Duality in speech signal processing

In fact, duality is a well-known property in disciplines that are seemingly not related to quantum mechanics. In signal processing, for example, duality is embedded in the frequency domain analysis. It is well known that given a signal in the time domain, there is a conjugate signal in the frequency domain that is the Fourier transform of the original signal. Time domain and frequency domain are therefore complementary to each other and can be considered a special case of duality. In the time domain the dynamic aspect of a signal is manifested. In the frequency domain, on the other hand, the stationary aspect is manifested. Not very surprisingly, this frequency-time duality can also be derived from quantum mechanics by considering a signal as electromagnetic properties. In quantum mechanics, an electromagnetic wave is treated as photons. Photons have properties of both waves and particles. Specifically, we have the following relation:

$$\Delta E\Delta t\ge {\hbar \over 2}$$

where $E$ is the energy of a photon and $t$ the time. Together with

$$E=h\nu =2\pi \hbar \nu$$

with $\nu$ being the frequency of the photon, we then have

$$\Delta \nu \Delta t\ge {1 \over {4\pi }}.$$

which is a well-known relation in signal processing.