Back Propagation

Back propagation uses the same structure as the feedforward and tap update algorithm using the following equation

$e_n = (H^T_n * e_{n+1})$

This algorithm differs in that it uses the transpose of the taps which requires some reordering to avoid the need of an adder tree. The block diagram is show below.

Data Flow

The data flow for this operation is as follows :

1. The vector version of the error is brought in through the tap input and latched into a delay line
2. The taps are brought in with an offset to order properly align the data and taps
3. The taps are multiplied by the error and fed to the accumulator
4. For each sample the data is rotated after each multiply to line up with the correct tap
5. When the calculation is complete the output of the accumulator is fed into a delay line and output serially

There is an issue with this algorithm in that it uses the transpose of the taps which slightly complicates the memory access. To get around this problem while keeping the same structure the taps are accessed from memory in an interleaved fashion as shown below. The table below shows this operation which does the following things :

1. Rotates the tap address on a per tap basis
2. Rotates the data as well using a delay line