Speed up your spectrum-analyzer measurements
Some little-known properties enable spectrum analyzers to provide useful results in a small fraction of the normal measurement time.
Morris Engelson, JMS Consulting -- EDN, January 20, 2000
A spectrum analyzer's calibrated default sweep-speed setting assumes a specific application—the accurate measurement of the magnitude of a sinusoidal signal. The analyzer displays an "uncalibrated" message when it sweeps faster than the default setting. But just because the instrument isn't calibrated doesn't mean that it can't make the measurement. All that the uncalibrated message means is that the sine-wave-measurement error exceeds the limits that the instrument's calibrated-accuracy specification defines. Frequently, users are happy to accept a slight loss in accuracy to obtain a quicker result. After all, measurement errors exist even when the instrument is fully calibrated. Furthermore, many signals, such as those in pulsed radar and digitally modulated communications systems, aren't really sine waves. Yet, the analyzer's default settings treat all signals as if they were sine waves. Rather surprisingly, the analyzer loses little, if any, accuracy when it measures the vast majority of common signals at five to 10 times the calibrated speed.
Swept-frequency-analysis instruments, such as spectrum analyzers, are subject to an inherent measurement-time constraint. The signals must pass through a bandpass filter, whose output amplitude responds rather slowly to changes in input amplitude; the narrower the filter's passband, the more slowly the filter responds. If the signal frequency sweeps too rapidly through the filter's passband, the signal amplitude at the filter output is too low. In most cases, the minimum time required for making calibrated measurements is proportional to the reciprocal of 0.5 times the filter bandwidth (B) squared—the so-called 0.5 relationship. Measurement times of tens of minutes are not unusual when the resolution bandwidth is 10 Hz or less. Even at a bandwidth of several kilohertz, the measurement time is frequently many seconds.
This article discusses the relationship between sweep (measurement) time and accuracy and the impact of different signal types on accuracy. The article also provides procedures and examples that permit much faster measurements with, at most, a small loss in accuracy.
Sweep-time limitations
Reference 1 shows that the amplitude error for a sinusoidal signal displayed on a spectrum analyzer is equal to 
=[1+(0.195(S/(BR2T))2)]–¼. In this equation, is the ratio of the display amplitude to a normalized level of unity, S is the displayed frequency span, T is the sweep time, and BR is the resolution bandwidth at the –3-dB points. Analyzer manufacturers usually choose an error default of 0.1 dB, which makes =1/1.0116. Choosing this amplitude level yields S/(BR2T)=0.49. Hence the generally accepted rule is that S/T=0.5BR2. For example, sweeping through a frequency span of 10 MHz with a 10-kHz resolution bandwidth calls for a sweep time of 200 msec to limit sine-wave errors to 0.1 dB.
The effect on measurement error is not linear with sweep time. For instance, in the previous example, a sweep time of 200 msec yields an of 1.012, which is equivalent to a 0.1-dB error. Reducing the sweep time to 100 msec yields an of 1.046, which represents nearly a 0.4-dB error. Still, given the spectrum analyzer's usual accuracy (really inaccuracy), a 0.4-dB amplitude-measurement error may be perfectly acceptable in return for halving the measurement sweep time. Table 1 provides the amplitude-error range for sinusoidal signals as a function of the dimensionless ratio S/(BR2T)=k.
In the expression for k, the sweep time, T, is in the denominator. Therefore, reducing the sweep time from 3.9 sec to 0.5 sec—a factor of nearly eight—multiplies k by eight and introduces a 3-dB error. At k=5, which corresponds to a factor-of-10 reduction in sweep time, the error is 3.8 dB. Although 3 dB isn't particularly remarkable in this context, it has something of a special quality that people tend to choose. This choice explains a statement that you sometimes see in the literature—that you can increase the sweep speed to about 10 times the calibrated value before the amplitude error becomes really objectionable.
Obviously, you can use the above relationship to make any computation you choose. For example, the predicted amplitude error is 13.4 dB at a sweep-time reduction of 100 times. But even if you could accept a 13-dB error, such a speed-up would be impractical because several limitations prevent such a large reduction in the measurement time. The maximum practical sweep-time reduction is a factor of approximately 20, which produces a 6.6-dB error. A good rule to follow is that a five-times reduction is almost always possible, 10 times is usually possible, and 20 times is sometimes possible. There are rare situations in which time is all that counts. Usually though, time is important—but not at the expense of all else. In such cases, the so-called optimum-sweep-time relationship becomes useful.
