Usually books (e.g. Oppenheim) use the notation X(jw) for the continuous time Fourier Transform and X(e^j$\Omega$) for the discrete time Fourier transform.

AFAIK the only parameter is w o $\Omega$, respectively. Why they include these other parameters in the representation?


1 Answer 1


The argument $j\omega$ in the Fourier transform comes from the Laplace transform, where the complex argument is $s=\alpha+j\omega$. If the imaginary axis is inside the region of convergence of the Laplace transform, the Fourier transform can be obtained by setting $s=j\omega$.

A completely analogous argument is true for the discrete-time Fourier transform and the $\mathcal{Z}$-transform: if the unit circle is inside the region of convergence of the $\mathcal{Z}$-transform, the discrete-time Fourier transform can be obtained from evaluating the $\mathcal{Z}$-transform on the unit circle, i.e, by setting $z=e^{j\Omega}$. This way of choosing the argument of the discrete-time Fourier transform also makes explicit its $2\pi$-periodicity.

These are just conventions, and it is perfectly possible, yet less common, to use $\omega$ (or $\Omega$) as the argument of the Fourier transform.

Also take a look at these two related questions over at DSP.SE: Q1, Q2.


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