr B w and (Ω in terms f the desired H a (s cut-ff frequencies Ω c and Ω c. 4. A prttype Butterwrth lwpass filter H (s with even rder n has n real ples. The ples therefre cme in cmplex cnjugate pairs. The first n/ stable ples are s = e jθ(. (a Give an expressin fr θ(, fr = t n/. (b Give an expressin fr b if H (s can be written as n/ H (s= = ( s + b s + (c H (s can be transfrmed int the bandpass filter n/ c s H a(s= 4 3 = ( s + d s + e s + f s + g Assume that H a (s is t be transfrmed t H(z using the bilinear transfrm, and that H(z is t have cut-ff frequencies w c and w c, in radians. Find expressins fr H a (s cut-ff frequencies Ω c and Ω c in terms f w c, w c, and T. Give expressins fr Ω and Bw in terms f Ωc and Ωc.
5. We want t explre an apprach fr generating the ples f the bandpass filter H a (s frm the ples f H (s. Assume that the ples f H (s have the frm exp(jθ( where θ( = θ + (- θ. (a Give expressins fr θ and θ if H (s is a Butterwrth prttype filter with an integer rder n. (b Given the th ple f H (s, what equatin must be slved t get the resulting tw ples f H a (s? The equatin's cefficients must be functins f B w and Ω. (c If H a (s is t be an dd-rdered Butterwrth bandpass filter, what can we say abut the rder f H (s? (d Is it pssible t design an dd-rdered Butterwrth bandpass filter, using a prttype filter H (s? 6. A prttype Butterwrth lwpass filter H (s with dd rder n can be transfrmed int the bandpass filter H a (s= ( s c + f s s + g (n-/ = which can then be transfrmed int the filter - ( j z + m H(z= - ( q z + r - z + n - z + s (n-/ = ( s ( h ( 4 + d c 3 s + e s s + f s + g -4-3 - - z +i z + j z + m z + n -4-3 - - z + p z + q z + r z + s using the bilinear transfrm with T=. (a Give expressins fr j, m, n, q, r, and s in terms f c, f, and g. (b Give expressins fr h, i, j, m, and n.
IIR Frequency Selective Filter Specificatins. A Butterwrth bandpass digital filter is t be designed with an upper cut-ff f w d radians and a lwer cut-ff f w d radians, using the bilinear transfrm. We want the amplitude respnse f the filter t be dwn by X db at a frequency f w d3 radians. (a Express Ω c, Ω c, Ω c3, Bw, and Ω fr H a (s as functins f w d, w d, w d3, and T. (b Express Ω, Ω, and Ω 3, fr the prttype filter H (s, in terms f Ω c, Ω c, Ω c3, Bw, and Ω. (c Find the rder n f the filter H(z, as a functin f X and any ther relevant quantities frm part (b.. A Butterwrth lwpass digital filter is t be designed with a cut-ff f w d radians, using the bilinear transfrm. We want the amplitude respnse f the filter t be dwn by at least X db at a frequency f w d radians. (a Express Ω c and Ω c fr H a (s as functins f w d, w d, and T. (b Find the minimum rder n f the filter H(z, as a functin f X and any ther relevant quantities frm part (a. 3. An IIR Butterwrth lwpass digital filter is t be designed with a cut-ff frequency f Ω dc = 5 radians/sec. Its amplitude respnse is t be dwn at least 40 db at a frequency f Ω d = 5 0 radians/sec. (a Find the crrespnding frequencies Ω c and Ω c fr the lwpass filter H a (s if the sampling rate in radians per secnd satisfies Ω s >> Ω d. (b Given the assumptin f part (a, what minimum rder shuld the filter have?
IIR Filters. H(z is written as H(z = (l + + j + + ( + + + + -4-3 - - z z z m z n -4-3 - - = z p z q z r z s where H(z can be factred initially as H (z H (z and s = s =. H (z is the part f H(z with =. As written abve, H(z is causal and stable, with H (e jw = H (e jw e jφ(w and H (e jw = H (e jw e jφ(w. Hwever, we will replace z - by z in H (z t get H (z -. H (z - is nw stable and anti-causal. (a Assuming that H (z has input x(n and utput y (n, write the difference equatin fr y (n in terms f x(n. (b Assuming that H (z - has input y (n and utput y(n, write the difference equatin fr y(n in terms f y (n. (c Give the phase f the nn-causal filter H(z = H (z H (z -, in terms φ (w and φ (w.. A digital filter h(n is t be designed as h(n = h (n h (n, where all three impulse respnses are real. (a Express H(e jw in terms f H (e jw and H (e jw (b If H (e jw and H (e jw have the same ideal lwpass amplitude respnse with cut-ff frequency w c < π/, and phases φ (w = φ (w = -.5(- w, setch the magnitude respnse and give the phase respnse φ(w f the filter H(e jw. (c Suppse that H (e jw and H (e jw exist, and are described as jw jw H (e =, jw (e = H jw.5e.5e Assuming again that h(n = h (n h (n, give the impulse respnse h(n. (d Cntinuing part (c, give the frequency respnse H(e jw.
