Chapter 1 - Signals and Systems
Signals, Continuous time vs. Discrete time, Frequencies/Sinusoids, Sampling and Reconstruction.
What is a signal?
"A signal is a function that conveys information about a phenomenon, any quantity that can vary over space or time, e.g., temperature."
Continuous time vs. Discrete time
Analog systems
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
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.
The best of both worlds
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.
High-level questions
- 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.
Frequencies/sinusoid in continuous time
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: \(x_a(t) = x_a(t + T_p)\).
Euler's identity
Write cosine using complex numbers.
$$\cos(\phi)=\frac{e^{j\phi}+e^{-j\phi}}{2}$$
Then we can rewrite our signal as:
$$\begin{aligned}
x_{a}(t)=& A\cos\left(\Omega t+\theta\right) \\
=& \frac{A}{2}e^{j(\Omega t+\theta)}+\frac{A}{2}e^{-j(\Omega t+\theta)}
\end{aligned}$$
Frequencies/sinusoid in discrete time
Discrete-time sinusodial
\(x(n) = A\cos{(\omega n + \theta)}\) 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".