2.7. Dirac delta and FTs of infinite-energy signals#
Similar to DTFT, we can use windowing to extend the FT toolset to some common infinite-energy continuous-time signals by considering the following counterpart of the window signal in continuous time:
\[\begin{equation*} w_{\mu} (t) = e^{-\mu |t|} ~~\stackrel{\text{FT}}{\longleftrightarrow} ~~ W_{\mu}(\omega) = \frac{2\mu}{\mu^2 + \omega^2} \end{equation*}\]for any \(\mu >0\).
Similar to before, we can define the following (aperiodic) Dirac delta “function”:
\(\displaystyle \delta(\omega) = \lim_{\mu \rightarrow 0} \frac{W_{\mu}(\omega)}{2\pi} = \begin{cases} \infty, & \omega = 0 \\ 0, & \omega \neq 0 , \end{cases}\)
\(\displaystyle \int_{-\infty}^{\infty} \delta(\omega) \, d\omega = 1\), and
If \(X(\omega)\) is bounded and continuous at \(\omega = \omega_0\), then \(\displaystyle \int_{-\infty}^{\infty} \delta(\theta) X(\omega_0 - \theta) \, d\theta = X(\omega_0)\),
as a mnemonic to shorthand the limiting processing of considering \(w_{\mu} (t) \stackrel{\text{FT}}{\longleftrightarrow} W_{\mu}(\omega)\) as \(\mu \rightarrow 0\).
For an infinite-energy signal \(x(t)\) that grows only polynomially in \(|t|\), the windowed signal \(w_{\mu} (t) x(t) \in L^1 \cap L^2\) and hence has FT \(X_{w_{\mu}}(\omega)\). Since \(\lim_{\mu \rightarrow 0} w_{\mu} (t) x(t) = x(t)\) for each fixed \(t \in \mathbb{R}\), we may define the limit of \(X_{w_{\mu}}(\omega)\) as \(\mu \rightarrow 0\) (note again that such a limit may not exist) to be the FT of \(x(t)\).
Examples: Calculation of the FTs of the following signals in terms of \(\delta(\omega)\) is similar to the DTFT examples given in Section 2.4.
\(\displaystyle ~~e^{j\omega_0 t} \stackrel{\text{FT}}{\longleftrightarrow} 2\pi \delta(\omega - \omega_0)\)
\(\displaystyle ~~\cos(\omega_0 t) \stackrel{\text{FT}}{\longleftrightarrow} \pi \delta(\omega - \omega_0) + \pi \delta(\omega + \omega_0)\)
\(\displaystyle ~~\sin(\omega_0 t) \stackrel{\text{FT}}{\longleftrightarrow} \frac{\pi}{j} \delta(\omega - \omega_0) - \frac{\pi}{j} \delta(\omega + \omega_0)\)
\(\displaystyle ~~u(t) = \begin{cases} 1, & t \geq 0 \\ 0, & t < 0 \end{cases} \stackrel{\text{FT}}{\longleftrightarrow} \pi \delta(\omega) + \frac{1}{j\omega}\).
Let \(x(t)\) be a finite-power periodic signal with period \(T\). Hence, \(x(t)\) has the FS expansion \(x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j\frac{2\pi k t}{T}}\), where \(a_k\) is the \(k\)th FS coefficient of \(x(t)\). By the linearity property of FS and Example 1. above, we may extend the FT mapping to \(x(t)\) as:
\[\begin{equation*} x(t) \stackrel{\text{FT}}{\longleftrightarrow} 2\pi \sum_{k=-\infty}^{\infty} a_k \delta\left(\omega - \frac{2\pi k}{T} \right) \end{equation*}\]Caution
We have assumed the linearity property of FT extends to the infinite series of the FS synthesis formula above. This assumption can be justified in this case by first considering the partial sum \(\sum_{k=-N}^{N} a_k e^{j\frac{2\pi k t}{T}}\) and then the \(L^2[-T/2,T/2]\)-convergence of the partial sum to \(x(t)\) as \(N \rightarrow \infty\).
Similarly, we may extend the FT toolset to handle some infinite-energy FTs by considering the following frequency-domain window:
\[\begin{equation*} \tilde{w}_{\mu} (t) = \frac{\mu}{\pi (\mu^2 + t^2)}~~\stackrel{\text{FT}}{\longleftrightarrow} ~~ \tilde{W}_{\mu}(\omega) = e^{-\mu |\omega|} \end{equation*}\]for any \(\mu >0\). The same Dirac delta mnemonics 1.-3. can be used
For an infinite-energy FT \(X(\omega)\) that grows only polynomially in \(|\omega|\), the windowed FT \(\tilde{W}_{\mu} (\omega) X(\omega) \in L^1 \cap L^2\) and hence has an inverse FT \(x_{\tilde{w}_{\mu}} (t)\). Since \(\lim_{\mu \rightarrow 0} \tilde{W}_{\mu} (\omega) X(\omega) = X(\omega)\) for every \(\omega \in \mathbb{R}\), we may define the limit of \(x_{\tilde{w}_{\mu}} (t)\) as \(\mu \rightarrow 0\) as the inverse FT of \(X(\omega)\).
Example:
\(\displaystyle ~~\delta(t-t_0) \stackrel{\text{FT}}{\longleftrightarrow} e^{-j\omega t_0}\)