# Training Fractionally Spaced Channel Estimator and Equalizer

If you are attempting least squares channel estimation with a fractionally spaced channel estimator, do you want the training sequence to also be fractionally spaced or symbol spaced? It looks like you could do either.

• In fractionally-spaced equalizer (FSE) implementations that I've seen before, the filter decimates, so that for a spacing of $\frac{T}{N}$, only one sample is output for each $N$ inputs. In an LMS equalizer, for example, the training sequence is used as the desired output $d[n]$ when calculating the error signal $e[n] = d[n] - y[n]$, where $y[n]$ is the filter output. In this formulation, it looks like you would want a symbol-spaced training sequence in order for its sample rate to line up with the equalizer's output. – Jason R Feb 26 '19 at 2:23
• Yes, but you could also estimate the channel taking into account mid-symbol values which are output by the correlator. In a sense this increases the information gained per training symbol, I'm not sure if this would screw something up or not? – FourierFlux Feb 26 '19 at 18:58

The training sequence should be at the same spacing as the equalizer when considering its sampling at the input to the equalizer. Adaptive algorithms converge to the least square solution based on the error between the received sequence and the transmitted sequence (known when a training sequence is used). Further, the equalizer can only determine a solution where channel energy is present (for example, if you transmitted a single tone, you would only find the single phase/magnitude weight for that frequency)

If you use only one sample per symbol to derive the channel, the solution will suffer from "band edge aliasing" as I depict in the figures below, and the channel solution will suffer from the folded energy at the band edges.  For further details see this post where I have derived the operation of the LMS Equalizer and the solution using the Wiener-Hopf equations (for when you have the luxury of block post-processing, but also helps illuminate the operation since this is the same solution that the adaptive algorithms would converge to).

Compensating Loudspeaker frequency response in an audio signal

• It seems like you're missing a link in the last paragraph. – Jason R Jan 13 at 13:06
• @JasonR Yes the link would be helpful, wouldn't it? Thanks for the catch, I just updated it. – Dan Boschen Jan 13 at 13:11