Time-Frequency Uncertainty Principle 2.18 2001/12/29 2005/10/05 15:45:38.862 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Elizabeth Chan lizychan@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Gaussian pulse time-frequency uncertainty principle This module introduces time-frequency uncertainty principle. Recall that Fourier basis elements bΩ t Ω t exhibit poor time localization abilities - a consequence of the fact that bΩ t is evenly spread over all t . By time localization we mean the ability to clearly identify signal events which manifest during a short time interval, such as the "glitch" described in an earlier example. At the opposite extreme, a basis composed of shifted Dirac deltas bτ t Δ t τ would have excellent time localization but terrible "frequency localization," since every Dirac basis element is evenly spread over all Fourier frequencies Ω . This can be seen via Bτ Ω t bτ t Ω t 1 ∀ Ω , regardless of τ. By frequency localization we mean the ability to clearly identify signal components which are concentrated at particular 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 the time 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 Ω . Define the energy of the waveform to be E, so that (by Parseval's theorem) E t b t 2 1 2 Ω B Ω 2 Next, define the temporal and spectral centers It may be interesting to note that both 1 E b t 2 and 1 2 E B Ω 2 are non-negative and integrate to one, thereby satisfying the requirements of probability density functions for random variables t and Ω. The temporal/spectral centers can then be interpreted as the means (i.e., centers of mass) of t and Ω. as tc 1 E t t b t 2 Ωc 1 2 E Ω Ω B Ω 2 and the temporal and spectral widths The quantities Δt 2 and ΔΩ 2 can be interpreted as the variances of t and Ω, respectively. as Δt 1 E t t tc 2 b t 2 ΔΩ 1 2 E Ω Ω Ωc 2 B Ω 2 If the waveform is well-localized in time, then b t will be concentrated at the point t c and Δ t will be small. If the waveform is well-localized in frequency, then B Ω will be concentrated at the point Ω c and Δ Ω will be small. If the waveform is well-localized in both time and frequency, then Δt ΔΩ will be small. The quantity Δt ΔΩ is known as the time-bandwidth product. From the definitions above one can derive the fundamental properties below. When interpreting the properties, it helps to think of the waveform b t as a prototype that can be used to generate an entire basis set. For example, the Fourier basis bΩ t Ω bΩ t can be generated by frequency shifts of b t 1 , while the Dirac basis bτ t τ bτ t can be generated by time shifts of b t δ t Δt and Δ Ω are invariant to time and frequency Keep in mind the fact that b t and B Ω t b t Ω t are alternate descriptions of the same waveform; we could have written ΔΩ b t Ω0 t in place of ΔΩ B Ω Ω0 above. shifts. t 0 Δt b t Δt b t t0 Ω 0 ΔΩ B Ω ΔΩ B Ω Ω0 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 Δt ΔΩ is invariant to time-scaling. The invariance property holds also for frequency scaling, as implied by the Fourier transform property ⇔ b a t 1 a B Ω a . Δt b a t 1 a Δt b t ΔΩ b a t a ΔΩ b t The above two equations imply a Δt ΔΩ b a t Δt ΔΩ b t Observe that time-domain expansion (i.e., a 1 ) increases the temporal width but decreases the spectral width, while time-domain contraction (i.e., a 1 ) does the opposite. This suggests that time-scaling might be a useful tool for the design of a basis element with a particular tradeoff between time and frequency resolution. On the other hand, scaling cannot simultaneously increase both time and frequency resolution. No waveform can have time-bandwidth product less than 1 2 . Δt ΔΩ 1 2 This is known as the time-frequency uncertainty principle. The Gaussian pulse g t achieves the minimum time-bandwidth product Δt ΔΩ 1 2 . g t 1 2 1 2 t 2 G Ω 1 2 Ω 2 Note that this waveform is neither bandlimited nor time-limited, but reasonable concentrated in both domains (around the points tc 0 and Ωc 0 ). Properties 1 and 2 can be easily verified using the definitions above. Properties 3 and 4 follow from the Cauchy-Schwarz inequality. Since the Gaussian pulse g t achieves the minimum time-bandwidth product, it makes for a theoretically good prototype waveform. In other words, we might 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 has doubly-infinite time-support, though, other windows are used in practice. Basis construction from a prototype waveform is the main concept behind Short-Time Fourier Analysis and the continuous Wavelet transform discussed later.