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Wireless Systems by Tom Lecklider, Senior Technical Editor Measurement values are important, but the shape of a power distribution function also has meaning. RF power meters with high-bandwidth diode sensors may provide statistical analysis functionality as well as basic power measurement capability. Such a meter is required, for example, to determine the effect on the peak-to-average power ratio when simultaneously transmitting multiple code division multiple access (CDMA) channels. In another example, a statistical view of peak power levels aids in assessing the severity of signal fading. Meters that are restricted to continuous wave (CW) signals typically use sensors that cannot respond to the peaks of fast pulses found in modern digital modulation schemes. Also, the sensor output may be averaged prior to further signal processing. In contrast, peak reading meters use diode sensors with sufficiently high video bandwidths to track fast modulation. In addition, the individual data samples are corrected before signal averaging. As an example, the Giga-tronics 8650A Series Universal Power Meters specify ≥3-kHz bandwidth in the CW mode but >10 MHz in the modulation mode. Meters from several manufacturers can compute and display the cumulative distribution function (CDF) or complementary CDF (CCDF), which shows the probability that the peak-to-average value will exceed a certain value. A two-channel meter that simultaneously displays an amplifier’s input and output CCDFs is an excellent tool for examining power-level related distortion.1 However, it is difficult to interpret the detailed shape of the probability distribution function (PDF) from the CCDF. Being the derivative of the CDF, the PDF is a sensitive indicator of specific changes to the distribution of received signal power. Histograms The superimposed graphs shown in Figure 1a are plots of 2,000 data points from three different 10,000-point, normally distributed populations. They show the value of three random variables at 2,000 successive points in time. Although all three plots are similar, the standard deviations of the three populations are significantly different. The greatest excursions correspond to the distribution with the highest s value, leading to the description of standard deviation as a measure of dispersion—the degree to which data points deviate from the mean. Figure 1b displays the PDFs corresponding to the data in Figure 1a. All 10,000 points from each population have been used to develop the histograms of Figure 1b. The PDFs have a mean value, μ, of 1.0, but σ, the standard deviation, has values of 1, 2, and 3. For a normal or Gaussian distribution, the PDF is given by: and the standard deviation by: where N = number of samples in the population The normal distribution has a symmetrical shape and is defined for both positive and negative values of x. A real power distribution would be centered on the mean average power, and all of the values making up the sample population would be positive. A power meter with statistical measurement capability builds histograms from captured data. Whether done automatically or manually, upper and lower limits are set that define the range of the horizontal axis. For a normally distributed variable, 99.7% of all samples will be within 3 σ of the mean. The limits used in Figure 1a were set to 5 σ, or 99.9999%, to increase the likelihood that all 10,000 samples of the distributions would be included. As an example of histogram capabilities, the Boonton Model 4500A Power Meter divides its entire measurement range into 4,096 bins with 0.02-dB resolution. Each bin can accumulate up to 2,100,000,000 readings. This capacity means that real CDMA signals can be monitored for a long time to determine their true statistics. Without this feature, a bin near the middle of the power distribution could fill totally before infrequently occurring extreme values were recorded. The PDFs in Figure 1b are histogram plots made using Excel’s data- analysis feature in the tools menu. Three columns of 10,000 normally distributed values were produced separately by the random-number generator. Different seeds were used for each run, and the standard deviation was successively set to 1, 2, and 3. A column of histogram bin values then was developed with 601 values from -15 to +15 and 0.05-bin width. The number of hits within each bin indicates the frequency of occurrence for values between the bin boundaries. For example, 184 of the 10,000 samples corresponding to a PDF with μ = σ =1 had values between 1.05 and 1.10. The graph of histogram hits, the PDF, is normalized by dividing the number of hits by the bin width and the total number of samples. Power meters perform this process to display a PDF curve. However, meter and sensor characteristics may affect histogram accuracy in several ways. John Kenneally, vice president of sales at Boonton Electronics, said that all samples ideally should be made with the same bandwidth and the same meter range. The Boonton peak power meters have only one power range and a constant bandwidth throughout the range. This avoids the possibility that the meter does not complete its range switching in time for the next sample or that the next sample must be delayed. The missed or delayed samples may be important data points that can skew the distribution because the same power level is missed each time the signal goes from one power range to another. If the bandwidth capability associated with low-level signals is less than that corresponding to higher level signals, signals rising or falling quickly at these low power levels can be missed. This effect also can distort the statistics of the distribution and the accuracy of the PDF. In a related development, Agilent Technologies has begun including peak flatness performance in the specifications of the EPM-P Series Power Meters and E9320 Series Sensors. According to Ian Messer, the company’s RF Power Meter product manager, “Peak flatness is the flatness of a peak-to-average ratio measurement for various tone separations of an equal-magnitude, two-tone RF input. CDMA peak power measurements are not yet traceable to national standards, so including a flatness specification helps to assure customers of the meter’s peak power measurement accuracy.” All PDFs Are Not Equal There are many types of signal-path impairments that can affect the distribution of received power measurements. Attenuation caused by foliage, hills, or buildings partially blocking the path between the transmitter and the receiver is termed slow fading. It is modeled by a log normal distribution. That is, a distribution in which the natural log of the variable is normally distributed. For the log normal distribution, the PDF is given by: where σ and μ are characteristics