Description
Spectral Representation of Finite Length Signals
One of the issues to deal with when interpreting the spectrum of a signal is the fact that
we will only be able to have a segment of it available to compute the spectrum. The most
obvious restriction is the finite memory size of the system. Therefore we can only have a
finite interval of the evolution of the signal versus its independent variable, say time t.
We can mathematically represent this effect using what is called a window in order to
truncate the signal in time. In this lab we use the default rectangular window w(t).
Review/study and run the MATLAB file “windowing.m” provided on the website.
It produces the plots attached at the end that correspond to Part 1.
1. a) Given the following signals for which t is given in [ms]:
)( )(
)5.0(2
1
x t e tw
j t
= ⋅
π
, this is a 10ms window of a 0.5kHz complex exponential,
)( cos(2 ))5.0( )( 2
x t = π t ⋅ tw , this is a 10ms window of a 0.5kHz cosine,
)2/( )( cos(2 ))5.0( 3
x t = π t ⋅ tw , this is a 20ms window of a 0.5kHz cosine,
where
>
<
=
t ms
t ms
tw
,0 5
,1 5
)( , this is a 10ms rectangular window.
Plot the nonzero segments of x2(t) and x3(t) using fS = 4 kHz.
b) Each signal above is sampled using the train of impulses shown below with a
sampling frequency fS = 4 kHz.
Plot the spectra
Ω of the sampled versions of the three signals above using
linear frequency F in kHz for the x-axis. Use a 2,000 points DFT.
c) For the MATLAB plots corresponding to the spectra comment on:
i. The peak value of each spectrum
ii. The frequency spacing between consecutive zeros, and
iii. The widths of the main-lobes.
∞
−∞=
= δ −
n
s
tp )( (t nT )
xi(t), i = 1, 2, 3 xP(t)
