FIR Hilber Transformer

I. R. Khan wrote:
> Thanks RBJ for explaining. I have another related question, and hope you
> or some one else can explain.
>
> We can transform an even length FIR differentiator to a Hilbert
> transformer by just making few changes to the signs of the impulse
> response coefficients. If we just assign negative sign to the first half
> of the coefficients and positive sign to the second half, we get a HT.
> Why does it happen? What actually happens to the frequency (magnitude)
> response of differentiator by making these changes to the impulse
> response coefficients?

i dunno.  is the sign change happening in some kind of pattern?  is it
every other coefficient that gets multiplied by -1?  perhaps, in the
frequency domain there is a sorta H1(f) = f getting added to H(f) = 1-f
which results in a sorta constant magnitude.  but none of this is
something i have thought of before.

> I can understand the transformation of differentiator to halfband
> lowpass filter, that we take derivative of the differentiator's
> frequency response (multiply impulse response coefficients with
> indices), squeeze to half (insert zeros in impulse response) and scale
> (set middle coefficient to 1/2 etc). How can we explain transformation
> of differentiator to HT?

boy, there are a lot of concepts here that i have never thunked about.
halfband design of the H.T. is something i have thought about (it needs
to be odd lengthed FIR and nearly half of the taps have zero
coefficients making it a little more efficient).  but this
transformation of differentiator to H.T. is not something i had thought
about.  but there is some "ring of truth" to what you are alluding to
(differentiating the linear gain will get you constant gain).  it does
not sound like crap.

> Ishtiaq.

now we know what the "I" stands for.

robert bristow-johnson wrote:

  ...

> in a sense, that's a better comparison since the integrator and H.T.
> are both 90 degree lag.

Is an inverter 180-degree lag, or 180-degree lead? Suppose I invert the
output of an HT; that sure looks like a lead if uninverted is a lag.

An inverted integrator makes negative output with DC in.  An inverted
differentiator makes negative spikes with positive steps in. Those
behaviors establish a reference. I know no comparable test for an
inverted HT, so it can lead or lag as one pleases.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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