of the normal distribution of ln(x). The actual mean and standard deviation of the log normal distribution shown in Figure 2a and 2b are 1.92 and 1.02, respectively. Slow fading is caused by attenuation beyond the expected reduction in signal power as the square of the distance between the transmitter and the receiver. Because a mobile receiver alters its position relative to fixed objects such as hills and buildings, the amount of attenuation also will change. Slow fading may be seasonal to the degree than foliage-related effects change during the year. In contrast, multipath fading results when no direct line of sight exists but many reflected signals impinge on the receiver. The Rayleigh distribution has been shown to be a good model for multipath fading when the number of reflected paths is at least six.2 For the Rayleigh distribution, the PDF is given by: where σ is termed the fading envelope of the distribution. In Figure 2a, the mean and standard deviation of the Rayleigh distribution are equal to those for the log normal distribution. However, as is seen from figures 2a and 2b, distributions are not completely defined by their standard deviations and means. The same values can correspond to PDFs with different shapes. The log normal and Rayleigh PDFs describe the distribution of only positive values of x, and these distributions are not symmetrical. In contrast to the normal distribution curves shown in Figure 1b, the average and median values in Figure 2b are not equal because a significant part of the area under the curve is contributed by large values of x relatively far out on the right-hand tail. In both distributions, the peak of the curve does not correspond to the mean value. From Figure 2a it can be seen that the frequency of large excursions is different for the log normal and Rayleigh data. Although the standard deviations for the total populations are equal, the log normal data used in this example has larger, less frequent peak values than does the Rayleigh data. Log normal and Rayleigh types of fading models commonly are used by wireless communications systems designers and often built into test generators. Other types of fading models also may be appropriate, such as Rician fading, which is similar to the Rayleigh multipath model but contains a direct, unobstructed path as well. For multipath scattering with relatively large delay-time spreads and different clusters of reflected waves, Nakagami fading is a better model than Rician and Rayleigh. The fine distinctions among many similar-appearing distributions are important to designers when modeling power distributions. As one reference explained, “Sometimes the Nakagami model is used to approximate a Rician distribution. While this may be accurate for the main body of the probability density, it becomes highly inaccurate for the tails. As bit errors or outages mainly occur during deep fades, these performance measures are mainly determined by the tail of the probability density function.”3 This is the reason that the PDF histogram data must be obtained in an unbiased manner and in large quantity. Many millions of samples may be acquired for the overall PDF before the tails become sufficiently defined. In addition, depending on the user’s experience and the exact nature of the application, having several display modes available can provide valuable clues to understanding a problem. Two Examples In a typical CDMA test setup, a cdma2000 forward channel signal was simulated by an Agilent ESG-D Signal Generator set to 1.9 GHz with 0-dBm average power. The generator output was measured by an Agilent EPM-P Series Single-Channel E4416A Power Meter and an E9327A Sensor with 5-MHz video bandwidth. The default acquisition conditions for this power meter ensure a 10-ms capture time at a 20-MS/s continuous sampling rate. As Figure 3 shows, the distribution of the generator power output in decibels referred to 1 mW is similar in shape to a normal distribution curve but not the same. If this type of difference in power distribution is important in your work, then a communications signal generator is called for rather than a random noise generator. The mean and standard deviation values of the actual data population and the reference normal curve are approximately equal. The linear values of the power samples approximately fit a log normal distribution because the dBm values are a close fit to a normal or Gaussian curve. This also is the case for the data plotted in Figure 4. In a separate test setup, a low-level, band-limited random noise signal was roughly simulated and used to drive an amplifier. The power measurements were made with a Boonton Model 4500A Power Meter and a Type 56518 Sensor. Although the generated data closely conforms to the reference normal curve, the statistics of the signal are skewed by the lack of meaningful data at very low levels. This is caused by the -40 dBm to +20-dBm dynamic range limitation of the peak power sensor. From about -40 dBm to -10 dBm, the generator’s output approximates a normally distributed variable. However, this 30-dB range does not appear in a recognizable form at the output of the amplifier. In fact, much of this range has been compressed by the amplifier. Instead of providing the same general shape of the generator signal, but shifted to the right, the amplifier limits at about +10 dBm. The data supporting Figure 4 was provided by Boonton as histogram values. The 4500A Meter performs statistical analysis onboard, although there also are modes in which the raw captured samples are available. In contrast, Agilent performs post-acquisition analysis in the EPM-P Analyzer software package that runs on a PC. The 60,000-sample data file Agilent provided comprised actual sample points that were entered into an Excel spreadsheet to generate the histogram of Figure 3. The EMP-P Analyzer program accepts all 200,000 samples captured by the meter, but Excel is limited to 64,000. The test applications for which you intend to use a power meter will greatly influence the type of features that are most appropriate. For example, Boonton’s self-contained approach is well-suited to both design and production, but the relatively large size and power requirements of the Model 4500A limit the meter’s use as a portable field tool. The Giga-tronics 8650A, Boonton’s 4530 Series, and the Agilent models are smaller and more easily transported, but the displays also are smaller and have lower resolution. Connecting to a PC can be a good solution, especially in a design environment where flexible software may be an advantage. References
Aknowledgement Thanks to Ian Messer of Agilent Technologies and Rick Theiss of Boonton Electronics for their help in preparing this article. FOR MORE INFORMATION on measuring
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