Question

We wish to implement an IIR filter with poles at z = +1,- and zeros at z = 2.-3. Assume that each real multiplier takes Tm = 2 ns and each 2-input real adder takes Ta = 1 ns to complete. In the following, no complex multipliers or adders are to be used. (a) Write down the transfer function H() of this IIR filter in terms of first and second order terms with real-valued coefficients. (b) Sketch the block diagrams of Direct Form-I, II, and the transpose structure and calculate their throughputs. (c) Sketch the block diagram of a 2-stage cascade form structure using a cascade of two Direct Form II structures, and calculate its throughput.

Answer 1

lets find the lIR filter transfer function #(2) = (1-22") (1+37') (1+ m2')(1-3993') (-') 1 - 22 +32° -62² ('+37')(-32 (94e44) +* 7') cos (0) = 20t@je ; i+2662 (i+ 32") (1 - 3 cos (25) + 23) 2 o cos(2) = -Ź +2) = 1+2'-632 In terms of ist & and order (I + ') (1 + Q za real coefficients = tal-622 #(2) = ita'-622

Direct form ® realiziahiom : yn 2 All zero part (numerator) и Pole pase (Denominator critical path includes I multiplies and 3 adders Tmin = 1x2 + 3x1 = 5ns Threghput = = t = 3x10 Direct form-ſ realizations - Tmio Is ain) -6 cristical path includes = imultiplier + Cadder.

Tomin = 1x2 + 6xl = 605 Throughput = =1067 *10%. 60s Transpose of DF-I realizations o change /reverse all arrow directions • Replace all summers by pickup points & Replace all peckup points by summers o Posterchange a,g. ("/p & 0/P). y[n] pe alor] ia rokorting op het critical pathe includes (from delasj to delay) = multiplier thadder - 1xans + 2x ins = cons Throughput = = 25 x 10 = 2.5 x 108 (increased

Transpose structure of DF-I realization y[o 2() t critical path includes = multiplier zadders from delay to delay element Tmes = 1X2 + ax1 . =&ns Throughput = firma = 2.5 xide (increased) T OTES (c) H(2) = (1-22') (1 + 3 2 ). x (1+02') (1+2 3 2 7 SADES 2²) ے > Hi(2) stage. I H₂(z) stage 1

ealisations (c) Cascade of a stages in of gro deos critical path has multeplies & Cadder Trin = 1x2 + 4xl =ons Through put = no = 1.67 x 10°