Fourier Transform

The Fourier transform is very similar to the Laplace transform. The Fourier transform uses assumption that any finite time-domain can be broken into a finite sum of sinusoidal (sine and cosine). Under this assumption, the Fourier transform converts a time domain signal into its frequency domain representation as a function of the radial frequency.

The Fourier transform of a signal is defined as follows:

(1)

It is possible to show hat the Fourier transform is equivalent to the Laplace transform when the following condition is true:

then

(2)

The formulas (1) and (2) represent direct Fourier transform.

The inverse Fourier transform is defined as follows:

(3)

The Fourier transform exists for all functions that satisfy the following condition

.

Simulink Toolbox permits to simulate white noise with the help of Band-Limited white noise block that located in a Library Sources. Band-Limited white noise block is given in Fig.3.

Fig.3 White noise generation block

This block includes the following parameters:

- N oise Power (intensity). This parameter is actually the height of the PSD of the white noise.

- Sample time. The correlation time of the noise.

- Seed. The starting seed for a random number generator.

The intensity on the output of the block is evaluated in the following way: Noise Power divided by tc, where tc – is a sample time.

Signal processing. Abovementioned that, in practice, the physical systems are disturbed both with deterministic and stochastic influences. Thus, during the investigations we obtain information about the process (process that we investigate) as the stochastic signal. The influence of deterministic component can be described via functional relations. Meanwhile, the stochastic signals give pretty enough information about the object (plant, system). Moreover, it is very difficult to describe the stochastic signals via mathematical relations. Thus, their description is utilized via probability characteristics such as: mathematical expectation, standard deviation, covariance function, spectral density. The stationary stochastic processes are of grate importance.

In a MatLab environment it is possible to determine some of them, for example, power spectral density. The power spectral density of the stochastic signal is estimated via psd command:

psd(x,nfft,Fs,window),

where x is a signal;

nfft is a number of points of a signal. Actually, it defines frequencies where the spectral density is estimated;

Fs is a sample time;

Window definestype of the window used during the PSD estimation. The following windows can be exploited: blackman(n), hamming(n) and returns n- point window, n is a positive integer.

In order to simulate the random process in a m. file the following initial data are necessary to initialize:

Simulation time: Tmax=100;

Sample time: dt=0.01;

Time interval: t=0:dt:100;

Maximum frequency: fs=1/dt;

Sample of frequency: df=1/Tmax;

Vector of frequency: f=0:df:fs;

White noise is generated via the following command: u=randn(1,120); u=randn(size(t));

Define a filter with its transfer function: W=tf([1 1],[1 0.01 1]) and perform simulation by applying the following command:

ffilter=lsim(sys,u,t,'zoh');

The cross spectral density of the stochastic signal is estimated via csd command:

csd(x, y,nfft,Fs,window),

where x, y are the signals;

nfft is a number of points of a signal. Actually, it defines frequencies where the cross spectral density is estimated;

Fs is a sample time;

Window definestype of the window used during the CSD estimation

To define and build the correlation function use he following syntax:

[c,lags]=xcorr(x,y)

plot(c, lags), grid on

Дата добавления: 2015-11-14