2.3. Periodic Dirac delta#
Common ideal signals, such as sinusoids and the unit step, do not have finite energy. The forward DTFT series on the RHS of (2.1) do not converge for these signals, and hence they do not have DTFTs strictly speaking.
These signals are mathematical idealizations that are much nicer to deal with than their practical counterparts. We may use the \(z\)-transform toolset described before to analyze these infinite-energy signals. However, doing so we lose the frequency-domain interpretation/intuition provided by DTFT because the ROCs of the \(z\)-transforms of these signals do not contain the unit circle.
A standard way to “extend” the DTFT toolset to cover these common infinite-energy signals is by way of the periodic Dirac delta function.
2.3.1. “Limit” of window signal#
Recall the window signal in Section 2.2.3:
\[\begin{equation*} w_{\lambda} [n] = \lambda^{|n|} ~~\stackrel{\text{DTFT}}{\longleftrightarrow} ~~ W_{\lambda}(e^{j\hat\omega}) = \frac{1 - \lambda^2}{1 - 2\lambda \cos \hat{\omega} + \lambda^2} \end{equation*}\]where \(\lambda \in (0,1)\).
Recall \(w_{\lambda} [n] \in \ell^1\). For each \(n \in \mathbb{Z}\), \(\lim_{\lambda \rightarrow 1} w_{\lambda} [n] = 1\). Intuitively, \(w_{\lambda} [n]\) becomes closer and closer to the constant signal \(1\) (not in \(\ell^1\)) as \(\lambda\) approaches \(1\).
Clearly, the DTFT \(W_{\lambda}(e^{j\hat\omega})\) is a positive, even, and periodic (with period \(2\pi\)) function in \(\hat\omega\). It also satisfies the following properties:
\(\displaystyle \lim_{\lambda \rightarrow 1} W_{\lambda}(e^{j\hat\omega}) = \begin{cases} \infty, & \hat\omega = 0 \bmod 2\pi \\ 0, & \hat\omega \neq 0 \bmod 2\pi. \end{cases}\)
Since \(\displaystyle \frac{1}{2\pi} \int_{-\pi}^{\pi} W_{\lambda}(e^{j\hat\omega}) \, d\hat\omega = w_{\lambda} [0] = 1\) for all \(\lambda \in (0,1)\), we may set \(\displaystyle \lim_{\lambda \rightarrow 1} \int_{-\pi}^{\pi} \frac{W_{\lambda}(e^{j\hat\omega})}{2\pi} \, d\hat\omega =1.\)
For any bounded \(X(e^{j\hat{\omega}})\), the circular convolution integral \(\displaystyle \frac{1}{2\pi} \int_{-\pi}^{\pi} W_{\lambda}(e^{j\theta}) X(e^{j(\hat{\omega}-\theta)}) \,d\theta\) exists. In addition, if the bounded \(X(e^{j\hat{\omega}})\) is continuous at \(\hat{\omega} = \hat{\omega}_0\), then it can be shown that
\[\begin{equation*} \lim_{\lambda \rightarrow 1} \int_{-\pi}^{\pi} \frac{W_{\lambda}(e^{j\theta})}{2\pi} X(e^{j(\hat{\omega}_0-\theta)}) \, d\theta = X(e^{j\hat{\omega}_0}). \end{equation*}\]For any \(x[n] \in \ell^1\), the DTFT \(X(e^{j\hat{\omega}})\) of \(x[n]\) is bounded and continuous on \([-\pi,\pi]\) as discussed in Section 2.1.1. In addition, the multiplication property of DTFT gives
(2.5)#\[\begin{equation} w_{\lambda}[n] x[n] ~~\stackrel{\text{DTFT}}{\longleftrightarrow} ~~ \frac{1}{2\pi} \int_{-\pi}^{\pi} W_{\lambda}(e^{j\theta}) X(e^{j(\hat{\omega}-\theta)}) \, d\theta. \end{equation}\]Taking limit on the LHS of (2.5), \(\lim_{\lambda \rightarrow 1} w_{\lambda}[n] x[n] = x[n]\) for each \(n \in \mathbb{Z}\) because \(\lim_{\lambda \rightarrow 1} w_{\lambda}[n] = 1\) as discussed above. On the RHS, we have \(\displaystyle \lim_{\lambda \rightarrow 1} \frac{1}{2\pi} \int_{-\pi}^{\pi} W_{\lambda}(e^{j\theta}) X(e^{j(\hat{\omega}-\theta)}) \, d\theta = X(e^{j\hat{\omega}})\) for each \(\hat{\omega}\) from 3. above. Thus in this sense, we can say that the DTFT mapping in (2.5) is preserved through the limiting process as \(\lambda \rightarrow 1\).
