To reduce or eliminate the ill effects of quantization noise in analog-to-digital (A/D) converters,. DSP practitioners can use two tricks to reduce converter quantization noise. Thoseschemesare called oversampling and dithering .
Oversampling. The process ofoversampling to reduce A/D converter quantization noise isstraightforward. We merely sample an analog signal at an fs sample ratehigher than the minimum rate needed to satisfy the Nyquist criterion (twice theanalogsignal's bandwidth), and then lowpass filter. What could be simpler?
The theory behind oversampling is based on the assumption that anA/D converter's total quantization noise power (variance) is theconverter's least significant bit (lsb) value squared over 12, or
The next assumption is: the quantization noise values are trulyrandom, and in the frequency domain the quantization noise has a flatspectrum. (These assumptions arevalid if the A/D converter is being driven by an analog signal thatcovers most of the converter's analog input voltage range, and is nothighly periodic. )
Next we consider the notion of quantization noise power spectraldensity (PSD), a frequency-domain characterization of quantizationnoise measured in noise power per hertz as shown in Figure 13″17 below.Thus we can consider the idea that quantization noise can berepresented as a certain amount of power (watts, if we wish) per unitbandwidth.
In our world of discrete systems, the flat noise spectrum assumptionresults in the total quantization noise (a fixed value based on theconverter's lsb voltage) being distributed equally in the frequencydomain, from “f s /2to + f s /2as indicated in Figure 13″17 below .The amplitude of this quantization noise PSD is the rectangle area(total quantization noise power) divided by the rectangle width (f s ), or
measured in watts/Hz.
|Figure13″17. Frequency-domain power spectral density of an ideal A/Dconverter.|
The next question is: “How can we reduce the PSDnoise level definedby Eq. (13″65)?”
We could reduce the lsb value (volts) in the numerator by using anA/D converter with additional bits. That would make the lsb valuesmaller and certainly reduce PSDnoise, but that's an expensivesolution. Extra converter bits cost money. Better yet, let's increasethe denominator of Eq. (13″65) by increasing the sample rate fs.
Consider a low-level discrete signal of interest whose spectrum isdepicted in Figure 13″18(a) below. By increasing the fs,old sample rate to some larger value fs,new(oversampling), we spread the total noise power (a fixed value)over a wider frequency range as shown in Figure 13″18(b).
The area under the shaded curves in Figure 13″18(a) and 13″18(b) areequal. Next we lowpass filter the converter's output samples. At theoutput of the filter, the quantization noise level contaminating oursignal will be reduced from that at the input of the filter.
The improvement in signal to quantization noise ratio, measured indB, achieved by oversampling is:
|Figure13″18. Oversampling example: (a) noise PSD at an fs,old samples rate;(b) noise PSD at the higher fs,new samples rate; (c) processing steps.|
For example: if f s ,old = 100 kHz, and f s,new = 400 kHz, the SNRA/D-gain = 10log10(4) = 6.02 dB. Thusoversampling by a factor of 4 (and filtering), we gain a single bit'sworth of quantization noise reduction.
Consequently we can achieve N+1-bit performance from an N-bit A/Dconverter, because we gain signal amplitude resolution at the expenseof higher sampling speed.
After digital filtering, we can decimate to the lower fs,old withoutdegrading the improved SNR. Of course, the number of bits used for thelowpass filter's coefficients and registers must exceed the originalnumber of A/D converter bits, or this oversampling scheme doesn't work.
With the use of a digital lowpass filter, depending on theinterfering analog noise in x(t), it's possible to use a lowerperformance (simpler) analog anti-aliasing filter relative to theanalog filter necessary at the lower sampling rate.
Dithering. Another techniqueused to minimize the effects of A/D quantization noise, dithering isthe process of adding noise to our analog signal prior to A/Dconversion.
