2018-10-13

2018-10-15 本文已影响0人 carpediemmlf

Lecture 4, Systematic errors and sampling

Null method: e.g. current bridge. The indicating device does not need to be linear or even calibrated.
Watch out for changes in time: e.g. do "ABC ABC ABC" instead of "AAA BBB CCC" because the experiment result may drift in time.
Differential measurements: e.g. measure the temperature w.r.t a standard and similar to the desired temperature, for example, thermal couple output ~ 100C compared to the output in boiling water.
whats the difference between this and the null method?

N.B. Always approach a measurement from the same side to avoid backlash.

Selection effects: make sure you are measuring the thing you want to measure. Avoid spurious correlation.

$f(t) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{- \infty} g(\omega) e^{i\omega t} {\rm d} {\omega}$

$g(\omega) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{- \infty} f(t) e^{- i\omega t} {\rm d} {t}$

Examples: cos, sinc, comb !!!! should have produced accompanying graphs

Minimum sampling rate: Nyquist Criterion basic version
For a band-limited function, need to sample at a minimum rate of twice the highest frequency Fourier component present in the signal.
If the sampling is noiseless, can recover the signal perfectly from its samples.
Aliasing
Present when the Nyquist criterion is not met.
$True\ freq = \frac{Sampling\ freq}{2} + e$
where
$e = v - v_a$
also notice that it can only represent an aliasing of an aliasing if the original signal is too high.
Convolution
$h(z) = f*g=\int^{+\infty}_{- \infty}f(t)g(z-t){\rm d}t$
Convolution with the comb function (a series of delta function) leads to replication of the original function at each delta point.
Convolution theorem

$FT(f(t) \cdot g(t))\propto F(\omega) *G(\omega)$

$FT(f*g) \propto F(\omega) \cdot G(\omega)$

Convolution in one domain is proportional to point-wise product in the other domain.