Consider now \(x[n]\) is an infinite-energy sinusoid (or the unit step). For every \(\lambda \in (0,1)\), the “windowed” signal \(w_{\lambda}[n] x[n] \in \ell^1\), and hence its DTFT \(X_{w_{\lambda}}(e^{j\hat{\omega}})\) exists.
Caution
However, (2.5) fails to hold, i.e., \(X_{w_{\lambda}}(e^{j\hat{\omega}})\) is not given by the RHS of (2.5), because the DTFT \(X(e^{j\hat{\omega}})\) of \(x[n]\) simply does not exist!
Nonetheless, we still have \(\lim_{\lambda \rightarrow 1} w_{\lambda}[n] x[n] = x[n]\) for each \(n \in \mathbb{Z}\). Thus, it makes intuitive sense to mimic the above limiting process in the case of absolutely summable signals to define a “DTFT” for the infinite-energy sinusoid \(x[n]\). That is, we want to call \(\lim_{\lambda \rightarrow 1} X_{w_{\lambda}}(e^{j\hat{\omega}})\) the DTFT of \(x[n]\), if the limit can be interpreted similar to that of \(W_{\lambda}(e^{j\hat\omega})\), even such interpretation may not exactly result in a mathematically meaningful function in \(\hat{\omega}\).
Tip
In order to avoid using more math machinery, we will interpret this limiting process as our way to extend the DTFT toolset to infinite-energy signals. Note that as long as the windowed signal \(w_{\lambda}[n] x[n] \in \ell^1\), \(w_{\lambda}[n] x[n] \stackrel{\text{DTFT}}{\longleftrightarrow} X_{w_{\lambda}}(e^{j\hat{\omega}})\) and all the DTFT properties in Section 2.2.2 apply to this proper DTFT mapping. As a result, all the DTFT properties also apply to the extended DTFT of \(x[n]\) defined through the limiting process.
2.3.2. Periodic Dirac delta “function”#
It is annoying however to keep referring to the above limiting process and writing down \(\lim_{\lambda \rightarrow 1}\) every time we apply DTFT tools to an infinite-energy signal. For convenience, we define the periodic Dirac delta “function” \(\delta(e^{j\hat{\omega}})\) as a mnemonic for the limiting process. Specifically, properties 1.-3. above are re-expressed in terms of \(\delta(e^{j\hat{\omega}})\):
\(\displaystyle \delta(e^{j\hat{\omega}}) = \begin{cases} \infty, & \hat\omega = 0 \bmod 2\pi \\ 0, & \hat\omega \neq 0 \bmod 2\pi \end{cases} ~~\left[ = \lim_{\lambda \rightarrow 1} \frac{W_{\lambda}(e^{j\hat\omega})}{2\pi} \right]\).
\(\displaystyle \int_{-\pi}^{\pi} \delta(e^{j\hat{\omega}}) \, d\hat{\omega} = 1\).
For any bounded \(X(e^{j\hat{\omega}})\) that is continuous at \(\hat{\omega} = \hat{\omega}_0\),
\[\begin{equation*} \int_{-\pi}^{\pi} \delta(e^{j\theta}) X(e^{j(\hat{\omega}_0-\theta)}) \, d\theta = X(e^{j\hat{\omega}_0}). \end{equation*}\]This is often referred to as the sifting property of the Dirac delta.
Caution
Note that \(\delta(e^{j\hat{\omega}})\) is not a well-defined function in \(\hat{\omega}\) by property 1. because its value is infinite when \(\hat\omega = 0 \bmod 2\pi\). This infinite value implies that \(a\delta(e^{j\hat{\omega}}) = \delta(e^{j\hat{\omega}})\) for any \(a > 0\), which in turn implies \(\delta(e^{j\hat{\omega}}) \equiv 0\) contradicting 1. above.
In fact, it is more appropiate to associate the scaled Dirac delta \(a\delta(e^{j\hat{\omega}})\) with property 2. such that \(\displaystyle \int_{-\pi}^{\pi} a \delta(e^{j\hat{\omega}}) \, d\hat{\omega} = a\), i.e., the area underneath the Dirac delta over a period, rather than the “function” value, is scaled by the factor \(a\). Property 2., not 1., provides the basis to perform linear algebraic operations with \(\delta(e^{j\hat{\omega}})\).
The somewhat hand-waving limiting process associated with \(\delta(e^{j\hat{\omega}})\) can be made more rigorous by considering a sequuence of math objects called distributions in place of \(\frac{W_{\lambda}(e^{j\hat\omega})}{2\pi}\). It turns out that there is a valid distribution, playing the role of \(\delta(e^{j\hat{\omega}})\), that is the limit (in some sense) of the sequence of distributions corresponding to \(\frac{W_{\lambda}(e^{j\hat\omega})}{2\pi}\) as \(\lambda \rightarrow 1\).