Glossary of Basic Terms Relating to FFT Analysis

When the sampling interval is Δt seconds (sampling is made every Δt seconds), the sampling frequency, fs, is 1/Δt (1/Δt points are sampled a second.) Sampling theory indicates the relationship between a temporally continuous signal and the least possible sampling rate with which the original information can be maintained. It prescribes that the sampling frequency must be at least two times the highest harmonic component contained in the signal. If the sampling frequency is lower than two times the signal frequency, aliasing (aliasing noise) occurs.

The sound intensity is the energy of sound which passes a unit cross-section area, including the sound field point, within a given unit of time. It is defined as a vector quantity which is the time average of the product of the sound pressure at point p (t) and the particle velocity, u_r (t), into direction r. If the sound intensity into direction r is I_r, the following expression is given.

In a steady medium with a density of ρ, the relationship between p(t) and ur(t) is represented by expression (2).

However, it is very difficult to measure the particle velocity directly and correctly. To solve this problem, a method to approximate the particle velocity from the difference between sound pressures of two points was considered. This method is the SI measurement method based on two microphones. Using sound pressures p₁(t) and p₂(t) of the two microphones which are separated by ⊿r into direction r, p (t) and can be obtained as an approximate value:

When substituting expression (4) into expression (2), the particle velocity into direction r, u_r(t), can be represented by expression (5).

Then, the sound intensity, I_r, is

This expression calculates I_r directly in the time domain and is referred to as the direct integral method.

In addition, in many cases expression (7) is used to obtain I_r into direction r at any desired frequency band from f₁ to f₂

where Im{G₁₂(f)} is the imaginary part of the (one-side) cross spectrum of p₁(t) and p₂(t). If you obtain the cross spectrum between sound pressure signals at two close points using a 2-channel FFT analyzer and then calculate the above expression using the imaginary part, you can obtain the frequency band I_r for any desired frequency band. This approach is referred to as the cross spectrum method.  Measurement errors of the SI measurement method include the limited differential error by limited ⊿r, the sensitivity between two microphones, and the error caused by mismatched phase. Various considerations are made on this correction method.

The following are sample applications using the SI measurement method.

(1) Measuring power level of sound source

The sound intensity represents the amount of sound energy passing in a unit area in a unit time. The total power, P, emitted from the sound source is given by

The above expression allows you to calculate the sound power based on sound intensity measurement at a split plane, on a hemisphere centering on the sound source and into the direction which is vertical to the surface of the hemisphere.

(2) Sound shield measurement

Through measurement of the transmission power for each section based on SI, you can make quantitative measurement of the sound shielding characteristics of walls consisting of multiple sections and the sound leakage from gaps. Therefore, the SI method is effective for sound shield measurement on site.

(3) Sound field analysis

The SI value is a vector quantity (a quantity with a magnitude and direction). Therefore, by displaying the propagating direction and the magnitude of the sound in 2D or 3D form, you can visualize and capture the flow of sound energy.

Differential

For calculation of the first and second order differential values, the 5th order Lagrange's interpolation formula is used. The data for a given point is obtained based on the value of the five points before and after the target point, including that point.

f₀, f₁, f₂, f₃, f₄, .... are sample data.

First order differential

Second order differential

Integral

For calculation of the single and double integral values, the trapezoidal formula is used.

Expressions for single integral value

Expressions for double integral value

The instantaneous waveform of the signal input from the input connector of the panel is displayed. The waveform for one frame is displayed. In this case, the X axis is assigned time (seconds) with the starting point of the frame set to 0 and the Y-axis the instantaneous value. The full scale of the X axis is set in conjunction with the frequency range setting.

(1) Mean value (MEAN)

(2) RMS value (RMS)

(3) Standard deviation (S.D.)

The 2nd order moment around the mean value is referred to as the dispersion, and the square root of the dispersion is referred to as the standard deviation. As for a signal excluding the DC component, the RMS value and standard deviation are equal.  The S.D. is obtained by

The relationship of expressions (1), (2), and (3) is

(4) Skewness (SKEWNESS)

Skewness denotes the 3rd order moment around the mean value normalized by σ³.  It is used as an index for asymmetry around the mean value. Skewness is obtained by

(5) Kurtosis (KURTOSIS)

Kurtosis denotes the 4th order moment around the mean value normalized by σ⁴.  It is used as an index for sharpness of waveform. Kurtosis is obtained by the following formula. Since the value of kurtosis in the time signal of a normal (Gaussian) distribution is 3, the value of kurtosis can be taken as the following equation minus 4.

(6) Crest factor (CRESTFACTOR)

Obtained by dividing the maximum value by the RMS value.

The trigger function is a function which starts sampling based on a certain point of an input signal or input of an external signal.

There are two types of trigger modes:  internal trigger mode and external trigger mode.  In the internal trigger mode, the input signal is used as a sampling start timing signal or trigger signal, and sampling starts when the input signal reaches a specified voltage.  In the external trigger mode, an external signal is input as a sampling start timing signal, and sampling starts when the external trigger signal is input.

This function makes it possible to efficiently sample and analyze a desired waveform portion. When averaging a time waveform, the trigger function is used for waveform synchronization.