Digital Signal Processing Reference
In-Depth Information
Ideal Lowpass analog filter
h
f
(
t
)
x
b
(
nT
)
∗
h
f
(
t
)
x
b
(
t
)
=
x
b
(
nT
)
X
b
(
e
j
ω
T
)
H
f
(
j
w
)
X
b
(
e
j
w
T
)
H
f
(
j
w
)
X
b
(
j
w
)
=
(
a
)
x(0)
Reconstructed signal
x(T)
−
T
0
T
2
T
3
T
4
T
(
b
)
Figure 3.5
Reconstruction of the bandlimited signal from its samples, using an ideal
lowpass analog filter.
This revolutionary theorem implies that the samples
x
b
(nT )
contain all the
information that is contained in the original analog signal
x
b
(t)
, if it is bandlim-
ited and if it has been sampled with a period
T<π/ω
b
. It lays the necessary
foundation for all the research and developments in digital signal processing
that is instrumental in the extraordinary progress in the information technology
that we are witnessing.
4
In practice, any given signal can be rendered almost
bandlimited by passing it through an analog lowpass filter of fairly high order.
Indeed, it is common practice to pass an analog signal through an analog lowpass
filter before it is sampled. Such filters used to precondition the analog signals
are called as
antialiasing filters
. As an example, it is known that the maximum
frequency contained in human speech is about 3400 Hz, and hence the sampling
frequency is chosen as 8 kHz. Before the human speech is sampled and input to
telephone circuits, it is passed through a filter that provides an attenuation of at
least 30 dB at 4000 Hz. It is obvious that if there is a frequency above 4000 Hz
in the speech signal, for example, at 4100 Hz, when it is sampled at 4000 Hz,
due to aliasing of the spectrum of the sampled signal, there will be a frequency
at 4100 Hz as well as 3900 Hz. Because of this phenomenon, we can say that
the frequency of 4100 Hz is folded into 3900 Hz, and 4000 Hz is hence called
the “folding frequency.” In general, half the sampling frequency is known as the
folding frequency (expressed in radians per second or in hertz).
4
This author feels that Shannon deserved an award (such as the Nobel prize) for his seminal contri-
butions to sampling theory and information theory.
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