All of the characteristics of the signals described in previous sections of this chapter are based on signals that switch instantaneously from one level to another. Actual input hardware will give signals that require a finite time to move from one level to another. The actual transition will usually be in the form of a ramp, an exponential, or a staircase as shown in Fig. 3.19. The ramp or exponential transition will apply for inputs in which the drive is continuous, such as a motor-driven, rack-and-pinion control-rod drive. The staircase transition will apply for discontinuous drives such as a magnetic jack or a lead screw with a stepping motor (see Chapter 6).

The finite transition time will cause the power spectrum of the system input to differ from the ideal spectrum. Since the actual input will usually be measured in a test, the actual spectrum will be obtained in the data analysis and the finite transition time will be accounted for. However, the effect of finite transition time should influence the planning of a test. This planning includes the selection of an input signal that contains sufficient

power in selected frequencies to give accurate results. Finite transition times will cause a reduction in available signal power at all of the frequencies. The selection of a signal with adequate power in measurement frequencies should be based on the power spectrum that will actually be achieved, rather than the ideal spectrum.

The loss of signal power due to finite transition times has been analyzed for the PRBS or the n sequence with ramp and staircase transitions. The results are shown below.

(1) Ramp Transition. Figure 3.20 shows a pulse with an instantaneous transition and one with a ramp transition that takes т seconds. The pulse is a member of the total pulse chain that starts at time P. The pulse is L seconds

long. The Fourier transform of the finite-transition pulse can be obtained as follows:

where F'(jco) is the Fourier transform of the finite-transition pulse. The result is

The second term is the error due to the finite transition. Each pulse in the pulse chain will have such an error, and the total error in the Fourier trans­form of the pulse chain is the sum of the errors at each transition. The fractional reduction in the power spectrum due to ramp transitions of the PRBS and n-sequence signals appears as the dashed lines in Fig. 3.21. The fractional loss is defined as follows:

Fractional loss = (P — P’)/P

where P is the power spectrum of the perfect signal, and P’ the power
spectrum of the finite-transition time signal. The loss is given as a function of the normalized harmonic number k/Z, where к is the harmonic number, and Z the number of bits in the sequence.

(2) Staircase Transition. Figure 3.22 shows a pulse with an instantaneous transition and one with a staircase transition. The staircase is characterized by the number of steps N, the step duration in terms of a fraction of a bit duration t, and the rise time Nr. The Fourier transform of this pulse is

The result is

(3.9.4)

The solid lines in Fig. 3.21 show fractional reductions in the power spectrum due to staircase transitions for PRBS and n sequences. As an example of how to use this information, let us consider the loss in magnitude and in power for the 56th harmonic (the half-power harmonic) of a 127-bit PRBS. In this case, k/Z = 0.44. If the transition is a ramp with a rise time equal to 0.6 of the bit time, then Fig. 3.21 indicates that the fractional loss in power is 0.21.

If the transition had been a staircase with two steps, the fractional power loss would have been 0.16. Also, any staircase transition with more than two steps would have a fractional power loss that lies between the value for the ramp transition and the value for the two-step staircase transition. Thus the solid and dashed lines in Fig. 3.21 give the range on the fractional energy loss for staircase transitions.

Another imperfection encountered in practical measurements is that the input hardware may cause the transition from one input level to another to have a shape that depends on the direction of travel. It has been found that this nonreversible transition can cause a pair of small spikes in the input signal autocorrelation function (18,23). This gives a ripple in the power spectrum of the signal. Neither of these effects is significant in determining the success of a measurement. However, this behavior might create concern until the cause is understood.

3.3. Summary

This chapter has presented all of the test signals of current practical im­portance. The MFBS signal is capable of providing the most accurate results in the shortest time. The input waveform can be accomplished as easily as any periodic, binary signal. The only disadvantage is the need to generate the signal off-line, load it into a signal storage device, and play into the system from this device. The PRBS or n sequence may be obtained from an easily constructed device that contains the logical elements needed to form the sequence. Since the PRTS offers no real benefits over other signals and is more difficult to use because of the three input levels, it will probably see little use in the future.

The nonperiodic pulses and steps suffer from signal-to-noise ratio prob­lems, but should be widely used for preliminary tests and for rough checks used between more accurate tests.

Effects of signal imperfections have been determined to allow realistic estimates of the spectral characteristics in pretest planning.