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This module introduces time-frequency uncertainty principle.

Recall that Fourier basis elements b Ω t Ω t exhibit poor time localization abilities - a consequence of the fact that b Ω t is evenly spread over all t . 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 τ t Δ t τ would have excellent time localization but terrible "frequency localization," since every Dirac basis element isevenly spread over all Fourier frequencies Ω . This can be seen via B τ Ω t b τ t Ω t 1 Ω , regardless of τ . 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 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 functionsfor 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 t c 1 E t t b t 2 Ω c 1 2 E Ω Ω B Ω 2 and the temporal and spectral widths
The quantities Δ t 2 and Δ Ω 2 can be interpreted as the variances of t and Ω , respectively.
as Δ t 1 E t t t c 2 b t 2 Δ Ω 1 2 E Ω Ω Ω c 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 Δ t Δ Ω will be small. The quantity Δ t Δ Ω is known as the time-bandwidth product .

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 Ω t Ω b Ω t can be generated by frequency shifts of b t 1 , while the Dirac basis b τ t τ b τ t can be generated by time shifts of b t δ t

  • Δ t 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 Δ Ω b t Ω 0 t in place of Δ Ω B Ω Ω 0 above.
    shifts. t 0 Δ t b t Δ t b t t 0 Ω 0 Δ Ω B Ω Δ Ω B Ω Ω 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 Δ t Δ Ω is invariant to time-scaling.
    The invariance property holds also for frequency scaling, as implied by the Fourier transformproperty b a t 1 a B Ω a .
    Δ t b a t 1 a Δ t b t Δ Ω b a t a Δ Ω b t The above two equations imply a Δ t Δ Ω b a t Δ t Δ Ω 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 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 1 2 . Δ t Δ Ω 1 2 This is known as the time-frequency uncertainty principle .
  • The Gaussian pulse g t achieves the minimum time-bandwidth product Δ t Δ Ω 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 t c 0 and Ω c 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, 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.

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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