Optimum sweep time
Sweep time and resolution bandwidth are independent of each other. Hence, you can take a different view of the relationships among sweep width, sweep time, and resolution. What happens if you fix the frequency span and sweep time and vary the bandwidth? A narrower bandwidth violates the 0.5 relationship and causes a loss in amplitude. But there is also an impact on the displayed, or apparent, bandwidth. At relatively low sweep speeds where the 0.5 relationship holds, the displayed or dynamic bandwidth, essentially equals the true, or static, bandwidth. But the dynamic bandwidth increases as you reduce the sweep time. Is there an optimum value of static bandwidth that yields the narrowest displayed, dynamic bandwidth? Yes, and for this combination of bandwidth and span, the sweep time is optimum. The optimum bandwidth, BO, is related to sweep time and span by a factor of 2.27 (Reference 2). This setting causes a 1.5-dB error in the measured amplitude.
The result is not intuitively obvious. Why does this setting give the narrowest displayed bandwidth? Why can't you get something narrower? The explanation is that a wider static bandwidth obviously provides a wider displayed dynamic bandwidth. Both static and dynamic bandwidths are essentially equal when the 0.5 relationship applies. However, when the 0.5 relationship no longer holds, a narrower static bandwidth results in an increase in dynamic bandwidth. The rate of widening of the dynamic bandwidth is not linear with the change in static bandwidth, and the increase in dynamic width eventually exceeds the decrease in the static value. The crossover point establishes the optimum bandwidth.
If you hold the sweep time constant, achieving the optimum bandwidth requires a slightly greater than 2-to-1 reduction in bandwidth—(2.27/0.5)½=2.1. The noise level is proportional to the bandwidth, so halving the bandwidth reduces the noise level by 3 dB. Hence, the noise drops by a bit more than 3 dB, the signal level drops by about 1.5 dB, and the SNR improves. Therefore, if you wish to sweep faster than usual, the ideal speedup is 2.27/0.5=4.5 times the preset calibrated value. This ratio provides the narrowest, or best, signal resolution and also produces the best SNR.
Yet, even a 1.5-dB measurement error can be significant. Surely, the situations in which such an error is acceptable must be rare. Not so. The reason is that fully half of spectrum-analysis measurements involve relative, rather than absolute, amplitude determination. Amplitude, frequency, and many other modulation measurements, as well as many spurious-response measurements, involve determining dBc, that is, signal-to-carrier ratio. In such measurements, the absolute amplitude error is of no consequence because both the carrier and the sidebands are similarly reduced in level, and the decibel difference remains unchanged. Another reason is that this error is systematic, rather than random. You can calculate a correction factor and adjust the final result accordingly. Hence, this procedure could enjoy much wider use—if more engineers knew about it.
The traces in Figure 1, taken from an Advantest 3132 spectrum analyzer, illustrates this idea. The two traces differ by 1.5 dB at a vertical setting of 2 dB per division. The span is 200 kHz, the bandwidth is 1 kHz, and the sweep time is 100 msec—all shown on the lower edge readout. This combination is not calibrated, as shown by the "uncal" message at the lower right. These settings result in the smaller trace, slightly to the right of center. The larger trace, to the left, was obtained at a calibrated sweep time of 400 msec. As required by the theory, the calibrated, S/B2T setting is 2×105/106×0.4=0.5. The sweep-time decrease is 400/100= 4 times, for a valuek=0.5×4=2. Note that the four-times sweep-time reduction is the closest time setting in a 1, 2, 5 sequence to the previously calculated optimum value of 4.5. The theoretical impact on accuracy should be 1.3 dB. The measured value is slightly more than that, in line with the fact that real filters are somewhat less forgiving than a theoretically perfect filter would be.
Nonsinusoidal signals
You can oversweep nonsinusoidal signals (that is, sweep them at a speed greater than the fastest calibrated value) to an even greater degree than you can oversweep sine waves. As with sine waves, faster sweeps do not affect measurements of nonsinusoidal signals' relative amplitudes. However, most absolute-amplitude results are also unaffected. Hence, most nonsinusoidal signals are good candidates for oversweeping. The following two illustrations explain why.
