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2-D FIR filter using 1-D window method

`h = fwind1(Hd,win)`

`h = fwind1(Hd,win1,win2)`

`h = fwind1(f1,f2,Hd,___)`

designs a two-dimensional FIR filter `h`

= fwind1(`Hd`

,`win`

)`h`

with frequency response
`Hd`

. (`fwind1`

returns `h`

as a computational molecule, which is the appropriate form to use with
`filter2`

.) `fwind1`

uses the one-dimensional
window `win`

to form an approximately circularly symmetric
two-dimensional window using Huang's method.

`fwind1`

designs two-dimensional FIR filters using the window
method. `fwind1`

uses a one-dimensional window specification to
design a two-dimensional FIR filter based on the desired frequency response
`Hd`

. `fwind1`

works with 1-D windows only;
use `fwind2`

to work with two-dimensional windows.

`fwind1`

takes a one-dimensional window specification and forms an
approximately circularly symmetric two-dimensional window using Huang's method,

$$w({n}_{1},{n}_{2})={w(t)|}_{t=\sqrt{{n}_{{}_{1}}^{2}+{n}_{2}^{2}}},$$

where *w(t)* is the one-dimensional window and
*w(n*_{1}*,n*_{2}*)*
is the resulting two-dimensional window.

Given two windows, `fwind1`

forms a separable two-dimensional
window:

$$w({n}_{1},{n}_{2})={w}_{1}({n}_{1}){w}_{2}({n}_{2}).$$

`fwind1`

calls `fwind2`

with `Hd`

and the two-dimensional window. `fwind2`

computes `h`

using an inverse Fourier transform and multiplication by the two-dimensional
window:

$${h}_{d}({n}_{1},{n}_{2})=\frac{1}{{\left(2\pi \right)}^{2}}{\displaystyle {\int}_{-\pi}^{\pi}{\displaystyle {\int}_{-\pi}^{\pi}{H}_{d}({\omega}_{1},{\omega}_{2}){e}^{j{\omega}_{1}{n}_{1}}{e}^{j{\omega}_{2}{n}_{2}}d{\omega}_{1}d{\omega}_{2}}}$$

$$h({n}_{1},{n}_{2})={h}_{d}({n}_{1},{n}_{2})w({n}_{2},{n}_{2}).$$

[1] Lim, Jae S., *Two-Dimensional Signal and Image
Processing*, Englewood Cliffs, NJ, Prentice Hall, 1990.

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