Civil Engineering Reference
In-Depth Information
kn
N
−
1
1
−
j2
π
∑
XXkf
=
(
Δ=
)
xe
N
k
=
, , , ,...,
N
−
1
,
(1.21)
k
n
N
n
=
0
where Δ
f
is 1/
T
. The
discrete inverse Fourier transform
(IDFT) is accordingly given by
N
−
1
kn
j2
π
∑
xxnt
=
(
Δ=
)
Xe
N
n
=
, , , ,...,
N
−
1
.
(1.22)
n
k
k
=
0
and not an approximation. This is true in spite of our starting point being the continuous
Fourier transform.
These equations form the basis of digital Fourier analysis. Looking at these
equations one will see that a straightforward analysis using an
N
-points transform will
require
N
2
complex multiplications. Here the
fast Fourier transform
(FFT) algorithm is
appearing as a saviour. Details on the procedure are outside the scope of this topic. It is
sufficient to point out the benefit of using this algorithm requiring
N
⋅
ln(
N
) operations
only, as opposed to
N
2
and the difference is quite formidable. Just for a “small” transform
using
N
= 2
10
= 1024, the ratio of these number of operations is 147. Using
N
= 2
16
=
65536, the ratio will be 5900.
It may at this stage be appropriate to point out some further important phenomena
appearing using DFT analysis. These are 1)
aliasing
and 2)
leakage
, where the first one
is a unique problem in digital analysis. As indicated above, the upper frequency limit
where reliable information can be extracted is 1/(2Δ
t
) = Δ
f
⋅
N
/2 =
f
s
/2, where
f
s
is the
sampling frequency. Any signal components with higher frequencies present will not be
detected but instead will be mistakenly assigned to lower frequency components. They
are being “folded back” to the lower frequency components, which is the reason behind
the term aliasing. There are two methods to escape this problem. One either uses a
sampling frequency much higher than the expected highest frequency component in the
signal or inserts sharp anti-aliasing filters which cut away or eliminate the components
having frequencies above the Nyquist frequency.
The second phenomenon, leakage, means that one may get frequency components
which in fact are not present in the actual signal. The main reason is that one is
performing an analysis using a limited time segment. Starting or/and stopping the
sampling at a time when the signal amplitude is not zero will create a discontinuity, i.e.
we get a step in the value of the function giving components that are not present in the
actual signal. The remedy is to apply various kinds of “window”; multiplying the time
signal with a form function before taking the transform. These windows are symmetrical
around
T
/2 and go to zero at each end. One then removes some of the energy but this
may be compensated for after performing the analysis. There are a number of such
windows in use, Hanning, Hamming and Kaiser-Bessel to mention a few.
1.4.5
Spectral analysis measurements
In practice, one will normally find that the sound or vibration phenomena to be analysed
cannot be distinctly classified as periodic, transient or stochastic. They may contain
elements of two or more types. Sound or vibration spectra from technical equipment such
as a motor, a pump etc. will contain periodic components due to the rotation of blade
