SYSC 3500 A Signals and Systems Fall 2020
Consider a DT periodic signal, x[n], with period: Nx=4. The values of x[n] in one period starting
from n=0 to n=3 are [0 -1 -1 1].
(a) Find the Fourier representation of x[n]
(b) Sketch the magnitude and phase of each frequency component of X
Now, create a new signal, y[n], which is also DT periodic but the period is: Ny=8. One period of
y[n] is given as:
y[n] = x[n]; 0 ≤ n ≤3; =0 otherwise
(c) Find the Fourier representation of y[n]
(d) Sketch the magnitude and phase of each frequency component of Y
(e) Compare the spectral lines of X and Y. In particular, comment on the number of
frequency components included in the spectrum of Y compared to the spectrum of X, and
comment on what would happen if we keep increasing the period of y[n] by adding zeros.
Consider two signals, m1(t) and m2(t). Both signals are band-limited (i.e. their Fourier transform is
zero outside a certain range of frequencies). The spectra of both signals are known and both of
them satisfy the condition:
Mx(ω) =0 when |ω| ≤5000 rad/s; x is either 1 or 2
We use m1(t) and m2(t) to form a third signal z(t) as follow:
z(t) = 2 cos (20000t) x [m1(t) +m2(t) x cos(10000t)]
Assume arbitrary shapes for M1(ω) and M2(ω). Show all the steps involved in developing the
spectrum of Z(ω).