Spectrum Scaling Options

This page provides an overview of the scaling options for spectral calculations. The following table shows symbols and important parameters for the scaling.

Symbol

Function block parameters

Meaning

N

nFFT_Length

Number of input values of the FFT

Fs

 

Sampling rate

Ʃwn

eWindowFunction, nWindowLength

Sum of the values of the window function

Ʃwn2

eWindowFunction, nWindowLength

Sum of the squared values of the window function

SQRT(x)

 

Square root of X

MAX(|Xn|)

 

Maximum of the spectral values Xn

RMS(xn) = SQRT([Ʃ (xn)2] / N)

 

Root Mean Square value of a signal

PSD(Xn)

 

Power Spectral Density

LSD(Xn)

 

Linear Spectral Density

A

 

Amplitude of a reference sine signal

The following table lists default scaling options (of type E_CM_ScalingType), which can be selected by the function blocks FB_CMA_PowerSpectrum and FB_CMA_MagnitudeSpectrum and function blocks derived from these. The resulting factors do not have to be evaluated by the user. They are given in the second column in order to be able to include further parameters if necessary. The values xn denote the input values of the function block and the values Xk the spectral value for the frequency channel k resulting from the scaling.

Scaling option

Contained factor

Description

Deterministic signals

eCM_PeakAmplitude

2 / Ʃwn

This scaling adapts the magnitude values in such a way that an input sine signal with the amplitude A reaches a maximum value of A. The result is independent of the type of window function. The unit of the magnitude value is the same as the unit of the input signal.

MAX(|Xk|) = A
However, the maximum values of the spectrum do not enable a robust estimation of the amplitude, since so-called Scalloping Losses may occur.

eCM_RootPowerSum

2 / SQRT(N * Ʃwn 2)

This scaling adapts the spectral values in such a way that for an input sine signal with the amplitude A, the square of the sum of the power values has the value A. Accordingly the square root of the sum of the squares of the magnitude values can also be used. The result is thus equal to the RMS value of the input signal multiplied by SQRT(2).

SQRT(Ʃ|Xk|2) = A

This scaling is suitable for the evaluation of narrow-band signals. Since the summing via neighboring frequency bands reduces scalloping losses, it is considerably more robust than eCM_PeakAmplitude.

eCM_RMS

SQRT(2/N * Ʃwn 2)

This scaling results in power values and the square root of their sum is equal to the RMS value of the input signal. A sinusoidal signal with the amplitude A results in a value of A/SQRT(2):

SQRT(Ʃ|X(k)|2) = RMS(xn) = A * SQRT(1/2)

Like eCM_ROOT_POWER_SUM this scaling is also robust and suitable for the evaluation of narrow-band signals. In addition the RMS value is also well-defined for broadband signals.

Stochastic and broadband signals

eCM_PowerSpectralDensity

SQRT(2 / Ʃwn2)

This scaling determines the Power Spectral Density (PSD). For broadband and stochastic signals this is independent of the parameters of the FFT and window function.

PSD(Xk) = |Xk|2/FS

In order to determine a physically correct power spectral density, the result must additionally be divided by the sampling rate of the input signal in Hertz. If the input signal has the unit Volt, then the unit 1 V/Hz is obtained for the magnitude and the unit 1 V2/Hz for the power density. Division by the root of the sampling rate must take place for the Linear Spectral Density; the unit is then 1 V/(1 Hz)1/2:

LSD(Xk) = |Xk|/ SQRT(FS)

Elementary

eCM_DiracScaling

sqrt(N / Ʃwn 2 )

This scaling normalizes the power spectrum in such a way that the broadband signal is equal to the unscaled FFT (with the definition given above). The influence of window type and window length is thus eliminated. However, the effect of the FFT-length N exists just as it does with the unscaled FFT.

eCM_NoScaling

1

No scaling. The result consists of the application of the window function (which always has a maximum of one in accordance with convention) followed by the FFT.