Questions about Signals

Question

A continuous-time signal x(t) is a sum of two sinusoids whose frequencies are known to lie in the range 500 Hz - 550 Hz. It is sampled at a rate of 100 samples/second starting at t=0. The resulting samples satisfy the equation:

1. Find an equation for x(t).
2. How would your answer to (i) differ if the frequencies of the two components of x(t) were in the range 550 Hz - 600 Hz instead?

Answer 1

the discrete time signal is given by: x[n]=3.4cos(0.3πn+1.7)+2.1cos(0.8πn+2.5)

sampling frequency F_s=100 samples/sec, the analog frequencies F₁, F₂ ∈ [500 550] Hz

digital frequency f is given by: f=F/F_s

0.3/2=F_1o/F_s⇒F_1o=15 Hz, 0.8/2=F_2o/F_s⇒F_2o=40 Hz

F₁=F_1o+kF_s, where k=±1, ±2, ⋅⋅⋅, and since F₁ ∈ [500 550] Hz ⇒k=5⇒F₁=515 Hz

similarly F₂=F_2o+kF_s, F₂ ∈ [500 550] Hz ⇒k=5⇒F₂=540 Hz

Therefore the corresponding analog signal x(t) is given by:

x(t)=3.4cos(2π515t+1.7)+2.1cos(2π540t+2.5)