The complex notation

A fundamental mathematical property of linear time-invariant systems such as seismographs (as long as they are not driven out of their linear operating range) is that they do not change the waveform of sinewaves and of exponentially decaying or growing sinewaves. The mathematical reason for this fact is explained in the next section, 2.2. An input signal of the form

$$f(t)=e^{\sigma t}(a_1 \cos \omega t + b_1 \sin \omega t)$$ (1)

will produce an output signal

$$g(t)=e^{\sigma t}(a_2 \cos \omega t + b_2 \sin \omega t)$$ (2)

with the same $\sigma$ and $\omega$ but possibly different a and b. Note that $\omega$ is the angular frequency, which is $2 \pi$ times the common frequency. Using Euler's identity

$$e^{j\omega t}=\cos \omega t + j \sin \omega t$$ (3)

and the rules of complex algebra, we may write our input and output signals as

$$f(t)=\Re[c_1 e^{(\sigma + j \omega)t}], \qquad g(t)=\Re[c_2 e^{(\sigma + j \omega)t}]$$ (4)

respectively, where $\Re$ denotes the real part, and c₁=a₁-jb₁, c₂=a₂-jb₂. It can now be seen that the only difference between the input and output signal lies in the complex amplitude c, not in the waveform. The ratio c₂/c₁ is the complex gain of the system, and for $\sigma=0$, i.e for pure sinewaves, it is the value of the complex frequency response at the angular frequency $\omega$ . What we have outlined here may be called the engineer's approach to complex notation. The sign $\Re[\dots]$ for the real part is normally omitted but always understood.

The mathematical approach is slightly different in that real signals are not considered to be the real parts of complex signals but the sum of two complex-conjugate signals with positive and negative frequency:

$$f(t)=c_1 e^{(\sigma+j \omega)t}+c_1^* e^{(\sigma-j \omega)t}$$ (5)

where the asterisk ^* denotes the complex conjugate. The mathematical notation is slightly less concise, but since for real signals only the c₁ term must be explicitly written down (the other one being its complex conjugate), the two notations become very similar. However, the c₁ term describes the whole signal in the engineering convention but only half of the signal in the mathematical notation! This may easily cause confusion, especially in the definition of power spectra. Power spectra computed after the engineer's method (such as the USGS Low Noise Model, see paragraph 5.1) concentrate all power at positive frequencies and are therefore by a factor of 2 larger than "mathematical" power spectra.