Signals, Continuous time vs. Discrete time, Frequencies/Sinusoids, Sampling and Reconstruction.

Dubulicious
2024-03-22

*"A signal is a function that conveys information about a phenomenon, any quantity that can vary over space or time, e.g., temperature."*

A device takes a continuous input signal and outputs a continuous output signal.

Straightforward to implement using analog circuits (resistors, capacitors, inductors, amplifiers, ...).

Continuous-time signals have a value at each time instant \(t\).

Digital systems take discrete inputs and give discrete outputs. Implemented using digital circuits (CPUs, FPGAs, custom digital designs, ...)

Discrete-time signals have a value at certain time intervals \(t = nT_s\).

**Discrete-time signals are at the core of this course.**

Assume inputs and outputs are analog, then inside the device you have **Sampling (ADC)** and **Reconstruction (DAC)**.

This way you can process the signal with **Digital Signal Processing (DSP)**.

*AnalogInput -> ADC -> DSP -> DAC -> AnalogOutput.*

**How do we sample**a continuous signal to a discrete signal without losing important data?**How do we process**the discrete signal to generate the discrete output signal?**How do we reconstruct**the desired continuous output signal from the discrete equivalent without losing important data?

Things we may **want** to do:

- Reduce noise in sound and images.
- Mimic echo-effects or otherwise manipulate audio signals.
- Create and decode continuous signals that carry digital information (radio, wires, optical fibers).

Theoretical limits on what we **can** do:

- With a certain sampling rate, we can only represent a limited range of continuous signals with equivalent discrete signals.
- An incorrectly designed DSP may result in an unstable system.

Efficiency depends on **how** a system is designed and implemented:

- Different types of process may require more/less memory, higher/lower energy consumption, influence cost, more/less flexibility.
- An important design goal is usually to minimize resources used for processing.

Frequency in hertz: \(\Omega = 2\pi F\) Hz.

$$

x_a(t) = A\cos{(2\pi Ft + \theta)}

$$

Period time: \(T_p = \frac{1}{F}\)

This is a periodic signal, for which: \(x_a(t) = x_a(t + T_p)\).

Write cosine using complex numbers.

$$\cos(\phi)=\frac{e^{j\phi}+e^{-j\phi}}{2}$$

Then we can rewrite our signal as:

$$\begin{aligned}

x_{a}(t)=& A\cos\left(\Omega t+\theta\right) \\

=& \frac{A}{2}e^{j(\Omega t+\theta)}+\frac{A}{2}e^{-j(\Omega t+\theta)}

\end{aligned}$$

\(x(n) = A\cos{(\omega n + \theta)}\) where \(n\) is a time sample and \(\omega\) is angular frequency.

Reconstruction using \(x_a(t) = \sum_n x(n)g(t-nT)\) where `g(t)` is some *"reconstruction function"*.