7.1 Representation of Complex Signals

Signals in their classical analogue form, represented by a suitable physical quantity, e.g. by electrical voltage at a certain point of a circuit, naturally acquire only real values in every time instant. Two such voltages would be needed to represent a signal, the values of which could be complex; nevertheless, to construct reliable circuits maintaining the proper relationship between the component signals would be extremely difficult. On the other hand, digital representation enables us to represent simply signals with samples of complex value such signals will be called complex signals. Subsequently, advanced methods using complex signals can be conceived that would not be feasible with analogue signal representation.

A pair of numbers, which are interpreted as the real and imaginary components, respectively, of the complex sample value, represents every sample of a complex signal,

(7.1)

Alternatively, the sample may be represented in polar form by its magnitude and phase,

(7.2)

In place of a single sequence of scalar values as in the case of a common scalar signal, the complex signal is therefore represented by a sequence of complex vectors or, in other words, by a pair of scalar sequences. Which of the above representations will be chosen depends on the prevailing type of operation to which the complex samples are to be submitted. Let us recall that the sum of two complex samples is

f _n + g _n = ( f _Rn + g _Rn) + j