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Time-Frequency Uncertainty Principle

Module by: Phil Schniter. E-mail the author

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Summary: This module introduces time-frequency uncertainty principle.

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Recall that Fourier basis elements bΩt=ⅇⅈΩt bΩ t Ω t exhibit poor time localization abilities - a consequence of the fact that bΩt bΩ t is evenly spread over all t∈-∞∞ 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 bτt=Δt−τ 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∈-∞∞ Ω . This can be seen via |BτΩ|=|∫-∞∞bτtⅇ-ⅈΩtdt|=1 Bτ Ω t bτ t Ω t 1 ∀Ω ∀ Ω , regardless of ττ. 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, bt b t . Its CTFT will be denoted by BΩ B Ω . Define the energy of the waveform to be EE, so that (by Parseval's theorem) E=∫-∞∞|bt|2dt=12π∫-∞∞|BΩ|2dΩ E t b t 2 1 2 Ω B Ω 2 Next, define the temporal and spectral centers 1 as tc=1E∫-∞∞t|bt|2dt tc 1 E t t b t 2 Ωc=12πE∫-∞∞Ω|BΩ|2dΩ Ωc 1 2 E Ω Ω B Ω 2 and the temporal and spectral widths 2 as Δt=1E∫-∞∞t−tc2|bt|2dt Δt 1 E t t tc 2 b t 2 ΔΩ=12πE∫-∞∞Ω−Ωc2|BΩ|2dΩ ΔΩ 1 2 E Ω Ω Ωc 2 B Ω 2 If the waveform is well-localized in time, then bt b t will be concentrated at the point t c t c and Δ t Δ t will be small. If the waveform is well-localized in frequency, then BΩ B Ω will be concentrated at the point Ω c Ω c and Δ Ω Δ Ω will be small. If the waveform is well-localized in both time and frequency, then ΔtΔΩ Δt ΔΩ will be small. The quantity ΔtΔΩ 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 bt b t as a prototype that can be used to generate an entire basis set. For example, the Fourier basis {bΩt|-∞<Ω<∞} bΩ t Ω bΩ t can be generated by frequency shifts of bt=1 b t 1 , while the Dirac basis {bτt|-∞<τ<∞} bτt bτ t τ bτ t can be generated by time shifts of bt=δt b t δ t

  1. Δt Δt and Δ Ω Δ Ω are invariant to time and frequency 3 shifts. ∀, t 0 ∈ℝ:Δtbt=Δtbt−t0 t 0 Δt b t Δt b t t0 ∀, Ω 0 ∈ℝ:ΔΩBΩ=ΔΩBΩ−Ω0 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 bt b t will inherit the temporal and spectral widths of bt b t .
  2. The time-bandwidth product ΔtΔΩ Δt ΔΩ is invariant to time-scaling. 4 Δtbat=1|a|Δtbt Δt b a t 1 a Δt b t ΔΩbat=|a|ΔΩbt ΔΩ b a t a ΔΩ b t The above two equations imply ∀,a∈ℝ:ΔtΔΩbat=ΔtΔΩbt a Δt ΔΩ b a t Δt ΔΩ b t Observe that time-domain expansion (i.e., |a|<1 a 1 ) increases the temporal width but decreases the spectral width, while time-domain contraction (i.e., |a|>1 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.
  3. No waveform can have time-bandwidth product less than 12 1 2 . ΔtΔΩ≥12 Δt ΔΩ 1 2 This is known as the time-frequency uncertainty principle.
  4. The Gaussian pulse gt g t achieves the minimum time-bandwidth product ΔtΔΩ=12 Δt ΔΩ 1 2 . gt=12πⅇ-12t2 g t 1 2 1 2 t 2 GΩ=ⅇ-12Ω2 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 tc 0 and Ωc=0 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 gt 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 gt 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.

Footnotes

  1. It may be interesting to note that both 1E|bt|2 1 E b t 2 and 12πE|BΩ|2 1 2 E B Ω 2 are non-negative and integrate to one, thereby satisfying the requirements of probability density functions for random variables tt and ΩΩ. The temporal/spectral centers can then be interpreted as the means (i.e., centers of mass) of tt and ΩΩ.
  2. The quantities Δt2 Δt 2 and ΔΩ2 ΔΩ 2 can be interpreted as the variances of tt and ΩΩ, respectively.
  3. Keep in mind the fact that bt b t and BΩ=∫-∞∞btⅇ-ⅈΩtdt B Ω t b t Ω t are alternate descriptions of the same waveform; we could have written ΔΩbtⅇⅈΩ0t ΔΩ b t Ω0 t in place of ΔΩBΩ−Ω0 ΔΩ B Ω Ω0 above.
  4. The invariance property holds also for frequency scaling, as implied by the Fourier transform property bat⇔1|a|BΩa ⇔ b a t 1 a B Ω a .

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