Biomedical Engineering Reference
In-Depth Information
What does it all mean? It has been proven by Landau that
(1) stable sampling can
not
be performed at a rate lower than the Nyquist rate; and
(2) data can
not
be transmitted as samples at a rate higher than the Nyquist rate.
To put it in simpler terms, Lessard's paraphrase of the sampling theorem is: “Sampling
rate must be at least twice the highest frequency of interest in the signal.”
The Fourier transform of the periodic analog signal,
x
(
t
), is given by (6.1):
∞
f
(
x
(
t
))
x
(
t
)
e
−
j
ω
t
dt
=
X
(
j
ω
)
=
(6.1)
−∞
When the analog signal is sampled, you have the resulting
sampled
signal (6.2):
x
s
(
t
)
=
x
(
t
)
p
(
t
)
(6.2)
where
p
(
t
) is the sampling function with the period
T
.
The Fourier Series of the sampling function is as shown in (6.3):
∞
c
n
e
jw
s
nt
p
(
t
)
=
(6.3)
n
=−∞
where
w
s
=
2
π/
T
(sampling rate in radians/second) and
c
n
is the Complex Fourier
coefficient.
The Fourier coefficients are evaluated by (6.4):
T
/
2
1
T
p
(
t
)
e
−
j
ω
nt
dt
c
n
=
(6.4)
−
T
/
2
The result is given by (6.5):
sin
n
s
P
2
⎛
⎞
ω
P
T
⎝
⎠
c
n
=
(6.5)
s
P
2
n
ω
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