Multifrequency Binary Sequence (MFBS)f

Even though two of the signals considered previously in this chapter are binary and have signal power in a number of harmonic frequencies, we will reserve the name multifrequency binary sequence for signals that can be generated with a spectrum whose shape is (within certain restrictions) specified by the user. This differs from the other signals whose harmonic amplitudes are predetermined once the number of bits is selected.

A continuous-level, periodic, multifrequency signal may be developed by simply adding sinusoids with selected harmonic frequencies:

N

I(t) = Y, at s’n (3.5.1)

i= 1

where N is the number of sinusoids, щ the selected amplitude of the ith sinusoid, and щ a harmonic frequency. A signal of this type is shown in Fig. 3.12. The complicated waveform even for this simple example would be difficult to implement with usual system hardware.

image55

Fig. 3.12. A continuous multifrequency signal and its binary approximation.

Jensen (62) proposed a binary multifrequency signal that is obtained simply by setting the input equal to +A when I(t) from Eq. (3.5.1) is positive and — A when I(t) is negative. That is,

I'(t) = A{sgn[I(t)]} (3.5.2)

where I'(t) is a binary signal with values 4-A and — A. A binary signal ob­tained this way is also shown in Fig. 3.12. This approach gives a signal that largely overcomes the hardware problem and still retains a large fraction of

t See the literature (49, 50, 52-54, 62-65).

the signal power in the selected frequencies (usually 40-60%). However, because of the quantization, the distribution of the energy among the selected harmonics may be far from the desired distribution.

Later work (52,65) provides MFBS signals that achieve both a concentra­tion of signal energy in selected harmonics and a good matching of the distribution to the desired distribution. This is accomplished by a computer optimization that modifies the polarities of the bits in a binary pulse chain until the difference between the desired spectrum and the obtained spectrum is minimized. Since antisymmetry is achieved simply by making the last half of a period the negative of the first half, this property is easily obtained.

A computer code (52) has been developed for generating MFBS signals and has been used to prepare a number of different signals. Experience shows that the procedure gives sequences that concentrate 70-80% of the total

TABLE 3.4

Selected MFBS Signals”

Signal number

1

2

3

4

5

6

3-

12 +

2 +

3 +

1 +

17-

6 +

11-

2-

3-

4-

5 +

1-

4 +

25 +

1 +

2 +

10-

6 +

13-

9-

4-

5-

7 +

2-

1 +

52 +

3 +

9 +

12-

21 +

13-

12-

2-

6-

17 +

10-

4 +

5 +

2 +

2 +

22-

2 +

26-

10-

1-

19-

8 +

6-

6 +

4 +

3 +

7 +

7-

2 +

3-

и —

14-

4-

41 +

5-

4 +

6+

2 +

3 +

1-

2-

24-

2-

5-

1 +

6 +

40 +

5 +

3 +

8-

3-

22-

2-

4-

8 +

12 +

8 +

2 +

5 +

8-

8-

24-

1-

8-

19 +

2 +

27 +

22-

3-

7-

8 +

7 +

4 +

9-

2-

3-

26 +

“ Only the first, half of each signal is presented. The second half is obtained by inverting the first half.

bThe notation J+ or J— means the signal is positive or negative over J minimum-width intervals.

signal power in the selected harmonics. The user specifies the number of bits in the sequence, the harmonics at which power is to be concentrated, and the magnitude desired at each of these harmonics. Of course, there is no need to specify harmonic amplitudes that are greater than the maximum possible amplitude given by Eq. (2.12.7). Several MFBS signals and their spectra are shown in Tables 3.4 and 3.5.

TABLE 3.5

Spectral Characteristics of Selected MFBS Signals

Fraction of signal power in indicated harmonic

Ftarmonic

Signal number: 1 Number of bits: 128 Number of frequencies: 6

2

256

6

3

512

6

4

128

12

5

256

12

6

512

12

1

0.119

0.146

0.134

0.0438

0.0636

0.0656

3

0.134

0.120

0.120

0.0685

0.0671

0.0713

5

0.146

0.125

0.133

0.0639

0.0684

0.0791

7

0.0850

0.0690

0.0696

9

0.114

0.117

0.124

0.0464

0.0651

0.0693

13

0.0754

0.0586

0.0666

17

0.120

0.139

0.132

0.0665

0.0682

0.0708

21

0.0694

0.0662

0.0756

25

0.0561

0.0553

0.0555

29

0.0388

0.0678

0.0737

33

0.102

0.109

0.110

0.0635

0.0750

0.0784

37

0.0756

0.0704

0.0545

Total

0.735

0.756

0.753

0.753

0.795

0.830

The MFBS must be generated on a computer using the optimization procedure. The sequence is then loaded into an input device for feeding into the system being tested. The input device might be an on-line computer, a paper-tape reader, or an electronic storage device such as a ring-tail counter.

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