Data acquisition
SAMPLING FUNDAMENTALS Digitisation of data involves the two fundamental processes of sampling and quantization, as shown in Figure 1. Sampling is the first step wherein a continuous-time varying analogue signal x(t) is converted into a discrete-time signal x(n) using
sampling frequency fS. The result is evenly separated by a period of 1/TS (fS = 1/TS).
AC AND DC DATA ACQUISITION SIGNAL CHAINS MADE EASY
Figure 1. Data sampling.
The second step is quantization, which approximates the value of these discrete-time samples to one of the finite possible values and is represented in digital code, as shown in Figure 1. This quantization to a finite set of values leads to error in digitisation called quantization noise. The sampling process also results in aliasing, in which we see foldback from input signals and its harmonics around sample and hold clock frequency. The Nyquist criterion requires that the sampling frequency be at least twice the highest frequency contained in the signal. If the sampling frequency is less than twice the maximum analogue signal frequency, a phenomenon known as aliasing will occur. In order to understand the implications of aliasing in both the time and frequency domain, first consider the case of a time domain representation of a single tone sine wave sampled as shown in Figure 2. In this example, the sampling
frequency, fS, is not at least 2fa, but only slightly more than the analogue input frequency, fa, thus failing to meet the Nyquist criterion. Notice that the pattern of the actual samples produces an aliased sine wave at a lower frequency equal to fS – fa.
Sampling phenomena in analogue-to-digital converters (ADCs) induce the problems of aliasing and capacitive kickback, and to solve these problems, designers use filters and driving amplifiers that introduce their own sets of challenges. This makes achieving precision dc and ac performance in medium bandwidth application areas a challenge and designers end up trading off system goals to do so. This article from Wasim Shaikh, applications engineer and Srikanth Nittala, lead technologist, both with Analog Devices, describes continuous-time sigma-
delta (∑-Δ) ADCs that inherently and dramatically solve the sampling problems by simplifying signal chains. They remove the need for antialiasing filters and buffers, and solve signal chain offset errors and drift issues associated with the additional components. These benefits shrink the solution size, ease solution design, and improve the phase matching and overall latency of the system. This article also draws a comparison with discrete-time converters and highlights system benefits, as well as the constraints of using continuous-time sigma-delta ADCs.
wave of frequency fa sampled at a frequency fS by an ideal impulse sampler (see Figure 1). Also
assume that fS > 2fa. The frequency domain output of the sampler shows aliases, or images, of the
original signal around every multiple of fS; that is, at frequencies equal to | ± KfS ± fa|, K = 1, 2, 3, 4, and so on.
Now consider the case of a signal that is outside of the first Nyquist zone in Figure 3. The signal frequency is only slightly less than the sampling frequency, corresponding to the condition shown in the time domain representation in Figure 2. Notice that even though the signal is outside the first
Nyquist zone, its image (or alias), fS – fa, falls inside. Returning to Figure 3, it is clear that if an unwanted signal appears at any of the image
frequencies of fa, it will also occur at fa, thereby producing a spurious frequency component in the first Nyquist zone.
Figure 2. Aliasing: representation in the time domain.
The corresponding frequency domain representation of this scenario is shown in Figure 3. The Nyquist bandwidth is defined to be the
frequency spectrum from dc to fS/2. The frequency spectrum is divided into an infinite number of Nyquist zones, each having a width
equal to 0.5fS. In practice, the ideal sampler is replaced by an ADC followed by an FFT processor. The FFT processor only provides an
output from dc to fS/2; that is, the signals or aliases that appear in the first Nyquist zone. Consider the case of a single frequency sine
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COMBATING CHALLENGES FOR PRECISION PERFORMANCE For high performance applications, system designers need to combat quantization noise, aliasing, and switched capacitor input sampling issues resulting from the sampling process. Both types of precision ADCs available in the industry - that is, successive approximation registers (SARs) and sigma-delta ADCs - are designed using switched capacitor-based sampling techniques.
QUANTIZATION NOISE
In an ideal Nyquist ADC, the LSB size of the ADC will determine the quantization noise that gets
Figure 3. Aliasing: representation in the frequency domain.
added to the input, while doing analogue-to- digital conversion. This quantization noise is spread
over the bandwidth of fS/2. To combat quantization noise, the first technique is oversampling, which is sampling the input signal at a much higher rate than the Nyquist frequency to increase the signal- to-noise ratio (SNR) and the resolution (ENOB). In oversampling, the sampling frequency is chosen to
be N times the Nyquist frequency (2 × fIN), and as a result the same quantization noise has to now spread over N times Nyquist frequency. This also relaxes the requirements on the antialiasing filter.
Oversampling ratio (OSR) is defined as fS/2fIN, where fIN is the signal BW of interest. As a general guideline, oversampling the ADC by a factor of
four provides one additional bit of resolution, or a 6 dB increase in dynamic range. Increasing the oversampling ratio results in overall reduced noise and the dynamic range (DR) improvement due to
oversampling is ΔDR = 10log10 OSR in dB. Oversampling is inherently used and implemented together with an integrated digital
September 2023 Instrumentation Monthly
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