About Signal Operators

This operator computes the arithmetic mean of the curve on the selected range. It is computed with the following formula (trapezoid approximation):

In the formula, n is the number of points of the curve in the selected range, and a_i is the value of each of these points.

This operator computes the average rectified mean of the curve on the selected range. It is computed by taking the mean value of the absolute values:

The function f takes the average of the absolute values and if both values have the same sign it is just , and if they have different sign: . The reason for the complex formula is to ensure that any piece-wise linear signal (including triangular waves) is correct.

In the case of a set of n values {x₁, x₂, x₃,..., x_n} representing the values of the point of the curve, the RMS (Root Mean Square) value is given by this formula:

Note that for the important case of equidistantly sampled sine signal (with arbitrary phase) this formula gives the exact result (below the Nyquist frequency, and if the simulated interval is a multiple of the period of the sine signal).

This property is related to the standard deviation in the statistics and can be computed in two different steps. (The first step is used to avoid numerical issues with round-off and overflow):

This signal operator is similar to the arithmetic mean signal operator above. Comparing with the first formula for that signal operator above, to get the value at x2, we change “Mean(a)” to “Mean(a, x2)”. In the integrations we change “min” to “x2-w/2”, “max” to “x2+w/2”. Finally, instead of integrating “a(x)”, we change to integrating “a(x)/w” where w is the width of the moving average “window”.

In general, the signal operator is implemented as a Centered Moving Average. If any endpoint is missing for the calculation, it is calculated by interpolation.

The amplitude of the first harmonic is the amplitude of the sine signal of the given period (the given period should correspond to the base frequency of the signal); in electrical applications the frequency/period of the signal is usually known.

The operator is based on a windowed Fast Fourier Transform (FFT), see last section.

Note that the corresponding block in Modelica.Blocks in the Modelica Standard Library has an output called rms, which is the rms value = amplitude/sqrt(2). For a periodic signal, the amplitude of the first harmonic can exceed the amplitude of the underlying signal.

The total harmonic distortion operator (THD) used here is called amplitude ratio THD on Wikipedia (and measures how distorted the signal is compared to an ideal sine signal):

In practice the sum is always truncated due to the sampling, and to get comparable result independent of the number of points stored the sum could be truncated before the sampling limit.

The operator is based on the windowed Fast Fourier Transform (FFT) described in the next section.

The Fast Fourier Transform (FFT) is a generalized FFT that can handle any input length, not only powers of 2. Windowed FFT means that the signal is premultiplied by a windowing function; this is customary in signal processing and gives better result for signals that are not actually periodic (for example the addition of random noise or a slight trend).

The FFT gives a decomposition of the signal as (computer efficiency):

The frequency-based computations are based on a number of equidistant sampled points; the distance between the points should correspond to output interval length in the execution setup.

The FFT can also be used to compute the period/frequency of the first harmonic; a simple formula is added here to take the period corresponding to the highest amplitude in the FFT (excluding the DC component).