2.4. DTFTs of infinite-energy signals

• For any infinite-energy signal \(x[n]\) whose signal magnitude increases at most polynomially with \(|n|\) (i.e., \(|x[n]| = \mathcal{O}(|n|^{\alpha})\) for some \(\alpha>0\)), its windowed version \(w_{\lambda} [n] x[n] \in \ell^1\) and hence the DTFT mapping \(w_{\lambda}[n] x[n] \stackrel{\text{DTFT}}{\longleftrightarrow} X_{w_{\lambda}}(e^{j\hat{\omega}})\) exists for all \(\lambda \in (0,1)\). As discussed in Section 2.3.1, we may define \(\lim_{\lambda \rightarrow 1} X_{w_{\lambda}}(e^{j\hat{\omega}})\) as the DTFT \(X(e^{j\hat{\omega}})\) of \(x[n]\).

• Note that the limit of \(X_{w_{\lambda}}(e^{j\hat{\omega}})\) may not exist for any general \(x[n]\). Even if it exists, calculating it may be difficult. Luckily, for some of the most common infinite-energy signals, we can express \(\lim_{\lambda \rightarrow 1} X_{w_{\lambda}}(e^{j\hat{\omega}})\) in terms of \(\delta(e^{j\hat{\omega}})\) as shown below.

2.4.1. Sinusoid \(x[n] = e^{j\hat{\omega}_0 n}\)

Because

\[\begin{align*} X_{w_{\lambda}}(e^{j\hat{\omega}}) &= \sum_{n=-\infty}^{\infty} w_{\lambda}[n] x[n] e^{-j\hat{\omega} n} \\ &= \sum_{n=-\infty}^{\infty} \lambda^{|n|} e^{j\hat{\omega}_0 n} e^{-j\hat{\omega} n} \\ & = \frac{1 - \lambda^2}{1-2\lambda \cos(\hat{\omega}-\hat{\omega}_0) + \lambda^2} \\ &= W_{\lambda}(e^{j(\hat{\omega}-\hat{\omega}_0)}), \end{align*}\]

\(\displaystyle X(e^{j\hat{\omega}}) = \lim_{\lambda \rightarrow 1} X_{w_{\lambda}}(e^{j\hat{\omega}}) = \lim_{\lambda \rightarrow 1} W_{\lambda}(e^{j(\hat{\omega}-\hat{\omega}_0)}) = 2\pi \delta (e^{j(\hat{\omega}-\hat{\omega}_0)})\). In other words, we have established the DTFT pair

\[\begin{equation*} e^{j\hat{\omega}_0 n} ~~ \stackrel{\text{DTFT}}{\longleftrightarrow} ~~ 2\pi \delta (e^{j(\hat{\omega}-\hat{\omega}_0)}) \end{equation*}\]

For the case of \(\hat{\omega}_0 = 0\), we have

\[\begin{equation*} 1 ~~ \stackrel{\text{DTFT}}{\longleftrightarrow} ~~ 2\pi \delta (e^{j\hat{\omega}}) \end{equation*}\]

By the Euler identity and linearity of DTFT, we get

\[\begin{align*} \cos(\hat{\omega}_0 n) ~~ &\stackrel{\text{DTFT}}{\longleftrightarrow} ~~ \pi \delta (e^{j(\hat{\omega}-\hat{\omega}_0)}) + \pi \delta (e^{j(\hat{\omega}+\hat{\omega}_0)}) \\ \sin(\hat{\omega}_0 n) ~~ &\stackrel{\text{DTFT}}{\longleftrightarrow} ~~ \frac{\pi}{j} \delta (e^{j(\hat{\omega}-\hat{\omega}_0)}) - \frac{\pi}{j} \delta (e^{j(\hat{\omega}+\hat{\omega}_0)}) \end{align*}\]

2.4.2. Unit step \(u[n]\)

The DTFT \(U_{w_{\lambda}}(e^{j\hat{\omega}})\) of \(w_{\lambda}[n] u[n]\) is given by

\[\begin{equation*} U_{w_{\lambda}}(e^{j\hat{\omega}}) = \sum_{n=-\infty}^{\infty} w_{\lambda}[n] u[n] e^{-j\hat{\omega} n} = \sum_{n=0}^{\infty} \lambda^{n} e^{-j\hat{\omega} n} = \frac{1}{1-\lambda e^{-j \hat{\omega}}}. \end{equation*}\]

Caution

• Note that we can’t directly take limit of \(U_{w_{\lambda}}(e^{j\hat{\omega}})\) as \(\lambda \rightarrow 1\) to obtain the DTFT \(U(e^{j\hat{\omega}})\) of the unit step \(u[n]\) because the limit doesn’t exist at \(\hat\omega = 0 \bmod 2\pi\).

• To overcome this difficulty, we will express \(U_{w_{\lambda}}(e^{j\hat{\omega}})\) in terms of \(W_{\lambda}(e^{j\hat{\omega}})\) before taking limit to let the Dirac delta function \(\delta(e^{j\hat{\omega}})\) absorb all problems associated with the limiting process.

