Digital Signal Processing Reference
In-Depth Information
TABLE 6.3
Key Equations of Chapter 6
Equation Title
Equation Number
Equation
q
n=-
q
x(nT
S
)e
-jnT
S
v
Fourier transform of sampled signal
(6.17)
X
S
(v) =
Frequency spectrum of
cosine-modulated signal
1
2
[M(v - v
c
) + M(v + v
c
)]
(6.20)
X(v) =
Notice that the original continuous-time signal can be approximately recov-
ered by filtering the sampled signal with a low-pass filter with cutoff frequency
v
M
6 v
c
6 v
s
/2.
In this chapter, we look at several ways that the Fourier transform can be applied to
the analysis and design of signals and systems. The applications considered here
demonstrate the use of the Fourier transform as an analysis tool.
We consider the
duration-bandwidth
relationship and find that the band-
width of a signal is inversely proportional to its time duration. We see that if a signal
changes values rapidly in time, it has a wide bandwidth in frequency.
Four basic types of
ideal filters
are presented. Applications are shown for the
concepts of the
ideal low-pass, ideal high-pass, ideal bandpass
, and
ideal bandstop
filters. Although these ideal filters are not physically realizable, it is shown that
the concept of an ideal filter can simplify the early stages of a system analysis or
design.
Butterworth
and
Chebyschev filters
are presented as standard filter designs
that provide physically realizable approximations of ideal filters. Examples show
how these filters can be realized by electrical circuits.
Signal reconstruction is presented as an application of filtering and as the
process of convolving the sample-data signal with an interpolating function.
Two techniques of sinusoidal modulation (DSB/SC-AM and DSB/WC-AM)
and two types of pulse-amplitude modulation (natural and flat-top) are presented
to demonstrate applications of the Fourier transform to the study of communication
systems and signals.
See Table 6.3.
6.1.
Show mathematically that the ideal high-pass filter is not physically realizable.
6.2.
Show mathematically that the ideal bandpass filter is not physically realizable.
