Part A | 9.4
216 Part A Fundamentals
The computation of FFT using decimation-in-frequency (DIF) is shown in Figs. 9.25 and 9.26. This
method requires that the bit reversal algorithm be applied to the output X.k/. Note that the butterfly for the
DIF algorithm differs slightly from the decimation-intime butterfly, as shown in Fig. 9.27.
The use of decimation-in-time versus decimationin-frequency algorithms is largely a matter of preference, as either yields the same result. System constraints may make one of the two a more optimal
solution. It should be noted that the algorithms required
to compute the inverse FFT are nearly identical to
those required to compute the FFT, assuming complex
FFTs are used. In fact, a useful method for verifying a complex FFT algorithm consists of first taking
the FFT of the x.n/ time samples and then taking
the inverse FFT of the X.k/. At the end of this process, the original time samples, Re.x.n//, should be
obtained and the imaginary part, Im.x.n//, should be
zero (within the limits of the mathematical round off
errors).
The FFTs discussed up to this point are radix-2
FFTs, i. e., the computations are based on two-point
butterflies. This implies that the number of points in
the FFT algorithms must be a power of 2. However,
non-radix-2 FFT algorithms have been developed and
are available in modern computational software packages like Matlab, Mathematica, Maple, etc., to be used
in a variety of applications in ocean engineering as well
as beyond [9.1, 2, 4–6].
9.4 Waveform Analysis
A waveform is one recording of a deterministic signal or that of an instance of a random signal (also
referred to in the literature as a stochastic process).
Actually, in practice we can make a recording of finite duration, which means that a waveform is the
set of signal recordings over possibly non-contiguous
time intervals. Typical examples of (lumped) waveforms in ocean engineering are significant wave height
at a point or average, oceanic temperature or salinity,
acoustic recordings of marine life, sonar or radar signals, etc. [9.3].
9.4.1 Definitions for Waveforms
and Random Signals
The time mean (or average) of a waveform is defined as
follows in the continuous time domain
x , lim
T!1
8
<
:
1
2T
CT
Z
T
x.t/ dt
9
=
;
:
(9.74)
The time-wise cross-correlation of two waveforms is
defined as follows
R y;x ../ , lim
T!1
8
<
:
1
2T
CT
Z
T
y.t/x.t C / dt
9
=
;
;
D Œy ˝ x .t/ ../ :
(9.75)
In the above, ˝ stands for the convolution operation applied in the square brackets between signal y and signal
x .t/ , which is the mirrored version of signal x about
origin of time (t D 0), with time shift , i. e.,
x .t/ .t/ D x.t/ :
Also, note that cross-correlation as defined above can be
used to introduce (time-wise) autocorrelation of a single
waveform if we set y Á x.
The limiting process to infinity for time variable T
introduced in the equations above is required in the case
when the waveform has infinite duration so that the definitions make sense and do not give rise to indeterminate
or infinite sums.
On the other hand, if the waveform (or at least one
of them in the cross-correlation case) is finite in duration, then the time interval (2T) needs to be set equal to
the full finite, and not infinite, duration of the signal. In
this case, the definitions need to look like the following
to avoid any issues
x ,
1
2T
C1 Z
1
x.t/ dt ;
(9.76)
R y;x ../ ,
1
2T
C1 Z
1
y.t/x.t C / dt :
(9.77)
The above equations are well defined in the case of
deterministic signals and waveforms that occur as instances of random signals. Specifically, a random signal, denoted with a capital letter, e.g., X.t/, in contrast to
lowercase latters used for deterministic signals, is associated with a probability density function (PDF), f X .x/
quantifying the probability of the signal value at any
216 Part A Fundamentals
The computation of FFT using decimation-in-frequency (DIF) is shown in Figs. 9.25 and 9.26. This
method requires that the bit reversal algorithm be applied to the output X.k/. Note that the butterfly for the
DIF algorithm differs slightly from the decimation-intime butterfly, as shown in Fig. 9.27.
The use of decimation-in-time versus decimationin-frequency algorithms is largely a matter of preference, as either yields the same result. System constraints may make one of the two a more optimal
solution. It should be noted that the algorithms required
to compute the inverse FFT are nearly identical to
those required to compute the FFT, assuming complex
FFTs are used. In fact, a useful method for verifying a complex FFT algorithm consists of first taking
the FFT of the x.n/ time samples and then taking
the inverse FFT of the X.k/. At the end of this process, the original time samples, Re.x.n//, should be
obtained and the imaginary part, Im.x.n//, should be
zero (within the limits of the mathematical round off
errors).
The FFTs discussed up to this point are radix-2
FFTs, i. e., the computations are based on two-point
butterflies. This implies that the number of points in
the FFT algorithms must be a power of 2. However,
non-radix-2 FFT algorithms have been developed and
are available in modern computational software packages like Matlab, Mathematica, Maple, etc., to be used
in a variety of applications in ocean engineering as well
as beyond [9.1, 2, 4–6].
9.4 Waveform Analysis
A waveform is one recording of a deterministic signal or that of an instance of a random signal (also
referred to in the literature as a stochastic process).
Actually, in practice we can make a recording of finite duration, which means that a waveform is the
set of signal recordings over possibly non-contiguous
time intervals. Typical examples of (lumped) waveforms in ocean engineering are significant wave height
at a point or average, oceanic temperature or salinity,
acoustic recordings of marine life, sonar or radar signals, etc. [9.3].
9.4.1 Definitions for Waveforms
and Random Signals
The time mean (or average) of a waveform is defined as
follows in the continuous time domain
x , lim
T!1
8
<
:
1
2T
CT
Z
T
x.t/ dt
9
=
;
:
(9.74)
The time-wise cross-correlation of two waveforms is
defined as follows
R y;x ../ , lim
T!1
8
<
:
1
2T
CT
Z
T
y.t/x.t C / dt
9
=
;
;
D Œy ˝ x .t/ ../ :
(9.75)
In the above, ˝ stands for the convolution operation applied in the square brackets between signal y and signal
x .t/ , which is the mirrored version of signal x about
origin of time (t D 0), with time shift , i. e.,
x .t/ .t/ D x.t/ :
Also, note that cross-correlation as defined above can be
used to introduce (time-wise) autocorrelation of a single
waveform if we set y Á x.
The limiting process to infinity for time variable T
introduced in the equations above is required in the case
when the waveform has infinite duration so that the definitions make sense and do not give rise to indeterminate
or infinite sums.
On the other hand, if the waveform (or at least one
of them in the cross-correlation case) is finite in duration, then the time interval (2T) needs to be set equal to
the full finite, and not infinite, duration of the signal. In
this case, the definitions need to look like the following
to avoid any issues
x ,
1
2T
C1 Z
1
x.t/ dt ;
(9.76)
R y;x ../ ,
1
2T
C1 Z
1
y.t/x.t C / dt :
(9.77)
The above equations are well defined in the case of
deterministic signals and waveforms that occur as instances of random signals. Specifically, a random signal, denoted with a capital letter, e.g., X.t/, in contrast to
lowercase latters used for deterministic signals, is associated with a probability density function (PDF), f X .x/
quantifying the probability of the signal value at any