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What are the different Plotter operations available in CFD-VIEW?

The Plotter operator in CFD-VIEW supports signal processing for Time History data. A previous user tip Digital Signal Processing using CFD-VIEW shows how time history date can be made available and how to perform a PSD on a periodic signal. It is recommended to review this user tip before going further.The aim of this new user tip is to list all options supported by the Plotter Operator.

This tip will discuss the following operations:
- Correlation
- Fast Fourier Transform
- Discrete Fourier Transform
- Integral
- Derivative
- Power Spectral Density
- Smoothing
- Filter
- Sound Pressure Level
- Window

If you are interested in these functions, please read on.

1. Correlation

This operation performs an auto or inter-correlation, depending on whether one or two entities are currently selected (no more than two entities are authorized). The user has to define two parameters, both in number of sampling points:
- Lag length: the length over which the correlation has to be estimated (by default, half the number of curve points).
- Samples number: allows the user to truncate the curves given the number of abscissa points. The default value is the number of points in the curve.

Inter-correlation assumes that both curves have the same number of points.

The result is a curve {Ri, i=1,n_sample}, where n_sample is the value of Samples number, and Ri is defined as:

Image

Where:
- x and y are the point-wise values of the single curve (auto-correlation) or of each of the two curves (inter-correlation),
- N is the lag length,
- <x>M1,M2 is the average of x over the sampling points ranking from M1 to M2

Image

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2. Fast Fourier Transform (FFT)

The algorithm used is the base 2 FFT, which performs a temporal permutation at the start of the computation. The input signal step must be quasi constant and increasing. Otherwise the transformation cannot be performed.

If the signal has not a number of points of the form 2n, it is re-sampled by linear interpolation. In any case (with or without re-sampling) the FFT result is scaled by the signal time step Δt.

An option is available to convert the FFT magnitude in dB:

Image

The phase, the imaginary and the real part of the FFT is also provided.

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3. Discrete Fourier Transform (DFT)

This option is applied on the selected data via the list Entities. It computes the Discrete Fourier Transform and normalizes it by 1/Fe (as with the FFT). The DFT has the same output options as the FFT (Magnitude ...) and performs (n N) operations (N is the number of frequencies and n is the number of samples). Its algorithm is not optimized as the FFT (n x ln(n) operations).

The user input are the minimum, maximum frequencies and the frequency step, so that the transform is computed with a limited required frequency bandwidth and a frequency resolution (therefore, this can reduce the time computation).

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4. Integral

Select one entity in the list of external entities, corresponding to {y1i, i=1,n}. Choose the Integral item in the Operation list, then Create at the bottom of the window: this will create a new curve defined on the set of mid-points of the original set of abscissa, defined as:

Image

where i =1,n-1.

The integral curve is defined on n-1 points only.

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5. Derivative

Select one entity in the list of external entities, corresponding to {y1i, i=1,n}. Choose the Derivative item in the Operation list, then Create at the bottom of the window: this will create a new curve defined on the set of mid-points of the original set of abscissa, defined as {y2i = (y1i+1 - y1i-1) / (x1i+1 - x1i-1) , i=2,n-1}.

The derived curve is defined on n-2 points only.

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6. Power Spectral Density (PSD)

The PSD is computed by the previously described FFT so that the signal step must be constant and increasing. If the number of signal points is not of the form 2n, it is re-sampled by linear interpolation.

The PSD is the power density per bandwidth of frequency:

Image

Where N is the number of signal points, Dt is the signal time step, and T is the total time of the signal.

Image

The result is then expressed in (Physical Unit)2/Hz.

The value of PSD magnitude in dB is given by

Image

REF_PRESSURE is equal to 2.0e-5

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7. Smoothing

Select 1 entity in the list of external entities, corresponding to {y1i, i=1,n}. Choose the Smooth item in the Operation list, then Create at the bottom of the window: this will create a new curve defined on the set of n-2 "interior" original X-points, defined as {y2i= (y1i-1 + y1i + y1i+1) / 3, i = 2,n-1}.

The smoothed curve is defined on n-2 points only.

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8. Filter

The Window frequency filter is used to block some frequency and pass others. For example, used as a low pass filter, it can erase the numerical noise which can appear in numerical computed results.

Two user defined frequency gives the window parameters. By default, the band pass is [0,υmax]; these parameters give back the original signal.

Time signal process:
The frequency product is performed: X(υ)H(υ). H(υ)= if υ1 < υ < υ2, else H(υ)=0 and X(υ) is the FFT of the input signal. The Inverse Fast Fourier Transform is then directly applied on the frequency product to give the time filtered signal. The IFFT is performed by the same algorithm than the FFT.

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9. Sound Pressure Level (SPL)

The Sound Pressure Level is computed by the formula:

Image

where the reference acoustic pressure is 2.10-5 Pa, Sυ is the power spectral density, and [υ1,υ2] is the bandwidth frequencies expressed in octave, 1/3 octave, or user defined band in Hz.

The power spectral density is computed as described previously.

In each band j, the integral

Image

is approximated by the trapezoid method. The SPL is then computed by:

Image

Acoustic filters can be applied to the result in order to obtain SPL A, B, or C.

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10. Window

The windowing operation is defined by its type and its size. The size is specified by the user through the Lower abscissa and the Upper abscissa input fields, whose default value are the min & max of the abscissa of each entity, separately. Windowing can be used, for instance, to truncate irrelevant parts of a signal before its Fourier analysis.

The size of the window identifies a list of N abscissa index k, where the weight Wk applied to the point-wise signal value will be different from 0. This weighting function Wk applied over the window size depends on the window type, and is defined from the literature as follows:

- Rectangular: Wk = 1

- Hamming: Wk = 0.54 - 0.46*cos(2πk/(N-1))

- Hanning: Wk = 0.50 - 0.50*cos(2πk/(N-1))

- Blackman: Wk = 0.42 - 0.50*cos(2πk/(N-1)) + 0.08*cos(4πk/(N-1))

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Last update: 2009-07-14 15:40
Author: ESI-CFD Support Team
Revision: 1.5

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