Question

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2. (i) Find the energy and the average power of the following signals. Also, state whether each signal is an energy or a power signal. (Remember: you need to give three answer to each signal: its energy, average power and its classification). (12 points) a. X[n] = eja/4n b. x[n] = 2-In (ii) Determine whether the following signals are periodic or not. If the signals are periodic, determine their fundamental period. (12 points) a. x[n] = (-1)" + ej311n/4 + ej7an/5 b. x[n] = ej0.6n

Answer 1

This is a problem of Signals and Systems whose solution has been provided below.

02 0 x [n] = ejton Energy of a signal is given by, bahal? 2 lejny E h- no = E= - ho UJ E = 0 Power of a signal is given by lim A when by- Pa lim (ew. 2. 1x6.31') Nis the Time period of sc [n], which can be abtained N = N K 25 = N N = 8 # Wo N=8 N (2N+1)= 1 p= lim. n=-N N>00 INA 2N+I N> 0 Power I Watts O PLO 4 Power is ite, Energy is infinite It is a Power Signal.

x [n] = 2 This can be written as x[n] = 2h -00 <n<-1 och Lo - n=0 Energy of signal , E= 5 xG)/2 Elen)? + E 6-)? Ž 22h E-11-27 o]+2+{2-2 +24 T E= x 2 + e-2n n=-00 ho +2 4+2 6+ +2 +---

=) - E= [1 + 2-2+2-4+26 _ _]+[1+2 ³+2 4+_o] Sum of with common factor & 8 E E= 1 1-2-2 [ a GP series - 2 % for stems = Joules. 월 3 Now, power of signal is Ž 1x[n]/? p = lim. N-00 ANH n=-N

-2n p= lim. I NO 2N+1 > - 2n N p=lim. NO 2N+I 21 + 2+ 3 2 2n h=1 > -2 -2 +2 " + p=lim. N 00 QN+1 N tems) + (1+2 +2 4+ N' tems) have a Cp but with Ntems, whose Again we sum 1-IN is given by 1-oh -ZN [ 1-2-20 P= lim. I No 2N+1 +1-2- 1-2-2 (1-2-2N) x4 p= lim. I No. 2N+1 p= lim Noo ANH I (1-2-2) Х 0 -- > P= X $(1-0) Power =0 Energy is finite & Power iso Energy Signal. It is an

jun x [n]= (1)h tejern tet n (COSTT + ED" can be written as, -jsin) x[n]= eith + ej 37 Note: for discrete signals, if individud signals periodic, then their sum also periodic. h tej zun are is 2T Wo, must be a rational value, 2 NL 2# TT zy J- ... ejth is periodic with period, N,=2. for, ejarah 2T Wor N₂ = 2# 37 K2 K2 Ny = 9 (which is rational) is periodic with period, N =P Nej 3 h for edun 2T Woz K3

e j71n is not periodic. x [n] is a non Hence persadic signal. 6 x [h] = ejooon Here, RT Wo N K 2T 0.6 N K (issational) = Therefore, a[n] is non-periodic signal.