2. Spatial and temporal
resolutions and limits
That is our second analogy : in both Quantum Mechanics
and Digital Signal Processing, there is a minimum amount of time and a minimum
amount of space, nothing can exist beyond those limits.
The main reason for these limits is that the physical
universe and the memory of a computer are not infinite. So they have to use
quanta, as if there was such a thing as infinitely small values, the physical
universe and the computers would have to be infinite, but they can't:
there are not enough resources available.
First there is a "real" wave in the physical universe.
An electronic chip, the analog to digital converter, is
converting this wave into numbers (samples), and there is a minimum number which
is called resolution.
Samples are created according to a regular rythm, the
"sampling frequency", so that there is a minimum amount of time (sampling
period).
When the amplitude of the wave is inferior to the
resolution, it cannot be represented.
Digital Signal Processing
Version 2
January 21, 2007
In Digital Signal Processing:
- Minimum time = sampling
period
- Minimum space =
resolution
Samples, at the intersections
of spatial and temporal grids
In Quantum Mechanics:
- Minimum time = Planck's
Time
- Minimum space = Planck's
Length
Where
- Planck's Time = 5.391 × 10-44 seconds
- Planck's Length = 1.616 × 10-35 metres
How do we know it?
The discovery of Planck's limits comes from the study of black
body radiations in the last years of the 19th century.
A black body absorbs all light that falls onto it, but does
produce thermal radiations such as light.
When measuring the radiations of black bodies, scientists
discovered that they were not in agreement with the classical view where
electromagnetic waves are continuous phenomena.
The following figure represents the curve predicted by the
classical theory of electromagnetism (where light and the universe are continous
entities) and the real curves (red / blue / green curves):
The real curves don't agree with the classical
view.
The only way to explain the experimental graphs was to
assume that electromagnetic radiation could propagate only in discrete packets,
or quanta.
As a consequence, both time and space are also based on
discrete packets, and the sizes of the spatial and temporal packets give us the
Planck's time and length.
By extending the analogy, we see that very short
events, very high frequencies or very small objects can exist in the upper
universe, when they cannot exist in the physical universe.
When the duration of an event is inferior to the
sampling period, it cannot be represented.
All samples have the same value, the real wave is invisible
All samples have the same value, the real wave is invisible
Short events and small signals and waves cannot be
represented in the digital domain, but they can exist in the real
world.
That is the curve we should have if the universe was continuous