The experimenter should select input signals that provide frequency response results over the range of interest with the desired frequency resolu­tion. In order to obtain an adequate signal-to-noise ratio in the shortest possible time, he must minimize the waste of available signal energy in frequencies other than those desired.

The MFBS signal has the greatest potential for minimizing signal-energy waste, since the energy can be concentrated in harmonics of interest. The spectra for the other signals (PRBS, PRTS, and n sequence) are completely predetermined once the number of bits is chosen. Their harmonics are evenly spaced on a linear scale, but frequency response results are usually inter­preted using Bode plots, which employ a logarithmic frequency scale. This means that the harmonics for the PRBS, PRTS, and n sequence fall closer together on a Bode plot as the frequency increases. This is shown in Fig. 3.13, which shows a PRBS spectrum on a logarithmic frequency scale. The bunch­ing of the points at higher frequencies is apparent. In most applications, much of the energy in these closely spaced harmonics would be used up in identifying frequency response points that are only slightly different from one another.

It is informative to compare the testing time required to obtain the same harmonic amplitude at desired frequencies in two separate tests instead of a single wide-band test. For simplicity, let us assume that all nonzero harmonics within the bandwidth in a signal have the same amplitude. Then the energy in each harmonic is

Ek = E/L

where Ek is the energy in harmonic к, E is the total energy, and L is the
number of harmonics in the bandwidth. Since the total energy is proportional to the test duration Tt, we obtain

Ek = C(TJL)

where C is a constant. This shows that a reduction in the number of fre­quencies in the bandwidth can be accompanied by a proportional reduction in the testing time without a loss of signal energy in the harmonics of interest. Thus an MFBS test with most of the energy concentrated in the desired harmonics can be completed in a much shorter time than a PRBS test that covers the same frequency range.

The same considerations are important even if the MFBS is not used. The experimenter usually has to decide whether to use a single test to cover the whole range of interest or to use several tests with frequencies that cover different portions of the range of interest. In this case, there is a trade-off between convenience and efficiency. For example, a PRBS that has a band­width of D decades will have 10й harmonics in the bandwidth. The harmonics will be distributed as follows:

Number of harmonics 9 90 900

We observe that PRBS signals that span more than about two decades concentrate an excessive fraction of the signal energy in the higher fre­quencies.

As an example, let us consider the design of a PRBS test to cover three decades of frequencies. The 2047-bit PRBS has a bandwidth that extends to the 900th (2047 x 0.44) harmonic. Thus this signal is the appropriate PRBS to cover approximately three decades. However, the test must be run long enough to build up enough signal energy in each of 900 harmonics. As an alternate selection, consider two 127-bit sequences. The first would have the same period as the 2047-bit sequence. This would cover the range shown in Fig. 3.14. Then a second test would use a 127-bit sequence with a period equal to one eighteenth of the first 127-bit sequence. Figure 3.14 shows that this test extends out to the highest frequency of interest. Also, we note that there is an overlap of the results from the two tests. This is important to check whether the tests were both run at the same conditions. The two-test procedure provides results at 110 frequencies (127 x 0.44 x 2) compared to 900 for the single-test procedure. If this resolution is suitable, then the two-test procedure would require 110/900 as much time to give approxi­mately the same energy in the useful harmonics.

A possible difficulty is that the low-frequency 127-bit signal will have longer runs with the signal at a fixed value. This can be a problem if it is necessary to insure that the output stays within prescribed limits. The bits in the 127-bit sequence must be 2047/127 as long as the bits in the 2047-bit sequence in order to obtain the same period. The longest runs are 7 bits long for the 127-bit PRBS and 11 bits long for the 2047-bit PRBS. Thus the longest run for the 127-bit PRBS will be longer than the 2047-bit PRBS by a factor (2047 x 7)/(127 x 11) = 10.2.

In general, the length of the longest run in a PRBS with a given period decreases as the number of bits increases. The period is (2" — 1) At, where n is an integer and At the bit duration. The duration of the longest run is n At. Then the ratio of the duration of the longest run to the period is n/(2" — 1). Table 3.6 shows this ratio for various PRBS signals. This demonstrates that problems caused by long runs can be eased by using longer sequences.