# [[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 $\omega_s$. 
![[Pasted image 20250109181319.png]]
 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
![[Pasted image 20250116160940.png]]
![[Pasted image 20250116161008.png]]