By the time reversal property, \(\displaystyle w_{\lambda}[-n] u[-n] \stackrel{\text{DTFT}}{\longleftrightarrow} U_{w_{\lambda}}(e^{-j\hat{\omega}}) = \frac{1}{1-\lambda e^{j \hat{\omega}}}\).

Since \(w_{\lambda}[n] u[n] + w_{\lambda}[-n] u[-n] = w_{\lambda}[n] + \delta[n]\), taking DTFT on both sides of this equation gives

(2.6)\[\begin{equation} U_{w_{\lambda}}(e^{j\hat{\omega}}) + U_{w_{\lambda}}(e^{-j\hat{\omega}}) = W_{\lambda}(e^{j\hat{\omega}}) + 1. \end{equation}\]

On the other hand,

(2.7)\[\begin{equation} U_{w_{\lambda}}(e^{j\hat{\omega}}) - U_{w_{\lambda}}(e^{-j\hat{\omega}}) = \frac{1}{1-\lambda e^{-j\hat{\omega}}} - \frac{1}{1-\lambda e^{j\hat{\omega}}} = \frac{-2j\lambda \sin\hat{\omega}}{1-2\lambda \cos\hat{\omega} + \lambda^2}. \end{equation}\]

Adding (2.6) and (2.7)gives

(2.8)\[\begin{equation} U_{w_{\lambda}}(e^{j\hat{\omega}}) = \frac{1}{2} W_{\lambda}(e^{j\hat{\omega}}) + \frac{1}{2} \left( 1 - \frac{2j\lambda \sin\hat{\omega}}{1-2\lambda \cos\hat{\omega} + \lambda^2} \right). \end{equation}\]

Since \(\displaystyle \lim_{\lambda \rightarrow 1} \frac{1}{2} \left( 1 - \frac{2j\lambda \sin\hat{\omega}}{1-2\lambda \cos\hat{\omega} + \lambda^2} \right) = \begin{cases} \frac{1}{2}, & \hat{\omega} = 0 \bmod 2\pi \\ \frac{1}{1-e^{-j\hat{\omega}}}, & \hat{\omega} \neq 0 \bmod 2\pi, \end{cases}\) taking limit as \(\lambda\rightarrow 1\) on both sides of (2.8) gives

\[\begin{equation*} U (e^{j\hat{\omega}}) = \pi \delta(e^{j\hat{\omega}}) + \frac{1}{1-e^{-j\hat{\omega}}}. \end{equation*}\]

Tip

Note the second term on the RHS above should be \(\begin{cases} \frac{1}{2}, & \hat{\omega} = 0 \bmod 2\pi \\ \frac{1}{1-e^{-j\hat{\omega}}}, & \hat{\omega} \neq 0 \bmod 2\pi. \end{cases}\) However, the exact value of the term at \(\hat{\omega} = 0 \bmod 2\pi\) is immaterial (as long as it is finite) because any such value can be abosrbed into the infinite value of \(\delta(e^{j\hat{\omega}})\) at the same \(\hat{\omega}\). Hence, for convenience, we may simply write \(\frac{1}{1-e^{-j\hat{\omega}}}\) as in above with the implicit understanding that the term’s value at \(\hat{\omega} = 0 \bmod 2\pi\) is, say, \(0\).

In summary, we have

\[\begin{equation*} u[n] ~~ \stackrel{\text{DTFT}}{\longleftrightarrow} ~~ \pi \delta(e^{j\hat{\omega}}) + \frac{1}{1-e^{-j\hat{\omega}}} \end{equation*}\]

2.4.3. Impulse train \(\delta_N[n] = \sum_{k=-\infty}^{\infty} \delta[n-kN]\)

Note that \(\delta_N[n]\) is periodic with period \(N\). Let \(\Delta_{N, w_{\lambda}}(e^{j\hat{\omega}})\) be the DTFT of \(w_{\lambda}[n] \delta_N[n]\). Then

\[\begin{align*} \Delta_{N, w_{\lambda}}(e^{j\hat{\omega}}) &= \sum_{n=-\infty}^{\infty} w_{\lambda}[n] \delta_N[n] e^{-j\hat{\omega} n} \\ &= \sum_{k=-\infty}^{\infty} \lambda^{|Nk|} e^{-j\hat{\omega} Nk} \\ &= \frac{1 - \lambda^{2N}}{1-2\lambda^N \cos(N\hat{\omega}) + \lambda^{2N}} \\ & = W_{\lambda^{N}}(e^{jN\hat{\omega}}). \end{align*}\]

It is easy to see that \(W_{\lambda^{N}}(e^{jN\hat{\omega}})\) behaves similar to \(W_{\lambda}(e^{j\hat{\omega}})\):

• \(W_{\lambda^{N}}(e^{jN\hat{\omega}})\) is periodic in \(\hat\omega\) with period \(\frac{2\pi}{N}\),

• \(\displaystyle \lim_{\lambda \rightarrow 1} W_{\lambda^{N}}(e^{jN\hat{\omega}}) = \begin{cases} \infty, & \hat\omega = 0 \bmod \frac{2\pi}{N} \\ 0 & \hat\omega \neq 0 \bmod \frac{2\pi}{N}, \end{cases}\) and

• \(\displaystyle \lim_{\lambda \rightarrow 1} \frac{1}{2\pi} \int_{-\pi}^{\pi} W_{\lambda^{N}}(e^{jN\hat{\omega}}) \, d\hat{\omega} = 1\).

