🚀 go-pugleaf

In article <1992Sep9.153835.13810@news2.cis.umn.edu>, patel@sparky.drad.umn.edu () writes:
|> I have a signal currupted by white additive Gausian noise.
|> y(t) = x(t) + n(t)  ; where n(t) is the additive Gaussian noise
|> and x(t) is the desired signal. x(t) is not a stochastic signal
|> but I know its characteristics like it is smooth everywhere
|> except at the edges. What is the best method to filter out the
|> noise from the observed signal y(t) to recover x(t)? I would
|> like to preserve the edges in the signal x(t) which are of utmost
|> importance to me. Any pointers to literature appreciated.
|> Thanks
|> 
|> Reply to   patel@sparky.drad.umn.edu


A good solution to this problem is to use an FIR Median Hybrid (FMH) filter.
The basic idea is (with n in place of t):


  y(n) =  median[1/M*sum(x(n-M),...,x(n-1)) , x(n) , 1/M*sum(x(n+1),...,x(n+M))]



this has the effect of using the average for the smooth parts but
with the advantage that the steps are preserved.  For more information see

\bibitem{Hei87b}{P.~Heinonen and Y.~Neuvo,
``FIR-median hybrid filters,''
{\sl IEEE Trans.\ Acoust., Speech, Signal Processing,}
vol.~ASSP-35, no.~6, pp.~832--838, June~1987.}


An extension of this works even better but is more complex:

  U_1    =  median[x(n-1) x(n) x(n+1)]

         :

  U_M/2  =  median[1/(M/2)*sum(x(n-M/2),...,x(n-1)), U_M/4, 1/(M/2)*sum(x(n+1),...,x(n+M/2))]
  y(n)   =  median[1/M    *sum(x(n-M),  ...,x(n-1)), U_M/2, 1/M*sum(x(n+1),...,x(n+M))]


see:

\bibitem{Wic90a}{R.~Wichman, J.~Astola, P.~Heinonen, and Y.~Neuvo,
``FIR-median hybrid filters with excellent transient response in noisy
conditions,''
{\sl IEEE Trans.\ Acoust., Speech, Signal Processing,}
vol.~ASSP-38, pp.~2108--2117, Dec.~1990.}

\bibitem{Nie89a}{A.~Nieminen, Y.~Neuvo, A.~V\"arri, and U.~Mitra,
``Algorithms for real-time trend detection,''
{\sl Signal Processing,}
vol.~18, pp.~1--15, Sept.~1989.}


Good luck

George
--