Existence of The Fourier Transform
Sources:
B. P. Lathi & Roger Green. (2018). Chapter 7: Continuous-Time Signal Analysis. Signal Processing and Linear Systems (3nd ed., pp. 704-708). Oxford University Press.
Because the Fourier transform is derived here as a limiting case of the Fourier series, it follows that the basic qualifications of the Fourier series, such as equality in the mean and convergence conditions in suitably modified form, apply to the Fourier transform as well. It can be shown that if \(x(t)\) has a finite energy, that is, if \[ \int_{-\infty}^{\infty}|x(t)|^2 d t<\infty \] then the Fourier transform \(X(\omega)\) is finite and converges to \(x(t)\) in the mean. This means, if we let \[ \hat{x}(t)=\lim _{W \rightarrow \infty} \frac{1}{2 \pi} \int_{-W}^W X(\omega) e^{j \omega t} d \omega \] then Eq. (7.10) implies \[ \int_{-\infty}^{\infty}|x(t)-\hat{x}(t)|^2 d t=0 \]
In other words, \(x(t)\) and its Fourier integral [the right-hand side of Eq. (7.10)] can differ at some values of \(t\) without contradicting Eq. (7.13). We shall now discuss an alternate set of criteria due to Dirichlet for convergence of the Fourier transform.
As with the Fourier series, if \(x(t)\) satisfies certain conditions (Dirichlet conditions), its Fourier transform is guaranteed to converge pointwise at all points where \(x(t)\) is continuous. Moreover, at the points of discontinuity, \(x(t)\) converges to the value midway between the two values of \(x(t)\) on either side of the discontinuity. The Dirichlet conditions are as follows: 1. \(x(t)\) should be absolutely integrable, that is, \[ \int_{-\infty}^{\infty}|x(t)| d t<\infty \]
If this condition is satisfied, we see that the integral on the right-hand side of Eq. (7.9) is guaranteed to have a finite value. 2. \(x(t)\) must have only a finite number of finite discontinuities within any finite interval. 3. \(x(t)\) must contain only a finite number of maxima and minima within any finite interval.
We stress here that although the Dirichlet conditions are sufficient for the existence and pointwise convergence of the Fourier transform, they are not necessary. For example, we saw in Ex. 7.1 that a growing exponential, which violates Dirichlet's first condition in Eq. (7.14), does not have a Fourier transform. But the signal of the form \((\sin a t) / t\), which does violate this condition, does have a Fourier transform.
Any signal that can be generated in practice satisfies the Dirichlet conditions and therefore has a Fourier transform. Thus, the physical existence of a signal is a sufficient condition for the existence of its transform.