Biomedical Engineering Reference
In-Depth Information
1.2 Discrete signals
In nature, most of the signals of interest are some physical values changing in time
or space. The biomedical signals are continuous in time and in space.
On the other hand we use computers to store and analyze the data. To adapt the
natural continuous data to the digital computer systems we need to digitize them.
That is, we have to sample the physical values in certain moments in time or places
in space and assign them a numeric value—with a finite precision. This leads to the
notion of two processes: sampling (selecting discrete moments in time) and quanti-
zation (assigning a value of finite precision to an amplitude).
1.2.1 The sampling theorem
Let's first consider sampling. The most crucial question is how often the signal
contains no frequencies
1
f
(
t
)
must be sampled? The intuitive answer is that, if
f
(
t
)
higher than
F
N
,
f
cannot change to a substantially new value in a time less than
one-half cycle of the highest frequency; that is,
1
(
t
)
2
F
N
. This intuition is indeed true. The
Nyquist-Shannon sampling theorem [Shannon, 1949] states that:
contains no frequencies higher than
F
N
cps, it
is completely determined by giving its ordinates at a series of points
spaced
If a function
f
(
t
)
1
2
F
N
seconds apart.
The frequency
F
N
is called the Nyquist frequency and 2
F
N
is the minimal sampling
frequency. The “completely determined” phrase means here that we can restore the
unmeasured values of the original signal, given the discrete representation sampled
according to the Nyquist-Shannon theorem
(Figure 1.1).
A reconstruction can be derived via sinc function
f
sinπ
x
π
x
. Each sample value
is multiplied by the sinc function scaled so that the zero-crossings of the sinc func-
tion occur at the sampling instants and that the sinc function's central point is shifted
to the time of that sample,
nT
,where
T
is the sampling period (Figure 1.1 b). All
of these shifted and scaled functions are then added together to recover the original
signal (Figure 1.1 c). The scaled and time-shifted sinc functions are continuous, so
the sum is also continuous, which makes the result of this operation a continuous sig-
nal. This procedure is represented by the Whittaker-Shannon interpolation formula.
Let
x
(
x
)=
be the
n
th
sample. We assume that the highest frequency
present in the sampled signal is
F
N
and that it is smaller than half of the sampling
frequency
F
N
[
n
]
:
=
x
(
nT
)
for
n
∈
Z
1
2
F
s
. Then the function
f
<
(
t
)
is represented by:
sinc
t
∞
∑
∞
∑
−
nT
T
sinπ
(
2
F
s
t
−
n
)
f
(
t
)=
x
[
n
]
=
x
[
n
]
(1.14)
(
−
)
π
2
F
s
t
n
n
=
−
∞
n
=
−
∞
1
Frequencies are measured in cycles per second—cps, or in Hz—[Hz]=
1
[
.
s
]
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