Exercise 6 Consider the signal
x(t)=cos(\pi t)+2cos(4\pi t)
The signal is sampled to obtain the signal
x[n]=x(nT)
with Fourier transform
x(e^(j\omega ))
. a) (5 pt) What is the minimum sampling frequency
\omega _(s)=(2\pi )/(T)
that can be used to avoid aliasing? In the following questions assume that the sampling frequency is
6\pi
b) (10 pt) Sketch the Fourier transform
x(e^(j\omega ))
of
x[n]
(or
x_(T)(\omega )
). c) (10 pt) The signal, sampled with sampling frequency is
\omega _(s)=6\pi
, is reconstructed with an ideal reconstruction filter with
H(j\omega )={(3,|\omega |<3\pi ),(0 otherwise ):}
Find an expression for the output
y(t)
of the filter.