Problem 5 (a) In order to figure out if a Ct signal a (t)is periodic, we need to find a finite, non-zero value of T such that a(t)=a(t+T) for all t. The smallest T that satisfies this is the fundamental period This function is quite straightforward. We know that the function sin(4t-1)is periodic with period 3. Since the positive and negative cycles of sinusoids have the same shape the square of this function, i.e. a(t)=[sin(4t-1)2 is periodic with fundamental period Also, we can use the relation 11 SIn T 22 which is periodic with period (b)For a DT function an], we need to find a finite, non-zero integer N such that a[nl cn +N for all n. The smallest integer N for which this holds is the fundamental period. If we cannot find such an N, then the function is not periodic We need cos 4(n+N)+ 4 cos「4n+ For the above to hold, the following has to be true for some integer(s)k AN 4n++2k 4 /I Since T is not a rational number, we cannot find an integer n that satisfies this. Thus, the function is not periodic (c) We can use the same steps as we did above but we can start with finding the funda- mental period of the simpler function y[n]=cos(2) We need the following to hold 2T(n +N) 7 7 So we need the following to hold for at least one integer value of k� � � � � � � � � � � � Problem 5 (a) In order to figure out if a CT signal x(t) is periodic, we need to find a finite, non-zero value of T such that x(t) = x(t + T) for all t. The smallest T that satisfies this is the fundamental period. This function is quite straightforward. We know that the function sin(4t−1) is periodic . Since the positive and negative cycles of sinusoids have the same shape, � with period 2 the square of this function, i.e. x(t) = [sin(4t−1)]2 is periodic with fundamental period . � 4 Also, we can use the relation 1 1 sin2 x = sin 2x, 2 − 2 � which is periodic with period 4 . (b) For a DT function x[n], we need to find a finite, non-zero integer N such that x[n] = x[n + N] for all n. The smallest integer N for which this holds is the fundamental period. If we cannot find such an N, then the function is not periodic. We need cos 4(n + N) + = cos 4n + 4 4 For the above to hold, the following has to be true for some integer(s) k. � � 4n + 4N + = 4n + + 2�k 4 4 � N = k 2 Since � is not a rational number, we cannot find an integer N that satisfies this. Thus, the function is not periodic. (c) We can use the same steps as we did above but we can start with finding the funda￾mental period of the simpler function y[n] = cos 2�n 7 . We need the following to hold 2�n 2�(n + N) cos = cos 7 7 So we need the following to hold for at least one integer value of k. 10