Capacity and Capacity-Achieving Input Distribution of the Energy Detector

Project Type: Master/Diploma Thesis
Student: Erik Leitinger


Non-coherent receiver structures, which include the energy detector, have great importance for ultra-wideband systems due to their low complexity. The channel capacity, i.e. the maximum amount of information that can be transmitted at a particular signal-to-noise ratio SNR represents a fundamental concept in information theory. In order to achieve the channel capacity, the input alphabet has to satisfy certain statistical properties. For a non-linear channel like the energy detector, such an alphabet shows a discrete amplitude distribution and can be represented by on-off keying for low SNR or amplitude-shift-keying for higher SNR. The difficulty lies in providing a solution for the underlying non-linear optimization problem for the continuous channel (energy detector). This master thesis presents a numerical algorithm to compute the channel capacity and the corresponding capacity-achieving input distribution over a wide range of SNR. Furthermore, it will be demonstrated that the input distribution has to be discrete with a finite number of mass points. A detailed literature research supports these conclusions.

Short Description

The energy detector, as depicted below, is a receiver architecture often used in ultra wide-band (UWB) communications due to its simplicity. It is a nonlinear channel, but we believe that for suffciently large integration times it can be modeled as an AWGN channel with signal-dependent noise variance.


The capacity of a channel is the maximum amount of information that can be transmitted at a particular SNR. To achieve this capacity, the input has to satisfy certain statistical properties. For nonlinear channels -- and many linear channels -- it is known that the input has to be a discrete random variable. Corresponding modulation schemes are, e.g., on-off keying for low SNR, amplitude- shift keying for higher SNR, etc.

In [1] a numerical procedure to compute the capacity of a channel together with its capacity-achieving distribution is described. In this work, this procedure shall be used to obtain the capacity of the energy detector for a wide range of SNR and to verify the modeling assumption. In addition to that, the results for the input statistics shall be compared to standardized UWB signal models.

[1] J. Dauwels, "Numerical computation of the capacity of continuous memoryless channels," in Sym. on Information Theory in the BENELUX, 2005

Fulltext & Additional Material

The full thesis can be downloaded here.

Leitinger12_DA.pdf3.8 MB