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The complex notation
A fundamental mathematical property of linear
timeinvariant 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

(1) 
will produce an output signal

(2) 
with the same
and
but possibly different
a and b. Note that
is the angular
frequency, which is
times the common frequency.
Using Euler's identity

(3) 
and the rules of complex algebra, we may
write our input and output signals as

(4) 
respectively, where
denotes the real
part, and
c_{1}=a_{1}jb_{1},
c_{2}=a_{2}jb_{2}. 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 c_{2}/c_{1}
is the complex gain of the system, and for
,
i.e for pure sinewaves,
it is the value of the complex frequency
response at the angular frequency
. What we
have outlined here may be called the engineer's
approach to complex notation. The sign
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
complexconjugate signals with positive and
negative frequency:

(5) 
where the asterisk ^{*} denotes the complex
conjugate. The mathematical notation is slightly less
concise, but since for real signals only
the c_{1} term must be explicitly written down (the
other one being its complex conjugate), the
two notations become very similar. However, the
c_{1} 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.
Next: The Laplace transformation
Up: Basic Theory
Previous: Basic Theory
Erhard Wielandt
20021108