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Re: [10GBT] Issues with solarsep_varlen7a.m

> Sailesh Rao <saileshrao@OPTONLINE.NET> writes:
> >
> > The folded SNR calculations in lines 443, 453 and 463 are not right.
> >
> > If f1 and f2 are mirror frequencies about fs/2, the formula being
> > used
> > is
> >
> > S/N = ABS(S1/N1) + ABS(S2/N2) ;
> >
> > However, the actual SNR at the folded frequency would be
> >
> > S/N = ABS(S1+S2)/ABS(N1+N2)
> >
> > where S1, S2, N1, and N2 are complex phasors. Therefore, in the
> > context of folding, the actual PSD of the signal becomes relevant,
> > whereas the original Salz formula for the optimum DFE SNR is
> independent of the PSD.

> George Zimmerman <gzimmerman@SOLARFLARE.COM> writes:
> >
> > On the folded SNR calculation, however, you are incorrect.  The
> > optimum DFE is based on a folded SNR which is the sum of the SNRs,
> > not
> > the sum of the signal over the sum of the noise.  You can check
> > either
> > Salz, or for a more direct representation, please check Pottie &
> > Eyuboglu, JSAC, August 1991, equation 6.
> >

The Salz DFE analysis assumes a prefilter prior to the baud sampler.
Following optimization (in AWGN) the prefilter turns out to be
equivalent to the cascade of a channel matched filter followed by
a one-sided synchronous tapped delay line.  The matched filter's phase
(conjugate to channel) ensures that the net transfer function (channel*MF)
lies on the positive real axis prior to the baud sampler.  Thus, all
folding translates (f0+k/T, k = -inf ... inf) add unidirectionally,
eliminating the effects of channel phase.  It is only because of this
phase alignment that the optimized integrand involves the sum-of-SNRs,
and not sum-of-signal/sum-of-noise.  (The same holds for a DFE with
fractionally spaced FFF.)

But for a synchronous DFE in the absence of a matched filter -- probably
the system of interest to most of us -- no special phase alignment of
the translates can be assumed, and the relevant folding expression (for
flat AWSS noise with variance N0) is

    abs(SUM H(f0+k/T)) ** 2 / N0 ,

H(f) being the net transfer function from the Tx to the Rx baud sampler
input.  Except for a missing "**2", this is essentially as Sailesh

The bottom line is that without a MF or fractionally spaced FFF, the
value of the summation depends on the channel and front-end phases at
the translate frequencies, which is the point I believe Sailesh was making.
The sum-of-SNRs folding is an upper bound.  Thus, the solarsep code yields
optimistic results, unless the assumed system model includes a fractionally
spaced or MF front end.

For our channel, as long as the rolloff is smooth, the 'optimism' will not
be very large, because even if translates are completely out of phase, the
in-band translate magnitudes dominate.  Similarly if the front-end rolls
off reasonably above 1/(2Fs).  But if there are large ripples near 1/(2Fs)
and shallow front-end rolloff, then significant dips in the folded spectrum
can be introduced which could result in non-negligible MSE differential
between the solarsep method and a more realistic (synchronous, no MF)

Glenn Golden
Principal Engineer
Teranetics, Inc.