The longperiod seismographs of the now obsolete WWSSN (worldwide standardized seismograph network) consisted of a longperiod electrodynamic seismometer normally tuned to a free period of 15 sec, and a longperiod mirrorgalvanometer with a free period around 90 sec. (In order to avoid confusion with the frequency variable of the Laplace transformation, we use the nonstandard abbreviation ``sec'' for seconds in the present section.) The WWSSN seismograms were recorded on photographic paper rotating on a drum. We will now derive three equivalent forms of the transfer function for this system. In our example the damping constants are chosen as 0.6 for the seismometer and 0.9 for the galvanometer. Our treatment is slightly simplified. Actually the free periods and damping constants are modified by coupling the seismometer and the galvanometer together; the above values are understood as being the modified ones.
As will be shown in paragraph 2.9 (Eq. 31),
the transfer function of an electromagnetic seismometer (input: displacement,
output: voltage) is
Here is
the responsivity (in meters per volt) of the galvanometer with the
given coupling network and optical path. We use a value of 393.5 m/V,
which gives the desired overall magnification. The overall transfer
function H_{d} of the seismograph is in our simplified treatment obtained as the product of the factors (16) and (17):
As the input and output signals are displacements, the absolute value
of the transfer function is simply the frequencydependent magnification of the seismograph. The gain factor C has the physical dimension
,
so H_{d}(s) is in fact a dimensionless quantity. C itself is however not the magnification of the seismograph! To obtain the magnification at the angular frequency ,
we have to evaluate
:
Eq. (18)
is a factorized form of the transfer function in which we still
recognize the subunits of the system. We may of course insert the
numerical constants and expand the denominator into a fourthorder
polynomial:
The poles and zeros of the transfer function are most easily determined from Eq. (18). We read immediately that a triple zero is present at s=0. Each factor
in the denominator has the zeros
so the poles of H(s) in the complex s plane are (Fig. 2):
In order to reconstruct H(s) from its poles and zeros and the gain factor, we write
It is now convenient to pairwise expand the factors of the denominator into secondorder polynomials:

Fig. 3 shows the corresponding amplitude response of the WWSSN seismograph as a function of frequency. The maximum magnification is 750 near a period of 15 sec. The slopes of the asymptotes (thin lines) are at each frequency determined by the dominant powers of s in the numerator and denominator of the transfer function. Generally, the lowfrequency asymptote has the slope m (the number of zeros, here =3) and the highfrequency asymptote has the slope mn (where n is the number of poles, here =4). What happens in between depends on the position of the poles in the complex s plane. Generally, a pair of poles s_{1},s_{2} corresponds to a secondorder corner of the amplitude response with and . A single pole at s_{0} is associated with a firstorder corner with . The poles and zeros do however not indicate whether the respective subsystem is a lowpass, highpass, or bandpass filter. In fact this does not matter; the corners bend the amplitude response downward in each case. In the WWSSNLP system, the lowfrequency corner at 90 sec corresponding to the pole pair s_{1}, s_{2} reduces the slope of the amplitude response from 3 to 1, and the corner at 15 sec corresponding to the pole pair s_{3}, s_{4} reduces it further from 1 to 1.
Looking at the transfer function H_{s} (Eq. 16) of the electromagnetic seismometer alone, we see that the lowfrequency asymptote has the slope 3 because of the triple zero in the numerator. The pole pair s_{1}, s_{2} corresponds to a secondorder corner in the amplitude response at which reduces the slope to 1. The resulting response is shown in a normalized form in Fig. 6, upper right panel. As stated in section 2.6 under point 3, this case of n<m can only be an approximation in a limited bandwidth. In modern seismograph systems, the upper limit of the bandwidth is usually set by an analog or digital cutoff (antialias) filter.
As we will see in section 2.9, the classification of a subsystem as a highpass, bandpass or lowpass filter may be a matter of definition rather than hardware; it depends on the type of ground motion (displacement, velocity, or acceleration) to which it relates. We also notice that interchanging with will change the gain factor C in the numerator of Eq. (19) from to and thus the gain, but will leave the denominator and therefore the shape of the response unchanged. While the transfer function is insensitive to arbitrary factorization, the hardware may be quite sensitive, and certain engineering rules must be observed when a given transfer function is realized in hardware. For example, it would have been difficult to realise a WWSSN seismograph with a 15 sec galvanometer and a 90 sec seismometer; the restoring force of a Lacostetype suspension cannot be made small enough without becoming unstable.

Fig. 4 illustrates the impulse responses of the seismometer, the galvanometer, and the whole WWSSNLP system. We have chosen a pulse of acceleration (or of calibration current) as the input, so the figure does not refer to the transfer function H_{d} of Eq. (18) but to H_{a}(s)=s^{2}H_{d}(s). H_{a} has a single zero at s=0 but the same poles as H_{d}. The pulse was slightly broadened for a better graphical display (the pulse is not plottable). The output signal (d) is the convolution of the input signal to the galvanometer (b) with the impulse response (c) of the galvanometer. (b) itself is the convolution of the broadband impulse (a) with the impulse response of the seismometer. (b) is then nearly the impulse response of the seismometer, and (d) is nearly the impulse response of the seismograph.