Question

1.4. Let x[n] be a signal with x[n] = 0 for n < -2 and n > 4. For each signal given below, determine the values of n for which it is guaranteed to be zero. (a) xịn - 3] (b) x[n+ 4] (c) x[-n] (d) x[-n+2] (e) x[-n-2] 1.5. Let x(t) be a signal with x(t) = 0 for t <3. For each signal given below, determine the values of t for which it is guaranteed to be zero. (a) x(1 - 1) (b) x(1 - 1) + X(2-1) (c) x(1 - 1)x(2-1) (d) x(36) (e) x(t/3) Determine the fundamental period of the signal r(t) = 2 cos(10t+1) - sin(41 - 1). .. . -1. j ? - J2710/5 Determine whether or not each of the following continuous-time signals is periodic. If the signal is periodic, determine its fundamental period. (a) x(t) = 3 cos(4t + }) (b) x(t) = e(1-1 (c) x(t) = (cos(2t - )] (d) x(t) = &v{cos(4tt)u(t)} 34. In this problem, we explore several of the properties of even and odd signals. (a) Show that if x[n] is an odd signal, then į x[n] = 0. 7(b) Show that if xi[n] is an odd signal and x2[n] is an even signal, then x[n]X2[n] is an odd signal. (c) Tau 1.48. Let zo be a complex number with polar coordinates (ro, ) and Cartesian coordi- nates (xo, yo). Determine expressions for the Cartesian coordinates of the following complex numbers in terms of xo and yo. Plot the points 20, 21, 22, 23, 24, and 2s in the complex plane when yo = 2 and 8. = 7/4 and when ro = 2 and 8 = 7/2. Indicate on your plots the real and imaginary parts of each point. (a) 21 = roe-jo b ) 22 = ro (C) 23 = roel(+) (d) 24 = roe - 8+) (e) zs = roel0+277)

Answer 1

14 Given, k(n) is a signal with aln)=0 for n<-2 and n>4 (as 2(n-3) x(1) is shifted to the right by 3 units to obtain x(n-3) We already know that a(n)= o 1. n<-2 and no4 Therefore for x(n-3), replace n by n-3 in above condition x(n-3)=0 for n-3<-2 and n-3 >4 1 x(n-3)=0 for n<1 and not. So, for nci and no7 , x(n-3) is guaranteed to be zero (b) 2(n+4) Replace n by nth in x(n+4)=0 &(n+4)=0 for for a(n)=0 for n<-2 and no4 nth <-2 and nt474 n<-6 and nzo

© 2[n] . Given that a(n)=0 for n4.2 and n74 To obtain xett. xl-n] ='o , suploce niby on x(-)=0 for an<-2 and an>4 &(-1) zo for n>2 and n<-4 @ -n+2) 1. Replace n by ant2 to obtain values of infor which signal al-nt2) will be zero . . Given x(n)=0 xinzo for for n-2 and n74 2 " n -n+2 for x(-n+2)=0 _nt2<-2 and antros X(-2+2)=0 for an> an<-4 1. n34 and and na-z X(-n+2)=0

6 xl-n-2) Replace in by -n-2 to obtain values of a for which signal xl-n-2) will be zero. Given. x(n)=0 x(n=o for n<-2 and no4 go h -n-2 x(-n-2)=0 for -nz c-2 and -h-274 26-n-2)=0 for an ao and 676 x(-n-2)=0 for no and n<-6