3.2 Complex Notation

The complex nature of the Fourier transform expressions necessitates a reminder of basic complex notation and some definitions. The representation of s(t) in frequency domain produces S(f), a complex function called the complex spectrum or complex spectral density of s(t). Therefore, in general, it can be expressed by real and imaginary parts in rectangular form as

(3.2.1)

or, in terms of an amplitude spectrum, A(f) and a phase spectrum, f(f) in polar form by

(3.2.2)

 

SAGE SAYS:

Here the amplitude means the absolute value of the complex spectral density at each frequency.

Here,

(3.2.3)

is the amplitude spectrum and

(3.2.4)

is the phase spectrum of s(t).

Because the arctangent function is multivalued and discontinuous, the phase spectrum is usually expressed within the bounds -180 degrees to +180 degrees (-p and +p radians) and jumps of 360 degrees (2p radians) are allowed. Sometimes the amplitude spectrum is squared yielding the energy density spectrum. Figure 3.2 geometrically defines the relationships between the terms of a complex quantity such as S(f) at a given frequency, f0.

Figure 3.2. Definitions of amplitude and phase spectra.