4.1. Uncertainty Principle#

  • Consider a finite-energy continuous-time signal \(x(t) \stackrel{\text{FT}}{\longleftrightarrow} X(\omega)\) with energy \(\displaystyle E_x = \int_{-\infty}^{\infty} |x(t)|^2 \, dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2 \,d\omega > 0\).

  • Without loss of generality, assume \(x(t)\) satisfies:

    • \(\displaystyle\int_{-\infty}^{\infty} t |x(t)|^2 \, dt = 0\), and

    • \(\displaystyle \frac{1}{2\pi} \int_{-\infty}^{\infty} \omega |X(\omega)|^2 \,d\omega = 0\).

    Note that these two assumptions mean that the signal energy is centered about \(t=0\) in the time domain and about \(\omega = 0\) in the frequency domain. If either assumption is not satisfied, we may simply shift the time or frequency axis to force the assumption to hold.

  • Define the quantities

    \[\begin{align*} \sigma_t &= \sqrt{\frac{1}{E_x} \int_{-\infty}^{\infty} t^2 |x(t)|^2 \, dt } \\ \sigma_{\omega} &= \sqrt{\frac{1}{2\pi E_x} \int_{-\infty}^{\infty} \omega^2 |X(\omega)|^2 \,d\omega } \end{align*}\]

    to measure the spreads of signal energy about the centers in the time and frequency domains, respectively.

    Example: For \(\displaystyle \tilde{w}_{\mu} (t) = \frac{\mu}{\pi (\mu^2+t^2)} \stackrel{\text{FT}}{\longleftrightarrow} \tilde{W}_{\mu}(\omega) = e^{-\mu |\omega|}\), \(\sigma_t = \sqrt{\pi} \mu\) and \(\sigma_{\omega} = \frac{1}{\sqrt{2} \mu}\).

  • Now, suppose \(\sigma_t < \infty\) (i.e., \(tx(t) \in L^2\)) and \(\sigma_{\omega} < \infty\) (i.e., \(\omega X(\omega) \in L^2\)). Then, we have

    • \(tx(t) \rightarrow 0\) and hence \(t|x(t)|^2 \rightarrow 0\) as \(t \rightarrow \pm \infty\)

    • \(tx(t) \stackrel{\text{FT}}{\longleftrightarrow} j \frac{dX(\omega)}{d\omega}\)

    Further, suppose \(\frac{dx(t)}{dt}\) exists on \(\mathbb{R}\). Then

    • \(\displaystyle \frac{d|x(t)|^2}{dt} = \frac{dx(t)}{dt} \cdot x^*(t) + x(t) \cdot \left( \frac{dx(t)}{dt} \right)^*\)

    • \(\displaystyle \frac{dx(t)}{dt} \stackrel{\text{FT}}{\longleftrightarrow} j\omega X(\omega)\)

    • \(\displaystyle \int_{-\infty}^{\infty} \left| \frac{dx(t)}{dt} \right|^2 \,dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} \omega^2 | X(\omega)|^2 \,d\omega\).

    As a result, integration by parts gives

    (4.1)#\[\begin{split}\begin{align*} E_x &= \int_{-\infty}^{\infty} |x(t)|^2 \, dt \\ &= t|x(t)|^2 \Big|_{-\infty}^{\infty} - \int_{-\infty}^{\infty} t \frac{d|x(t)|^2}{dt} \, dt \\ &= - \int_{-\infty}^{\infty} t \frac{d|x(t)|^2}{dt} \, dt \\ &= - \int_{-\infty}^{\infty} \frac{dx(t)}{dt} \cdot (tx(t))^* \, dt - \int_{-\infty}^{\infty} (tx(t)) \cdot \left( \frac{dx(t)}{dt} \right)^* \,dt \\ &\leq 2 \int_{-\infty}^{\infty} \left| \frac{dx(t)}{dt} \right| \cdot |tx(t)| \,dt \\ &\leq 2 \sqrt{\int_{-\infty}^{\infty} \left| \frac{dx(t)}{dt} \right|^2 \,dt} \cdot \sqrt{\int_{-\infty}^{\infty} t^2 |x(t)|^2 \,dt} & \text{(Cauchy-Schwartz inequality)} \\ &= 2 \sqrt{ \frac{1}{2\pi} \int_{-\infty}^{\infty} \omega^2 | X(\omega)|^2 \,d\omega} \cdot \sqrt{\int_{-\infty}^{\infty} t^2 |x(t)|^2 \,dt}. \end{align*}\end{split}\]

    Expressing (4.1) in terms of \(\sigma_t\) and \(\sigma_{\omega}\), we get:

    Uncertainty Principle

    For any (non-trivial) finite-energy, differentiable signal \(x(t)\), its time-bandwidth product \(\displaystyle \sigma_t \sigma_{\omega} \geq \frac{1}{2}\).

  • The uncertainty principle inequality dictates that the time-bandwidth product of a practical continuous-time signal can not be arbitrarily small. That is, if the signal’s energy is concentrated in time, its energy must be spread out in frequency, and vice versa.

    Example: Recall for \(\tilde{w}_{\mu} (t) \stackrel{\text{FT}}{\longleftrightarrow} \tilde{W}_{\mu}(\omega)\), \(\sigma_t = \sqrt{\pi} \mu\) and \(\sigma_{\omega} = \frac{1}{\sqrt{2} \mu}\). As \(\mu \rightarrow 0\), \(\sigma_t \rightarrow 0\) but \(\sigma_{\omega} \rightarrow \infty\) with \(\displaystyle \sigma_t \sigma_{\omega} = \sqrt{\frac{\pi}{2}} > \frac{1}{2}\).

  • For a discrete-time signal \(x[n]\), the intuitive extension of the uncertainty principle inequality by defining \(\sigma_t = \sqrt{\frac{1}{E_x} \sum_{n=-\infty}^{\infty} n^2 |x[n]|^2}\) is problematic because for \(x[n]=\delta[n]\), the value of such defined \(\sigma_t\) is \(0\). Thus, no meaningful lower bound on the time-bandwidth product can be obtained.

    Tip

    Nonetheless, we may still obtain a meaningful lower bound on the spread (resolution) \(\sigma_t\) in the time domain by treating \(\delta[n]\) as obtained from oversampling some continuous-time signal \(x(t)\) whose FT is \(X(\omega)\). Since \(\delta[n]\stackrel{\text{DTFT}}{\longleftrightarrow} 1\), we have \(X(\omega) = \begin{cases} \frac{1}{f_s}, & \text{if} -\pi f_s \leq \omega < \pi f_s \\ 0, & \text{otherwise} \end{cases}\) from (3.7). Simple calculation then shows that \(\displaystyle \sigma_{\omega} = \frac{\pi f_s}{\sqrt{3}}\) for this \(X(\omega)\). Applying the uncertainty principle inequality, we get \(\displaystyle \sigma_t \geq \frac{1}{2\sigma_{\omega}} = \frac{\sqrt{3}}{2\pi f_s}\). We may now treat the lower bound \(\displaystyle \frac{\sqrt{3}}{2\pi f_s}\) as the finest possible time resolution that we can obtain from the sampled signal \(\delta[n]\). The finest possible time resolution for a general discrete-time signal \(x[n]\) can be obtained in a similar manner.

  • The overall implication provided by the uncertainty principle is that we can not simultaneously achieve arbitrarily fine resolution in time and in frequency for any practical continuous- or discrete-time signal.