Scaling the Peaks of EDGE Modulation Quality

EDGE power measurements were studied in a previous CommsDesign article, titled "Measuring Power on the EDGE". This paper looks at a second noise-like parameter, the EDGE root mean square (RMS) error vector magnitude (EVM) measurement.

RMS EVM is one of a number of modulation quality parameters specified for EDGE. All of these parameters have noise-like behavior, but fortunately most are specified in terms of their average values. Averaging greatly simplifies the measurement challenge; an average value can be estimated from a small number of samples within a quantifiable confidence interval. In contrast, RMS EVM is specified in terms of a peak value. Mobile phone conformance testing requires that no burst in a 200-burst test exceeds the 9% RMS EVM specification. How can this conformance test case be translated into a volume-manufacturing environment?

To optimize the manufacturing test of this noise-like parameter with its peak specification, its statistics must be well understood. This paper derives a statistical distribution to model the behavior of the RMS EVM measurement. It then uses this model to show how the peak value specified in the standards can be estimated by averaging a small number of RMS EVM measurements.

RMS EVM Defined
The RMS EVM value of an EDGE burst is defined in the standards by the formula:

The terms in the numerator sum represent the powers of the error components evaluated at the centre of each symbol. The terms in the denominator sum represent the powers of the ideal signal at the same points. The EVM measurement must therefore take the received signal and resolve it into an ideal signal component and an error signal component.

The RMS EVM is then calculated as the square-root of the ratio of average error power to average signal power at the centre of each symbol. For convenience, ε is adopted in this paper to represent RMS EVM, and P_n and P_s to represent the average powers of the error or noise component and signal components respectively, during the burst, as shown in the following equation:

The quantities P_n and P_s will vary from burst-to-burst and so must be treated as random variables. The modulating data causes burst-to-burst variation in Ps; wideband noise, signal compression, I/Q modulator imbalances and in-band spurs cause burst-to-burst variation in P_n. The analysis below will consider the case where the noise power, P_n , results entirely from wideband or thermal noise. In this particular case, the two random variables, P_n and P_s, are uncorrelated.

Building a Statistical Model
If P_n and P_s are noted from many burst RMS EVM measurements; and, for each measurement a dot is plotted using the values of P_n and P_s as rectangular coordinates, then a scatter plot similar to Figure 1 would be obtained.

The density of dots in a particular region of the scatter plot indicates how likely the combination of P_n and P_s values bounding that region are to occur. If the P_n and P_s axes are divided into finite intervals of length ΔP_n and ΔP_s, and the proportion of dots in each square of this grid determined and plotted as a number in that square of the grid, an approximation to the joint probability density function (pdf) of P_n and P_s would be obtained.

The histogram approximation to the joint pdf of P_n and P_s becomes exact as the number of dots tends to infinity and the intervals ΔP_n and ΔP_s tend to zero length. Graphically the joint pdf is represented as a surface in three dimensions, as illustrated in Figure 2. The height of this surface above any pair of P_n and P_s coordinates represents the probability of that combination of P_n and P_s occurring. The total volume under this surface is unity.

The marginal pdfs of P_n and P_s are obtained using a similar analysis. In the case of P_n for example, the probability of a given ΔP_n interval occurring is the sum of probabilities of all elements in the grid with that ΔP_n coordinate. Again, the histogram approximation to the pdf becomes exact in the limit. The marginal pdfs of P_n and P_s are also illustrated in Figure 2 above. The area under each of these curves is unity.

Figure 3a shows a histogram for P_s obtained from 40,000 measurements. Figure 3b, on the other hand, shows the histograms for three instances of P_n, again each obtained from 40,000 measurements. The ideal EDGE signal is modulated by a pseudorandom bit sequence (PRBS) and the noise component created using AWGN. Superimposed on each histogram of real data is an ideal Gamma pdf.

The Gamma distribution models both P_n and P_s histograms well. This distribution reflects the asymmetry present in the histograms (the positive tail is longer than the negative tail). The gamma distribution is only defined for positive values, just as the powers P_n and P_s can only have positive values. Furthermore, as can be seen from Figure 3b, P_n exhibits a behaviour whereby the ratio of standard deviation to mean value remains constant for all mean values. That is to say, as the mean value of P_n increases so too does the spread in values about that mean. The Gamma distribution captures this behaviour.

