D-A Convertor

Sampling Theorem: If a signal is constructed in the way of a D/A it is possible to be comlpetely reconstructed Reconstruction Theorem

Reconstruction Formula:

The continuous signal $x(t)$ can be reconstructed from its discrete samples $x[n]=x(nT)$, where $T=\frac{1}{fs}$​ is the sampling period, using the following interpolation formula:

$$\begin{array}{r}x(t)=\sum _{n=-\infty }^{\infty }x(nT)\text{sinc}(\frac{t-nT}{T})\end{array}$$

Here:

• sinc$(x)=\frac{sin(\pi x)}{\pi x}$ is the sinc function
• Each term $x(nT)\text{sinc}(\frac{t-nT}{T})$ represents a weighted sinc function centered at the sample time $t=nT$ with amplitude $x(nT)$ This formula effectively reconstructs $x(t)$ by summing scaled sinc functions centered at each sampling point.

How?

• Sampling: When we sample a bandlimited signal at a rate equal to or greater than twice its highest frequency, we retain all frequency information without overlap.
• Aliasing Prevention: Sampling below the Nyquist rate leads to overlapping frequency components in the spectrum of the sampled signal, causing aliasing, where higher frequencies are indistinguishable from lower ones. The Reconstruction Theorem prevents aliasing by specifying the minimum sampling rate.
• Sinc Interpolation: The sinc function acts as an ideal interpolation function for reconstructing the continuous signal from its discrete samples. It ensures that each sample affects only its surrounding area without altering the values at other sample points, leading to an exact reconstruction of the original signal.