As a matter of fact, we may express \(\lim_{\lambda \rightarrow 1} W_{\lambda^{N}}(e^{jN\hat{\omega}})\) in terms of the Dirac delta function \(\delta(e^{j\hat{\omega}})\):

\[\begin{equation*} \lim_{\lambda \rightarrow 1} W_{\lambda^{N}}(e^{jN\hat{\omega}}) = \frac{2\pi}{N} \sum_{k = 0}^{N-1} \delta\left(e^{j(\hat{\omega} - \frac{2\pi k}{N})} \right). \end{equation*}\]

Caution

This however only holds at the limit. That is, \(W_{\lambda^{N}}(e^{jN\hat{\omega}}) \neq \frac{2\pi}{N} \sum_{k = 0}^{N-1} W_{\lambda}\left(e^{j(\hat{\omega} - \frac{2\pi k}{N})} \right)\) for any \(\lambda \in (0,1)\).

As a result, the DTFT \(\displaystyle \Delta_N (e^{j\hat{\omega}}) = \lim_{\lambda \rightarrow 1} W_{\lambda^{N}}(e^{jN\hat{\omega}}) = \frac{2\pi}{N} \sum_{k = 0}^{N-1} \delta\left(e^{j(\hat{\omega} - \frac{2\pi k}{N})} \right)\). In other words, we have the DTFT mapping

\[\begin{equation*} \delta_N[n] ~~ \stackrel{\text{DTFT}}{\longleftrightarrow} ~~ \frac{2\pi}{N} \sum_{k = 0}^{N-1} \delta\left(e^{j(\hat{\omega} - \frac{2\pi k}{N})} \right) \end{equation*}\]

2.4.4. General periodic \(x[n]\) with period \(N\)

Note that we can write \(\displaystyle x[n] = \sum_{k=0}^{N-1} x[k] \delta_N[n-k]\). Hence, by the linearity and time shifting properties, the DTFT \(X(e^{j\hat{\omega}})\) of \(x[n]\) is given by

\[\begin{align*} X(e^{j\hat{\omega}}) &= \sum_{k=0}^{N-1} x[k] e^{-j \hat{\omega} k} \cdot \frac{2\pi}{N} \sum_{l = 0}^{N-1} \delta\left(e^{j(\hat{\omega} - \frac{2\pi l}{N})}\right) \\ &= 2\pi \sum_{l = 0}^{N-1} \bigg( \underbrace{\frac{1}{N} \sum_{k=0}^{N-1} x[k] e^{-j\frac{2\pi l k}{N}} }_{\displaystyle a_l} \bigg) \delta\left(e^{j(\hat{\omega} - \frac{2\pi l}{N})}\right). \end{align*}\]

In summary, we have just established the following DTFT mapping for any periodic \(x[n]\) of period \(N\):

\[\begin{equation*} x[n] ~~ \stackrel{\text{DTFT}}{\longleftrightarrow} ~~ 2\pi \sum_{l = 0}^{N-1} a_l \delta\left(e^{j(\hat{\omega} - \frac{2\pi l}{N})} \right) \end{equation*}\]

DIscrete Fourier Series

• The coefficients \(a_0, a_1, \ldots, a_{N-1}\) are referred to as the discrete Fourier series (DFS) coefficients of the periodic \(x[n]\).

• The inverse DTFT formula (2.2) gives

  \[\begin{align*} x[n] &= \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\hat{\omega}}) e^{j\hat{\omega}n} \, d\hat{\omega} \\ &= \sum_{l=0}^{N-1} a_l \int_{-\pi}^{\pi} e^{j\hat{\omega}n} \delta\left(e^{j(\hat{\omega} - \frac{2\pi l}{N})}\right) \, d\hat{\omega} \\ &= \sum_{l=0}^{N-1} a_l e^{j \frac{2\pi ln}{N}} & (\text{sifting property of periodic Dirac delta}). \end{align*}\]

• The formulas

  \[\begin{align*} a_l &= \frac{1}{N} \sum_{k=0}^{N-1} x[k] e^{-j\frac{2\pi l k}{N}}, & l=0,1,\ldots, N-1 \\ x[n] &= \sum_{l=0}^{N-1} a_l e^{j \frac{2\pi ln}{N}}, & n=0,1,\ldots, N-1 \end{align*}\]

  are the DFS analysis and synthesis formulas, respectively. They provide a shortcut to go directly between the periodic signal \(x[n]\) and its DTFT \(X(e^{j\hat{\omega}})\)