The Gamma distribution pdf for P_s is given by:

where

Similarly, the Gamma distribution pdf for P_n is given by:

where

The μ and σ terms represent the means and standard deviations of the distributions.

Determining RMS EVM pdf
The RMS EVM pdf is obtained from the joint pdf of P_n and P_s. Consider the definition of RMS EVM in Equation 2. By rearranging:

In terms of the scatter plot of Figure 1, Equation 5 represents a line passing through the origin and having a slope of ε². Extending this idea to the 3D graphic of Figure 2, this equation defines a plane, passing through the origin and parallel to the third axis. This plane divides the volume under the 3D surface that defines the joint pdf of P_n and P_s.

With ε equal to zero, the plane stands above the P_s axis. As ε increases, this plane rotates counter-clockwise, slicing off an increasing proportion of the volume under the joint pdf of P_n and P_s.

For a given value of RMS EVM, ε_x, the volume sliced off by the corresponding plane represents the probability of an RMS EVM value less than or equal to ε_x occurring. Because P_n and P_s are uncorrelated, this probability can be calculated from the double integral of the product of the P_n and P_s marginal pdfs, as shown in:

F(ε_x) is the cumulative distribution function (cdf) of ε evaluated at &epsilon_x. F(ε) may be rewritten in the form:

The derivative term is obtained from Equation 5, and is computed as:

Expressing the P_n terms in terms of P_s and ε, we obtain:

By differentiating F(ε) with respect to ε, the probability density function, f(ε), is obtained:

With some simplification, Equation 10 can be represented as:

After some further manipulation, Equation 11 can be represented as:

The integrand is in the form of a Gamma distribution, which integrates to unity between zero and infinity. Hence, we obtain:

Using the definitions of the b and c coefficients in Equations 3a and 4a above, f(ε) can finally be written in the form:

Where B(c_s,c_n) is the Beta function:

And the quantity ε_o is defined as:

The RMS EVM pdf for a signal with AWGN is therefore defined in terms of three quantities: ε_o, c_s, and c_n . From Equations 3a and 4a, c_s and c_n are derived from the ratio of the mean and standard deviation values for the P_s and P_n distributions. In the case of P_s, the mean and standard deviation can be determined from the impulse responses of the EDGE transmit filter and the EDGE modulation quality measurement filter. It has the value 197.6.

The mean and standard deviation of P_n are functions of the additive white Gaussian noise (AWGN) bandwidth and the measurement filter. However, if the AWGN bandwidth is wide compared to the measurement filter bandwidth (90 kHz), its effect on the ratio of the mean to standard deviation of P_n is negligible. Under these circumstances, c_n has the value 68.4. Thus, only ε_o remains to be determined, which will be shown below.

Estimating ε_o
The average signal and noise powers, μ_s and μ_n, are generally not available from a measuring instrument, but clearly the RMS EVM, ε is available. The RMS EVM sample average is:

The random variables P_s and P_n can be redefined in terms of their average values and a new random variable, Δ, representing the difference between each P sample and the mean value:

Equation 17 can be then be rewritten as:

By rearranging, this equation can be used to estimate ε_o from the sample average of ε. The result is show in Equation 20.

The composite term multiplying the sample average is a random variable. It has been determined empirically that the distribution of this random variable becomes normal after taking four or more RMS EVM averages. This normal distribution has a mean value of 1 and a standard deviation, as shown in Equation 21.

The estimate of ε_o can then be expressed in terms of the sample average of K RMS EVM measurements and a normal random variable, X_k, with zero mean and a standard deviation given by Equation 21:

Using the statistics of normal random variables, for a sample size of four, the 3-sigma limits on the estimate of ε_o are 0.9 and 1.1 times the sample average. More certainty in the estimate of ε_o can be obtained by increasing the sample size, K.

Figure 4 illustrates the RMS EVM histograms for three different signal-to-noise ratios (SNRs). Superimposed are the pdfs calculated using Equation 14. In these graphs the average of the 40,000 RMS EVM samples is used as the estimate of ε_o.

Compression and Other Mechanisms
The analysis above assumed that the signal power, P_s, and the noise power, P_n, are uncorrelated. This will not generally be the case with real EDGE signals. Other mechanisms will also contribute to the RMS EVM value — most likely in combination. These other mechanisms include signal compression, gain and phase imbalances in the I/Q (quadrature) modulator, or in-band spurs.

