> But the delay in b[n] is fractional, 1/7 sample, and the resulting
> windowed and sampled sinc does not have linear phase response. It is
> interesting that with infinite many samples, strict symmetry of the
> coeffcients is not needed anymore for linear phase.
Silly me! I took it to be 7 samples. 1/7 is so bizarre, I filtered it out.
Engineering is the art of making what you want from things you can get.
Reply by Greg Berchin●December 15, 20052005-12-15
On 14 Dec 2005 20:44:03 -0800, "robert bristow-johnson"
>but it's not so much the ripples ramping up to the main lobe of the
>impulse response, Al. it's a consequence of using the Parks-McClellan
>algorithm to design an "equi-ripple" filter. those ripples in the
>pass-band are like multiplying by a cosine which has the effect of
>translation in the other domain.
Harris describes an analogous phenomenon in his discussion of the
Dolph-Chebyshev window; "On the Use of Windows for Harmonic Analysis
with the Discrete Fourier Transform", Proceedings of the IEEE, Vol. 66,
No. 1, January 1978. This window is characterized by equal-amplitude
sidelobes (Harris describes them as "almost sinusoidal!") that result in
displaced impulses when transformed into the other domain.
Reply by ●December 15, 20052005-12-15
robert bristow-johnson wrote:
> email@example.com wrote:
> > You claim that the infinitely long sequence
> > b[n] = sinc(n + 1/7),
> > for all n, is linear-phase. That could be true. Do you have an idea for
> > a proof?
> it appears to me that you already have it proved. at least for
> frequencies between -Nyquist to +Nyquist.
> since the acausal continuous-time impulse response
> h(t) = sinc(t - d)
> has frequency response as
> H(f) = exp(-j*2*pi*f*d) for |f| < 1/2 , zero otherwize,
> then H(f) is clearly linear phase for any constant delay, d. and since
> H(f) is 0 for |f| >= 1/2, h(t) can be sampled at a sampling rate,1/T,
> of 1 and have no aliasing:
> h[n] = h(n*T) = sinc(n - d)
> according to the sampling theorem the frequency response of the sampled
> function is the same as the frequency response of the continuous-time
> function getting sampled for frequencies with magnitude less than
> Nyquist (in this case, Nyquist is 1/2) and that frequency response is
> periodically extended for frequencies above Nyquist.
Sounds good to me. The truncated sinc in Randy's example was not
bandlimited, and the aliasing when sampling messed up the phase
linearity. I think Randy's proof outline was somewhere on along your
lines as well.
What do you think of that theorem that I posted earlier from this
(at the bottom of the first page).
I have a feeling that anti-symmetric linear-phase sequences are not
covered in the statement of the proof. This would make the "iff" claim
> does that do it for you, Abariska?
In case you hadn't noticed, "abariska" stands for Andor Bariska. It's
my old and inactive student account (go spam that) :-).