**UA Dr. Robert Legenstein, WiMAus Prashant Joshi, M.S.
Institute for Theoretical Computer Science
Technische Universität Graz
A-8010 Graz, Austria
{legi, joshi}@igi.tugraz.at **

NACHNAME | Vorname | Matrikelnmr | Team |

In this exercise you are going to do some pattern generation using a network made of 4 perceptrons which receives output feedback. More precisely, the task is to generate two different periodic sequences as the network output, if the network is started from two different initial conditions.

Theory

A perceptron is a kind of threshold neuron, which receives inputs
and gives the output *y*.
The relation between input and output is given by equation 1:

where the neuron is receiving inputs, and the input is connected to this neuron by a synapse of weight , and is the bias term. Intuitively, if the weighted sum of the inputs to a neuron is greater then the bias , then the neuron outputs a value , otherwise it outputs a value .

The *hardlims* function
is defined by equation
2:

Detailed task description

Suppose you have a network made of 4 perceptrons. Each of these perceptrons is receiving a 4 bit binary input (each bit is either or ). More precisely, the value of input bit at time-step is the output of the neuron at time-step (please see figure 1). Also given are two periodic sequences , and .

Design a network whose perceptron's output is given by equation 3

or in matrix notation shown as equation 4:

where
and
.

Find the weight matrix and the bias vector
,
such that when the network is started at time with input
, the network produces the sequence A, and when on
the other hand the network is started at time with input
, the network produces the sequence B.

The two sequences and desired network behavior are given below:

x(t) for sequence A | x(t) for sequence B | |

0 | -1 -1 -1 +1 | -1 +1 +1 -1 |

1 | +1 -1 -1 -1 | +1 -1 -1 +1 |

2 | -1 +1 -1 -1 | -1 +1 +1 -1 |

3 | -1 -1 +1 -1 | +1 -1 -1 +1 |

4 | -1 -1 -1 +1 | -1 +1 +1 -1 |

5 | +1 -1 -1 -1 | +1 -1 -1 +1 |

6 | -1 +1 -1 -1 | -1 +1 +1 -1 |

7 | -1 -1 +1 -1 | +1 -1 -1 +1 |

: | : | : |

: | : | : |

- Find the weight matrix and the bias
. For
this, you can:
- Use some learning algorithm
- Calculate the weights and biases needed
- Intuitively find them (descrive your thoughts precisely)

- Write clearly the numerical values of and that you obtained.
- Examine the network behavior when it is initialized at time
with a different inital condition.

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- ... SS06
^{1} - Class Website: http://www.igi.tugraz.at/lehre/NNB/SS06/

Joshi Prashant 2006-03-13