Küpfmüller's uncertainty principle by Karl Küpfmüller in the year 1924 states that the relation of the

rise time In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratiosSee for example , and . or, equiva ...

of a

bandlimited Bandlimiting is the process of reducing a signal’s energy outside a specific frequency range, keeping only the desired part of the signal’s spectrum. This technique is crucial in signal processing and communications to ensure signals stay cl ...

signal to its bandwidth is a constant. :

\Delta f\Delta t \ge k

with

k

either

1

\frac

Proof

A bandlimited signal

u(t)

with

fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

\hat(f)

is given by the multiplication of any signal

\underline(f)

with a

rectangular function The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname\left(\frac\right) = \Pi\left(\frac\ri ...

of width

\Delta f

in frequency domain: :

\hat(f) = \operatorname \left(\frac \right) =\chi_(f)
      := \begin1 & , f, \le\Delta f/2 \\ 0 & \text \end.

This multiplication with a rectangular function acts as a

Bandlimiting Bandlimiting is the process of reducing a signal’s energy outside a specific frequency range, keeping only the desired part of the signal’s spectrum. This technique is crucial in signal processing and communications to ensure signals stay cl ...

filter and results in

\hat(f) =\hat(f) \underline(f)=:.

Applying the

convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time dom ...

, we also know :

\hat(f) \cdot \hat(f) = \mathcal((g * u)(t))

Since the fourier transform of a rectangular function is a

sinc function In mathematics, physics and engineering, the sinc function ( ), denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatorname(x) = \frac. Alternatively, ...

\operatorname

and vice versa, it follows directly by definition that :

g(t) =\mathcal^(\hat)(t)=\frac1  \int \limits_^ 1 \cdot e^ df = \frac1 \cdot \Delta f \cdot \operatorname \left( \frac \right)

Now the first root

g(\Delta t) =0

is at

\Delta t= \pm \frac

. This is the

\Delta t

of the

pulse In medicine, the pulse refers to the rhythmic pulsations (expansion and contraction) of an artery in response to the cardiac cycle (heartbeat). The pulse may be felt ( palpated) in any place that allows an artery to be compressed near the surfac ...

g(t)

. Since the rise time influences how fast g(t) can go from 0 to its maximum, it affects how fast the bandwidth limited signal transitions from 0 to its maximal value. We have the important finding, that the rise time is inversely related to the frequency bandwidth: :

\Delta t =  \frac,

the lower the rise time, the wider the frequency bandwidth needs to be. Equality is given as long as

\Delta t

is finite. Regarding that a real signal has both positive and negative frequencies of the same frequency band,

\Delta f

becomes

2 \cdot \Delta f

, which leads to

k = \frac

instead of

k = 1

Proof

See also

References

Further reading