From my (not 100%) memory of signal processing, to sythesise an FIR filter
with a pure delay of a whole number of samples, you can get the impulse response by sampling a sinc function with the correct delay and sample rate.
It happens that sampling a sinc with a whole number of samples delay, you get an IR with one sample of amplitude 1 at the sinc's centre peak, and the rest of the sample points fall on zeros of the sinc function, so the IR has an amplitude of 0 at all other points. Thus you get the single unit impulse function for the impulse response.
I recall that it is likewise possible to synthesise a fractional delay filter by sampling a sinc function with the correct delay- which if fractional, gives you an infinite impulse response of mostly non-zero values, which of course is impossible to use as a real filter. Then you can go about creating variously optimised bandlimited methods of approximating this response for example by using different windows to sample the sinc.
There are also other methods such as using polynomial interpolation or splines.
There are some notes on the mathematics of this here and on other pages linked from here: http://ccrma.stanford.edu/~jos/Interpolation/sinc_function.html
Nick
-----Original Message----- From: pd-list-admin@iem.at [mailto:pd-list-admin@iem.at]On Behalf Of Larry Troxler Sent: Saturday, 15 May 2004 1:33 AM To: Roman Haefeli Cc: post pd-msg Subject: Re: [PD] smallest possible value of a delay-time
On Fri, 14 May 2004, Roman Haefeli wrote:
hi
i had a discussion with a teacher today. the topic was the smallest delay-time possible. in his opinion one sample is the atom of signal and cannot be divided anymore. in my opinion it should be possible to get shorter delays than 1 samples with interpolation. my argument
was: it should
be possible to set the values of each sample so, that the
resulting signal
would be similar to a digitized analogue signal with a shorter
than 1 sample
delay.
It's certainly possible to get bandlimited fractional sample delays, using all pass filters. I don't remember if the minimum possible delay of the filter (if it can give a fractional delay less than one sample). But this is the technique used to tune physical waveguide models.
Try searching the web for "fractional" "waveguide" "allpass" or the like.
Larry Troxler
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