The Uncertainty Principle - Signals, Processes, and Systems - page 60

Illustration 47: δ -function in the time and frequency domain

A δ -pulse in one of the two domains ( Δ t −> 0 and Δ f −> 0) always implies an infinite extension in the

complementary domain ( Δ f −> ∝ and Δ t −> ∝ ).

On closer examination it emerges that the spectral line of the sine (above right) is not a line in the true
sense ( 0) but in a certain sense is blurred i.e. uncertain. The sine was also evaluated only within the

segment illustrated from

Δ f −>

Δ t = 1s. According to the Uncertainty Principle UP this results in Δ f ≥ 1, i.e. a

blurred stroke of at least 1 Hz band width.

A (one-off) δ -pulse produces an "infinite" bandwidth and Δ f −> ∝ as a result of Δ t −> 0. It contains all the

frequencies with the same amplitude; see also Illustration 36. This makes the -pulse an ideal test signal

from a theoretical point of view, because - see the FOURIER Principle - the circuit/ system is tested at the
same time with all the frequencies (of the same amplitude).

δ

Why ideal filters cannot exist

Filters are signal technology components which allow frequencies - i.e. certain sinusoidal
signals within a frequency range to pass through (conducting state region) or block them
(blocking state region). If only the low frequencies up to a certain limiting frequency are
to be allowed to pass, this is called a lowpass filter. As we wish to demonstrate, the
transition from a conducting state region to a blocking state region and vice versa must
always take place with a certain uncertainty.