Half-Band Filter Length

"Randy Yates" <yates@ieee.org> wrote in message 
news:llcwzrnt.fsf@ieee.org...
..................

>>
>> It is not exactly true that the middle coefficient will be equal to
>> the sum of the other coefficients. The sum of the coefficients is
>> equal to magnitude response at the zero frequency and for the
>> half-band filters the magnitude response at zero frequency is either
>> 1-delta or 1+delta where delta is the ripple in the pass- and stopband.
>
> Why? Why is the response at DC necessarily one of these two values? Why
> couldn't it be the case that 1-delta < H(0) < 1+delta?

Randy,

I was surprised at this once upon a time - having fiddled quite a bit with 
the Remez algorithm on potentially more general functions.  However, I 
believe that Rabiner & Gold showed in their book and Parks & McClellan took 
advantage of the fact that peaks of a minimax approximation appear at the 
band edges ( i.e. H(0)= |delta| ).  So, Juha implies a minimax solution 
here - and there will be a peak of the error at f=0; H(0)=+/-delta where 
delta is the peak minimax error or ripple.

It is possible to force H(0) to be 1.00000 (or at whatever scale you like) 
and, thus, H(fs/2)=0  and this comes at the expense of the peak ripple - 
which will be larger if you force this equality.

This zero-forcing constraint may be of some advantage if you plan on using 
zero stuffing in the frequency domain around fs/2 to get temporal 
interpolation.  That is because there won't be any sharp edges created by 
the "abrupt" introduction of zeros in frequency - because there will be a 
double zero at fs/2 after applying this type of half-band filter.

Fred