ENEE 241 02* HOMEWORK ASSIGNMENT 3 Due Thu 02/12
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Two periods of the sinusoid x(t) = Acos(Ωt+φ) are plotted above. The value of x(0) equals
−3 sin(π/5).
(i) (6 pts.) Determine A, Ω and φ. Express φas an exact rational multiple of πin the range
[0,2π).
In what follows: Aand φare as found in part (i) and x[n] = x(nTs), where Tsis a suitably chosen
sampling period.
(ii) (3 pts.) For what values of Tsis it true that x[n] is constant in n?
(iii) (3 pts.) For what values of Tsis it true that x[n] = −x[n−1] for all n?
(iv) (4 pts.) Determine all values of Tssuch that x[n] = Acos((5π/8)n+φ).
(v) (4 pts.) Determine all values of Tssuch that x[n] = Acos((π/6)n−φ).
Solved Examples
S3.1 (P1.19 in textbook). The continuous-time sinusoid
x(t) = cos(150πt +φ)
is sampled every Ts= 3.0 ms starting at t= 0. The resulting discrete-time sinusoid is
x[n] = x(nTs)
(i) Express x[n] in the form x[n] = cos(ωn +φ) i.e., determine the value of ω.
(ii) Is the discrete-time sinusoid x[n] periodic? If so, what is its period?
(iii) Suppose that the sampling rate fs= 1/Tsis variable. For what values of fsis x[n] constant
for all n? For what values of fsdoes x[n] alternate in value between −cosφand cos φ?
S3.2. The first period of the sinusoid x(t) = Acos(Ωt+φ) is plotted below, where x(0) = 3√2/2.
(i) Determine the values of A, Ω and φ.
(ii) The sinusoid is sampled every Ts= 0.05 seconds starting at t= 0 to produce
x[n] = x(nTs)
