# What windowed Sinc might be closest to a Parks–McClellan low-pass filter?

In another question on dsp.stackexchange, a statement was made that the output of MATLAB firpm() or firls() for an anti-alias low-pass filter was "close" to the coefficients of a windowed-Sinc low-pass filter.

If so:

What window (applied to a Sinc "perfect" reconstruction waveform) would provide low-filter kernel coefficients closest (In least squared error or other commonly used metric) to those of a same-length FIR filter designed by the Parks–McClellan or Remez-exchange methods? Is it one of the well-known named window functions (Nuttall et.al.)? Or is there a good numerical or tractable equation approximation to this window?

• Perhaps you can derive some optimally-matched window to what you would get from Parks-McClellan (I doubt it), but the "classic" window that seems to have the most similar behavior to me would be the Dolph-Chebyshev window. It has equiripple behavior in the stopband, although not in the passband. – Jason R May 18 '15 at 12:28
• one thing i tried to do about 25 years ago was, by using the Remez exchange algorithm attached to a different set of basis functions, to design an optimal window that would multiply a $\text{sinc}(\cdot)$ function to get minimax error on a brick-wall target. assuming zero-phase (fix the delay later) it was like $$\text{sinc}(t) w(t) = \text{sinc}(t) \sum\limits_{n=0}^{N} a_n t^{2n} \text{rect}(Bt)$$ where $\frac{1}{B}$ is the length of the FIR kernel and $B$ is related to the transition bandwidth of the brick-wall LPF kernel. my attempt was not successful. – robert bristow-johnson May 19 '15 at 18:26
• above, the basis functions for the Remez exchange algorithm would be $$H_n(f) = \mathcal{F} \left\{ \text{sinc}(t) t^{2n} \text{rect}(Bt) \right\}$$ for $0 \le n \le N$ and the $a_n$ would be the coefficients that Remez would optimally be looking for (to fit the sum to an ideal brickwall $\text{rect}(f)$ function) like $$H(f) = \sum\limits_{n=0}^{N} a_n H_n(f) \approx \text{rect}(f)$$ i actually asked this of Jim McC once and he had an explanation for why it wouldn't work that i did not understand. – robert bristow-johnson May 19 '15 at 18:45
• it seems to me that Tchebyshev polynomials could be set up to work for this, and i don't remember exactly how this failed, but i remember i was not successful. – robert bristow-johnson May 19 '15 at 18:49