Biomedical Engineering Reference
In-Depth Information
where
T
-sampling period and
n
-number of samples, equally spaced with
T
as the sam-
pling interval.
Then the Discrete Fourier Transform equation is given by (13.2):
∞
x
(
n
)
e
−
j
ω
n
x
(
ω
)
=
(13.2)
n
=−∞
where
ω
ranges from 0 to 2
π
and
x
(
ω
) is periodic with 2
π
13.3 DEFINITION OF SAMPLING RATE
(OR SAMPLING FREQUENCY )
Let us review highlights of Chapter 6 on Sampling Theory before discussing the Fast
Fourier Transform (FFT). The basis of the Nyquist sampling theorem is the premise
that in order to successfully represent and/or recover an original analog signal, the analog
signal must be sampled at least twice the highest frequency in the signal. If the highest
frequency in the analog signal is known, then it can be sampled at twice the highest
frequency,
f
n
·
I
n
terms of time, the spacing between samples is called time period or
sampling interval,
T
. Thus,
T
≤
1(
/
2
f
n
) r
f
sampling
≥
2
f
n
=
1
/
T
where
T
is the sampling period (or sampling interval), and
f
sampling
is the sampling
frequency or sampling rate.
It should be noted that the maximum frequency,
f
n
,
is also called the
Cutoff
Frequency
, the
Nyquist Frequency
,orthe
Folding Frequency
.
If the highest frequency in the signal is not known, then the signal should be band-
limited with an analog low-pass filter, and then the filtered signal should be sampled at
least twice the upper frequency limit of the filter bandwidth. If the low-pass filtering
procedure is not followed, then the problem of “Aliasing” arises. The Aliasing problem
is usually caused when the spacing between samples is very large, which corresponds to
sampling at a rate less than twice the highest frequency in the signal.
Aliasing
is often defined as “the contamination of low frequencies by unwanted
high frequency signals/noise” (Fig. 13.1), which by itself is an incorrect statement without
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