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Fig. 1. Conventional Sampler Fig. 2. Binary Sampler

Introduction

Noise is a fundamental problem in metrology, and causes unwanted variation from one measurement to the next, as shown in Fig. 1. Averaging is often used to smooth the data and improve the signal to noise ratio. It provides good results at the beginning, but each 3 dB improvement requires double the number of samples, which reduces the throughput. A limit is reached when averaging takes too long, or the signal drifts during the measurement.

A new method called Binary Sampling reduces the effect of noise on a repetitive signal, as shown in Fig. 2. It is most effective against broadband Gaussian and impulse noise. The process is simple, fast, and accurate. It gives much higher throughput than conventional systems. Unlike conventional sampling or digitizing techniques, the performance improves as the frequency increases.

How It Works

When we average a signal to improve the SNR, the signal increases linearly but the noise increases as the square root of the number of samples. This means the ratio of signal to noise improves as the square root of the number of samples. The exponential increase in the number of samples required places a limit on the amount of improvement that can be obtained by averaging.

Another way to look at it is to calculate the standard deviation. The process is:

1. Find the average value (mean)
2. Subtract the mean from each sample
3. Square the deviations
4. Add the squares
5. Divide by total number of samples less one
6. Square root of the result is the Standard Deviation

Say we have 6 samples: 1, 3, 4, 6, 9, 19. We find the mean, or average value:

mean: (1 + 3 + 4 + 6 + 9 + 19) / 6 = 42 / 6 = 7

list of deviations: -6, -4, -3, -1, 2, 12

We square each value:

squares of deviations: 36, 16,  9,  1, 4, 144

Add them up:

sum of squares: 36 + 16 + 9 + 1 + 4 + 144 = 210

Divide by one less that the number of items in the list:

variance: 210 / 5 = 42

Take the square root:

sqrt(42) = 6.48

Thus the standard deviation, or rms value of the noise, is 6.48.

Note how the last sample value of 19 dominates the result. The deviation is 12, which is twice as large as the next largest value of 6. And yet this sample is only about 2 standard deviations away from the mean, and can occur about 5% of the time. This shows that a great deal of averaging is needed to get a stable value, but this takes time and increases the cost of data acquisition. Of couse, if the averaging is continued, the system will eventually drift and render the measurement invalid.

Standard Deviation Using Binary Sampling

Lets repeat the calculation, but ignore the amplitude of the sample and only record the direction away from the mean. Instead of the 8 to 12 bit conversion used in conventional sampling, this quantizes the noise to a single bit.

Say we have the sequence 0, 1, 1 0, 0, 1. The average is

mean: (0 + 1 + 1 + 0 + 0 + 1) / 6 = 3 / 6 = 0.5
list of deviations: -0.5, +0.5, +0.5, -0.5, -0.5, +0.5,
squares of deviations: 0.25, 0.25, 0.25, 0.25, 0.25, 0.25
sum of squares: 0.25 +0.25 +0.25 +0.25 +0.25 +0.25 = 1.5
variance: 1.5 / (6 - 1) = 0.3
StDev: sqrt(0.3) = 0.547

Note the deviation is always 0.5 and the square is always 0.25. This shows that no individual sample can dominate the result like the previous example. This illustrates how Binary Sampling eliminates the random fluctions due to noise and quickly converges on the mean, which is the value we are looking for.

Other Advantages of Binary Sampling

As well as eliminating noise from the measurement, the Binary Sampler can sample at very high frequencies - hundreds or thousands of times faster than conventional samplers. This improves the signal-to-noise ratio and allows a heterodyne technique that eliminates the timing jitter and nonlinearities of conventional timebases.

Fig 2 above shows the ability of Binary Sampling to capture a 1 MHz square wave with less than 1 ps rms jitter in 1 second. Conventional sampling technology would require more than 169 seconds to achieve the same performance. However, the system is likely to drift during this time, making the measurement impossible with current technology.