Phase digitizing: a new method for capturing and analyzing spread-spectrum signals - includes related article on a method of reading a counter on the fly - technical

Phase Digitizing: A New Method for Capturing and Analyzing Spread-Spectrum Signals

PRECISE AMPLITUDE MEASUREMENT has been the basis of many traditional instruments. Oscilloscopes, spectrum analyzers, power meters, and voltmeters all focus on precise analog voltage as their basic measurement. Even a vector analyzer, which measures both amplitude and phase, does so by measuring two analog voltages: the I and Q modulation components.

This preoccupation with precise analog amplitude in instrumentation is at variance with modern modulation methods, which tend to deemphasize analog amplitude modulation in favor of frequency, phase, and time modulation for more reliable communication. Not only ordinary FM, PM, and pulse width modulation, but also modern FSK, PSK, and QPR in communications in Barker (binary phase), polyphase, and chirp modulation in radar, all deemphasize amplitude in favor of time-based parameters. Even QAM signals contain more information in the phase than in the amplitude. For these signals, fidelity is characterized by precision in frequency, phase, and time. Interest in amplitude is often connected with studying dropouts only.

The frequency agile signal, which is active over a wide frequency range in a short period of time, poses another challenge to many established practices in instrument design. Techniques such as quasistatic range switching to cover different frequency bands, searching for a signal with a sweeping narrowband receiver, or taking time to phaselock to a signal are no longer appropriate with the agile signal. Another difficulty is that some signals are not repeatable, at least not well enough to be measured by equivalent-time techniques. The output of a pulsed VCO and the waveform of on oscillator acquiring lock with another are examples of signals that are different on each occurrence. Often, it is statistical variations, such as jitter, and not just an average value, that are of interest. These can only be measured with repeated single-pass measurements, even though the signal is nominally repetitive. Measurement methods that require multiple passes for one measurement are not applicable.

In short, traditional narrowband, amplitude-based instruments requiring repetitive signals are suddenly found wanting when confronted with frequency agile, time-encoded or phase-encoded signals that do not repeat.

Amplitude Digitizing

The waveform recorder (or high-sampling-rate digitizing oscilloscope) is wideband and can capture a transcient spread-spectrum signal by digitizing its voltage, taking samples regularly spaced over time. While this is a good way to view the waveform, it is a difficult and inefficient method of characterizing the signal's modulation fidelity.

Consider a modulated signal with an agile frequency carrier randomly hopping about. To capture the waveform without aliasing requires sampling above the Nyquist rate. For such a signal, the sampling rate must be more than twice (closer to four times in practice) the highest carrier frequency plus modulation. Since the modulation (information) bandwidth is always less, and often much less, than the waveform bandwidth, sampling for full waveform recovery is grossly inefficient, resulting in unnecessary data bulk, if only the information is of interest. Apart from this, processing the data to uncover the modulation is a complex and difficult process since the data is in the form of voltage, that is, v(t.sub.i.), where v(t) may take on the form v(t) = [V.sub.o + V.sub.a.(t)]sin[2[pi]f(t)t + [theta](t)] and any of the three--the amplitude V.sub.o at V.sub.a.(t), the frequency f(t), and the phase [theta](t), may change with time depending on the modulation type. Selecting the optimum band-limited functions V.sub.a.(t), f(t), and [theta](t) to fit a set of v(t.sub.i.) means iterative curve fitting to trigonometric functions with complicated and possibly discontinuous arguments, an incredibly messy operation "prone to undersirable convergence behavior because of the nonlinear behavior relationship between parameters." This is true when V.sub.a., f, and [theta] are constants. The complexity of fitting to variable parameters is simply staggering.

A less sophisticated but more effective approach is to find the locations of peaks and zero crossings. For a properly band-limited signal, interpolation between samples using digital signal processing allows computation of a voltage given a time. Computing a zero crossing time is the reverse; one computes a time given a voltage (zero). For slow-slewing signals, a combination of (sin x)/x interpolation and linear approximation may enhance the resolution of zero-crossing time measurements below one sampling period. Using this technique and counting the crossings in software, Nichols simulated a frequency counter with a variable gate for frequency profiling. Unfortunately, for binary-voltage fast-switching signals, there is no timing resolution enhancement, the precision voltage measuring capability of the waveform recorder is not really being used, and the resulting counter performance is poor.