Figure 5 illustrates the RMS EVM histograms of two signals with increasing compression. For each histogram, the corresponding pdf as defined by Equation 14 is calculated using the sample RMS EVM average as the estimate for ε_o. Although a signal in compression will have significant correlation between P_s and P_n, the AWGN (or uncorrelated) model provides a reasonable approximation to the histograms.

Closer examination of Figure 5 shows that the uncorrelated model consistently predicts that the extreme values on the upper-side of the distribution have a higher probability of occurrence. This behaviour is consistently observed and can be explained using the scatter plot of Figure 6 and drawing comparisons with the scatter plot in Figure 1.

In Figure 6, the correlation between P_s and P_n samples for a signal in compression is obvious. Higher values of P_s generally produce higher values of P_n; and lower values of P_n generally result from lower values of P_s. This relationship is the nature of the compression mechanism.

As already discussed, a line drawn from the origin joins points with RMS EVM values equal to the square-root of the slope of the line. The effect of compression, and subsequent correlation between the P_s and P_n samples, is to rotate the distribution counter-clockwise. This means the line that delimits the most extreme point will have a smaller slope and consequently a lower RMS EVM value than the equivalent AWGN case.

Figures 7 and 8 illustrate RMS EVM histograms for a signal with I/Q imbalances and a signal with an in-band spur respectively. Superimposed are the pdfs for the uncorrelated model, calculated using the average values from all the samples.

The I/Q impairments result from 1dB gain imbalance and 2-degrees of quadrature skew. The in-band spur is at -20 dBc. In both cases the pdf for the uncorrelated model over estimate the probability of extreme values occurring.

These observations allow the uncorrelated model to be viewed as the limiting case for estimating the probability of a peak RMS EVM value occurring, regardless of the underlying mechanism or combination of mechanisms.

Replacing the Peak Test Limit
In the discussion above, we discussed a probability density function that can be used to model the distribution of RMS EVM values. Although this pdf was developed for signals with AWGN as the sole impairment mechanism, it can be regarded as a worst-case distribution for predicting the probability of a given peak RMS EVM value occurring under a combination of different signal impairment mechanisms. The pdf is defined in terms of a single variable, ε_o. ε_o can be determined from measurement by averaging a sample of RMS EVM values while transmitting a random data pattern.

By assuming a particular failure rate for the 9-percent peak RMS EVM limit called out in the standards, a point is established on the pdf curve. From this single point the whole pdf can be determined — in particular, the value of ε_o can be determined. As ε_o can also be determined by averaging a sample of RMS EVM measurements, there is an equivalence between the limit value for an average measurement and the limit value for a peak measurement.

It follows then, that the limit value for an average RMS EVM test is derived through a two-step process. In the first step the theoretical value of ε_o is determined from the peak RMS EVM test limit and an acceptable failure rate. Equation 14 is used for this step.

In the second step, a decision is made about the number of averages used to estimate ε_o from a sample of RMS EVM values. Using Equations 21 and 22 and their associated statistics, the average RMS EVM test limit is determined by backing off the theoretical value of ε_o.

For the first step, assume an initial value of 1 for Ε_o in Equation 14. This creates a normalized pdf, f_o (ε). Using numerical techniques, determine the value of that integrates f_o (ε) to 1 minus the failure rate. Divide the peak RMS EVM limit by this value to obtain the theoretical value for ε_o.

For the second step, the number of RMS EVM measurements to be made and the desired confidence interval for the estimation of must first be decided. These two factors applied to Equations 21 and 22 determine the back-off from the theoretical value for ε_o required to set the limit for an average RMS EVM test.

As an example, assume that a failure rate of 1 burst in 10,000 at the 9% RMS EVM limit is acceptable. Measurement time constraints dictate that only four RMS EVM measurements can be made, but 99.7% confidence is required.

Step 1: For a failure rate of 1/10000, the normalised value of εo is 1.287. Dividing the 9% peak EVM limit by 1.287 produces a value of 7%, the required value of ε_o.

Step 2: Four samples produces a value of 0.032 for σ_x in Equation 21. A 99.7% confidence interval requires setting X_K equal to 3σ_x in Equation 22. Solving Equation 22 for the average value of ε, an average RMS EVM limit value of 6.36% is calculated.

About the Author
Dave Dunne is an RF systems engineer in the Wireless Division of Agilent Technologies. He received his BEng from Limerick University, Ireland, and can be reached at dave_dunne@agilent.com.