The signal input to the filter causes a delay when output. How
much delay time each frequency of the output signal has with
respect to the input is referred to as the group delay
characteristic. In concrete, the group delay is obtained by
integrating the phase characteristic in terms of the frequency
and used to evaluate the characteristic of a filter circuit.
With this characteristic, the value (delay time) depends on the
filter circuit and frequency. When using a filter with different
delay time for the frequency, waveform distortion arises in the
output signal with respect to the input signal. Therefore, FFT
analyzers correct this value, remove waveform distortion, and
display an exact waveform.

Harmonic Distortion

Vibration waveform observed in the mechanical vibration system
usually contains various harmonic components in addition to the
fundamental component. When a sine wave is input to a
transmission system, harmonic components, referred to as
distortion components, of the input signal appear in the output
signal because of the non-linear characteristic of the
transmission system. By focusing this distortion, harmonic
components of vibration waveform or output signal are analyzed
and the vibration characteristics and the fidelity of the
transmission system are considered.

When the waveform under
observation, generally the output waveform, consists of
fundamental frequency f_{1}, 2nd order harmonic
frequency f_{2}, 3rd order harmonic frequency f_{3},
and other harmonic components and if the RMS value of each
component is |E_{1}|, |E_{2}|, |E_{3}|,
..., the total harmonic distortion is defined as

Using the power spectrum of each frequency component (p1, p2,
p3, ...), the total harmonic distortion becomes

When the harmonic distortion in the n-th harmonic component
is focused, the following n-th order relative harmonic content
is used:

Hilbert Transform

The Hilbert transform, g (t), and the inverse Hilbert transform
of real function f (t) are defined by expressions (1) and (2),
respectively.

The * mark denotes convolution.

Using Hilbert transform g (t) of real function f (t), analysis
signal (complex number) Z (t) is defined by the following
expression (3).

Since Z (t) is a complex number, it can be
represented as a vector.

where

r (t) is referred to as the amplitude (envelope)
of f (t) and θ (T) as the instantaneous phase. Therefore, any
real function f (t) can be represented by

Thus, the use of the Hilbert transform makes it
possible to obtain the envelope of f(t) and calculate the
logarithmic damping ratio (and then the damping factor).

The envelope represents the temporal variation
of the instantaneous energy of a system. In addition, f(t) can
be observed not only by the amplitude but also by another
parameter, i.e., the instantaneous phase.

When θ (t) is differentiated,

it can be recognized as the instantaneous
angular frequency.

The X axis is assigned the frequency and the Y
axis is assigned a logarithmic scale where 1 V^{2}rms
corresponds to 0 dB Vrms.

Impulse Response

Response h (t) output by the system when unit impulse δ (t) is
applied to it is referred to as impulse response. Impulse
response is a system characteristic expressed in the time
domain. On the other hand, a system characteristic expressed in
the frequency domain is referred to as the transfer function.

If the impulse response of a system is known, output y (t) when
x (t) is input to the system can be calculated through
convolution of input x (t) and impulse response.

FFT analyzers from Ono Sokki perform the inverse Fourier
transform of the frequency response function to obtain the
impulse response.

Inverse Fourier
Transform

The relationship between the Fourier transform and the inverse
Fourier transform is shown below.

The inverse Fourier transform of the cross spectrum is the
cross-correlation function. The inverse Fourier transform of the
frequency response function is the impulse response.

Isolation

Insulation of electrical signals is referred to as isolation. It
is required for measurement when the common line of the signal
source differs from the ground potential. Measurement is
possible when the difference between the signal common line and
the ground potential, common mode voltage, is small. When it is
100 V or higher, measurement is not possible because of
excessive input. In this case, an isolation amplifier or the
like may be used and, as for the FFT analyzer, the analog input
section including the amplifier, anti-aliasing filter, and A-D
converter is isolated from the chassis ground using a
photo-coupler on a channel basis. Since the digital circuit and
analog circuit are insulated, isolation is advantageous when
eliminating the ground loop or removing connection with the
common line of the signal source.

Liftered Envelope

The envelope of the power spectrum is obtained by performing
inverse Fourier transform of the short quefrency. The liftered
envelope is specific to the signal transmission system and does
not depend on the spectrum of the input signal.

Applications of the liftered envelope include extraction of the
fundamental frequency and power spectrum envelope from sound
waves, bio-waves, etc.