Linear Networks with Delays

Tapped Delay Line

You need a new component, the tapped delay line, to make full use of the linear network. Such a delay line is shown below. There the input signal enters from the left and passes through N-1 delays. The output of the tapped delay line (TDL) is an N-dimensional vector, made up of the input signal at the current time, the previous input signal, etc.

Linear Filter

You can combine a tapped delay line with a linear network to create the linear filter shown.

The output of the filter is given by

• 

The network shown is referred to in the digital signal processing field as a finite impulse response (FIR) filter [WiSt85]. Look at the code used to generate and simulate such a network.

Suppose that you want a linear layer that outputs the sequence T, given the sequence P and two initial input delay states Pi.

• P = {1 2 1 3 3 2};
  Pi = {1 3};
  T = {5 6 4 20 7 8};
  

You can use newlind to design a network with delays to give the appropriate outputs for the inputs. The delay initial outputs are supplied as a third argument, as shown below.

• net = newlind(P,T,Pi);
  

You can obtain the output of the designed network with

• Y = sim(net,P,Pi)
  

to give

• Y = [2.7297] [10.5405] [5.0090] [14.9550] [10.7838] [5.9820]
  

As you can see, the network outputs are not exactly equal to the targets, but they are close and the mean square error is minimized.

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Linear System Design (newlind)

LMS Algorithm (learnwh)

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