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# fwind2

2-D FIR filter using 2-D window method

## Syntax

h = fwind2(Hd, win)
h = fwind2(f1, f2, Hd, win)

## Description

Use fwind2 to design two-dimensional FIR filters using the window method. fwind2 uses a two-dimensional window specification to design a two-dimensional FIR filter based on the desired frequency response Hd. fwind2 works with two-dimensional windows; use fwind1 to work with one-dimensional windows.

h = fwind2(Hd, win) produces the two-dimensional FIR filter h using an inverse Fourier transform of the desired frequency response Hd and multiplication by the window win. Hd is a matrix containing the desired frequency response at equally spaced points in the Cartesian plane. fwind2 returns h as a computational molecule, which is the appropriate form to use with filter2. h is the same size as win.

For accurate results, use frequency points returned by freqspace to create Hd. (See the entry for freqspace for more information.)

h = fwind2(f1, f2, Hd, win) lets you specify the desired frequency response Hd at arbitrary frequencies (f1 and f2) along the x- and y-axes. The frequency vectors f1 and f2 should be in the range -1.0 to 1.0, where 1.0 corresponds to half the sampling frequency, or π radians. h is the same size as win.

## Class Support

The input matrix Hd can be of class double or of any integer class. All other inputs to fwind2 must be of class double. All outputs are of class double.

## Examples

Use fwind2 to design an approximately circularly symmetric two-dimensional bandpass filter with passband between 0.1 and 0.5 (normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians):

1. Create a matrix Hd that contains the desired bandpass response. Use freqspace to create the frequency range vectors f1 and f2.

```[f1,f2] = freqspace(21,'meshgrid');
Hd = ones(21);
r = sqrt(f1.^2 + f2.^2);
Hd((r<0.1)|(r>0.5)) = 0;
colormap(jet(64))
mesh(f1,f2,Hd)```

2. Create a two-dimensional Gaussian window using fspecial.

```win = fspecial('gaussian',21,2);
win = win ./ max(win(:));  % Make the maximum window value be 1.
mesh(win)```

3. Design the filter using the window from step 2.

```h = fwind2(Hd,win);
freqz2(h)```

expand all

### Algorithms

fwind2 computes h using an inverse Fourier transform and multiplication by the two-dimensional window win.

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

$h\left({n}_{1},{n}_{2}\right)={h}_{d}\left({n}_{1},{n}_{2}\right)w\left({n}_{1},{n}_{2}\right)$

## References

[1] Lim, Jae S., Two-Dimensional Signal and Image Processing, Englewood Cliffs, NJ, Prentice Hall, 1990, pp. 202-213.