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Engineers work very hard to reduce the effects of noise. We use directional antennas, error-correcting codes, matched filters or whatever it takes to enhance the desired signal and attenuate noise. In some situations, however, adding a controlled amount of noise to a signal actually improves performance.

Consider, for example, an ideal quantizer. Its output equals the input plus quantization error, which is usually modeled as spectrally flat (or "white") random noise. This model isn't bad if the input signal is broadband and random, but what happens when it's periodic? In this case, the error signal is also periodic, so you can represent it as a Fourier series. Because a Fourier series is nothing more than a weighted sum of sinusoids, the error signal shows up on a spectrum analyzer as a series of peaks (sometimes called spurious components or "spurs") instead of the ideal flat noise floor.

If you sample a sinewave with an A/D converter and take the FFT of the result, you'll see exactly what I just described. Because the error signal is sampled, some of the error harmonics get aliased back into the range between zero and half the sample rate. The number and amplitudes of these harmonics depend on many factors including the frequency of the sinewave relative to that of the sample clock. When these two frequencies are related or nearly related by a simple fraction (such as 1/4 or 3/8), then the quantization error tends to end up concentrated in fewer harmonics.

To see a computer simulation of this effect, examine Fig 1a. It shows the magnitude of the FFT of a sampled sinewave processed by an ideal quantizer in a computer simulation. As you can see, the quantization noise is present at a handful of discrete frequencies and not as spectrally flat noise. In this case the Spur-Free Dynamic Range (SFDR) is only about 40 dB. This system would obviously make a very poor spectrum analyzer; you wouldn't know which components were real and which were artifacts of the quantizer. This problem isn't limited to A/D converters. Numerically controlled oscillators (NCOs) that synthesize a sinewave using a counter, lookup table and D/A converter also produce spurious output components.

To alleviate the problem, you can make the troublesome periodic quantization-error signal nonperiodic by adding a small amount of random noise before the quantization takes place. For an ideal quantizer, the required RMS noise level is roughly equal to the quantization step size. Real quantizers, particularly subranging A/Ds, exhibit deviations from the ideal so that higher levels of noise are required to suppress the spurious components. You can prevent this noise from degrading a signal by using a filter to restrict the noise to an unused frequency band as shown in Fig 2. Depending on the application, this frequency might be near zero or half the sample rate.

Fig 1b shows the effect of adding noise in the band between 0.4 and 0.5 before quantization in the computer simulation that produced Fig 1a. As you can see, the spurious components have vanished, and a spectrally flat-noise floor has taken their place. The total amount of quantizer error hasn't changed, but it has been evenly redistributed in frequency.

Acknowledgment

Thanks to Brad Brannon at Analog Devices (Greensboro, NC) for a preview of Application Note AN-410, "Overcoming Converter Non-Linearities with Dither," which describes the benefits of added noise and presents experimental measurements for the AD9042, a 12-bit 41-MHz A/D converter.

Mark Sullivan (dalek@radix.net) is Chief Scientist at SkyBitz Inc (Herndon, VA), a developer of tracking and communications services based on the GLS (Global Locating Systems) technology it invented. Mark received a PhD in Information Technology from George Mason Univ.

This article originally appeared in Personal Engineering & Instrumentation News, February 1996, pgs 61-62. Reprinted with permission of PEC Inc; all rights reserved.

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