Phase Progression Digitizing

The signal digitizing method used in the HP 5371A frequency and Time Analyzer is based on continuous counting and sampling only at signal zero crossings. The method bypasses the two extra steps, voltage digitizing and voltage-to-phase conversion, and directly digitizes the phase progression of the signal. The procedure may therefore be appropriately called "phase progression digitizing," or "phase digitizing" for short. Amplitude information is discarded. Because the data is already in the form of phase and time, trigonometric functions are totally avoided, replaced by simple functions like straight lines and parabolas, making analysis simple for even moderately complex modulation.

Phase digitizing is illustrated in Fig. 1a, which shows the sample locations on a sinusoidal signal of changing frequency. Samples occur at occasional upcrossings at a relatively constant rate. Each sample produces two increasing numbers, the total cycle count and the time stamp at that point. The cycle count comes from reading, on the fly, a counter counting the signal. Every cycle is counted, not just those on which a sample occurs. When a sample does occur, that particular cycle's upcrossing is gated by a synchronizer for time stamping, which is accomplished by reading (also on the fly) another counter counting a 500-MHz time base clock. With interpolation, described in the article on page 35, the resolution is improved from 2 ns to 0.2 ns. In counter jargon, the cycle count is called the event number and the time stamp is simply called time.

In contrast, amplitude digitizing is illustrated in Fig. 1b. The samples are regular in time and they fall on the signal wherever they may, not necessarily at zero crossings. There are always more samples in voltage digitizing than phase digitizing. There must be several samples per signal carrier cycle in voltage digitizing, regardless of the modulation. In phase digitizing, it is the other way around. There are usually several cycles per sample. The sampling rate must only exceed the Nyquist rate for the mudulation, regardless of the carrier frequency.

The event number in phase digitizing is a measure of the phase progression of the signal. Because every cycle is counted, each event means the signal has progressed by one cycle, that is, 2[pi] radians or 360 degrees. Consider a modulated signal s(t) of the form v(t) = Asin(2 [pi][phi](t)) where [phi](t) is monotonically increasing. Sampling at up-crossings means samples are taken only at integer values of [phi](t). The ith event sample e.sub.i and the ith time sample t.sub.i bear a simple mathematical relationship: [phi](t.sub.i.) = e.sub.i..

Both [phi](t.sub.i.) and e.sub.i are integer valued, and t.sub.i is quantized to 200 ps. In short, the function [phi](t) is digitized, although not uniformly in time as is customary.

Phase Progression Plot

In Fig. 2a, the samples obtained from phase digitizing a BPSK signal are shown with their t.sub.i and e.sub.i values. In Fig. 2b they are plotted using t.sub.i and e.sub.i as the X and Y coordinates. The two different frequencies are manifested in the apparently piecewise linear figure. The slope d[phi](t)/dt of the line shown in the figure is the derivative of the phase function [phi](t) and therefore indicates the frequency of the signal. Notice that the sudden change in signal frequency is effortlessly handled by the phase digitizing process. The sample rate is relatively constant, and the only clue indicating a frequency increase is a larger difference between consecutive event numbers.

A "modulation-domain" plot, a term coined by an enthusiastic colleague for a plot of instantaneous frequency as a function of time, can be obtained by digital differentiation of the event/time function. The simplest instantaneous frequency estimate is obtained by dividing the change in event number by the change in time from two adjacent samples, that is, f.sub.1.(t.sub.1.) = e.sub.i+1 - e.sub.i./t.sub.i+1 - t.sub.i

A more sophisticated estimate is obtained by fitting a parabola through three consecutive points. The derivative of the parabola at the middle point is the instantaneous frequency there. f.sub.2.(t.sub.i.) = (t.sub.i+1 - t.sub.i.)f.sub.1.(t.sub.i-1.) + (t.sub.i - t.sub.i-1.)f.sub.1.(t.sub.i.)/t.sub.i+1 - t.sub.i-1

Neither of these digital differentiation methods depends on uniformly sampled data. Other differentiation algorithms can be used, generally trading bandwidth of the f(t) function for noise reduction.

