2.6. Continuous-Time Fourier Transform (FT)#

For a continuous-time signal \(x(t)\):

  • Forward FT:

    (2.12)#\[\begin{equation} X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} \, dt \end{equation}\]

  • Inverse FT:

    (2.13)#\[\begin{equation} x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} \, d\omega \end{equation}\]

Notation

  1. We use the notation \(x(t) \stackrel{\text{FT}}{\longleftrightarrow} X(\omega)\) to denote the mapping between the signal \(x(t)\) and its FT \(X(\omega)\) according to the forward and inverse FT formulas above.

  2. A signal \(x(t)\) is absolutely integrable, denoted by \(x(t) \in L^1\), if \(\int_{-\infty}^{\infty} |x(t)| dt < \infty\).

  3. A FT function \(X(\omega)\) is absolutely integrable, denoted also by \(X(\omega) \in L^1\), if \(\frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)| d\omega < \infty\).

  4. A signal \(x(t)\) has finite energy, denoted by \(x(t) \in L^2\), if \(\int_{-\infty}^{\infty} |x(t)|^2 dt< \infty\).

  5. A FT function \(X(\omega)\) is square-integrable, denoted also by \(X(\omega) \in L^2\), if \(\frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2 d\omega < \infty\).

Tip

  • Like before, when considering a signal \(x(t) \in L^p\) (\(p=1,2\)), we do not make any distinction between \(x(t)\) and another signal \(\tilde{x}(t)\) such that \(\int_{-\infty}^{\infty} \left|\tilde{x}(t) - x(t) \right|^p dt = 0\). We group all such signals in an equivalence class and consider the equivalence class as a unique member of \(L^p\).

  • The same equivalence class consideration applies to FTs in \(L^p\) (\(p=1,2\)) as well.

  • Similar to the case of DTFT, we have to consider the conditions for existence of the integrals on the RHS of both the forward and inverse FT formulas, (2.12) and (2.13). Similar to the case of DTFT, a simple sufficient condition can be obtained as follows:

    • If \(x(t) \in L^1\), then the integral on the RHS of (2.12) exists. Let \(X(\omega)\) be the FT given by (2.12).

    • If this \(X(\omega) \in L^1\), then the integral \(\frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega) e^{j\omega t} d\omega\) on the RHS of (2.13) exists and is continuous for all \(t \in \mathbb{R}\).

    • In addition, the inverse FT formula (2.13) holds almost everywhere on \(\mathbb{R}\).

    These results indeed provide a meaning to the mapping \(x(t) \stackrel{\text{FT}}{\longleftrightarrow} X(\omega)\).

  • Again, as in the case of DTFT, restricting \(x(t) \in L^1\) and its FT \(X(\omega) \in L^1\) is rather limiting. Extension of FT to cover all finite-energy signals is trickier than in the case of DTFT because \(L^2 \not\subset L^1\). That is, there may be finite-energy FT \(X(\omega)\) such that the integral \(\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} d\omega\) does not exist (hence the inverse FT formula cannot hold)! Nevertheless, this difficulty can be resolved by an approach similar to the windowing idea used in DTFT. The resulting extension is often called the Plancherel theorem [Rud87]:

    Plancherel

    For every finite-energy signal \(x(t) \in L^2\), there is a unique square-integrable FT \(X(\omega) \in L^2\) such that both the forward and inverse FT formulas, (2.12) and (2.13) hold with equality interpreted respectively as

    \[\begin{align*} \lim_{T \rightarrow \infty} \frac{1}{2\pi} \int_{-\infty}^{\infty} \left| X(\omega) - \int_{-T}^{T} x(t) e^{-j\omega t} \, dt \right|^2 \, d\omega &= 0 \\ \lim_{\Omega \rightarrow \infty} \int_{-\infty}^{\infty} \left| x(t) - \int_{-\Omega}^{\Omega} X(\omega) e^{j\omega t} \, d\omega \right|^2 \, dt &= 0. \end{align*}\]

    In addition, the mapping \(x(t) \stackrel{\text{FT}}{\longleftrightarrow} X(\omega)\) is an isomorphism between the (Hilbert) space of finite-energy signals and the (Hilbert) space of finite-energy FTs, satisfying the Parseval theorem:

    \[\begin{equation*} \int_{-\infty}^{\infty} x(t) y^*(t)\, dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) Y^*(\omega) \, d\omega \end{equation*}\]

    for every \(x(t) \stackrel{\text{FT}}{\longleftrightarrow} X(\omega)\) and \(y(t) \stackrel{\text{FT}}{\longleftrightarrow} Y(\omega)\).

  • As a special case of the Parseval theorem, we have \(\int_{-\infty}^{\infty} |x(t)|^2\, dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2 \, d\omega\), i.e., the energy of a signal can be calculated in the time or frequency domain.