Reducing ADC Quantization Noise Two techniques, oversampling and dithering, have gained wide acceptance in improving the performance of commercial ADCs.
Richard Lyons Consultant Besser Associates, 201 San Antonio Circle, Bldg B, Suite 115, Mountain View, CA 94040; (650) 949 3300, e-mail:
[email protected]
Randy Yates Senior Staff Engineer Sony Ericsson Mobile Communications, 7001 Development Drive, Research Triangle Park, NC 27709; (919) 472-1124, email:
[email protected] Analog-digital converters (ADCs) are useful, and interesting, devices. ADCs perform amplitude quantization of an analog input signal into binary output words of finite length, normally in the range from 6-bits to 18-bits, and this amplitude quantization makes their operation inherently nonlinear. This nonlinearity manifests itself as a wideband noise in the ADC's binary output, called quantization noise, limiting the ADC's dynamic range. This article describes the two most popular methods for improving the quantization noise performance in practical ADC applications: oversampling and dithering. To understand quantization noise reduction methods, first we recall that the signal to quantization noise ratio, in dB, of an ideal N-bit ADC is SNRQ = 6.02N + 4.77 + 20log10 (LF) dB, where LF is the loading factor measure of the ADC's input analog voltage level. (A derivation of SNRQ is provided in Reference 1.) LF is defined as the analog input rms voltage over the ADC's peak input voltage. When an ADC's analog input is a sinusoid driven to the converter's full scale voltage, LF = 0.707. In that case, the last term in the SNRQ equation becomes -3 dB and, the ADC's maximum output signal to noise ratio is SNRQ-max = 6.02N + 4.77 -3 = 6.02N + 1.77 dB. Microwaves & RF magazine, June 2005.
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The SNRQ-max expression, which we often encounter in the literature, is why engineers use the rule of thumb that the SNR of an ADC is 6 dB/bit. As a practical matter, the SNRQ-max equation is unrealistically optimistic. First, that SNR expression is for an ideal ADC, and we can't build or buy ideal components. Not even at WalMart. Second. rarely in practice do we drive an ADC's input to its full scale voltage. Real-world analog signals are often impulsive in nature and driving the ADC's input into saturation causes signal clipping that drastically reduces the ADC's output SNR. So, in practice, we avoid driving an ADC's analog input to full scale. However, not to be too pessimistic, in this article we assume we're using a high quality ADC and utilizing most of its input analog voltage range. With that assumption we'll stick to the venerable notion that an ADC's SNRQ is 6 dB/bit. Next we consider the oversampling method for improving SNRQ. The process of oversampling, to reduce ADC quantization noise, is straightforward. We merely sample an analog signal at an fs sample rate higher than the minimum rate needed to satisfy the Nyquist criterion (twice the input analog signal's bandwidth), and then lowpass filter. What could be simpler? The theory behind oversampling is based on the assumption that an ADC's total quantization noise power (variance) is the converter's least significant bit (lsb) voltage value squared over 12, or total quantization noise power = 2 =
(lsb value)2 . 12
The next assumption is: the quantization noise values are truly random, thus in the frequency domain the quantization noise has a flat spectrum. (This assumption is valid if the ADC is being driven by an analog signal that covers a significant of the converter's analog input voltage range, and is not highly periodic.) Next we consider the notion of quantization noise power spectral density (PSD), a frequency-domain characterization of quantization noise measured in noise power per hertz as shown in Figure 1. Thus we can consider the idea that quantization noise can be represented as a certain amount of power (watts, if we wish) per unit bandwidth. PSD PSDnoise
Total quantization noise -fs/2
0
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fs/2
Freq
2
Figure 1 Frequency-domain power spectral density of an ideal ADC.