In general, for differentiation using 2M points, one can use the following mathematical structure, choosing a set of weights [[lambda].sub.i.] to achieve the desired filter response. The average frequency * at the average time * can be expressed as: where the average time * is given by:

The weights [lambda].sub.j sum to unity. For odd numbers of points, 2M+1, one can simply change i-j+1 to i-j in the subscripts.

Lenear Curve Fitting

Sometimes the average frequency is taken over the entire data array for an overall average.

Fig. 3 shows a phase progression plot of corrupted data. The corruption may be from many sources, such as quantization noise or modulation. The average frequency is to be determined from the corrupted data. One method, used traditionally by frequency counters, is to find the slope of the line joining the two end points, since intermediate points are not available. The estimate * in phase digitizing notation is:

Better methods are available from statistics to find a linear best fit to the data. The slope of the fit is the average frequency estimate, and the Y intercept gives the phase relationship with the other signals. A least-squares linear fit, which minimizes the sum of the squared error terms is given by:

This estimate is computation intensive, involving sums of products and of squares.

Another linear fit that is almost as good, but much easier to compute than the least-squares fit is the bicentroid method. It does not involve squares or products. It partitions the number of samples into three roughly equal groups chronologically. The first group and the last group have the same number, say Q, of samples. The middle group is ignored. One then proceeds to compute the average e and average t for the two groups. The average frequency is estimated by the line joining these two "centroids." With equal numbers in each group, the normalizing factor (1/Q) for each term cancels out. The calculation is reduced to sums of data and one division,

In the general equation given above, this corresponds to a choice of [lambda].sub.j proportional to t.sub.i+j - t.sub.i-j+1 at the two ends of the data points and zero at the middle, where i is at the array center.

The average phase is best estimated by the average of all of the e.sub.i and t.sub.i., that is,

The equation for the curve fit *(t) becomes *(t) = e.sub.0 + * X (t - t.sub.0.).

The frequency estimate * is obtained by one of the above methods or by any other method the user feels appropriate. We use *.

How Good Are the Estimates?

The standard deviations of these estimates can be computed by making some assumptions. We assume that the random time quantization error q is the predominant contributor of error, and that the sampling rate is approximately uniform. The standard deviation of these estimates can be expressed in terms of q, the gate time g (measurement time), and the number N of samples taken. Standard deviations of the fractional frequency for the three estimates--traditional counter, bicentroid, and least-squares--are: Traditional counter: [sigma].sub.[delta]f/f = (q [square root of]2)/g Bicentroid: [sigma].sub.[delta]f/f = (q [square root of]13.5)/(g [square root of]N) Least squares: [sigma].sub.[delta]f/f = (q [square root of]12)/(g [square root of]N)

The timing quantization error q for the HP 5371A is 200 ps.

The smaller the standard deviation, the letter the estimate. Notice a dramatic reduction of standard deviation for the bicentroid method over the traditional counter method for large N. Note also the meager improvement of the least-squares fit over the bicentroid fit, considering the extra amount of computation it requires.

Modulation Computation

Computing average parameters is one function of the curve fit. Another very important function is the computation of deviations, both intentional (modulation), and unintentional (error). Deviation is the difference between the data points and the curve fit. Three main kinds of deviation are of interest (see Fig. 3):

* Phase. The vertical distance from a data point to the curve is phase deviation. By computing this for every point, phase deviation as a function of time is obtained: [delta].sub.[phi].(t.sub.i.) = e.sub.i - *(t.sub.i.).

* Time. The horizontal distance from a point to the curve is a time deviation. By computing this for every point, a graph of time deviation as a function of time is generated: [delta].sub.t.(t.sub.i.) = - [delta].sub.[phi].(t.sub.i.)/*

* Frequency. Subtracting the derivative of the curve fit from the instantaneous frequency produces frequency deviation as a function of time. When measuring a frequency modulated (FM) signal, the slope of the linear curve fit gives the carrier frequency and the frequency deviation is the modulation. Frequency deviation as a function of time is: [delta].sub.f.(t.sub.i.) = f.(t.sub.i.) - *.

