Obsidian/.archive/300s School/ME 2046 - Digital Control Theory/2025-01-09 Sampling Theory.md

ME2046_Sampled_Data_Analysis_Reading_Chapter_2pdf2254ME

Impulse Sampling

How do we represent a sequence of numbers? Impulse sampling does it by

1. having a continuous signal
2. having an impulse train (impulses at sampling frequency)
3. multiply em together

[!info] Functionals Laurent Schwartz (1950):

\int_{-\infty}^\infty \phi(x) f(x) dx = z

Shift Property:

\int_{-\infty}^\infty \phi(t-\tau) f(t) dx = FINISH

Laplace Transform

Pulse Train $\delta_t(t)$

\delta_T(t) = \sum_{k=-\infty}^\infty \delta(t-kT) x^*(t) = x(t)\delta_T(t) = \sum_{k=-\infty}^\infty x(t)\delta(t-kT)

Where the sampled signal is x^* What about in Laplace domain?

X^*(t) = \int \left[ \sum_{k=-\infty}^\infty x(t)\delta(t-kT)\right] e^{-st} dt X^*(t) =\sum_{k=-\infty}^\infty \left[\int x(t)\delta(t-kT) e^{-st} \right] dt X^*(t) =\sum_{k=-\infty}^\infty \left[\int x(t)e^{-st} \delta(t-kT) \right] dt

Now using the shift property...

X^*(t) =\sum_{k=-\infty}^\infty x(kT)e^{-kTs}

Some Observations

If we change the variable z = e^{Ts}

[!important] The Z-Transform

X^*(t) = X(z) = \sum_{k=-\infty}^\infty x(kT) z^{-k}

Z transform can be viewed as short hand of the Laplace transform

Sampling is a time varying process. If x(t) is time shifted by a small amount, the sampled signal x(kT) will be different.

Frequency Domain Interpretation

\delta_T(t) is periodic, so we can turn it into a Fourier series...

\delta_T(t) = \sum_{N=-\infty}^N C_N e^{j(\frac{2\pi}{T})Nt} C_n = \frac{1}{T}\int_{-T/2}^{T/2} \delta_T(t) e^{-j(\frac{2\pi}{T})Nt} dt

Apply a shift and do some stuff...

C_n = \frac{1}{T} \int_{-T/2}^{T/2} e^{-j \frac{2\pi}{T} Nt} dt C_n = \frac{1}{T}

So then...

\delta_T(t) = \frac{1}{T} \sum_{N=-\infty}^N e^{j(\frac{2\pi}{T})Nt} \delta_T(t) = \frac{1}{T} \sum_{N=-\infty}^N e^{j \omega_s T Nt}

Insert Steps from Class to get to $X^*(Z)$

The spectrum of the sampled signal is also a periodic function of frequency with period \omega_s. ! A lot of times, we need to filter the high frequency stuff out, or else we'll get some issues with aliasing.

Band Limited |X(j\omega)|=0 \forall |\omega|>\omega_0

Shannon's Sampling Theorem

! !