6C: Removing Uncertainty

6C: Removing Uncertainty

qmp140

qmp141

engineering, see [Couch, p.93]. Clearly (6.19) is a characteristics of Fourier transform pair as we have seen in the proofs given above. Since the spectrum function is derived from the time function, the two functions are related, and (6.19) represents one such relation. More details of such relationship are described in a later section.

Dimensionality Theorem

The dimensionality theorem is stated in [Couch, p.93], as:

“When BT is large, a real waveform may be completely specified by N=2BT independent pieces of information that will describe the waveform over a T interval. N is said to be the number of dimensions required to specify the waveform, and B is the absolute bandwidth of the waveform”.

Comparing with equation (6.19) we can see that Δt = T and Δw = 2B. Thus the dimensionality theorem is same as uncertainty principle. We show that the dimensionality theorem is derived from FT and requires infinity assumption. Thus we can say, dimensionality theorem = time bandwidth product = uncertainty principle. They are all equivalent to FT and are derived from it, in different branches of science and engineering, by different people, at different periods.

However these theories violate a fundamental property of continuous functions – all such functions are infinite dimensional over finite or infinite domain. But the dimensionality theorem clearly says they have dimension N.

Before we show this relationship, we state the Nyquist Theorem, without proof.

“The minimum sample rate allowed to reconstruct a band limited waveform without error is given by fs = 2B”.

Here B is the bandwidth and fs is the sampling rate also known as the Nyquist rate.

The following proof of dimensionality theorem is based on [Shannon]. In case of a band limited waveform, that is, a waveform whose

qmp143

qmp144

qmp145

qmp146

qmp147