For people familiar with quantum physics, it’s a known fact that a ‘waveform’ can be essentially ‘measured’ in either ‘position’ or ‘momentum’ basis. We define the inner products using the following notations:
The point to be noted is that momentum is the time derivative of position. In such a case,
This can be proved as follows. A transformation from the basis to the basis can be made as follows.
The operator is
To find eigenstates of , we can call
which yields to .
Now, to pass from a basis to the other we can write
which is a Fourier transform.
Thus and form a Fourier pair. Simply speaking, a waveform that is measured in the time derivative of a certain basis will be the Fourier transform of the waveform being measured in that basis.
Now, coming to the signal processing scenario in which the representation of a ‘signal’ is effectively a ‘measurement’ in either ‘time’ or ‘frequency’ basis.
The correspondence between these two models:
Classically, time and frequency are inverses of each other.
where is the time period (measured in time domain).
Similarly, going by de Broglie wave equation,
where is the wavelength (measured in position domain).
In the next post, I will elaborate on the uncertainty relation in both signal processing and quantum physics. The whole idea behind creating an analogy between the two is to see if certain tools that have already been developed in quantum physics, can be adapted to enhance signal processing techniques.