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Time-Frequency Uncertainty Principle

Module by: Phil Schniter. E-mail the author

Summary: This module introduces time-frequency uncertainty principle.

Recall that Fourier basis elements bΩ ⁢t=ej⁢Ω⁢t bΩ t Ω t exhibit poor time localization abilities - a consequence of the fact that bΩ ⁢t 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 bτ ⁢t=Δ⁢t−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 | Bτ ⁢Ω|=|∫−∞∞ bτ ⁢t⁢e−(j⁢Ω⁢t)dt|=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, b⁢t b t . Its CTFT will be denoted by B⁢Ω B Ω . Define the energy of the waveform to be EE, so that (by Parseval's theorem) E=∫−∞∞|b⁢t|2dt=12⁢π⁢∫−∞∞|B⁢Ω|2dΩ E t b t 2 1 2 Ω B Ω 2 Next, define the temporal and spectral centers 1 as tc =1E⁢∫−∞∞t⁢|b⁢t|2dt tc 1 E t t b t 2 Ωc =12⁢π⁢E⁢∫−∞∞Ω⁢|B⁢Ω|2dΩ Ωc 1 2 E Ω Ω B Ω 2 and the temporal and spectral widths 2 as Δt =1E⁢∫−∞∞t− tc 2⁢|b⁢t|2dt Δt 1 E t t tc 2 b t 2 ΔΩ =12⁢π⁢E⁢∫−∞∞Ω− Ωc 2⁢|B⁢Ω|2dΩ ΔΩ 1 2 E Ω Ω Ωc 2 B Ω 2 If the waveform is well-localized in time, then b⁢t 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 B⁢Ω 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 Δt ⁢ ΔΩ Δt ΔΩ will be small. The quantity Δt ⁢ ΔΩ 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 b t as a prototype that can be used to generate an entire basis set. For example, the Fourier basis bΩ ⁢t bΩ ⁢t −∞<Ω<∞ bΩ t Ω bΩ t can be generated by frequency shifts of b⁢t=1 b t 1 , while the Dirac basis bτ ⁢t −∞<τ<∞ bτ ⁢t bτ t τ bτ t can be generated by time shifts of b⁢t=δ⁢t b t δ t

  1. Δt Δt and Δ Ω Δ Ω are invariant to time and frequency 3 shifts. Δt ⁢b⁢t= Δt ⁢b⁢t−t0  ,   t 0 ∈R    t 0 Δt b t Δt b t t0 ΔΩ ⁢B⁢Ω= ΔΩ ⁢B⁢Ω−Ω0  ,   Ω 0 ∈R    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 b t will inherit the temporal and spectral widths of b⁢t b t .
  2. The time-bandwidth product Δt ⁢ ΔΩ Δt ΔΩ is invariant to time-scaling. 4 Δt ⁢b⁢a⁢t=1|a|⁢ Δt ⁢b⁢t Δt b a t 1 a Δt b t ΔΩ ⁢b⁢a⁢t=|a|⁢ ΔΩ ⁢b⁢t ΔΩ b a t a ΔΩ b t The above two equations imply Δt ⁢ ΔΩ ⁢b⁢a⁢t= Δt ⁢ ΔΩ ⁢b⁢t  ,   a∈R    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 . Δt ⁢ ΔΩ ≥12 Δt ΔΩ 1 2 This is known as the time-frequency uncertainty principle.
  4. The Gaussian pulse g⁢t g t achieves the minimum time-bandwidth product Δt ⁢ ΔΩ =12 Δt ΔΩ 1 2 . g⁢t=12⁢π⁢e−(12⁢t2) g t 1 2 1 2 t 2 G⁢Ω=e−(12⁢Ω2) 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 g⁢t 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 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⁢|b⁢t|2 1 E b t 2 and 12⁢π⁢E⁢|B⁢Ω|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 Δt 2 Δt 2 and ΔΩ 2 ΔΩ 2 can be interpreted as the variances of tt and ΩΩ, respectively.
  3. Keep in mind the fact that b⁢t b t and B⁢Ω=∫−∞∞b⁢t⁢e−(j⁢Ω⁢t)dt B Ω t b t Ω t are alternate descriptions of the same waveform; we could have written ΔΩ ⁢b⁢t⁢ej⁢Ω0⁢t ΔΩ b t Ω0 t in place of ΔΩ ⁢B⁢Ω− Ω0 ΔΩ B Ω Ω0 above.
  4. The invariance property holds also for frequency scaling, as implied by the Fourier transform property b⁢a⁢t⇔1|a|⁢B⁢Ωa ⇔ b a t 1 a B Ω a .

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