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The matrix formulation of the (discrete) Fourier transform for a signal 5 terms long, can be illustrated as follows:

Signal or time domain

$\left( \begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \end{array} \right)$

Dicrete Fourier Cosine Transform

$\left( \begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) \\ 1 & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\ 1 & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\ 1 & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) \end{array} \right)$

Spectrum or frequency domain

$\left( \begin{array}{c} 5 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right)$

Where the signal or time domain is multiplied element wise with each row in the Fourier transform matrix. Taking the row sums of the resulting matrix gives the spectrum or frequency domain.

Is there a corresponding matrix formulation for the Laplace transform? And if so what does it look like for a 5 times 5 matrix?

Hopefully this is not a waste of space on the math SE site with yet another question on the Laplace transform.

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    (cont., sorry for the ramble): At the end of the day, however, you need to ask what operations are convenient for you (convolution, computational speed, etc.) and the choose accordingly.2012-12-19

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