On 2/2/2011 1:51 PM, dbd wrote:
> On Feb 2, 6:16 am, "bharat pathak"<bharat@n_o_s_p_a_m.arithos.com>
> wrote:
>> Hello,
>>
>>       Half band filters are good for interpolation and decimation by
>>       2 as they have alternating 0's as their coefficients.
>>
>>       Is there something like 1/3rd band filters for doing interpolation
>>       and decimation by 3 with in-between 2 coeffcients being 0?
>>
>> Regards
>> Bharat
>
> Nth band filters seem to have one zero coefficient in N, not N-1 zero
> coefficients in N.
>
> See:
> On half-band, third-band, and Nth-band FIR filters and their design
> Mintzer, F.
> IBM Thomas J. Watson Research Center, Yorktown Heights, NY
> Acoustics, Speech and Signal Processing, IEEE Transactions on
> Oct 1982, Volume: 30, Issue: 5, page(s): 734 - 738
>
> Dale B. Dalrymple

Dale,

Interesting.  I'd not seen this one before.  It makes me imagine this:

As we well know, given some fs, the perfect reconsctruction filter has a 
sinc/Dirichlet time response with periodic zeros.  This corresponds to a 
lowpass filter with cutoff at fs/2.

A classic halfband filter has the same zero periodicity at twice the 
interval (2T=2/fs) and a cutoff at fs/4.

A narrower filter with the same *type* of response will have those zeros 
further apart.
So, it's not much of a stretch to think there are filters with periodic 
zeros in the coefficient structure - directly related to the frequency 
response.

This also suggests a relationship with the Gibby-Smith criterion for 
bandlimited functions that aren't "brick wall" bandlimited and still 
have periodic zeros.R. A. Gibby and J. W. Smith, &#4294967295;Some extensions of 
Nyquist&#4294967295;s telegraph transmission theory,&#4294967295; Bell Syst. Tech. J., vol. 
XLIV, pp. 1487-1510, Sept. 1965.

But, I'm afraid that lots of the neat properties of the classic 
half-band filter are lost.  And, the number of zero coefficients 
dimishes pretty quickly as I>2.

Fred

On 2/2/2011 6:11 PM, bharat pathak wrote:
> Thanks Fred.
>
Well, I said"
Or a compromise might be to have a "transition/stop" band with the first 
cutoff frequency at fs/6 but just keeping the normal half-band structure 
with half the coefficients zero.  After all, this is pre-decimation or 
interpolation at fs, right?

But that's not sensible because I was thinking about the fs/2 rate 
filter and not the fs rate filter with the center coefficient nonzero.

When the frequency response is "lifted" with the center coefficient, any 
original stop band becomes a filter with value 0.5 or thereabouts.  And 
that's obviously not what you want.

Fred

On Feb 2, 6:16&#4294967295;am, "bharat pathak" <bharat@n_o_s_p_a_m.arithos.com>
wrote:
> Hello,
>
> &#4294967295; &#4294967295; &#4294967295;Half band filters are good for interpolation and decimation by
> &#4294967295; &#4294967295; &#4294967295;2 as they have alternating 0's as their coefficients.
>
> &#4294967295; &#4294967295; &#4294967295;Is there something like 1/3rd band filters for doing interpolation
> &#4294967295; &#4294967295; &#4294967295;and decimation by 3 with in-between 2 coeffcients being 0?
>
> Regards
> Bharat

Nth band filters seem to have one zero coefficient in N, not N-1 zero
coefficients in N.

See:
On half-band, third-band, and Nth-band FIR filters and their design
Mintzer, F.
IBM Thomas J. Watson Research Center, Yorktown Heights, NY
Acoustics, Speech and Signal Processing, IEEE Transactions on
Oct 1982, Volume: 30, Issue: 5, page(s): 734 - 738

Dale B. Dalrymple

On 2/2/2011 6:16 AM, bharat pathak wrote:
> Hello,
>
>       Half band filters are good for interpolation and decimation by
>       2 as they have alternating 0's as their coefficients.
>
>       Is there something like 1/3rd band filters for doing interpolation
>       and decimation by 3 with in-between 2 coeffcients being 0?
>
> Regards
> Bharat

I don't think so....  let's see:

Properties of the half-band filter are that it's always odd-ordered and 
that it's frequency response is real because it's even in time.

If you leave out the center coefficient it's an even-ordered filter with 
sample rate at fs/2 and it's antisymmetric

H(0>fs/4) = -H(fs/2>fs/4)
and
H(fs>3fs/4) = -H(fs/2>3fs/4)

looking sorta like a "square wave" which obviously has zero values at 
fs/4 and 3fs/4

Then, reintroducing the center coefficient brings the sample rate to fs 
and "dc biases" the frequency response (it being purely real) so that 
the center frequencies go from negative values to zero .. becoming the 
stop band.  It's the antisymmetric quality that ties with the periodic 
zeros in the time response.

So, looking at the "filter" without the center coefficient, the filter 
sample rate is fs/2 and you get those zeros in time that you mentioned.

If you had a filter structure with those zeros in time that you want 
then, for starters, the underlying filter would have a sample rate of 
fs/3.  If that's the case then the filter couldn't be antisymmetric and, 
it would also be difficult to figure out what to do in the center of the 
coefficient structure because now there'd be two zeros to fill instead 
of just one and I guess the filter order would remain even, eh?

I'm sorry if this isn't a more robust description of the relationships 
betweem time and frequency but I'm in a bit of a hurry right now.

One question might be then is there a similar sorta lowpass filter where 
the sample rate is fs/3?  I just can't imagine... But there may be.

Perhaps a viable approach would be to interpolate by 6 and then decimate 
by 2 (I have no idea .. it's just a thought as there are similar 
approaches I've read about) - or to just bite the bullet and forget 
about the zero coefficients once and for all.

Or a compromise might be to have a "transition/stop" band with the first 
cutoff frequency at fs/6 but just keeping the normal half-band structure 
with half the coefficients zero.  After all, this is pre-decimation or 
interpolation at fs, right?

Fred