Uncertainty principle for the short-time Fourier transform

There are many things one can do to signals to study them. However, if one do something to a signal that modifies it in some way, one should not confuse uncertainty principle

Uncertainty principle

In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

applied to the modified signal with the uncertainty principle

Uncertainty principle

as applied to the original signal. One of the methods used to estimate properties of a signal is to take only a small piece of the signal around the time of interest and study that piece while neglecting the rest of the signal. In particular, one can take the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of the small piece of the signal to estimate the frequencies at that time. If one make the time interval around the time t small, it will have a very high bandwidth. This statement applies to the modified signal; that is, to the short interval that one have artificially constructed for the purpose of analysis. What does the uncertainty principle

Uncertainty principle

as applied to a small time interval have to do with the uncertainty principle

Uncertainty principle

of the original? Very often nothing and statements about the chopped up signal should not be applied to the original signal. The process of chopping up a signal for the purpose of analysis is called the short-time Fourier transform

Short-time Fourier transform

The short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....

procedure.

Characteristic of time wave forms: averages, mean time, and duration

as a density in time is considered, the average time can be defined in the usual way any average is defined:

(1)

The reason for defining an average are that it may give a gross characterization of the density and it may give an indication of where the density is concentrated. Many measures can be used to ascertain whether the density is concentrated around the average, the most common being the standard deviation,

, given by

(2)

The standard deviation is an indication of the duration of the signal.

The average of any function of time, g(t), is obtained by

(3)

Mean frequency, bandwidth, and frequency averages

represents the density in frequency then it can be used to calculate averages, the motivation being the same as in the time domain, namely that it gives a rough idea of the main characteristics of the spectral density. The average frequency,

, and its standard deviation,

(commonly called the root mean square bandwidth and signified by B), are given by

(4)

(5)

and the average of any frequency function, g(

), is

(6)

The covariance of a signal

If one want a measure of how time and instantaneous frequency are related, it can be done it calculating the covariance or the correlation for signals. Consider the quantity

(7)

where

is the phase of

is the instantaneous frequency,

which may be the thought of as the average of time multiplied by the instantaneous frequency. Now if time and frequency have nothing to do with each other then it is expected that

to equal

. Therefore, the excess of

over

is a good measure of how time is correlated with instantaneous frequency.
Define the covariance of a signal by

(8)

The uncertainty principle

T and B are standard deviations defined by the previous equations:

(9)

(10)

The uncertainty principle

Uncertainty principle

(11)

Therefore one cannot have or construct a signal for which both T and B are arbitrarily small.

A more general uncertainty principle. A stronger version of the uncertainty principle

Uncertainty principle

(12)

Proof of the uncertainty principle

First, let us note that no loss of generality occurs if signals are taken that have zero mean time and zero mean
frequency. The reason is that the standard deviation does not depend on the mean because it is defined as the broadness about the mean, If one have a signal

, then a new signal defined by

(13)

has the same shape both in time and frequency as

except that it has been translated in time and frequency so that the means are zero. Conversely, if one have a signal

that has zero mean time and zero mean frequency and one want a signal of the same shape but with particular mean time and frequency, then

(14)

The bandwidth expressed in terms of the signal is

(15)

The duration is

(16)

and therefore

(17)

The fact that s and S are Fourier transform

Fourier transform

pairs is reflected in Eq.(15)

Now, for any two functions (not only Fourier transform

Fourier transform

pairs)

(18)

which is commonly known as the Schwarz inequality. Taking

and

gives

(19)

The integrand, written in terms of amplitude and phase (

), is

(20)

The first term is a perfect differential and integrates to zero. The second term gives one half since it is assumed that the signal is normalized and the third term gives j times the covariance of the signal. Hence

(21)

Therefore the uncertainty principle

Uncertainty principle

is proved as given in Eq. (12). Since

is always positive, it can, if one choose, be dropped to obtain the more usual form, Eq. (11).

The uncertainty principle for the short-time Fourier transform

From the original signal s(t) one defines a short duration signal around the time of interest, t, by multiplying it by a window function that is peaked around the time, t, and falls off rapidly. This has the effect of emphasizing the signal at time, t, and suppressing it for times far away from that time. In particular, Define the normalized short duration signal at time, t, by

(22)

where h(t) is the window function, t is the fixed time on which is focused , and

is now the running time. This normalization ensures that

(23)

for any t. Now

as a function of the time

is of short duration since presumably a window function has been chosen to make it so. The time, t, acts as a parameter. The Fourier transform

Fourier transform

of the small piece of the signal, the modified signal, is

(24)

gives us an indication of the spectral content at the time t. For the modified signal one can define all the relevant quantities such as mean time, duration, and bandwidth in the standard way, but they will be time dependent. The mean time and duration for the modified signal are

(25)

(26)

Similarly, the mean frequency and bandwidth for the modified signal are

(27)

(28)

Time-dependent and window-dependent uncertainty principle

Since a normalized signal have been used to calculate the duration and bandwidth,

can be immediately written down. This is the uncertainty principle

Uncertainty principle

for the short-time Fourier transform. It is a function of time, the signal, and the window. It should not be confused with uncertainty principle

Uncertainty principle

applied to the signal. It is important to understand this uncertainty because it places limits on the technique of the short-time Fourier transform

Short-time Fourier transform

procedure. However, it places no constraints on the original signal.

It is known that for infinitely short duration signals the bandwidth becomes infinite. Hence it is expected that

as the window is narrowed , which is indeed the case. It is true that if the signal is modified by the technique of the short-time Fourier transform

Short-time Fourier transform

, the abilities in terms of resolutionis are limited and so forth. This is a limitation of the technique. The uncertainty principle

Uncertainty principle

of the original signal does not change because it has been decided to be modified by windowing.

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