2.13 Fourier transforms as unitary operators

Fourier transforms as unitary operators

We have just seen that the DTFT can be viewed as a unitary operator between ${\ell }_2\left(\mathbb{Z}\right)$ and $L_2[-\pi ,\pi ]$ . One can repeat this process for each Fourier transform pair. In fact due tothe symmetry between the DTFT and the CTFS, we have already established this for CTFS, i.e.,

is a unitary operator. Similarly, we have

is a unitary operator as well. The proof of this fact closely mirrors the proof for the DTFT. Finally, we also have

This operator is also unitary, which can be easily verified by showing that the DFT matrix is actually a unitary matrix: $U^{H}U=U{U}^H=I$ .

Note that this discussion only applies to finite-energy ( ${\ell }_2/L_2$ ) signals. Whenever we talk about infinite-energy functions (things like theunit step, delta functions, the all-constant signal) having a Fourier transform, we need to be very careful about whether we are talking about atruly convergent Fourier representation or whether we are merely using an engineering “trick” or convention.