Biomedical Engineering Reference
In-Depth Information
For the Sampling Function, where
P
is the pulse width of
p
(
t
), the Fourier Transform
of the Sampled function (6.6) is written in the integral format as (6.7),
∞
x
(
t
)
c
n
e
jw
s
nt
x
s
(
t
)
=
x
(
t
)
p
(
t
)
=
(6.6)
n
=−∞
∞
x
(
t
)
c
n
e
jw
s
nt
e
−
j
ω
t
dt
∞
f
(
x
s
(
t
))
=
(6.7)
n
=−∞
−∞
which results in the expression given by (6.8):
∞
c
n
X
(
ω
−
n
ω
s
)
(6.8)
n
=−∞
where for the ideal delta function,
c
n
nT
.
The Fourier Transform of the sampled function is often called “Discrete Fourier
Transform of a Time Series.” It should be noted that the Discrete Fourier Transform
DFT is a transform in its own right as is the Fourier integral transform, and that the
DFT has mathematical properties analogous to the Fourier Integral Transform. The
usefulness of the DFT is in obtaining the Power Spectrum analysis or filter simulation
on digital computers. The Fast Fourier Transform, which will be covered, is a highly
efficient procedure of a time series. Restrictions on the period
T
of the pulse train
p
(
t
)
to avoid ambiguity in the sampled version. The Nyquist Sampling Theorem must be
followed.
Let us examine the 1st step in converting from an analog signal to a digital signal.
Consider an analog signal
v
(
t
)
=
1
/
T
and
t
=
t
). The sampled values will give the exact value
of
v
(
t
) only at the sampled time; thus, a discrete-time sample of
v
(
t
). If the period or
time between samples is constant,
T
(the sampled period), the discrete approximation
is expressed as
v
(
nt
), where
n
is an integer index. Sampling with a constant period
is referred to as “uniform sampling,” which is the most widely used sampling method
because of the ease of implementing and modeling. As is known, the sample period,
T
,
effects how accurately the original signal is represented by the sampled approximation.
The ideal sampled signal
X
s
(
t
) is zero except at time
t
=
V
sin(
ω
=
nT
and the signal is represented
Search WWH ::
Custom Search