The expansion coefficients can be obtained using the orthogonality of the Laguerre polynomials by evaluation of the appropriate integral of the signal with the k-th basis functions. For details see http://mathworld.wolfram.com/OrthogonalPolynomials.html. Laguerre polynomials are also important in the context of numerical integration: They essentially give Gauss quadrature rules for the semiinfinite interval and the weight function exp(-x).

The parameter s>0 can in principle be chosen arbitrarily. (One can scale the integral of the orthogonality using this parameter, leaving the integration interval unchanged. However, it may be important as regards the convergence velocity of the expansion. In applications, normally one has some sort of natural time scale T of the signal and then, the choice s=1/T may be a reasonable starting value for further investigations.

I remember having seen even complex s. But then, the integration theory becomes more complicated.

http://mathworld.wolfram.com/OrthogonalPolynomials.html

The expansion coefficients can be obtained using the orthogonality of the Laguerre polynomials by evaluation of the appropriate integral of the signal with the k-th basis functions. For details see http://mathworld.wolfram.com/OrthogonalPolynomials.html. Laguerre polynomials are also important in the context of numerical integration: They essentially give Gauss quadrature rules for the semiinfinite interval and the weight function exp(-x).

The parameter s>0 can in principle be chosen arbitrarily. (One can scale the integral of the orthogonality using this parameter, leaving the integration interval unchanged. However, it may be important as regards the convergence velocity of the expansion. In applications, normally one has some sort of natural time scale T of the signal and then, the choice s=1/T may be a reasonable starting value for further investigations.

I remember having seen even complex s. But then, the integration theory becomes more complicated.

http://mathworld.wolfram.com/OrthogonalPolynomials.html

Hello Herbert H H Homeier !

Thank you very much for your answer!

I think I am already  able to start choosing the s parameter. 

However, it is not 100% clear to me how to calculate the expansion coefficients.

According to the reference you sent me, should I use equation (2) to determine such coefficients?

I have already seen in some references, that the coefficients calculation could be performed by:

ak = int(0, Inf) [e(t) * exp(-s*t/2) * Lk(s*t)  d(s*t)]

Where:

ak = coefficient of order k;

Lk(t) = Laguerre Polynomial of order k;

s = time scale.

Is that suitable? I am not getting suitable results trying to expand a signal e(t) by this way. [just for example, lets assume e(t) = sin(w*t)]

Regards!

Hello Roy R. Lederman,

I thank you very much for your answer. It helped me a lot.

I am calculating the integral by means of MATLAB. This software offers the possibility of solving the integral in a analytically way. However, aiming to improve the code, I will consider to solve it numerically in the future.

The idea I have to approximate the singnal e(t) = sin(w*t) by means of laguerre function basis is just multiply the function e(t) by a descending exponential, i.e.:

e*(t) = sin(w*t) * exp(-a*t)

If I choose a suitable value for the constant a, the function e(t) won't be different from e*(t) inside a desired interval. In that way, I could reproduce the signal e(t) [or e*(t)] by means of the laguerre basis.