Several nonperiodic signals such as single pulses or steps are very easy to input into a system. However, we shall see that the total available energy is limited, and these signals have difficulty in providing a high enough signal — to-noise ratio to give accurate frequency response results. Nevertheless, they may be used to give approximate results very easily. These tests can comple­ment more accurate methods involving periodic inputs. One important use would be for frequent checks for major changes in system behavior. More accurate methods would be used on a less frequent schedule or when the tests with nonperiodic signals indicate a need for the more accurate tests.

(a) Pulse Tests t

Frequency response tests using a single pulse as the input have been used in several nuclear reactor tests and in numerous tests on chemical process equipment. The simplest pulse shape is a square pulse. The square pulse and its energy density appear in Fig. 3.15. Note that the signal energy is given in terms of an energy density (energy per unit frequency), since the signal distri­butes a finite amount of energy over a continuous frequency range. The energy density e for a square pulse of duration T is given by

_ T2T2rsin(coT/2)

Є 2n (x>T/2

t See Hougen and Walsh (66) and Hougen (67).

0.1 1 10 100 Dimensionless Frequency (<t)T)

Fig. 3.15. A square pulse and its energy density.

It is important to note that the energy density in a nonperiodic signal is simply the power density times the duration of the input (PT). In contrast, the energy Ek in the kth harmonic of a periodic signal is given by

Ek = PknT (3.8.2)

where n is the number of cycles and T the period. Thus the signal energy can be made as large as desired by increasing n for a periodic signal, but the energy for a nonperiodic signal with a fixed amplitude and duration is fixed. The only way to improve the results from tests with a given nonperiodic signal is to average the results from multiple tests. This procedure may be difficult to use, since it may be difficult to achieve the same initial conditions for each test.

Since the signal energy relative to the noise energy determines the accuracy of the test results, it is useful to use the energy density results and determine the total energy in a specified frequency range for a square pulse.

The energy in frequency range cu, — w2 is given by

Eicot — to2) = A2T

Since A2 T represents the total signal energy, the term in brackets represents the fraction of the total signal energy in the range coj — co2. The integral

1 f« (sm(a}T/2)2 M ^

2nJaT (oT/2 I d{(° ] (3.8.5)

has been evaluated and is shown in Fig. 3.16 (50). This allows the evaluation of E(tol — co2) as follows :

E(tol — a>2) = £(«! — oo) — E(co2 — oo) (3.8.6)

This type of information is useful for comparing different signals.

DIMENSIONLESS FREQUENCY (tuT) Fig. 3.16. Energy above a specified frequency for a square pulse.

For an example, let us determine the signal energy between 0.1 and 0.2 rad/sec in a pulse with an amplitude of 2 and a duration of 10 sec. From Fig. 3.16, we obtain

and

£(0.2 — oo) = 40(0.43) = 17.2

Then

£(0.1 — 0.2) = 27.6 — 17.2 = 10.4 energy units

(b) Step Tests t

The frequency reponse can also be obtained from tests using a step change in the input. A step input may be used if the output settles to a constant value (including zero) after the step input. Stated differently, the frequency response gain at zero frequency must be finite.

The Fourier transform is not defined for a function that does not return to its original value. The output of a system with a finite, nonzero gain at zero frequency does not return to its initial value in a step test. However, the derivative of the signal does return to its initial value. The ratio of the Fourier transforms of the derivatives of the output and input gives the system frequency response:

It is convenient to express the input and output signals relative to their final values:

30* = 30(oo) — 30 31* = 3I(oo) — 31

Thus

d 30*/dt = — d 30/dt and d 31*/dt = — d 31 dt

and we obtain

OUTPUT

INPUT

Fig. 3.17. A typical step test.

t See Nyquist, Schindler, and Gilbert (68).

This may be integrated by parts to give

The energy in a true step input signal would be infinite. In an actual test that is terminated after the system settles out, the useful energy is finite. The total energy in a step test that is terminated at time T is half the useful energy in a square pulse of duration T. As an example, consider the step test shown in Fig. 3.17 and the pulse test shown in Fig. 3.18. The pulse test con­tains the same useful energy as two step tests.

T 2T

Fig. 3.18. A typical pulse test.