Frequency Agile Carrier

The measurement of modulation in an agile carrier is demonstrated in Fig. 4, which shows FSK modulation on a carrier hopping between 10 and 500 MHz. The top graph shows the instantaneous frequency within the 140-[mu]s time frame. The dehopped FSK signal is shown in the bottom graph. The signal was phase digitized by the HP 5371A Frequency and Time Analyzer in one pass into 1000 data points, and analyzed by an external controller in accordance with the method described here.

Pulsed Frequency Bursts

Fig. 5 shows phase progression plots for two pulse bursts separated by a region of no activity. This signal generates two lines separated horizontally by a gap. This example illustrates a significant advantage of phase digitizing over waveform digitizing: there are no samples where there is no signal--a direct consequence of signal-triggered sampling. No memory is wasted on empty regions. This is particularly important for narrow pulse bursts at low pulse repetition rates, or for sparse transient capture with limited memory. A waveform recorder would fill the memory up with mostly zero data. On the other hand, should noise be present in supposedly quiet periods, samples will be generated.

Frequency Chirp

A chirp is a signal whose frequency varies linearly with time. Since phase is the integral of frequency, the phase progression of a chirp is quadratic with time. An ideal chirp would produce a parabola on the phase progression plot (Fig. 6). To measure chirp nonlinearity, the data can be compared with a best-fit parabola. The least-squares quadratic fit is well-described in the literature. It requires even more lengthy computations than the linear least-squares fit. A considerably simpler method, suggested by a colleague, partitions the data into five approximately equal groups chronologically, making the first, third, and fifth groups equal in number, say Q, where Q is about N/5. Assuming that the quadratic fit is given by: e(t)=at.sup.2 + bt + c, the coefficients a, b, and c can be computed from the following three equations:

There are sums of squares, but no cross products. The solution is straightforward and is not reproduced here. The second and fourth groups are ignored.

Once the coefficients a, b, and c are computed, the usual deviation computation can be performed to generate chirp nonlinearity parameters.

* Phase nonlinearity [delta].sub.[phi].(t.sub.i.) is given by e.sub.i - e.sub.i., or [delta].sub.[phi].(t.sub.i.) = e.sub.i - (at.sub.u.sup.2 + bt.sub.i + c).

* Time nonlinearity [delta].sub.t.(t.sub.i.) is given exactly by: [delta].sub.t.(t.sub.i.) = t.sub.i - t'.sub.i where t'.sub.i is the smaller root of the equation ax.sup.2 + bx + c = e.sub.i., or approximately by -[delta].sub.phi.(t.sub.i.)/(curve-fit slope), that is, [delta].sub.t.(t.sub.i.) [approx is.] (at.sub.i.sup.2 + bt.sub.i + c - e.sub.i.)/(2at.sub.i + b).

Advanced phase means earlier in time, so [delta].sub.[phi].(t.sub.i.) and t.sub.i are always opposite in sign. Time nonlinearity is interpreted as the difference in time between the zero crossings of the signal and the zero crossings of the ideal chirp.

* Frequency nonlinearity is given by the difference between instantaneous frequency and the derivative of the quadratic fit: [delta].sub.f.(t.sub.i.) = f(t.sub.i.) - (2at.sub.i + b) where instantaneous frequency is computed as before. Fig. 7 shows a chirp analysis from a measurement with an HP 5371A. The signal was sampled 700 times in one pass and analyzed. Fig. 7a shows the instantaneous frequency superimposed on the derivative of the fitted parabola, which is not really distinguishable since the chirp is near ideal. The signal chirped from 210 MHz to 370 MHz in just under 70 [mu]s. Fig. 7b shows frequency deviation, Fig. 7c shows phase deviation, and Fig. 7d shows time deviation. Nonlinearity is clearly visible in the phase and time deviation plots.

