Originally posted Jul 2, 2013
My measurements are usually more noisy than I expect. Are yours?
I know I’m misusing the word a little but I think it’s clear enough when I say that I want measurements that are deterministic, not stochastic. That is, when I measure a signal I want a number I can rely on, with variance small enough to be ignored, and not some thorny statistical distribution that demands processing or interpretation.
Alas the universe shows no sign of caring about what I want and instead gives me just what I deserve. I have two choices about how to respond, and they’re not mutually exclusive:
- Set up my measurements to deserve better, to get answers closer to the pure values I want
- Be realistic about what sort of noise or variance will be inherent in the answers I can expect.
This post will explore the second choice, examining a little of the essentials of realism in signal measurements, using a familiar signal in several different forms. Other posts have already (and will in the future) discuss techniques to make the measurements themselves better.
The example for this discussion is an original GSM mobile phone signal, with a modulation scheme known as Gaussian filtered minimum shift keying or GMSK. While in the real world this signal would be bursted or framed as part of a time division multiple access (TDMA) technique, that is not the case here. The measurements in this example would behave the same way if appropriate time-gated measurements were made on a bursted signal.
Unlike many digitally modulated signals the GMSK signal is “constant envelope” and information is transmitted through frequency or phase changes only. The RF carrier changes phase by ±90 degrees over each symbol interval and the signal amplitude or RF envelope is essentially constant. That makes it compatible with efficient power amplifiers.
I generated a GSM signal with three different signal/noise ratios (SNRs) and measured the complementary cumulative distribution function (CCDF) to explore the peak/average power ratio, or how noisy amplitude measurements would be. CCDF is a two-dimensional, log-log scaled plot of peak/average power ratio vs. frequency of occurrence. Specifically the CCDF shows, for a given peak/average ratio, how often that peak occurs. Here are the three measurements, in two traces:
CCDF measurement of a constant-envelope GSM (GMSK) signal at 3 different SNRs. Note how reducing the SNR drives the CCDF curve dramatically to the right, indicating a much more noisy measurement of signal amplitude.
CCDF plots are generated from thousands or millions of power measurements and the Y-axis is a log-scaled measure of how frequently certain peak/average power values are observed. The scale covers from 0.001% to 100%. The X-axis indicates peak/average power ratio in 1 dB/div. so the CCDF curve is normalized to average signal power.
Describing CCDF can be clumsy (and I just proved that in the paragraphs above!) so let’s take an example from the lower measurement: Following the dotted red line to the blue “SNR 10 dB” curve you can see that power peaks of 5 dB or more occur in about 1% of the measurements.
That’s the sort of result that gets my attention. I’m measuring a constant-amplitude signal with an SNR that places it well above the noise floor and yet in one measurement out of 100 the reading will be 5 dB high due to noise. Ten percent of the measurements will read 3 dB or more above average.
Increasing the SNR to 20 dB reduces the measured peaks, though they are still substantial, at around 2.5 dB or greater in 1% of the measurements. Of these three measurements it’s only the one with the very high SNR (better than 60 dB) that creates a CCDF curve where peaks never exceed average values by more than a small fraction of 1 dB.
What I learned from this experiment: The universe is more noisy than I expected, and even measurement configurations that look favorable can yield readings with a surprisingly wide variance. I learned that if I really want the truth I should make friends with the most efficient averaging techniques (more in a future post) or take advantage of measurement applications where the averaging techniques and statistical analysis are built in.
Lastly, I should mention a couple of things that will be useful if you try this experiment yourself. First, the results vary with frequency span, so your comparisons should all use the same span. Second, any pulsing of signals will dramatically affect CCDF so it’s best to measure continuous signals or to make time-gated measurements. By the way, the AWGN reference line provided in many CCDF measurements refers to the curve for additive white Gaussian noise, a signal with wider amplitude variations than most digitally modulated signals.