ftrans2

2-D FIR filter using frequency transformation

Syntax

h = ftrans2(b,t)

h = ftrans2(b)

Description

h = ftrans2(b,t) produces the two-dimensional FIR filter h that corresponds to the one-dimensional FIR filter b using the transform t. The transform matrix t contains coefficients that define the frequency transformation to use.

h = ftrans2(b) uses the McClellan transform matrix t.

t = [1 2 1; 2 -4 2; 1 2 1]/8;

Examples

Design Circularly Symmetric 2-D Bandpass Filter

Use ftrans2 to design an approximately circularly symmetric two-dimensional bandpass filter with passband between 0.1 and 0.6 (normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians). Since ftrans2 transforms a one-dimensional FIR filter to create a two-dimensional filter, first design a one-dimensional FIR bandpass filter using the Signal Processing Toolbox™ function firpm.

colormap(jet(64))
b = firpm(10,[0 0.05 0.15 0.55 0.65 1],[0 0 1 1 0 0]);
[H,w] = freqz(b,1,128,'whole');
plot(w/pi-1,fftshift(abs(H)))

Use ftrans2 with the default McClellan transformation to create the desired approximately circularly symmetric filter.

h = ftrans2(b);
freqz2(h)

Input Arguments

b — FIR filter

numeric matrix

FIR filter, specified as a numeric matrix. b must be a 1-D Type I (even symmetric, odd-length) filter such as can be returned by fir1 (Signal Processing Toolbox), fir2 (Signal Processing Toolbox), or firpm (Signal Processing Toolbox).

Data Types: double

t — Transform matrix

numeric matrix

Transform matrix, specified as a numeric matrix. t contains coefficients that define the frequency transformation to use. By default, ftrans2 uses a McClellan transform matrix.

Data Types: double

Output Arguments

h — 2-D FIR filter

numeric matrix

2-D FIR filter, returned as a numeric matrix. ftrans2 returns h as a correlation kernel, which is the appropriate form to use with filter2. If t is m-by-n and b has length Q, then h is size ((m-1)*(Q-1)/2+1)-by-((n-1)*(Q-1)/2+1).

Algorithms

The transformation below defines the frequency response of the two-dimensional filter returned by ftrans2.

$${H_{({\omega }_1,{\omega }_2)}=B(\omega )|}_{\cos \omega =T({\omega }_1,{\omega }_2)},$$

where B(ω) is the Fourier transform of the one-dimensional filter b:

$$B(\omega )={\displaystyle \sum _{n=-N}^{N}b(n)e^{-j\omega n}}$$

and T(ω₁,ω₂) is the Fourier transform of the transformation matrix t:

$$T({\omega }_1,{\omega }_2)={\displaystyle \sum _{n_2}{\displaystyle \sum _{n_1}t(n_1,n_2)e^{-j{\omega }_1{n}_1}{e}^{-j{\omega }_2{n}_2}}}.$$

The returned filter h is the inverse Fourier transform of H(ω₁,ω₂):

$$h(n_1,n_2)=\frac{1}{{\left(2\pi \right)}^2}{\displaystyle {\int }_{-\pi }^{\pi }{\displaystyle {\int }_{-\pi }^{\pi }H({\omega }_1,{\omega }_2)e^{j{\omega }_1{n}_1}{e}^{j{\omega }_2{n}_2}d{\omega }_{1}d{\omega }_2}}.$$

References

[1] Lim, Jae S., Two-Dimensional Signal and Image Processing, Englewood Cliffs, NJ, Prentice Hall, 1990, pp. 218-237.

Version History

Introduced before R2006a