Inverse Laplace Transform

Ilaplace command used to perform the inverse laplace transform for the given function. Let us consider an example;

F(s) =

 

F= 5*(s+2)/(s*(s^2+9*s+5));

 ilaplace (F)

ans = 9*exp(-2*t)*cos(t)+3 *exp(-2*t)*sin(t)+5

In inverse laplace transform function hyperbolic expression can solved

Sinh(x)=

Cosh(x)=

Fourier Transform In Matlab:

Some numerical methods are used for solving continuous time signals in matlab with Fourier transform. Let us choose an example to plot fourier transform function

f=-6:.01:6;

 X=9*sinc(9*f);

 plot(f,X)

Fast Fourier Transform:

Depending on the length of the sequence being transformed with the DFT the computation of this transform are often time consuming. The Fast Fourier Transform (FFT) is an algorithm for computing the DFT of a sequence during a more efficient manner. MATLAB provides a inbuilt command for computing the FFT of a sequence.

For example, using the same two-frequency signal x(t) used above we can produce a sequence of samples of length N = 150 spaced every Ts = .0003 seconds as shown previously.

Whilethe frequency components are spaced every 1/(N*Ts) Hz and the corresponding frequency values  are from 0 to (N-1)/(N*Ts) Hz as shown below.

In order to plot a fast fourier transform (fft) by using fft command

clear

 N=150;

 ts=.0003;

 deltaf=1/(N*ts);

 t=[0:N-1]*ts;

 x=cos(2*pi*100*t)+cos(2*pi*500*t);

 Xf=fft(x);

 f=[0:N-1]*deltaf;

 plot (f,abs(Xf)

In order to plot a fast fourier transform (fft) we should follow two rules

In order to plot a fast fourier transform (fft) we should follow two rules
Select Ts as large as possible but in order that the very best frequency component in your signal is a smaller amount than 1/2Ts. 2.

After determining the worth of Ts, select N in order that 1/NTs, the frequency resolution is little enough to accurately display your frequency components.