Question

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26. [M] Example 3 in Section 4.8 displayed a low-pass linear filter that changed a signal y) into y+) and changed higher-frequency signal winto the zero signal, where y cos (Tk/4) and wcos(3 k/4). The following cal- culations will design a filter with approximately those prop- erties. The filter equation is a ao y&+2+ay+1+ azyk= Zk for all k Because the signals are periodic, with period 8, it suffices to study equation (8) for k 0..7. The action on the two signals described above translates into two sets of eight equations, shown below: y+2 +1 k=0 .7 7 0 -.7 -.7 -1 0 do 7 -7 -1 -1 7 0 k 7 7 Wk+2 W+ k=0 -7 .7 do 7 -1 .7 -1 .7 7 k 7 0 Write an equation Ax b, where A isa 16x 3matrix formed from the two coefficient matrices above and where b in R is formed from the two right sides of the equations. Find ae, a1 and as given by the least-squares solution of Ax b. (The 7 in the data above was used as an approximation for 2/2, to illustrate how a typical computation in an applied problem might proceed. If .707 were used instead, the resulting filter coefficients would agree to at least seven decimal places with 2/4, 1/2, and 2/4, the values produced by exact arithmetic calculations.)

Answer 1

Ans:- flter with the Paoperty to design The Calculation's а for all k aoyк. *аgуК-ZК Нечe , y ТК — СОS FOY K-O ал сOS A- - сo-(" C Уhal СoS Ук- со5(0) is hen the equation

ind the Co-efticients of the Ik terms in he eq Similaty and wite ithe mainia form as. 0.7 0.7 -0-7 0-1 C37 -O.7 -67 -01 O 7 a-7 O.7 O.7 O O.7 -O.7 -O.1 -O.1 -0.7 o-7 1 O1 O. Use Least Squaeg method in MAPLE to Solve the system in the mataix and theo Now take the value Of / compute the sesult. o T

(with linear Algebya = (0. -07, A - 16 x3 mmaiia any thing Daio type Stoige Recłangul ar OYdex Fartyan oydeN c anspose CA) 3 x 16 mati Data type any thing Storage: sectanguka OYde: Faxtian oide СА 7.98 00000 o 0 O00 00 6.6 0000000 0 000O 5.600000O000000 7.9000000000000 5.600000000o000 5-6000000 000000 poo00000000 08 bL Eо-тъ o, o.,

I. 6 vectox column Data tyPe :any thing Storoge rectangulal Ordex: Fortran.OYde 5.6 o00 000000000o 798 O0000o00 OO00O 5.60000ooo000000 CCA) Matai inve1se 618.686868688448 - 7500000000a833 G18.81313131 u71 -815.6000000oa33 1837 5000000 0316 -875.00060o0oS833 G8.B131313(u7\ - 875.060000 o0aa33 B18.686868 688 448 7 P. CCB) 13 lo ų.5u TU7350 8864 64 13 4.541473 50886u64 lo

Now, take 4he Value of - 0.07 in the matria and oith (Linear algebva hen Соорине thе хesult A = -0.707,-,~0-707, O, 0-101, 1, 0-O7, 0 ,-0.707, -1, о-т0731t - 0.107, o ,0-707, 1> -0-707, - 6707, 6 0.107 O о707, 0, - о707, -t , -о701, о, О-л07. -4,-0 107, o, -O707 O.707, -i оло3) -O.107, 0, 6 x 3 matvia Data ty pe - any thing Storage sectangula Fartran ordel OydeY c TYanspose (A) 3x l6 matix Data type- anything Storage : vecłangulal : For tran- ordel Oide

CA 56560000000O 000 7.9a879800000000 5 65600000000000 5.5600000000000 1,a980ooo00o 7.9987a a0000000o 5.650000000000 ET07, , o, -0-707,-1, -0-107, O, o.707, l, o 107, o, -07o7 o-701 ...16 vectoY column Data type any thing Storage Rectangulal Fortyan,Οrdey OYde 5656 00o O OO O O00 00 1.998792 00 000 O o o o00 mataix inNerse ( CA) P 2.74069399167466 1o 3 8759 3648073&1 lo a1u69386665578 -3.8199 96u8073 891 10 5.48138785833043 16 3815996480S8a 2-1 4o6938666 S518 lo6 -3-8759 8648073gar 16 2.7u06939167UL-

P CCB) 3.1859639846191 16 t600000074508 3-725990a9846191 10 equal Theve-fore he valve of Go and ag