Helper variable Hi

Here the function of the helper variable 'Hi' can be set. Hi can be a sum counter, a parameter, a variable or be used for computiong the impulse response of functions. It may appear in the function term.

Sum counter The calculation fo the function is repeated for each defined step, with the function results for each X value are summed up. This option does NOT work together with the diff() and s() functions !
Parameter As before, but the results are not summed up but written to a separate graph for each value of 'Hi'.
Parameter list As before, but the parameters can be individually defined by entering the values in the text field. Values must be separated with blanks or windows separators.
Variable 'Hi' has the value of the 'Start value' field.
Fourier For the calculation of the impulse response (time domain) of a function defined in the frequency domain, the fourier theory is used. The response of the function is calculated for the base frequency and harmonics of the defined impulse function, and the results are overlayed for the total response.

A rectangular impulse can be computed by summing up the results of

N

Y = Σ sin((1+hi*2)*2*π*f*t)/(1+hi*2)

hi=0

with f as the frequency of the rectangular pulse. For a perfect rectangular pulse an infinite number of harmonics needs to be computed, but for practical purposes 10-50 harmonics give a already a very good approximation.

When using 'Hi' in the fourier mode, each value of 'Hi' is read as the frequencies of the to be overlapped sine waves. They are summed up with an amplitude of 1/(1+(number of harmonic)*Step/Start value), which gives for example

Start value : 2000

End value : 18000

Step : 4000

a rectangulare pulse with base frequency of 2000Hz. For this example the following frequencies are computed:

1. frequency: f= 2000 Hz Amplitude=1
2. frequency: f= 6000 Hz Amplitude=1/3
3. frequency: f=10000 Hz Amplitude=1/5
4. frequency: f=14000 Hz Amplitude=1/7
5. frequency: f=18000 Hz Amplitude=1/9

The current functions is computed for each value of 'Hi' and the results saved. Now the frequency components of the rectangular pulse is multiplied with the function values and all summed up, which results in the response function for the rectangular pulse. The X-value range is in the time domain for this calculation.

The values for 'Hi' can be modified such that other pulse forms such as sawtooth or triangle are computed.

The menu option fourier mode opens a window that simplifies the definition of a rectangular pulse.