orthfilt

Orthogonal wavelet filters

Syntax

[LoD,HiD,LoR,HiR] = orthfilt(W)

Description

[LoD,HiD,LoR,HiR] = orthfilt(W) computes the four lowpass and highpass, decomposition and reconstruction filters associated with the scaling filter W corresponding to a wavelet.

Examples

Compute Orthogonal Wavelet Filters of Scaling Filter

Create a scaling filter associated with the Daubechies db8 wavelet.

W = dbwavf("db8"); 
stem(W)
title("Original Scaling Filter")

Compute the four filters associated with the scaling filter.

[LoD,HiD,LoR,HiR] = orthfilt(W); 

Plot the decomposition lowpass and highpass filters.

subplot(2,1,1)
stem(LoD)
title("Decomposition Lowpass Filter")
subplot(2,1,2)
stem(HiD)
title("Decomposition Highpass Filter")

Plot the reconstruction lowpass and highpass filters.

subplot(2,1,1)
stem(LoR)
title("Reconstruction Lowpass Filter")
subplot(2,1,2)
stem(HiR)
title("Reconstruction Highpass Filter")

Check for orthonormality in the decomposition filters.

df = [LoD;HiD];
rf = [LoR;HiR];
id = df*df'

id = 2×2

    1.0000   -0.0000
   -0.0000    1.0000

Check for orthonormality in the reconstruction filters.

id2 = rf*rf'

id2 = 2×2

    1.0000    0.0000
    0.0000    1.0000

Check for orthogonality by dyadic translation.

df = [LoD 0 0;HiD 0 0]; 
dft = [0 0 LoD; 0 0 HiD]; 
zer = df*dft'

zer = 2×2
10-12 ×

   -0.1895   -0.0000
    0.0000   -0.1895

Plot the low-frequency transfer modulus.

fftld = fft(LoD); 
freq = [1:length(LoD)]/length(LoD); 
figure
plot(freq,abs(fftld),"x-")
title("Transfer modulus: Lowpass")

Plot the high-frequency transfer modulus.

ffthd = fft(HiD);
plot(freq,abs(ffthd),"x-")
title("Transfer modulus: Highpass")

Input Arguments

W — Scaling filter

real-valued vector

Scaling filter corresponding to a wavelet, specified as a real-valued vector.

Output Arguments

LoD — Decomposition lowpass filter

real-valued vector

Decomposition lowpass filter associated with the scaling filter W, returned as a real-valued vector.

HiD — Decomposition highpass filter

real-valued vector

Decomposition highpass filter associated with the scaling filter W, returned as a real-valued vector.

LoR — Reconstruction lowpass filter

real-valued vector

Reconstruction lowpass filter associated with the scaling filter W, returned as a real-valued vector.

HiR — Reconstruction highpass filter

real-valued vector

Reconstruction highpass filter associated with the scaling filter W, returned as a real-valued vector.

Algorithms

For an orthogonal wavelet in the multiresolution framework, start with the scaling function ϕ and the wavelet function ψ. One of the fundamental relations is the twin-scale relation:

$$\frac{1}{2}\varphi \left(\frac{x}{2}\right)={\displaystyle \sum _{n\in Z}{w}_n}\varphi (x-n)$$

All the filters used in the dwt and idwt functions are intimately related to the sequence $${(w_n)}_{n\in Z}$$. If ϕ is compactly supported, the sequence (w_n) is finite and can be viewed as an FIR filter. The scaling filter W is a lowpass FIR filter of length 2N, with the sum 1, and with the norm of 1/√2.

For example, for a db3 scaling filter,

w = dbwavf("db3") 
w = 0.2352 0.5706 0.3252 -0.0955 -0.0604 0.0249

sum(w)
 = 1.000
norm(w)
 = 0.7071

Define four FIR filters from filter W of length 2N and norm 1.

Filters	Low-Pass	High-Pass
Decomposition	LoD	HiD
Reconstruction	LoR	HiR

The function computes the four filters using the following scheme.

HiR and LoR are quadrature mirror filters: HiR(k) = (-1)kLoR(2N + 1 - k), for k = 1, 2, … , 2N. Because wrev reverses vectors, HiD and LoD are also quadrature mirror filters.

References

[1] Daubechies, Ingrid. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. Philadelphia, Pa: Society for Industrial and Applied Mathematics, 1992.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced before R2006a