Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then $0,fc,fs,\pi$ all get mapped from $[0,1]$. Since FIR filter is a polynomial, we can think of the filter as a polynomial map from $[0,1] \rightarrow [0,1]$. Kaiser's formula gives FIR filter length or the degree of the polynomial in terms of $fs-fc, ds$ and $dp$.

Now consider you are looking for a polynomial map from $[0,1]^n\rightarrow[0,1]$ so that a fixed subset of $[0,1]^n$ is the pass band and the complementary subset is the stop band. Is there a way to generalize Kasier's formula in this settings so that we can get an estimate on the total degree of the multivariate polynomial in this setting? Also what is design of such $n$-dimesnional filters (that is find the actual polynomials)?

One simple case is the following boundary conditions:

$(1)$ From frequencies $(0,0)$ to $(0,1)$ you have a low pass structure (that is $(0,0)$ has gain $1$ while $(0,1)$ has gain $0$).

$(2)$ From frequencies $(1,0)$ to $(1,1)$ you have a high pass structure (that is $(1,0)$ has gain $0$ while $(1,1)$ has gain $1$).

$(3)$ From frequencies $(0,0)$ to $(1,0)$ you have a Low pass structure (that is $(0,0)$ has gain $1$ while $(1,0)$ has gain $0$).

$(4)$ From frequencies $(0,1)$ to $(1,1)$ you have a high pass structure (that is $(0,1)$ has gain $0$ while $(1,1)$ has gain $1$).

We cannot have a direct product structure inside $[0,1]^2$ for the passband or the stopband since the boundary conditions flip on the opposite edges of the rectangle.

Even in this simple case Kaiser Bound does not work (even if interior conditions are disregarded).

Cross-posted http://dsp.stackexchange.com/questions/19759/basics-of-multidimensional-filter-design

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then $0,fc,fs,\pi$ all get mapped from $[0,1]$. Since FIR filter is a polynomial, we can think of the filter as a polynomial map from $[0,1] \rightarrow [0,1]$. Kaiser's formula gives FIR filter length or the degree of the polynomial in terms of $fs-fc, ds$ and $dp$.

Now consider you are looking for a polynomial map from $[0,1]^n\rightarrow[0,1]$ so that a fixed subset of $[0,1]^n$ is the pass band and the complementary subset is the stop band. Is there a way to generalize Kasier's formula in this settings so that we can get an estimate on the total degree of the multivariate polynomial in this setting? Also what is design of such $n$-dimesnional filters (that is find the actual polynomials)?

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by pi, then $0,fc,fs,\pi$ all get mapped from $[0,1]$. Since FIR filter is a polynomial, we can think of the filter as a polynomial map from $[0,1] \rightarrow [0,1]$. Kaiser's formula gives FIR filter length or the degree of the polynomial in terms of $fs-fc, ds$ and $dp$.

Now consider you are looking for a polynomial map from $[0,1]^n\rightarrow[0,1]$ so that a fixed subset of $[0,1]^n$ is the pass band and the complementary subset is the stop band. Is there a way to generalize Kasier's formula in this settings so that we can get an estimate on the total degree of the multivariate polynomial in this setting? Also what is design of such $n$-dimesnional filters (that is find the actual polynomials)?

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then $0,fc,fs,\pi$ all get mapped from $[0,1]$. Since FIR filter is a polynomial, we can think of the filter as a polynomial map from $[0,1] \rightarrow [0,1]$. Kaiser's formula gives FIR filter length or the degree of the polynomial in terms of $fs-fc, ds$ and $dp$.

Now consider you are looking for a polynomial map from $[0,1]^n\rightarrow[0,1]$ so that a fixed subset of $[0,1]^n$ is the pass band and the complementary subset is the stop band. Is there a way to generalize Kasier's formula in this settings so that we can get an estimate on the total degree of the multivariate polynomial in this setting? Also what is design of such $n$-dimesnional filters (that is find the actual polynomials)?

One simple case is the following boundary conditions:

$(1)$ From frequencies $(0,0)$ to $(0,1)$ you have a low pass structure (that is $(0,0)$ has gain $1$ while $(0,1)$ has gain $0$).

$(2)$ From frequencies $(1,0)$ to $(1,1)$ you have a high pass structure (that is $(1,0)$ has gain $0$ while $(1,1)$ has gain $1$).

$(3)$ From frequencies $(0,0)$ to $(1,0)$ you have a Low pass structure (that is $(0,0)$ has gain $1$ while $(1,0)$ has gain $0$).

$(4)$ From frequencies $(0,1)$ to $(1,1)$ you have a high pass structure (that is $(0,1)$ has gain $0$ while $(1,1)$ has gain $1$).

We cannot have a direct product structure inside $[0,1]^2$ for the passband or the stopband since the boundary conditions flip on the opposite edges of the rectangle.

Even in this simple case Kaiser Bound does not work (even if interior conditions are disregarded).

Cross-posted http://dsp.stackexchange.com/questions/19759/basics-of-multidimensional-filter-design