Assuming L as infinite theoritally means a rectangular window with
infinite width. Here H(W) would become FT(IFT(H[W]) = H[W].
Fred Marshall wrote:
> "Gert Baars" <
[email protected]> wrote in message
> news:
[email protected].. .
>
>>Nothing is wrong with the unwindowed sinc function if the #taps
>>are infinite and Ws >> Wc. Then the result is the exact H(W).
>>
>>
>>
>>
>>Fred Marshall wrote:
>>
>>>"Gert Baars" <
[email protected]> wrote in message
>>>news:
[email protected].. .
>>>
>>>
>>>>Hello,
>>>>
>>>>I'm trying understand designing a FIR filter from scratch
>>>>because I want to experiment with home-made windows.
>>>>
>>>>With H(W) = 1 for -W0 < W < W0
>>>> = 0 else
>>>>
>>>>After IFT(F[W]) the result f[t] is a sinc function.
>>>>This function is symmetrical to t=0
>>>>
>>>>Turning this function into h(n) without a window
>>>>
>>>>is the translation t = Ts(n-(L-1)/2)
>>>>
>>>>( so h(n) = f[Ts(n-(L-1)/2] )
>>>>
>>>>correct?
>>>
>>>
>>>Well, you really need a window if that's how you're going about it.
>>>The transition region can't be of zero width as in going from 1 to zero
>>>abruptly at W0.
>>>
>>>If you convolve the frequency domain function with a narrow "gate" you'll
>>>get a linear transition that corresponds to a wide sinc window in time.
>>>Other shapes, other time windows.....
>>>
>>>Fred
>
>
> Oh, OK - so you are assuming that H(w) is a continuous and periodic
> function.
> So, the IFT is effectively the computation of a Fourier Series ...
> and, it has an infinite number of terms as usual so h(n) is an infinite
> series.
>
> If H(w) isn't a continuous function, but rather a discrete sequence, then
> h(t) will be periodic as well - so not treated as infinite.
>
> However.....
> With L as the length of the filter, it is *not* infinite. With "n" the time
> index, then a causal filter of length L would normally be defined such that
> the beginning of the impulse response of the filter is at time zero (so I
> suppose you mean n=0??) and the end of the impulse response is at time
> (L-1)*T where T is the sampling interval.
>
> This means the center of the filter is at (L-1)*T/2
> If L is odd, this is an integer multiple of T.
> If L is even, this is an (integer + 1/2)*T
>
> Taking the center "L" samples out of an infinite sequence, *is* a
> windowing - it's just that the window is rectangular with no otherwise
> "interesting" shape.
> If you rectangularly window a discrete sequence in time then the result is
> still periodic in frequency. The truncation causes Gibb's phenomenon at the
> sharp transitions in frequency. Normally these are viewed as undesirable
> trillies - thus the use of more gradual windows as in the "Windowing Method"
> of filter design.
>
> I'm following this but I remain unclear as to your objective. It can't be
> both infinite in time and not infinite in time.
>
> Fred
>
>