What is it all about ?

The aim of the applet is to show from simple examples the effect of down sampling, windowing and noise in the time and frequency domains.

You have to select a function of time among the proposed ones and select the parameters (sampling period, range of representation, window type, window...) according to what you want to observe.

The results obtain will be plotted of the sampled function in the time domain and of its frequency image calculated over the selected window. The frequency representation can be given under several forms : Modulus, Phase, Real and Imaginar parts.

- sinus, cosinus.
- sinc.
- Rectangle , Triangle.
- Impulse function (represented with $(t)).
- Ramdom function, use to simulate noise.

The applet proposes several simple functions commonly used in signal processing:

To select a function you can either type it directly in the text box (X(T)= ...) or select it from the list. In the latter case the program will ask you for parameters (such as amplitude or frequecy of the signal).

The input function variable must be T or t (which are representing the same thing).

You can use several functions at the same time, the possible operations are addition, substraction, multiplication and division.

All operations must be written explicitly (i.e.5sin(10t) must be write 5*sin(10*t))

You can as well change the signal frequency (for example sin(5*t) or rect(12*t) or shift the function (for example : sinc(t-5))

The sample period range is free; by the way you are advised not to choose it too low (high sample frequency) in order to limit the number of calculations

The range choice is free also. This range of the signal representation is in the time domain. Once again you are advised not to choose a too wide range in order to limit the number of computations and keep a good representation.

A window is always chosen. Implicitely, the rectangular window is selected (which means extracting a set of samples without any alteration in order to calculate the FFT). Some other windows (less "natural") have proved themselves to have a good behaviour in the frequency domain, it is the reason why they could be chosen.

The window will appear in green in the time domain plotting

The window size is the width of the window. It corresponds to the number of samples used for the computation of the Fourier Transform (FT). Consequently since the sample period has been chosen , the window size determines the range of time over which the FT is performed; this range is equal to (sample period)*(size of the window); it is not necessary to include the range of representation in the time domain. The window size need to be a power of 2 since the program use a FTT (Fast Fourier Transform) Algorithm.

The window type has to be chosen among four types (Rectangular, Tringular, Hann , Hamming).

The Frequency spectrum can be obtain under different forms. You have the choice between a representation with the Modulus and the Phase of the Fourier transform or a representation with the real and imaginar parts of it.

The results appear in a separate window after clicking on the button "plot".

The first graph is the plotting of the function in the temporal domain. Each yellow vertical line corresponds to a sample. The green plot represents the window.

The second graph is the plotting of the Discrete Fourier Transform over one period (i.e. from 0 to the sampling frequency). Angles are given in radian for the Phase representation.

Created by Stéphane Verlet student at ESSTIN, last updated February 8th 1998