For a continuous time signal x(t) = 8 cos 200 ∏t.
Find: (1) Minimum sampling rate. (2) If f_s= 400Hz,what is discrete time signal? (3) If f_s= 150Hz,what is discrete time signal? (4) Comment on result obtained in 2 and 3 with proper justification.

Subject : Signals & Systems

Topic : Continuous Time Fourier Transform (CTFT) and Discrete Time Fourier Transform (DTFT).

Difficulty: Medium

(1) w_m= 200π 2 πF_m = 200 π F_m= 200π/2π = 100
∴ Minimum sampling rate (F_s) = 2F_m = 200Hz

(2) If F_s = 400Hz, then T = 1/F_s = 1/400. Since, x(t) = 8 cos200 πt
∴ put t = nT for discrete time signal x(nT) = 8 cos200 πnT
= 8 cos200 πn*1/400 = 8 cos 0.5πn.

(3) If F_s = 150Hz, then T = 1/F_s = 1/150. Since, x(t) = 8 cos200 πt
∴ put t = nT for discrete time signal

x(nT) = 8 cos200 πnT
= 8 cos200 πn*1/150 = 8 cos 1.333πn.

(4) If reconstruct the signal in (2) by putting n=ft we get original signal and hence there is no aliasing effect. Similarly, If reconstruct the signal in (3) by putting n=ft we do not get original signal and hence there is aliasing effect. To overcome this we use LPF and choosing the freq. above minimum rate and avoid aliasing effect.