Page 185 - Applied Numerical Methods Using MATLAB

174 INTERPOLATION AND CURVE FITTING
Successively, use the time shifting property (P3.17.3) to get the CtFT of

x(t) = (t + 2) − (t − 2) (P3.17.5)
as
ω
(P3.17.3, 4) j2ω −j2ω 2
X(ω) = T(ω)e − T(ω)e = j8sin(2ω) sin c
π
(P3.17.6)
Get the CtFT Y(ω) of the triangular wave that is generated by repeating
x(t) two times and described as below.

y(t) = x(t + 4) + x(t − 4) (P3.17.7)

Plot the spectrum X(ω) for 0 ≤ ω ≤ 2π and check if the result is the
same as depicted in a solid line in Fig. P3.17.2a or P3.17.2c. You can
also plot the spectrum X(ω) for 0 ≤ ω ≤ 4π and check if the result
is the same as the solid line in Fig. P3.17.2b. Additionally, plot the
spectrum Y(ω) for 0 ≤ ω ≤ 2π and check if the result is the same as
the solid line in Fig. P3.17.2d.
(b) The definition of DtFT, which is used for getting the spectrum of a
discrete-time signal x[n], is
∞
−j n
X( ) = x[n]e (P3.17.8)
n=−∞
Use this formula to compute the DtFTs of the discrete-time signals
x a [n],x b [n],x c [n],x d [n] and plot them to see if the results are the
same as the dotted lines in Fig. P3.17.2a–d. What is the valid ana-
log frequency range over which each DtFT spectrum is similar to the
corresponding CtFT spectrum, respectively? Note that the valid analog
frequency range is [−π/T, +π/T ] for the sampling period T .
(c) Use the definition (3.9.1a) of DFT to get the spectra of the discrete-time
signals x a [n],x b [n], x c [n], and x d [n] and plot them to see if the results
are the same as the dots in Fig. P3.17.2a–d. Do they match the samples
of the corresponding DtFTs at k = 2kπ/N? Among the DFT spectra
(a), (b), (c), and (d), which one describes the corresponding CtFT or
DtFT spectra for the widest range of analog frequency?

3.18 Windowing Techniques Against the Leakage of DFT Spectrum
There are several window functions ready to be used for alleviating the
spectrum leakage problem or for other purposes. We have made a MAT-
LAB routine “windowing()” for easy application of the various windows.