Causal Signals—Analytic Signals
It was shown in Chap. 8 that the impulse response of a physical system is zero for negative time. This follows from the principle of causality; the output of the filter cannot precede the signal that created it, in this case, the Dirac distribution which is zero for negative time. The effect cannot precede the cause. The physical system, which satisfies the principle of causality, is said to be causal. More generally, we will call causal any function that is null for negative time. The general properties of these functions are discussed here starting from the properties of the Fourier transform of the Heaviside function. In the first paragraph, the Fourier transform of the pseudo-function 1/t is carried out as a preliminary calculation that leads to the FT of the Heaviside function. We then show that the real and imaginary parts of the Fourier transform of a causal system are related by integration relationship formulas called the Hilbert transform. Analytic signals are defined as having a zero FT at negative frequencies. This notion brings an efficient tool to study several signal modulations and band-pass filtering.