by Leonard Staller, Applications Engineer,
Cygnal Integrated Products, Inc., Austin, Texas
Oversampling and averaging are techniques you can use to boost the resolution and the signal-to-noise (SNR)
ratio of A/D conversions. You can also increase the resolution of a measurement when using one of today's highly
integrated data-acquisition chips—without having to resort to off-chip A/D converters. In this second part
of our ongoing ChipCenter tutorial series, we'll take a closer look at the theory of noise and oversampling. We'll
focus on in-band noise, and show how to calculate oversampling requirements to obtain a desired SNR, or to get
better resolution.
Oversampling and averaging are done to accomplish two things: improve the SNR, and increase the
effective resolution (i.e., increase the effective number of bits) of an A/D converter measurement. Both of these
are really the same entities.
Let's assume you have a 12-bit A/D converter and want to generate codes with 16 bits of resolution. We can use
oversampling and averaging to get the SNR of a 16-bit A/D converter. This will increase the effective number of
bits (ENOB) of the measured data, which is another measure of the SNR. Producing a lower noise-floor in the signal
band, oversampling, and averaging filters let us realize 16-bit output words.
Oversampling and In-Band Noise
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A sampling frequency fs will permit signals of interest
to be reconstructed at one-half the sampling frequency (according to the Nyquist Theorem). If the sampling rate is
100 kHz, for example, then signals below 50 kHz can be reconstructed and analyzed reliably.
Along with the input signal, there will be a noise signal (present in all frequencies as white noise) that
will fold or alias into the measured frequency band of interest (frequencies less than one-half
fs).
The energy spectral density of in-band noise is
 |
(5) |
where erms
is the average noise power, fs is the sampling frequency, and
E(f) is the in-band ESD.
Equation (5) shows that the energy spectral density (ESD), or noise floor of the sampled noise,
will decrease in the signal band as the sampling frequency is increased.
Increased Resolution
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Given the fixed noise-power due to quantization noise, you can calculate the amount of oversampling
required to increase the effective resolution. For example, if you want to increase the ENOB of a
parameter measured with a 12-bit A/D converter to a 16-bit measurement, then you'll want to establish
a relationship that permits you to calculate the oversampling requirement. To do so, first define the
characteristics of the noise.
To understand the effects of oversampling and averaging on noise, you must first define what the
quantization noise will be. The distance between adjacent A/D converter codes determines the quantization
error. The A/D converter will round to the nearest quantization level, or A/D converter code:
 |
(6) |
where N is
the number of bits in the A/D converter code and Vref
is the reference voltage.
The quantization error (eq) is
 |
. |
Assuming the noise approximates white noise,
the random variable representing the noise is equally distributed with a zero mean between A/D converter
codes. Thus, the variance is the average noise power, calculated
 |
(7) |
A measure of the sampling frequency compared to the
Nyquist frequency (see Equation (1) in Part 1
of this feature article) is the oversampling ratio (OSR).
This is defined as
 |
(8) |
where fs
is the sampling frequency and fm is the highest frequency
component of the input signal.
If the noise is white, then the in-band noise power at the output of the low-pass filter is
 |
(9) |
where n0
is the noise power output from the filter.
Equation (9) shows that you can lower the in-band noise power by increasing
OSR. Oversampling and averaging doesn't affect the signal power.
You increase the SNR because oversampling lowers noise the power and doesn't affect the signal power.
From Equations (6), (7), and (9), you can derive the following expression relating the noise power
to the oversampling ratio and resolution.
 |
(10) |
where OSR is
the oversampling ration, N is the number of A/D converter bits, and
Vref is the reference voltage.
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