# Digital signal processing

## Introduction to Digital Signals

### Continuous vs Discrete

In order to understand the benefits and limitations of Digital Signal Processing (DSP) it is important to understand the distinction between **Continuous** and **Discrete** signal representations. In the Synth DIY world, much use is made of the terms **Analog** and **Digital** without any clear explanation of the very different assumptions that underpin those terms.

A *signal* can be defined as any function that conveys information about the state of a physical system. This is usually represented as a variation of values over time or space. Signals are represented mathematically as functions of one or more independent variables.

*Continuous-time* signals are defined across a continuum of time and reflect a continuously variable value. So, for example

**f(t) = sin(2wt)**

is continuously defined for any and all values of **t**.

*Discrete-time* signals are only defined at specific times and the independent variables can therefore only take on discrete values. Discrete-time signals are represented by a sequence of discrete values. In the context of Synth applications, the specific times at which the signals are defined are regular and evenly spaced.

In real world applications, the values a signal takes on as it varies usually represents an amplitude. The signal amplitude can also be either *continuous* or *discrete*.

When we talk about an *analog* signal, we are usually referring to a signal that is both a *continuous-time* and *continuous-amplitude* signal. A *digital* signal is both a *discrete-time* and *discrete-amplitude* signal, which translates into the reality that it is both sampled (discrete-time) and quantised (discrete-amplitude).

This all sounds very academic, of course, but is important to understand the fundamental differences between the two domains when making design decisions around sample rates, bit resolution and cost when selecting components for implementing DSP hardware.