Effective bits testing for digitising performance
23 April 2009
When you are buying an oscilloscope or digitising system you need a way to determine real-life digitising performance. How closely does the output of the waveform digitiser match the analogue input signal?
The effective bits concept
At the most basic level, digitising performance is a simple matter of resolution. For the desired amplitude resolution, pick a digitiser with the requisite number of “bits”. For the desired time resolution, run the digitiser at the required sampling rate. Those are simple enough answers. Unfortunately, they don’t tell the whole story and can be quite misleading, too.
While an “8-bit digitiser” might provide close to eight bits of accuracy and resolution at DC, that will not be the case at higher frequencies. Dynamic digitising performance can drop markedly as signal speeds increase. An 8-bit digitiser can drop to 6-bit, 4-bit, or even fewer effective bits of performance well before reaching its specified bandwidth.
Effective bits may be stated as part of an instrument specification. However, where effective bits are not specified it may be necessary to do an effective bits evaluation for comparison purposes.
Essentially, effective bits is a means of specifying the ability of a digitising device or instrument to represent signals of various frequencies. The basic concept is illustrated in Figure 1, which shows a plot of effective bits of two digitisers versus frequency.
The plot in Figure 1 shows that as the signal being digitised increases in frequency, digitising performance drops to lower values of effective bits.
This decline in digitiser performance is manifested as an increasing level of noise on the digitised signal. This noise on a digitised signal can be expressed in terms of a Signal-to-Noise Ratio (SNR).
SNR = rms_signal/rms_error [Eq.1]
where rms_signal is the root-mean-square value of the digitised signal and rms_error is the root-mean-square value of the noise error. The relationship to effective bits (EB) is given by,
EB = log_2(SNR) – ½log_2(1.5) – log_2(A/FS) [Eq.2]
where A is the peak-to-peak input amplitude of the digitised signal and FS is the peak-to-peak full-scale range of the digitiser’s input. Other commonly used formulations include,
EB = N – log_2 (rms_error/ideal_quantisatization_error) [Eq.3]
where N is the nominal (static) resolution of the digitiser, and,
EB = -log_2(rms_error * sqrt(12/FS)) [Eq.4]
Notice that all these formulations are based on a noise generated by the digitising process. In the case of Equation 3, the “ideal quantisation error” term is the rms error in ideal, N-bit digitising of the input signal. Both Equations 2 and 3 are defined by the IEEE Standard for Digitising Waveform Recorders. Equation 4 is an alternate form for Equation 3.
An important thing to notice about these equations is that they are based on full-scale signals (FS). In actual testing, test signals at less than full scale (e.g., 50% or 90%) may be used. Any comparisons of effective bits specifications or testing must take account of the test signal amplitude as well as frequency.
Error sources in the digitising process
Noise, or error, related to digitising can come from a variety of sources. Even in an ideal digitiser, there is a minimum noise or error level resulting from quantising. This “quantising error” amounts to ± ½ LSB (Least Significant Bit). As illustrated in Figure 2, this error is an inherent part of digitising. It is the resolution limit, or uncertainty, associated with ideal digitising. Additionally, a real-life digitiser adds further errors including those shown in Figure 3.
Many of the errors encountered in digitisers are the classical error types associated with any amplifier or analogue network.
On the other hand, aperture uncertainty and time base inaccuracies are phenomena associated with the sampling process that accompanies waveform digitising. The basic concept of aperture uncertainty is illustrated in Figure 4.
Aperture uncertainty results in an amplitude error whose magnitude is slope dependant. The steeper the slope of the signal, the greater the error magnitude resulting from a time jittered sample.
The effective bits measurement process
Rather than trying to distinguish and measure each individual error source within a digitising system, it is easier to measure overall performance. A good place to start is determining the digitising system’s SNR and the resulting effective bits as defined by Equations 2, 3 or 4. This provides an easily understood and universal figure of merit for comparisons.
The basic test process is illustrated in Figure 5. A high-quality signal is applied to the digitiser and the digitised waveform is then computer analysed. A sine wave is used as the test signal because high-quality sine waves are relatively easy to generate and characterise. The sine wave generator’s performance must significantly exceed that of the digitiser under test, otherwise the test will not be able to distinguish digitising errors from signal source errors.
To obtain an effective bits number, a perfect (idealised) sine wave is computed and fitted to the digitised sine wave. This perfect sine wave is described by,
A*sin(2*pi*f*t + θ) + C [Eq. 5]
where A is the sine wave’s amplitude, f is its frequency, θ is phase, t is time and C is DC offset. This result is considered to be a description of the analogue input to the digitiser. Because the analogue signal parameters are computed from the digitiser’s output, DC offset, gain, phase and frequency errors are not included. These excluded errors need to be measured by separate tests, such as a histogram test or other test appropriate to the specific error of interest.
After computing a model of the ideal input sine wave, further computations are done to determine ideal sampling and digitising of the sine wave. The difference between the computed ideal sine wave and the perfectly sampled and digitised version is then computed. The rms value of this provides the ideal quantisation error used in Equation 3. The rms error value used in the effective bits Equations (3 and 4) is obtained by subtracting the ideal sine wave from the actual digitised sine wave and finding the rms value of the result.
The final computation (using Equations 2, 3 or 4) results in an effective bits number for the digitiser. By keeping input signal amplitude constant for various frequencies, further effective bits numbers can be computed. These numbers can then be plotted against frequency to obtain a digitiser performance curve such as illustrated in Figure 1.
Other dynamic performance tests
Beyond effective bits testing, there are still other test methods that can be used to evaluate the dynamic performance of digitisers. These methods include FFT tests, spectral average tests and histogram tests. Generally, these tests are used to augment the results obtained by effective bits testing or to obtain specific information about some particular aspect of the digitiser’s performance.
Trevor Smith is the High Speed Serial Solutions Manager at Tektronix EMEA
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