Phase-Shift Keying

A BPSK signal is a constant-frequency signal, but it phase shifts by 180 degrees with a binary modulating signal. On the phase progression plot, such a signal generates a straight line of a constant slope which is periodically shifted down (or up) by half a count (see Fig. 8). Since BPSK signals phases are modulo 2[pi], 360 degree is the same as zero degrees, and the analysis treats it as such. The curve fit for BPSK is a system of parallel lines alternating between 0 and [plus-or-minus] 180 degrees. Data points are mapped to the nearest line for demodulation. Deviation parameters are computed from differences between the data and the curve fit as usual. Generalization to MPSK is immediate: for QPSK, the parallel lines are 90 degrees apart, and so on. Fig. 9 shows an 8PSK signal switching among the eight allowable phase states. The phase deviation graph shows the switching to be within the resolution of the instrument. Phase resolution is computed as 360qf degrees, where q is 200 ps and f is the carrier frequency. The signal was captured in one pass, taking 1000 phase digitizing samples. The 10-MHz carrier frequency is derived from the data by curve fit. The phase resolution is 0.72 degree.

PSK signals may also be pulsed, as in the case of a pulsed Barker-coded (BPSK) radar signal. Such signals can be captured and analyzed in a similar manner.

Frequency Coverage Extension

The counting range for the HP 5371A is 0 to 500 MHz, and this represents the dynamic frequency range an agile carrier can operate over and still be measurable. With a frequency down-converter, this frequency agility range can be moved to microwave regions. Heterodyning preserves phase and translates frequency, so frequency-coded and phase-coded signals can be demodulated after down-conversion. The signal illustrated in Fig. 4 was down-converted from 3 GHz. The HP 5364A Microwave Mixer/Detector, for example, allows measurement from 2 to 18 GHz with a 500-MHz dynamic range by heterodyning with a stable local oscillator. The frequency of the local oscillator is arbitrary as long as the agility range is brought to within the relatively wide counting range of the HP 5371A. Efficient digitizing can begin immediately, before demodulation, with no data loss. Actual demodulation is accomplished by software homodyning with a computed IF carrier, operating on captured data. A hardware IF local oscillator is not needed.

In contrast, conventional carrier-recovery demodulation requires heterodyning to a narrow IF band. Using a Costas phase-locked loop,.sup.3 a coherent IF carrier is generated. Only by hardware-homodyning the signal with the generated carrier can digitizing of the demodulated isgnal begin. With an agile carrier, it is almost impossible to heterodyne to a narrow IF window for all the channels. Also, significant data loss may result during the process of carrier-recovery phase locking.

The measurable frequency agility range is limited by the maximum count rate of the HP 5371A to 500 MHz. Using a high-frequency prescaler, the range can be extended. For example, a high-spped divide-by-four prescaler can quadruple the range of the instrument, that is, the 500-MHz range of the HP 5371A can be extended to 2 GHz. In contrast, extending the frequency range of a voltage digitizer is considerably more difficult.

The prescaler divides both frequency and phase by a fixed factor, four in this case, so both the carrier and the deviation results must be multiplied by four. For modulo 2[pi] phase modulation, such as BPSK, the data should be treated as 8PSK for analysis. Then all phases are multiplied by four before the modolu 2[pi] operation. For example, 8PSK phases of 1, 91, 181, and 271 degrees will all be corrected to the same 4-degree of the BPSK prescaler input signal.

Of course, prescaling and down-conversion can be combined to give a wide frequency agility range in the microwave region.

Conclusion

Phase digitizing is an effective method of capturing and analyzing many spread-spectrum signals. Compared with voltage digitizing, it offers great economy and precision and a wide frequency dynamic range in signal capture. Analyses of frequency-coded and phase-coded signals are vastly simplified because the data is in the form of phase and time. The agility range is extended relatively easily by prescaling. Microwave signals can be measured by downconversion using a stable but arbitrary local oscillator. However, phase digitizing ignores amplitude and is not suitable for amplitude modulated signals.

Acknowledgments

Mike Ward worked with me in demonstrating the first operating continuous counters, and he later designed the first ASIC version with Dave DiPietro and Grady Hamlett. The contribution of John Flowers to the analysis software was indispensable. Sherry Read suggested the quadratic curve-fit method described in this paper. The support of management to develop such a nontraditional instrument took courage. Lastly, I would like to salute the entire HP 5371A development team for ably transforming a caterpillar into the beautiful butterfly the HP 5371A is.

COPYRIGHT 1989 Hewlett Packard Company
COPYRIGHT 2004 Gale Group