# Linear Equalizers Weights

For linear equalizers, the weights of the equalizer are set to achieve minimum error between the equalized signal and the original input signal. First, a training sequence is sent as input, and the weights that achieve the minimum error between the equalized and the original training sequence are obtained. My question is:

1. In case of non-adaptive equalizers, are the weights obtained for the training sequence used for any other input signals?

2. In case of adaptive equalizers, how the weights are updated according to the new input signals? The error should be minimized between the equalized signal and what? There is no known sequence in this case.

• Hi! There are so many different variants, tehcniques, methods, inventions, patents etc on designing and implementing adaptive equalizers over practical data networks. In theory you are free to chose whatever technique you find suitable, but if your focus is on the practically existing techniques and algorithms then your question shoulde be much more specific to indicate that. Dec 30 '18 at 16:07

In an initial phase, the training sequence is used to determine the optimal equalizer weights. There is usually no reason to leave the weights fixed because you need an adaptation algorithm to compute the initial weights from the training sequence, so you might as well continue running the adaptation algorithm as soon as actual data are transmitted. If the channel remains constant, the weights will not change significantly, but if there are changes in the channel, the weights will be adjusted to minimize intersymbol interference.

After the training sequence, the equalizer usually works in a decision-directed mode, i.e., the decisions are assumed to be correct and they are used to compute the error signal. This is shown in the figure below: from: Digital Communication, E.A. Lee, D.G. Messerschmitt, 2nd ed., p. 518.

• I know that in the training mode, the error is computed between the training sequence and the equalized ouput. In the decision directed mode, how the error is computed? the error is of course the difference between two signals, one of them is the equalized signal, what is the second? this is my question. @Matt L. Dec 30 '18 at 19:08
• @KaremAdam: The second is the output of the slicer, i.e., the decisions. Dec 30 '18 at 20:00
• @KaremAdam: I've added figure for clarification. Dec 31 '18 at 9:14
• Could you please send me a source that simply explains the technique, since unfortunately I didn't pick the idea yet. @Matt L. Dec 31 '18 at 15:36
• Thanks a lot for your help. It seems that the slicer acts somewhat like a regenerative repeater in digital communication systems, right? I think that without the training sequence in the training mode, the output of the slicer can not be considered a correct decision in the decision directed mode, right? The training stage sets the weights of the equalizer, which are assumed not to change significantly with other input signals, right? @ Matt L. Dec 31 '18 at 15:43

The error should be minimized between the equalized signal and what? There is no known sequence in this case.

But there is: if the channel is still good enough so you can decode a packet successfully, you can reconstruct the original signal by simply re-encoding the packet.

• I know that in the training mode, the error is computed between the training sequence and the equalized ouput. In the decision directed mode, how the error is computed? the error is of course the difference between two signals, one of them is the equalized signal, what is the second? this is my question. @Hilmar Dec 30 '18 at 19:11
• I'm not talking about training. During regular operation you decode data packets. From the data packet (which typically has error checking & recovery built in) you can reconstruct the original signal that the transmitter sent. That's what you compare to your equalized signal. Dec 30 '18 at 19:39
• Could you please send me a source that simply explains the technique, since unfortunately I didn't pick the idea yet. @Hilmar Dec 31 '18 at 15:35