Biomedical Engineering Reference
In-Depth Information
it or perform the transform differently. In the following sections, we examine
several different transforms, which have been well used for this.
2.3.1 The Fourier Transform
In previous sections we represented the signal, whether analog
x
(
t
) or digital
x
(
n
), in terms of values, which occur at different integer intervals. This is
known as the
time-domain
representation of the signal. The Fourier trans-
form is a way of representing the signal in terms of frequencies, thereby trans-
forming the signal into the
frequency-domain
representation. The crux of the
Fourier transform is that any signal can be represented as a sum of sinusoids
of different frequencies as long as the signal is of finite amplitude.
For a continuous time signal
x
(
t
), the Fourier transform is given by the
integral
∞
x
(
t
)e
−jωt
d
t
X
(
jω
)=
(2.19)
−∞
and
x
(
t
) can be recovered by the inverse Fourier transform
∞
1
2
π
X
(
jω
)e
jωt
d
ω
x
(
t
)=
(2.20)
−∞
The discrete Fourier transform (DFT) for digital signals
x
(
n
) is similar to
the transform for the continuous analog signal
x
(
t
), except that the angular
frequency
ω
is divided into
N
equal parts on the interval [0
,ω
]. The transform
for the digital sequence is then a summation of
N
parts written as
N
x
(
n
)e
−jω
k
n
X
(
k
)=
(2.21)
n
=1
where the angular frequency
ω
k
=2
πk/N
for
k, n
=1
,...,N
. The inverse dis-
crete Fourier transform is just
N
x
(
n
)=
1
N
X
(
k
)e
jω
k
n
(2.22)
k
=1
Note here that both
x
(
n
) and
X
(
k
) are complex digital signals and the
transform is sometimes known as the
N
-point DFT due to the number of
frequencies. Since there are
N
samples
X
(
k
) each involving
N
summations,
calculating the DFT requires
N
2
operations, which becomes an increasingly
slow operation as the number of samples increases.
2.3.2 Fast Fourier Transform
One method to increase the eciency in calculating the DFT is the fast
Fourier transform (FFT) algorithm described by Cooley and Tukey (1965).
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