Part 3: Detection theory

Introduction

Noise makes it impossible to predict instantaneous values of received signals. Instead, we can define a probability density function that can tell us how likely it is that the received signal is within a certain interval. This probability density function is defined by the deterministic component of the stochastic signal, i.e. the noise-less signal, and the probability distribution of the noise signal. A discrete-time stochastic signal $x[n]$ can therefore be split into a deterministic signal $s[n]$ and a stochastic noise component $\epsilon[n]$ as \begin{equation} x[n] = s[n] + \epsilon[n]. \end{equation}

Now imagine the case where the noise power is close to or larger than the signal power. This situation might occur if the received deterministic signal has suffered from significant attenuation in its transmission path. Fig. 1 shows three received signals where the deterministic component is a block-wave pulse. The noise power in all three cases is kept constant, whereas the signal power is consecutively decreased as a result of attenuation.

One would conclude from the left and middle plot of Fig. 1 that the deterministic block-wave pulse signal can visually be detected. However, this conclusion is not as apparent in the case presented by the right plot. We could even ask ourselves the question of whether there actually is a deterministic signal present in the noise. The principles that focus on providing the answer to this question are the underlying foundation of detection theory.

The detection problem

The importance of detection theory might become evident in the case of telecommunication, where we are interested in the transmitted signal. By being able to distinguish between transmitted bits (0's and 1's) the original message can be reconstructed leading to successful communication. However, the field of telecommunication is not the only field where detection plays an important role.

Another example of detection theory involves radar or sonar, where the goal is to identify flying or submerged objects. In the case of radar, we might, for example, be interested in detecting a hostile aircraft. The detection of such an aircraft is of vital importance. In case of its presents, an appropriate reaction needs to be taken. However, if an aircraft is detected by the system and there actually is not one, then the consequences of performing a military counter-operation might cause a disastrous international conflict.

The field of detection theory also extends to the field of medicine, where the detection of cancerous tissue provides an example. The early detection of malignant (Dutch: kwaadaardig) tissue proves to be of vital importance in the treatment process. Failing to detect such a malignant tissue might damage the chances of successful treatment. However, when a malignant tumor is detected, whereas the tumor is actually benign (Dutch: goedaardig), the patient will undergo unnecessary treatment that can be harmful to the health of the patient.

Module overview

This module will cover the field of detection theory.

1. Hypothesis testing - This section will cover the decision theory concerning whether a signal present it is absent.
2. Matched filter - When the decision criterion is established, a practical application can be created. The idealized filter for testing whether a signal is present is called a matched filter.
3. Statistical tests - A single observation is usually not enough to accept or reject a hypothesis. Usually, a decision is based on multiple observations to limit the effect of measurement noise. This section will discuss how decisions can be made when dealing with limited information.