Question

I Need Help with 4,6,8,10,15,18

Problems 123 If f(n) is a periodic sequence with period N, it is also periodic with period 2N. Tet 8(k) denote the DFS coefficients of X(n) considered as a periodic sequence with period N and X,(k) denote the DFS coefficients of x(n) considered as a periodic sequence with period 2N. X,(k) is, of course, periodic with period N and X2(k) is periodic with period 2N. Determine 8(k) in terms of X (k). 5. Consider two periodic sequences *(n) and (n). A(n) has period N and yon) has period M. The sequence (n) is defined as w(n) = *(n) + y(n). (a) Show that w(n) is periodic with period MN. (b) Since *(n) has period N, its DFS coefficients (k) also have period N. Similarly, since y(n) has period M, its DFS coefficients Y(k) also have period M. The DFS coefficients of w(n), Wik), have period MN. Deter- mine (k) in terms of X(k) and Y(k). You may find it helpful to refer to the results of Problem 4 above. 6. *(n) denotes a periodic sequence with period N and X(k) denotes its discrete Fourier series coefficients. The sequence X(k) is also a periodic sequence with period N. Determine, in terms of x(n), the discrete Fourier series coefficients of (k). 7. Compute the DFT of each of the following finite-length sequences considered to be of length N. (a) x(n) = (n). (b) x(n) = (n - no), where 0 <n<N. c) x(n) -a", o sn SN-1. 8. In Fig. P3.8 is shown a finite-length sequence x(n). Sketch the sequence x((-n)) x(n) 4 5 6 -3 -2 -1 0 1 2 3 Fig. P3.8 10. Analog data *(n) denotes a finite-length sequence of length N. Show that x((--))x = *((N - ))x alog data to be spectrum-analyzed are sampled at 10 kHz and the DFT of 4 samples computed. Determine the frequency spacing between spectral Samples. Justify your answer. DFT of a finite-duration sequence corresponds to samples of its z-trans- m on the unit circle. For example, the DFT of a 10-point sequence x(n) 11. The DFT 01

Answer 1

44 let aim is a periodic sequence with peñod n Alm+N) = alo) +) also periodic will period an w laten) = alm) (2) let Fick) denotes ofs coefficients of am) Diserte fouries sexies (OFs). NA j kwon m- am] = RFO sa o wo= 2A E cke 3) .. Hore N= 2N. Nal meny ck= { " ] e Ñ a[n where h E ok= ск – ор се јаnt cka * LA periodicity -[cg= ekan Fim) = # 3 x(K) € Ñ for all on where the coefficients xck) are XIK) = { neo seen er all is

Eqbs ☺ and © are called DFS synthesis and analysis pår. Hence x(K), and nen) are periodic sequences FI) - Na zen) - 12 ok for all ik MED -6 k 2 am 1 for all is a TCK) = neo of Xics) is periodic will perod ^N' FCK) is periodic with period in! žem best tylk) To, (k+ v) = wick) hins vegar er HD Žolk+2m) = Fg (K) K) + ) = a ktas) tõlm) = vo (KAN) polo (k+n) is also peñodic 60 1516+ M) = 5)]

ñ (n+N) = ñ in penodic, sequence D=0 FK) = in de mim e os for alle a Slik) is also periodic with periodN xưK+ M) = FCK' . Fiktn) Stehe ato. )o(k+0) no Fm 249) om 2017 - E no 1 for all in values = X(K) . 3 -2 -1 0 1 3 4 7 5 6 The time reversal of a N-point sequence here N=4 ) acho is obtained by wrapping the sequence sing around the circle in the clock wise direction. It is denoted as a [(n) (mod n] alan), med n] = n(Nm] 09 N-) 053 a[1.17 od n] = a[m] = (4-) nao, 11213.

3 1309 aly_n) = aly) nelo al41) = 163) 2=) 1(4-2) = 912) *)=57) a(4-3) = a(lol) to a/m) is given below fs = 10KHz NE 1024 samples? prequency sparing Af= 2 = f Afa lokty Sofafz 9-764 TOKH2 1024 find Ny ft) DK Broto N-) | XSK)- Koi) )? We o to N-) t of .. ,10 n-point sean DET : Of alm o Wlk) DET NY DAT DAT {Alk) y Eyed an ver

gan) = in e xK) --(03. My(n) = DFT X(K) = No 327 OK z e xeln) en - neo L Nt Pint -3(27)mk 2 -32TTOK ar(n) = { } { xD) e W ye hao L320 .. no no 9 20 1. for all valles 8110) = p X(K)