5.2 Time-frequency uncertainty principle

Recall that Fourier basis elements $b_{\Omega }(t)=e^{i\Omega t}$ exhibit poor time localization abilities - a consequence of the fact that $b_{\Omega }(t)$ is evenly spread over all $t\in \left\{()\right\}$∞ ∞ . By time localization we mean the ability to clearly identify signal events which manifest duringa short time interval, such as the "glitch" described in an earlier example .

At the opposite extreme, a basis composed of shifted Dirac deltas $b_{\tau }(t)=\Delta (t-\tau )$ would have excellent time localization but terrible "frequency localization," since every Dirac basis element isevenly spread over all Fourier frequencies $\Omega \in \left\{()\right\}$∞ ∞ . This can be seen via $\left|B_{\tau }(\Omega )\right|=\left|\int_{()} \,d t\right|$∞ ∞ b τ t Ω t 1 $\forall (\Omega )$ , regardless of $\tau$ . By frequency localization we mean the ability to clearly identify signal components which are concentrated atparticular Fourier frequencies, such as sinusoids.

These observations motivate the question: does there exist a basis that provides both excellent frequency localization and excellent time localization? The answer is "not really": there is a fundamental tradeoff between thetime localization and frequency localization of any basis element. This idea is made concrete below.

Let us consider an arbitrary waveform, or basis element, $b(t)$ . Its CTFT will be denoted by $B(\Omega )$ . Define the energy of the waveform to be $E$ , so that (by Parseval's theorem) $$E=\int_{()} \,d t$$∞ ∞ b t 2 1 2 Ω ∞ ∞ B Ω 2 Next, define the temporal and spectral centers

From the definitions above one can derive the fundamental properties below. When interpreting the properties, it helpsto think of the waveform $b(t)$ as a prototype that can be used to generate an entire basis set. For example, the Fourier basis $\{b_{\Omega }(t)\colon ()\}$∞ Ω ∞ b Ω t can be generated by frequency shifts of $b(t)=1$ , while the Dirac basis $\{b_{\tau }(t)\colon ()\}$∞ τ ∞ b τ t can be generated by time shifts of $b(t)=\delta (t)$

• ${\Delta }_t$ and ${\Delta }_{\Omega }$ are invariant to time and frequency
  Keep in mind the fact that $b(t)$ and $B(\Omega )=\int_{()} \,d t$∞ ∞ b t Ω t are alternate descriptions of the same waveform; we could have written ${\Delta }_{\Omega }(b(t)e^{i{\Omega }_{0}t})$ in place of ${\Delta }_{\Omega }(B(\Omega -{\Omega }_0))$ above.
  shifts. $$\forall , t_0\in \mathbb{R}\colon {\Delta }_t(b(t))={\Delta }_t(b(t-t_0))$$ $$\forall , {\Omega }_0\in \mathbb{R}\colon {\Delta }_{\Omega }(B(\Omega ))={\Delta }_{\Omega }(B(\Omega -{\Omega }_0))$$ This implies that all basis elements constructed from time and/or frequency shifts of a prototype waveform $b(t)$ will inherit the temporal and spectral widths of $b(t)$ .
• The time-bandwidth product ${\Delta }_t{\Delta }_{\Omega }$ is invariant to time-scaling.
  The invariance property holds also for frequency scaling, as implied by the Fourier transformproperty $\iff (b(at), \frac{1}{\left|a\right|}B(\frac{\Omega }{a}))$ .
  $${\Delta }_t(b(at))=\frac{1}{\left|a\right|}{\Delta }_t(b(t))$$ $${\Delta }_{\Omega }(b(at))=\left|a\right|{\Delta }_{\Omega }(b(t))$$ The above two equations imply $$\forall , a\in \mathbb{R}\colon {\Delta }_t{\Delta }_{\Omega }(b(at))={\Delta }_t{\Delta }_{\Omega }(b(t))$$ Observe that time-domain expansion ( i.e. , $\left|a\right|< 1$ ) increases the temporal width but decreases the spectral width, while time-domain contraction( i.e. , $\left|a\right|> 1$ ) does the opposite. This suggests that time-scaling might be a useful tool for the design of abasis element with a particular tradeoff between time andfrequency resolution. On the other hand, scaling cannot simultaneously increase both time and frequency resolution.
• No waveform can have time-bandwidth product less than $\frac{1}{2}$ . $${\Delta }_t{\Delta }_{\Omega }\ge \frac{1}{2}$$ This is known as the time-frequency uncertainty principle .
• The Gaussian pulse $g(t)$ achieves the minimum time-bandwidth product ${\Delta }_t{\Delta }_{\Omega }=\frac{1}{2}$ . $$g(t)=\frac{1}{\sqrt{2\pi }}e^{-(\frac{1}{2}t^2)}$$ $$G(\Omega )=e^{-(\frac{1}{2}\Omega ^2)}$$ Note that this waveform is neither bandlimited nor time-limited, but reasonable concentrated in both domains(around the points $t_c=0$ and ${\Omega }_c=0$ ).

Since the Gaussian pulse $g(t)$ achieves the minimum time-bandwidth product, it makes for a theoretically good prototype waveform. In other words, wemight consider constructing a basis from time shifted, frequency shifted, time scaled, or frequency scaled versions of $g(t)$ to give a range of spectral/temporal centers and spectral/temporal resolutions. Since the Gaussian pulse hasdoubly-infinite time-support, though, other windows are used in practice. Basis construction from a prototype waveform is themain concept behind Short-Time Fourier Analysis and the continuous Wavelet transform discussed later.