Question 1 of 7: Noise, Errors and Signal Conditioning

Question 1: Noise, Errors and Signal Conditioning (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) Differentiation multiplies a signal's frequency-domain content by $j\omega$, so its gain rises linearly with frequency. Measurement noise is typically broadband (Johnson/shot noise extend flat to very high frequencies) while the signal of interest occupies a comparatively narrow, low-frequency band; the derivative amplifies the noise components far more than the signal, so the differentiated output can be noise-dominated even when the original signal had an acceptable SNR.

b) Low-pass filtering attenuates everything above its cutoff frequency. Because most of a noise source's energy is spread broadband while the wanted signal is concentrated at lower frequencies, an LPF set just above the signal bandwidth removes the noise energy sitting outside that band while passing the signal essentially unattenuated — trading response speed (bandwidth) above cutoff for reduced RMS noise in the passband.

c) Shot noise originates from the discrete, quantized nature of electric charge: carriers (electrons/holes) cross a potential barrier, such as a p-n junction depletion region, at random and statistically independent instants (a Poisson arrival process), so the instantaneous current fluctuates about its DC mean. The RMS shot noise current is $i_n=\sqrt{2qI_{DC}B}$, where $q$ is the electron charge, $I_{DC}$ the DC bias current and $B$ the measurement bandwidth.

d) Noise is a random, unpredictable, intrinsic disturbance (thermal, shot, flicker) generated within the measurement chain itself — it has no coherent waveform and cannot be predicted or cancelled by synchronous techniques. Interference is a deterministic disturbance coupled in from an identifiable external source (mains hum, RF pickup, switching transients) at a known frequency/waveform; because it is coherent it can in principle be shielded against, filtered at its known frequency, or cancelled by synchronous subtraction — none of which works on true random noise.

e) The first (buffer) stage gives very high input impedance at both $V_1$ and $V_2$ — each drives a non-inverting op-amp input directly, drawing negligible current, so the sensor source is not loaded — and the overall gain is set by the single resistor $R_g$ without disturbing the closely-matched resistor network of the difference-amplifier output stage. Because the buffer stage isolates the source from the difference stage, the configuration achieves high input impedance and high CMRR together with easily adjustable gain, a combination a single (one op-amp) difference amplifier cannot provide.

f) The anti-aliasing filter's cutoff frequency must not exceed half the sampling frequency, $f_{c,\max}=f_s/2$ (the Nyquist frequency). Any signal energy above $f_s/2$ that the filter fails to remove folds back (aliases) into the baseband on sampling and is indistinguishable from genuine low-frequency content, so the filter must have removed essentially all of that energy by the time it reaches the Nyquist frequency.

g) Bias (systematic) error is a fixed, repeatable offset that shifts every reading by the same amount. Statistical techniques (repeated sampling, averaging, standard deviation) act on the random scatter between readings and reduce random error, but they leave a constant bias completely unchanged — a biased instrument simply repeats the same wrong answer more precisely. Bias can only be found by comparing the instrument's readings against an independent, traceable reference standard, not by analysing the instrument's own repeated output.

h) Temperature change is generally the most common source of instrument drift: ambient or self-heating temperature shifts alter a sensor's and its signal-conditioning electronics' zero (offset) and span (gain) over time, producing a slow change in reading at a fixed input. Long-term component aging (electrolytic capacitor drift, resistor/semiconductor parameter shift) is the next most common contributor.

i) When $X=A-B$ with $A\approx B$, the absolute error in $X$ is roughly the combination of the absolute errors in $A$ and $B$ (comparable in size to each input's own absolute error), but the result $X$ itself is small. The relative error in $X$ is $\delta X/X$, and because $X$ has shrunk toward zero while the propagated absolute error has not, this ratio blows up. For example, with $A=100.0\pm0.1$ and $B=99.0\pm0.1$ (each 0.1 % relative error), $X=1.0\pm0.14$ — about 14 % relative error, roughly a hundred-fold worse than either input alone.

j) Given. A 13-bit ADC (Intersil 7109) digitizes a $\pm10\text{ V}$ input span. Find. The voltage resolution (one LSB). A 13-bit converter resolves $2^{13}$ discrete codes across the full input span of $20\text{ V}$ (from $-10\text{ V}$ to $+10\text{ V}$): $$\Delta V=\frac{V_{FS}}{2^{N}}=\frac{20\text{ V}}{2^{13}}=\frac{20\text{ V}}{8192} =\boxed{2.44\times10^{-3}\text{ V}\approx2.44\text{ mV}}$$ so the smallest voltage change the converter can distinguish is about 2.44 mV.