In our world of discrete systems, the random noise assumption results in the total quantization noise (a fixed value based on the converter's lsb voltage) being distributed equally in the frequency domain, from -fs/2 to +fs/2 as indicated in Figure 1. The amplitude of this quantization noise PSD is the total quantization noise power divided by the total bandwidth f s over which it is present, or PSDnoise =
(lsb value)2 1 (lsb value)2 = 12 fs 12fs
measured in watts/Hz. Note that the units of this derivation are appropriate: power (watts) divided by bandwidth (Hz) yields watts/Hz. The next question is: "How can we reduce the PSDnoise level?" We could reduce the lsb value (volts) in the numerator by using an ADC with additional bits. That would make the lsb value smaller and certainly reduce PSDnoise, but that's an expensive solution. Extra converter bits cost money. Better yet, let's increase the denominator of PSDnoise by increasing the fs sample rate. Consider a low-level discrete signal of interest whose spectrum is depicted in Figure 2(a). By increasing the ADC's fs,old sample rate to some higher value fs,new (oversampling) we spread the total noise power (a fixed value) over a wider frequency range as shown in Figure 2(b). The area under the shaded curves in Figure 2(a) and Figure 2(b) are equal because the total quantization noise power of a converter depends only on the number of bits and not the sample rate. Next we lowpass filter the converter's output samples. At the output of the filter, the quantization noise level contaminating our signal will be reduced from that at the input of the filter.
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PSD
Signal of interest Quant. Noise
(a) -fs,old/2
0
fs,old/2 PSD
Shaded regions have the same area.
lowpass filter response (b)
Quant. Noise 0
-fs,new/2
(c)
Freq
x(t)
Analog anti-aliasing lowpass filter
fs,new/2
Digital lowpass filter
A/D converter
Freq
x(n) Decimation
fs,new
Figure 2 Oversampling: (a) noise PSD at fs,old rate; (b) noise PSD fs,new rate; (c) processing steps.
The improvement in signal to quantization noise ratio, measured in dB, achieved by oversampling is: SNRQ-gain = 10log10(fs,new/fs,old). A derivation of the SNRQ-gain expression is provided in Reference 1. Because the number of bits of an N-bit ADC as a function of SNR is approximately SNR/6, the total number of effective bits is 10log(M)/6 + N, where M = fs,new/fs,old. This means that if the oversampling ratio M is two, the effective number of ADC bits is Nos = 0.5 + N. We gain a half a bit by oversampling by a factor of two! It can be shown that the oversampling ratio M required to achieve a specific number of K extra effective bits is given by M = 4K, so that the effective number of bits is Nos = K + N. For example: if fs,old = 100 kHz, and fs,new = 400 kHz, the SNRQ-gain = 10log10(4) = 6.02 dB. Thus oversampling by a factor of 4 (and filtering), we gain a single bit's worth of quantization noise reduction. Consequently we can achieve N+1-bit performance from an N-bit ADC, because we gain signal amplitude resolution at the expense of higher sampling speed. After digital filtering, we can decimate to the lower fs,old without Microwaves & RF magazine, June 2005.
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degrading the improved SNR. Of course, the number of bits used for the lowpass filter's coefficients and registers must exceed the original number of ADC bits, or this oversampling scheme doesn't work. With the use of a digital lowpass filter, depending on the interfering analog noise in x(t), it's possible to use a lower-performance (simpler) analog anti-aliasing filter in Figure 2(c) relative to the analog filter necessary at the lower sampling rate. Next we discuss a second common method for reducing a specific kind of quantization noise anomaly that occurs under special circumstances. Dithering, the second technique used to minimize the effects of A/D quantization noise, is the process of adding noise to our analog signal prior to A/D conversion. This scheme, which doesn't seem at all like a good idea, can indeed be useful and is easily illustrated with an example. Consider digitizing the low-level analog sinusoid shown in Figure 3(a), whose peak voltage just exceeds a single ADC least significant bit (lsb) voltage level, yielding the converter output x1(n) samples in Figure 3(b). The x1(n) output sequence is clipped and this generates spectral harmonics in its spectrum. Another way to explain the spectral harmonics is to recognize the periodicity of the quantization noise in Figure 3(c).
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Analog input to converter
1 lsb 0.5 lsb (a)
0 Time -0.5 lsb -1 lsb x1(n) 1 lsb
Converter output code
0.5 lsb (b)
0 Time -0.5 lsb -1 lsb
1 lsb
Converter quantization error
0.5 lsb (c)
0 Time -0.5 lsb -1 lsb
Figure 3 Dithering: (a) a low-level analog signal; (b) ADC output sequence; (c) quantization error.