This scheme, which doesn't seem at all like a good idea, can indeedbe useful and is easily illustrated with an example. Considerdigitizing the low-level analog sinusoid shown in Figure 13″19(a) below, whose peakvoltage just exceeds a single A/D converter least significant bit (lsb)voltage level, yielding the converter output x1(n) samples in Figure13″19(b).
|Figure13″19 Dithering: (a) a low-level analog signal; (b) the A/D converteroutput sequence; (c) the quantization error in the converter's output.|
The x1(n) output sequence is clipped. This generates all sorts ofspectral harmonics. Another way to explain the spectral harmonics is torecognize the periodicity of the quantization noise in Figure 13″19(c).
We show the spectrum of x1(n) in Figure 13″20(a) below where the spuriousquantization noise harmonics are apparent. It's worthwhile to note thataveraging multiple spectra will not enable us to pull some spectralcomponent of interest up above those spurious harmonics in Figure13″20(a).
|Figure13″20 Spectra of a low-level discrete sinusoid: (a) with no dithering;(b) with dithering.|
Because the quantization noise is highly correlated with our inputsinewave – the quantization noise has the same time period as the inputsinewave – spectral averaging will also raise the noise harmoniclevels. Dithering to the rescue.
Dithering is the technique where random analog noise is added to theanalog input sinusoid before it is digitized. This technique results ina noisy analog signal that crosses additional converter lsb boundariesand yields a quantization noise that's much more random, with a reducedlevel of undesirable spectral harmonics as shown in Figure 13″20(b).
Dithering raises the average spectral noise floor but increases oursignal to noise ratio SNR2. Dithering forces the quantization noise tolose its coherence with the original input signal, and we could thenperform signal averaging if desired.
Dithering is indeed useful when we're digitizing
1) low-amplitude analogsignals,
2) highly periodic analogsignals (like a sinewave with an even number of cycles in the sampletime interval), and
3) slowly varying (very lowfrequency, including DC) analog signals.
The standard implementation of dithering is shown in Figure 13″21(a) below.
|Figure13″21 Dithering implementations: (a) standard dithering process; (b)advanced dithering with noise subtraction.|
The typical amount of random wideband analog noise used in the thisprocess, provided by a noise diode or noise generator ICs, has a rmslevel equivalent to 1/3 to 1 lsb voltage level.
For high-performance audio applications, engineers have found thatadding dither noise from two separate noise generators improvesbackground audio low-level noise suppression.
The probability density function (PDF) of the sum of two noisesources (having rectangular PDFs) is the convolution of theirindividual PDFs.
Because the convolution of two rectangular functions is triangular,this dual-noise-source dithering scheme is called triangular dither.Typical triangular dither noise has rms levels equivalent to, roughly,2 lsb voltage levels.
In the situation where our signal of interest occupies some welldefined portion of the full frequency band, injecting narrowband dithernoise having an rms level equivalent to 4 to 6 lsb voltage levels,whose spectral energy is outside that signal band, would beadvantageous.
(Remember though: the dithersignal can't be too narrowband, like a sinewave. Quantization noisefrom a sinewave signal would generate more spurious harmonics! )
That narrowband dither noise can then be removed by follow-ondigital filtering. One last note about dithering: to improve ourability to detect low-level signals, we could add the analog dithernoise and then subtract that noise from the digitized data, as shown inFigure 13″21(b).
This way, we randomized the quantization noise, but reduced theamount of total noise power injected in the analog signal. This schemeis used in commercial analog test equipment.
Usedwith the permission of the publisher, Prentice Hall, this on-goingseries of articles is based on copyrighted material from “UnderstandingDigital Signal Processing, Second Edition” by Richard G. Lyons. Thebook can be purchased on line.
Richard Lyons is a consultingsystems engineer and lecturer with Besser Associates. As alecturer with Besser and an instructor for the University of CaliforniaSanta Cruz Extension, Lyons has delivered digitasl signal processingseminars and training course at technical conferences as well atcompanies such as Motorola, Freescale, Lockheed Martin, TexasInstruments, Conexant, Northrop Grumman, Lucent, Nokia, Qualcomm,Honeywell, National Semiconductor, General Dynamics and Infinion.