Discrete-Time Signal Processing

This is an introductory article on (one-dimensional) signal processing. It follows How to Record a Lion in a Concert Hall, which was meant to raise your interest. Did it manage to do that?

The title of this article is Discrete-Time Signal Processing, although the term digital signal processing (with the abbreviation DSP) is much more common. Both types of signals are discrete in the time dimension, in the sense that they are sequences of numbers. The difference is that a digital signal is a quantized, in that it consists of numbers from a limited set. For example, the samples of a signal that was recorded at 16-bit depth can have 65536 (\(2^{16}\)) possible values.

Although what you do with a computer is always digital signal processing, most of the theoretical stuff is actually about discrete-time signal processing. One of the standard reference texts is, after all, called “Discrete-Time Signal Processing” (Oppenheim & Schafer). For example, the famous Nyquist sampling theorem (also sometimes called Shannon sampling theorem, or, why not, Nyquist–Shannon sampling theorem) is only exactly true for discrete-time signals, not for digital ones.

Signal

A (one-dimensional) discrete-time signal is defined as a sequence of numbers, written as \(x[n]\), with \(n\in\mathbb{Z}\). It is written with square brackets to clearly differentiate it from a continuous signal \(x(t)\), with \(t\in\mathbb{R}\). Often, the discrete-time signal is a sampled version of a “real” continuous signal.

System

A system is a transformation that takes an input signal \(x[n]\) and produces an output signal \(y[n]\) from that. This is written as

\[y[n]=\mathcal{T}\{x[n]\}.\]

This notation is not very precise from a mathematical point of view, since it suggests that it is about the single numbers \(x[n]\) and \(y[n]\), while it are actually the complete sequences \(x[n]\) and \(y[n]\) that are involved in the transformation. But the notation is quite common, and I’ll use it anyway.

An example of a very simple system is the ideal delay system, defined as

\[y[n]=x[n-n_d],\]

where \(n_d\in\mathbb{Z}\) is the delay of the system in samples.

LTI System

The class of linear time-invariant (LTI) systems is very important. A system is LTI if it is linear and time invariant (also called shift invariant if the signal is not a function of time). What does it mean for a system to be linear and time-invariant?

A system is linear if it has the properties of additivity and homogeneity.

  • Addititivy means that \(\mathcal{T}\{x[n]+y[n]\}=\mathcal{T}\{x[n]\}+\mathcal{T}\{y[n]\}\).
  • Homogeneity means that \(\mathcal{T}\{ax[n]\}=a\mathcal{T}\{x[n]\}\).

Fun fact: the linear equation \(y=mx+b\) (the equation of a straight line) is not linear in this sense! This is because \(max+b\neq a(mx+b)=max+ab\). The equation is linear when \(b=0\).

A system is time invariant if a shift of the input signal results in a corresponding shift of the output signal. This means that a system that transforms \(x[n]\) into \(y[n]\) must transform \(x[n-n_d]\) into \(y[n-n_d]\) for each \(n_d\in\mathbb{Z}\) to be time invariant. This property simply states that the output of the system does not depend on the particular time that the input is applied.

An important property of an LTI system is that it’s completely characterized by its impulse response. I introduce that all important concept in a separate article.

Submitted by Tom Roelandts on 23 June 2013

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