At the SETI Science and Technology Working Group meetings there has been some discussion about the number of bits required in an A/D sampler to achieve adequate dynamic range in the presence of strong interference. We know that, in the absence of noise, an A/D sampler introduces quantization errors which produce harmonics and intermodulation products of sinusoidal signals. The debate has been about whether the addition of white noise to the sampler input will suppress these spurious products. This note presents a few simulation results that, indeed, show that white noise does have a linearizing effect on a sampler.
The simulation was performed the tools available in the AIPS++ data analysis package under development at the NRAO. The test functions were written in the glish scripting language using tools in the AIPS++ mathematics module.
The simulated signal to be sampled is a floating point data array which is the sum of one or more pure sinusoids of specified amplitudes and a chosen amount of band-limited white noise. The noise component is generated by a random number generator with a normal (Gaussian) amplitude distribution. This noise data array is convolved with the square root of the Fourier transform of a lowpass filter function to produce the noise spectrum shown in Figure 1.
To investigate the generation of harmonics by quantization errors the test signal is a 20 Hz sine wave intentionally centered on one of the FFT spectrum channels to eliminate any windowing sidelobes that would obscure the sine wave's harmonics. The test results are essentially the same with the test signal offset from a channel center. The harmonic content of the digitized test signal depends on the number of A/D sample intervals within the sine wave's peak-to-peak amplitude. For illustration we'll use a 4-bit sampler with the sine wave spanning 4 intervals (peak amplitude of 2.0 with sample interval boundaries at +/-0.5, +/-1.5, +/-2.5, ... +/-6.5). Since the test signal is symmetrically sampled, only odd harmonics will be produced. Figure 2 shows the spectrum of the digitized sine wave in the absence of noise.
Notice that the strongest harmonics are the sixth and tenth instead of the third and fifth. This is because A/D level transitions occur eight times within one sine wave cycle. If the test signal span only one or two transitions, the lower harmonics will be much stronger as shown in Figure 5.
To maximize the signal to noise ratio in one spectral channel when noise is added to the test signal, a very long FFT was computed on 512K data points. Only the first 512 spectral points will be shown since they contain the strongest harmonics. Figure 3 shows the harmonic intensities when band-limited noise is added with a an rms amplitude of three tenths of one sample interval. This noise level is considerably lower than one would normally operate any sampler, but we can already see ten dB or more harmonic suppression.
Figure 4 shows the test signal spectrum with an rms noise amplitude of six tenths of a sample interval, still below where one would operate a 4-bit sampler. All harmonics are now below the noise peaks at -55 dBc. The fundamental signal level changes less than 0.2 dB when noise is added to the test signal.
Maybe a slightly more challenging example is to digitize the sine wave as coarsely as possible. Figure 5 shows the spectrum of the noiseless test signal with a peak amplitude equal to one sample interval (sine wave peak equals 1.0 with sample interval boundaries at +/-0.5, +/-1.5, +/-2.5, ... +/-6.5). The harmonic intensities decrease monotonically with harmonic number, but harmonics 3, 9, 12, etc. are missing. Figure 6 shows the same signal with noise whose rms amplitude is six tenths of a sample interval as in Figure 4. Again, all harmonics are below the noise peaks. The noise level appears higher in Figure 6 than in Figure 4 because the vertical scale is normalized to a lower fundamental intensity.
An interesting aside is that, if we decrease the sine wave amplitude another 6 dB, it will disappear between two sample levels until we add noise.
My intuitive picture for where the harmonic power goes in the presence of noise is that the harmonics are totally phase or amplitude modulated by broadband noise. Hence, their spectra become the same as the noise and disappear into the noise floor. The fundamental is not modulated significantly because the noise is only additive to this term in the harmonic series, so the fundamental intensity is essentially unaffected.
An intermodulation products test was simulated in a manner very similar to the harmonics tests above. In this case a second sine wave at 24 Hz of the same amplitude as the first was added to the test signal. This signal combination is called a two-tone intermod test in receiver engineering jargon. The sum of the two sine waves now spans five quantization intervals, but the quantization error vector relative to each sinusoid is still the same as for the tests displayed in Figures 5 and 6. The noiseless intermodulation products spectrum is shown in Figure 7. The number of spurious signals is far greater than is seen with a single sine wave, although the peak spurious signal intensity is about 7 dB lower that the strongest harmonic in Figure 5.
Adding band-limited noise at an rms amplitude of 0.3 of a quantization interval to the two-tone test signals reduces the intermodulation products by about 15 dB as shown in Figure 8. This is roughly the same amount of harmonic signal reduction that we saw with the same noise level in Figures 2 and 3.
Figure 9 shows at least 27 dB of intermodulation product suppression with a noise rms amplitude of 0.6 quantization intervals. This is comparable to the harmonic reduction seen in Figure 6 with the same noise level.
Before making the simulations described above I measured the harmonic content of the A/D convertor output of the Green Bank spectral processor. This is a dual FFT spectrometer with 1024 channels in each half. The sampler for this spectrometer is a six-bit A/D whose input white noise rms amplitude normally spans about 2.5 quantization intervals.
The test setup is shown in Figure 10. The frequency synthesizer was set to 245.9765 MHz, which after conversions in the spectrometer's IF section produced a baseband CW signal at 0.9765 MHz in the 10 MHz passband. This signal was centered exactly on channel 100 of the output spectrum. The amplitude was set to span about three A/D quantization intervals. The combined noise and CW signals were sent to two independent IF's, A/D convertors and FFT engines in the spectrometer.
With the noise attenuator set to its maximum value the resulting spectrum from spectral processor channel A is shown in Figure 11. Even with no noise injected externally there is still a small white noise component to the spectrum from the spectrometer's IF amplifiers. There are also a few internally generated low level signals from the digital electronics that are exposed when the input noise is removed. The strongest one is at the center of the spectrum at 5 MHz.
Unlike the simulated spectra, we see the even harmonics of the CW test signal, the strongest of which is the second harmonic. This is presumably due to a DC offset or some other non-linearity in the A/D convertor which causes an asymmetry in the digitized signal. Both A/D's had a strong second harmonic content. All of the harmonics are definitely produced by A/D quantization because the strength of the harmonics relative to the fundamental decreases with increasing fundamental amplitude.
To test the effect of white noise on the intensity of the harmonics produced by A/D quantization, the external noise signal was increased in steps of about 6 dB and the harmonic strengths remeasured. The results for the second and third harmonics are shown in Figures 12 and 13. In these two plots a noise level to the A/D sampler of about -8 dB corresponds to an rms noise amplitude of one quantization interval.
The trend of decreasing harmonic content with increasing added noise amplitude is clear. The third harmonic intensity drops to -50 dB below the fundamental with a noise rms amplitude of a little less than one quantization interval. This agrees reasonably well with the simulations.
The second harmonic is a bit more subborn. It doesn't drop to -50 dBc until the noise is increased to an rms amplitude of about 2 quantization intervals. Note that the measurement errors on points near -11 and -1 dB noise level in Figure 12 are smaller than you would expect from the noise seen in Figure 11. These two measurements were made with 2000 seconds integration. All of the others are made with 20 seconds integration.
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