3. The transfer functin H(z f prblem 5 can be written as H (z H (z H (z H (n-/ (z where H (z is the secnd rder sectin and the H (z sectins are 4th rder. (a The 0-rder denminatr cefficients s and s must equal befre the filter can be applied. Give the cnstant K which, when multiplied by all numeratr and denminatr cefficients in H (z, cnverts s t. Give the cefficient K which des the same thing fr s in H (z. (b The input and utput transfrms fr the cascade sectin H (z can be dented as X (z and Y (z respectively. Give the difference equatin that relates x (n and y (n. Give the difference equatin that relates x (n and y (n. (c Give x (n in terms f x(n and x (n in terms f y m (n, and the final utput y(n in terms f yi(n. 4. H(z is designed by applying the bilinear transfrm t H a (s, where T=. (a Give the expressin fr Ω in terms f w. (b If H a (jω can be expressed as H a (jω exp(jφ a (Ω, give the phase respnse φ(w f H(z. (c Assuming that an allpass digital filter A(z has the phase respnse φ ap (w and an amplitude respnse f fr all w, give the allpass filter s frequency respnse. (d If A(z crrects the phase f H(z s that the phase respnse f A(zH(z is φ d (w, give φ ap (w in terms f φ d (w and φ a (. 5. The impulse respnse f an analg filter is h c (t = e -t sin(tu(t. A digital filter H(z is t be designed using impulse invariance. (a Give h(n as a functin f T. (b Give H(z in its final frm such that all cefficients are real.
Bilinear Transfrm. The standard bilinear transfrmatin, s = T + z z can map circles in ne dmain t circles in the ther dmain. Given values n the unit circle in the s-dmain, we want t find parameters f the resulting z-dmain circle. (a Give z as a functin f T and s. (b Give tw real values f s which are n the unit circle in the s-plane. (c Given the tw values f s in part (b, find the crrespnding values f z. (d Assume that a line, cnnecting the tw z values f part (c, passes thrugh the z-plane circle's center. Find the z value at the circle s center. (e Assume that a line, cnnecting the tw z values f part (c, passes thrugh the z-plane circle's center. Find the radius f the z-plane's circle.. The transfrmatin, z=(d+s/(-s, can map a digital filter H(z int an analg filter H a (s. (a Slve fr s in terms f d and z. (b What part f the s-plane is mapped t the unit circle f the z-plane by the transfrmatin? (c If d=/, des this transfrmatin map stable digital filters int stable analg filters? (answer always, never, r smetimes (d If d=, des this transfrmatin map stable digital filters int stable analg filters? (answer always, never, r smetimes 3. The transfrmatin, z=(5+s/(-s, can map a digital filter H(z int an analg filter Ha(s. (a What part f the s-plane is mapped t the unit circle f the z-plane by the transfrmatin? (b Des this transfrmatin map stable digital filters int stable analg filters? (answer always, never, r smetimes
4. The cntinuus-time lwpass filter H c (s has cut-ff frequency Ω c. This filter is transfrmed t a lwpass discrete-time filter with cut-ff w p by substituting (-z - /(+z - fr s. H c (s is transfrmed t a highpass discrete-time filter with cut-ff w p by substituting (+z - /(-z - fr s. (a Find Ω c in terms f w p. (b Find Ω c in terms f w p. (c Find a simple relatinship between w p and w p. 5. The standard bilinear transfrmatin, s = T + z z can map circles in ne dmain t circles in the ther dmain. Given values n a circle f radius R in the s-dmain, as s = R e jθ, we want t find parameters f the resulting z-dmain circle. (a Give z as a functin f T and s. (b If s = R e jθ, where θ can vary, give tw real values f s. (c Given the tw values f s in part (b, find the crrespnding values f z. (d Assume that a line, cnnecting the tw z values f part (c, passes thrugh the z-plane circle's center. Find the z value at the circle s center. (e Assume that a line, cnnecting the tw z values f part (c, passes thrugh the z-plane circle's center. Find the radius f the z-plane's circle.
Advanced Material. A straight line y = a + b x is t be fit t sme data pints {( x p, t p } where p v. The mean-squared errr t be minimized is v E = ( t p ( a + bxp v p = (a Give expressins fr g a = E/ a and g b = E/ b. (b Given symbls m t, m x, r, and c defined as m t = m x = c = v v v v p= v v p= p= t p x x t, p p p,, v r = xpxp v p= refine yur expressins fr g a and g b in terms f these symbls. (c If steepest descent is used t update estimates fr a and b, give the equatins fr the updates in terms f symbls g a, g b and B.