Systems, Sampling and Quantization
7
main lobe
frequency leakage
rectangular window
and its magnitude is drawn in dashed line in fig. 3 for F
s
= 50Hz. We can
see that sampling induces a periodic replication in the spectrum and that the
periodicity is established by the sampling rate. The fact that the spectrum is
not identically zero for frequencies higher than the Nyquist limit determines
aliasing. This can be seen, for instance, in the heightening of the peak at the
frequency of the damped sinusoid.
If we consider only the sampled signal lying within a window of N = 7
samples, we can compute the DTFT by means of (12) and obtain the third
curve of fig. 3. Two important artifacts emerge after windowing:
· The peak is enlarged. In general, we wave a main lobe for each relevant
spectral component, and the width of the lobe might prevent from resolv-
ing two components that are close to each other. This is a loss of frequency
resolution due to the uncertainty principle.
· There are side lobes (frequency leakage) due to the discontinuity at the
edges of the rectangular window. Smaller side lobes can be obtained by
using windows that are smoother at the edges.
Unfortunately, for signals that are not known analytically, the analysis can
only be done on finite segments of sampled signal, and the artifacts due to
windowing are not eliminable. However, as we will show in sec. 4.1.3, the tradeoff
between width of the main lobe and height of the side lobes can be explored by
choosing windows different from the rectangular one.
(a)
(b)
0
0.5
1
-1
0
1
Exponentially-decayed sinusoid
t [s]
y
0
50
100
-60
-50
-40
-30
-20
-10
Frequency response of a damped sinusoid
f [Hz]
|Y| [dB]
Figure 3: (a): Exponentially-decayed sinusoid, obtained as the real part of the
complex exponential y(t) = e
s
0
t
, with s
0
= -10 + j100; (b): Frequency analysis
of the complex exponential y(t) = e
s
0
t
. Transform of the continuous-time signal
(continuous line), transform of the signal sampled at F
s
= 50Hz (dashed line),
and transform of the sampled signal windowed with a 7-sample rectangular
window (dash-dotted line)
To conclude the example we report the Octave/Matlab code (see the ap-
pendix B) that allows to plot the curves of fig. 3. The computation of the DTFT
is particularly instructive. We have expressed the sum in (12) as a vector-matrix
multiply, thus obtaining a compact expression that is computed efficiently. We
also notice how Matlab and Octave manage vectors of complex numbers with
the proper arithmetics.