Question 5 of 7: Analog-to-Digital Conversion and Line-Frequency Rejection

Question 5: Analog-to-Digital Conversion and Line-Frequency Rejection (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

a) Both schemes effectively average (integrate) the input continuously over a window of exactly $T=1/60\,\text{s}$, the period of North American 60 Hz power-line interference. The integral (or discrete average) of a pure sinusoid at $60\,\text{Hz}$ — or any of its integer harmonics ($120,180,\ldots\,\text{Hz}$) — taken over exactly one full period (or an integer number of periods) is identically zero, because the positive and negative half-cycles contribute equal and opposite area/sum. Since the window length is locked to exactly $1/60\,\text{s}$, any $60\,\text{Hz}$ (or harmonic) interference riding on the signal integrates/averages to zero and is rejected completely, regardless of its phase or amplitude.

b) Given. 256 samples spaced equally over exactly $T=1/60\,\text{s}$.

Find. The anti-aliasing filter cutoff frequency required before this sampling.

1. Effective sample rate. $$f_s=\dfrac{256}{1/60\,\text{s}}=256\times60\,\text{Hz}=15{,}360\,\text{Hz}.$$
2. Nyquist cutoff. To avoid aliasing, any input content above half the sample rate must be removed before sampling: $$\boxed{f_c=\dfrac{f_s}{2}=7{,}680\,\text{Hz}}.$$

c) The ICL7109's dual-slope converter integrates the input continuously in the analog domain over the whole $1/60\,\text{s}$ window — there is no discrete sampling instant at all, only a continuous running integral, so there is no sample-rate-related aliasing mechanism to protect against in the first place. Aliasing is specifically an artifact of representing a continuous signal by discrete, periodically-spaced samples; a true continuous integrator has effectively infinite "sampling density" and simply has no gap between samples for a high-frequency component to be mis-represented into.

d) Both schemes are, mathematically, a rectangular time-averaging window applied to the input. The frequency response of a rectangular window of length $T$ is a sinc function, $|H(f)|\propto\left|\dfrac{\sin(\pi fT)}{\pi fT}\right|$, which has zeros not only at $f=60\,\text{Hz}$ but at every integer multiple of $1/T=60\,\text{Hz}$, and its overall envelope rolls off (falls toward zero) with increasing frequency between those nulls. So the same integration/averaging window that rejects the 60 Hz fundamental also attenuates higher-frequency noise generally, not just the power-line frequency and its exact harmonics.

e) The conversion (sampling) rate required is set by the highest frequency component present in the physical signal being measured — per the Nyquist criterion the sample rate must be at least twice that highest frequency ($f_s\ge2f_{max}$) to avoid aliasing, and in practice a comfortable margin above that bare minimum (several times $f_{max}$, after any anti-aliasing filter's roll-off) is used so residual energy above the filter's cutoff cannot alias back into the sampled band.

f) Averaging $N$ independent samples improves the signal-to-noise ratio by a factor of $\sqrt{N}$ (random, uncorrelated noise partially cancels while the repeated true signal component adds coherently) — for $N=256$, that is a $\sqrt{256}=16\times$ ($24\,\text{dB}$) noise reduction. It is efficient because this substantial noise reduction, plus the same line-frequency comb-filter rejection as continuous integration (part a/d), is achieved with only a simple digital summation of already-acquired samples — no extra analog filtering hardware, precision components, or additional acquisition time beyond the $1/60\,\text{s}$ window is required. It is also cheap in the converter's own logic: $256=2^8$, so dividing the accumulated sum by $N$ is a plain 8-bit right shift, needing no divider at all.