Figure 2 shows the frequency spectrum of a pulsed-carrier signal. The span is 100 MHz, the bandwidth is 30 kHz, and the sweep time is 230 msec. All of these values appear on the lower line of the readout. The sweep relationship yields a value of 0.48, so the result is calibrated, as shown by the absence of an "uncal" message. The second trace, whose readout is not shown, is overlayed on the first trace. The second trace is not recognizable in the dense distributed spectrum because it is nearly identical to the first trace. In the figure, the sweep time is 20 msec, for a sweep-time speedup of more than 10-to-1. A sine-wave signal would show an amplitude difference of nearly 4 dB under the same conditions. Why is the pulsed signal not affected?
The lobe spacing of the spectrum displays as 10 MHz at one-division width. The inverse is a pulse width of 100 nsec. The filter is being subjected to a 100-nsec-wide pulse and responds in a transient mode. The response of a 30-kHz-wide filter occupies 30 kHz/100 MHz=3×10–4of a screen width. This value corresponds to a sweep time of 20 msec×3×10–4=6 µsec at the faster 20-msec full-screen sweep. Hence, the narrower 100-nsec pulse width determines the filter's amplitude response regardless of the sweep time. You would have to speed up the sweep by more than 100 times to make the sweep time approach the pulse duration. Other factors would interfere with the measurement well before then. Hence, you can obtain pulsed-signal spectra 10 and sometimes more than 20 times as fast as the spectrum-analyzer calibration suggests.
Figure 3 shows two traces of a wideband code-division multiple-access (WCDMA) signal. The calibrated time setting is 2 sec for the trace that is slightly to the left. The second trace, shifted to the right, was obtained at an uncalibrated sweep time of 100 msec. The display shows an "uncal" reading. The only effect of the 20-times speedup is to shift the trace slightly to the right. The amplitude levels of the two traces are identical, and the occupied-bandwidth values are virtually the same, though one is shifted slightly in frequency. The spectrum analyzer's total-channel-power measurement shows an integrated power level of –17.4 dBm for the calibrated settings and not quite 0.1 dB less with a faster sweep.
The reason for the lack of sweep-time impact on this signal's displayed level is different from that for pulsed signals. The sweep-time effect manifests itself during the rising and falling edges of the spectrum display. The sine-wave-based spectrum is so narrow that there is just not enough time for it to reach an equilibrium state. The same effect occurs for the WCDMA signal, but there is time to get to a full-level equilibrium condition. The result shows as a shift of the signal to the right, but the shape and magnitude are unaffected.
More than just sweep time
Sweep time is just one of the factors that affect a spectrum analyzer's total measurement time. Various control and processing functions introduce significant dead time. Sometimes the sweep time dominates the measurement time, and sometimes the dead time does. Which one dominates depends on several factors (Reference 3). This article deals only with issues that relate to sweep time.
The discussion shows that you don't degrade amplitude-measurement accuracy when you sweep broadband spectra as much as 20 times as fast as the spectrum analyzer's fastest calibrated sweep. Broadband spectra are typical of pulsed and digitally modulated signals. With a similar sweep speedup, sinusoidal-signal spectra show a loss in level, but you can calculate and correct for it. Moreover, faster sweeps do not affect relative measurements. You can achieve sweep-speed increases of five to 10 times and, on occasion, even 20 times. The ideal sweep-speed increase is 4.5 times the calibrated speed. The higher speed yields optimum signal-resolving power and SNR.
Author info
Morris Engelson has been named an IEEE Fellow for his contributions to the field of spectrum analysis. He is a consultant for JMS Consulting (Portland, OR). You can reach him at jms_consulting@pcez.com. Additional tutorial materials on spectrum analysis appear at the JMS Consulting Web site, www.pcez.com/~jmsc/.
Handbook of Spectrum Analyzer Techniques, Polarad Electronics, 1955.
REFERENCE
1. Engelson, Morris, Modern Spectrum Analyzer Theory and Applications, pg 93 to 95, JMS Consulting.
2. "Optimizing spectrum analyzer measurement speed," Agilent Technologies, Application Note 1318.
