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Question 3 Soalan 3) Evaluate the signal in Figure 1 and express its Fourier coefficient to filter the signal by its k- th harmonic State the characteristics of the obtained signal. Plot the magnitude and phase spectrum for -2 <k < 2. Maikan isyarat dalam Rajah 1 dan wykapkan pekali Fouriennya untuk menapis isyarat melalui harmonikke- k. Nyatakan ciri-ciri syarat yang dihasilkan. Perkar spektruwwwwapnil dan fase-2 <k <2./ x[] 0 Figure 1 Rajah 11

Answer 1

2[n] 2 1 0 Tu to 4 2 2 -9 - 4 A 5 3 -- - 7 16-5 0 N56 As we can see that a given signal an] repeat aottern after every 6 internal (As we can see o-05 (same palten) forom 6) (pattern repeat) Similarly o 85 (same pattern) from -67 (pattern repeat) There forf Given signal Ind is periodic with period was y[u]s-1, 6 Nowy As we can see from signal x [0] =.), x [1] = 2 , & [2] = 1 x[3] = 0; x [5] = Zoage Signal repeats with some value. x[n] = {1, 2, -1, 0, -1, 2} Periodically repeats Fouries series expresentation of discrete time signalen] j? 2 Il nok x[n Where = farier Coefficient Ckit & x[n] e 12on Echel no For N=6

t k ܠܬ ܓܝܛܪܠܓܘ :ܝ 2 3 नाम --- wo e [cas (TT- 11/2) = -1/3 & sin (T-T/3) = 1/3] 0 b=0/19 siuf!4)_ - Jiak e W cos cock 14 -3 sin ( 4 € From eg. 3 z [C 6 1 I f2 cos 6 sf [ f + 8 2 (W/ + W5 I c w B taw) (43 k) - j sin (1/3 k) + cos (75 k) tj sin (5) est - cos (1/3 k) - I sin (1/3 k) - cos (15 k tisinco Cas & [+ + 2 (2 cos ( 153 k) 6 - 1.8 (-2cos (173 K ] & [st u cos (IK) + 2 cos (tk)] Ga I [1 + 6 cos (1 k) 6 6 6 Ck s It Cos 6 cas (14) (k = . C-k

* Magnitude and phase spectrum for -2<k<2 Magnitude Phase k satcos (II k) 1 tcos (ORT (2) = 1 -0.5 k LOPK -2 3 6 1-3 = -2 6 6 3 1 tcos/I (-1) O lk - 1 w/ro 3 ŞI t065 = 1+3 6 6 2 +0.5 su 6 뽐 2 3 1 'Is 7 6 f cost لمعمعع 3 1t1 6 s I t6 6 7 6 1 I tcos 6 I 3 IN 3 n 1+0.5 s 2 6 2 Itcos (21 6 آیا 1-0.5 = 6 3

Magnitude Plot 2/3- 22 न/ A हेर 2 --2 -1