Preliminaries
Contents
1.1. Preliminaries#
Before we get too far into digital signals, we’ll need to establish some basic notation and concepts. This section covers the following
What is a signal?
Mathematical notation
Standard conventions
It does not cover mathematical fundamentals. For the first few chapters, we will not need much math beyond basic algebra (variables and rules of arithmetic), and a bit of geometry. Later on, we’ll make use of more advanced concepts (complex numbers, polynomials, and exponentials), but those will be introduced as needed.
Note
I sometimes use the term basic when describing certain ideas. By this, I mean that these ideas are the base level upon which we will build. This is not the same thing as being easy. Many basic ideas can be complicated, and take quite some time to thoroughly understand.
If the mathematical concepts start to seem overwhelming at some point, don’t worry. Just take it slow, and don’t move on to the next section too soon. None of the contents here are magical: it all builds on basic ideas.
1.1.1. What is a signal?#
At a high level, a signal is a way of conveying information. There unfortunately isn’t a simple, concise, technical definition of signal that provides us with much insight. The most intuitive example of a signal, at least for acoustically inclined people, is the voltage on a wire connected to a microphone. If you were to watch the voltage change over time, you might see something like the figure below:
Fig. 1.1 An example signal, showing voltage changing over time.#
The blue curve above represents the voltage measured at every time
If we want to know the voltage at a specific time, say, 4 seconds, we would substitute that value for
To be a bit more precise, the signal
1.1.2. Notation#
The first bit of notation that we’ll need is for signals.
denotes a continuous-time signal. can be any real number: 0, 1, -53, , etc. We read this as “signal at time .” denotes a discrete-time signal. must be an integer: , etc. We read this as “the sample of signal .”
For this first chapter, we’ll deal only with continuous signals
Because we often use the letter “x” to refer to signals, we’ll need a different symbol to represent multiplication.
We’ll instead use the center dot *
) and in text, but this is for good reason: the
Usually, we’ll let
would define a pure tone oscillating at 1 cycle per second.
Most of what we do in signal processing amounts to modifying a signal in some way, for example, applying a low-pass filter to remove high-frequency content.
We can think of this as applying some function
1.1.3. Standard conventions#
We’ll often use
To keep things simple, time will always be referred to by the letter
Discrete (i.e., integer-valued) quantities will be referred to by the letters
Unless otherwise stated, we will by default assume that signals are 0-valued when
You can think of this as pretending that signals are silent until you start recording.
Fig. 1.2 An example signal showing that voltage
Angles will be denoted by the Greek letters
Fig. 1.3 Illustrations of angles measured in radians, denoted by
We’ll use the letter 1j
to denote the imaginary unit, so translating from text to code should be fairly straightforward.
Complex numbers — numbers with both a real and imaginary component — will be denoted generally by the letter
Finally, we will occasionally see snippets of code that implement the abstract concepts being described in the text. This code will be written in the Python programming language, and will be typeset as follows:
for n in range(len(x)):
y[n] = 2*x[n] # Hurray! Now y is twice as big as x
More specifically, I will assume Python 3.6 or later, and will make use of the NumPy
package for numerical computation.
If you’re new to Python, I recommend getting started with the Anaconda distribution.