We show the spectrum of x1(n), in dB, as in Figure 4(a) where the spurious quantization noise harmonics are apparent. It's worthwhile to note that averaging multiple spectra will not enable us to pull some spectral component of interest up above those spurious harmonics in Figure 4(a). Because the quantization noise is highly correlated with our input sinewave—the quantization noise has the same time period as the input sinewave—spectral averaging will also raise the noise harmonic levels. So, dithering to the rescue. Dithering is the technique where random analog noise is added to the analog input sinusoid before it is digitized. This technique results in a noisy analog signal that crosses additional converter lsb boundaries and yields a quantization noise that's much more random, with a reduced level of undesirable spectral harmonics as shown in Figure 4(b). Dithering raises the average spectral noise floor but increases our signal to noise ratio Microwaves & RF magazine, June 2005.
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SNR2. Dithering forces the quantization noise to lose its coherence with the original input signal, and we could then perform signal averaging if desired. |X1(m)| 0
|X2(m)|
Signal of interest
-10
0 SNR1
-10
-20
SNR2
-20
-30
-30
-40
Noise floor -40 increase
-50
-50 0
0
Freq (a)
Freq (b)
Figure 4 Spectra of discrete sinusoid: (a) no dithering; (b) with dithering.
Dithering is indeed useful when we're digitizing: low-amplitude analog signals, highly periodic analog signals (like a sinewave with an even number of cycles in the sample time interval), and slowly varying (very low frequency, including DC) analog signals. The standard implementation of dithering is shown in Figure 5(a). The typical amount of random wideband analog noise used in this process, provided by a noise diode or noise generator ICs, has a peak-to-peak value of 1/3-to-1 lsb voltage level. x(t) x(n)
x(t)
+
+
D/A
(a)
Digital pseudo-random noise generator
(b)
Figure 5 Dithering: (a) standard dithering; (b) subtractive dithering.
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+ -
ADC
Random analog noise
x(n)
+
ADC
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Wannamaker [2] has shown that dither with a triangular probability distribution function (TPDF) results in quantization noise that has a constant average of zero and a constant (non-zero) power independent of the input signal characteristics. These are both highly desirable characters for quantization noise; the first guarantees that the output of the quantizer is, on average, equal to the input; the second guarantees that there will be no “noise modulation.” Noise modulation occurs when the power of the quantization noise depends on, or is modulated by, the signal. This can be perceptually significant in audio signals and is generally undesirable. For demanding high-performance audio applications, engineers have found this type of dither to be ideal. It can be generated by adding dither noise from two separate, and independent, uniformly-distributed (also known as rectangular PDF) noise generators. The probability density function (PDF) of the sum of two independent noise sources is the convolution of their individual PDFs [3]. Because the convolution of two rectangular functions is triangular, this dual-noise-source dithering scheme generates the required TPDF. The ideal TPDF dither noise has a peak-to-peak level of exactly 2 lsb voltage levels. In the situation where our signal of interest occupies some well-defined portion of the full frequency band of 0 -to- fs/2, injecting narrowband dither noise, having a peak-topeak value equivalent to 4 -to- 6 lsb voltage levels, whose spectral energy is outside that signal band would be advantageous. (Remember though, the dither signal can’t be too narrowband, like a sinewave. Quantization noise from a sinewave signal would generate more spurious harmonics!) That narrowband dither noise can then be removed by followon digital filtering. The type of dithering discussed thus far is “non-subtractive dither” (NSD). Figure 5(b) illustrates another method of applying dither known as “subtractive dither” (SD). An SD system has all the benefits of dither (it randomizes the quantization noise) but none of the disadvantages (it does not increase the overall noise power). Wannamaker shows how subtractive dither of the proper character will result in a total quantization noise that is spectrally white and uniformly distributed.
References 1. R. Lyons, Understanding Digital Signal Processing, 2nd Edition, Prentice Hall, Upper Saddle River, New Jersey, 2004.
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2. Wannamaker, R., et al, "A Mathematical Theory of Non-Subtractive Dither," IEEE Trans. Signal Process, Vol. 48, 2000 Feb., pp. 499-516. 3. Y. Viniotis, Probability and Random Processes for Electrical Engineers, McGrawHill, 1998.
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