Küpfmüller's uncertainty principle by Karl Küpfmüller in the year 1924 states that the relation of the rise time of a bandlimited 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

\hat(f)

in frequency space is given by the multiplication of any signal

\underline(f)

with

\hat(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(t) = \Pi(t) = \left\{\begin{array}{rl ...

of width

\Delta f

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

as (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 pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...

) :

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

Since the fourier transform of a rectangular function is a sinc function and vice versa, follows :

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

Now the first root of

g(t)

is at

\pm \frac

, which is the rise time

\Delta t

of the

pulse In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the n ...

g(t)

, now follows :

\Delta t =  \frac

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