symaux

(To be removed) Symlet wavelet filter computation

symaux will be removed in a future release. Use daubfactors instead. (since R2026a) For more information, see Version History.

Syntax

w = symaux(n)

w = symaux(___,sumw)

Description

The symaux function generates the scaling filter coefficients for the "least asymmetric" Daubechies wavelets.

w = symaux(n) is the order n Symlet scaling filter such that sum(w) = 1.

Note

• Instability may occur when n is too large. Starting with values of n in the 30s range, function output will no longer accurately represent scaling filter coefficients.

• As n increases, the time required to compute the filter coefficients rapidly grows.

• For n = 1, 2, and 3, the order n Symlet filters and order n Daubechies filters are identical. See Extremal Phase Wavelet.

w = symaux(___,sumw) is the order n Symlet scaling filter such that sum(w) = sumw.

w = symaux(n,0) is equivalent to w = symaux(n,1).

Examples

Unit Norm Scaling Filter Coefficients

In this example you will generate symlet scaling filter coefficients whose norm is equal to 1. You will also confirm the coefficients satisfy a necessary relation.

Compute the scaling filter coefficients of the order 10 symlet whose sum equals$\sqrt{2}$.

n = 10;
w = symaux(n,sqrt(2));

Confirm the sum of the coefficients is equal to $\sqrt{2}$ and the norm is equal to 1.

sqrt(2)-sum(w)

ans = 
2.2204e-16

1-sum(w.^2)

ans = 
-3.3307e-14

Since integer translations of the scaling function form an orthogonal basis, the coefficients satisfy the relation ${\sum }_{\mathit{n}}\mathit{w}\left(\mathit{n}\right)\mathit{w}\left(\mathit{n}-2\mathit{k}\right)=\delta \left(\mathit{k}\right)$. Confirm this by taking the autocorrelation of the coefficients and plotting the result.

corrw = xcorr(w,w);
stem(corrw)
grid on
title('Autocorrelation of scaling coefficients')

stem(corrw(2:2:end))
grid on
title('Even-indexed autocorrelation values')

Symlet and Daubechies Scaling Filters

This example shows that symlet and Daubechies scaling filters of the same order are both solutions of the same polynomial equation.

Generate the order 4 Daubechies scaling filter and plot it.

wdb4 = dbaux(4);
stem(wdb4)
title('Order 4 Daubechies Scaling Filter')

wdb4 is a solution of the equation: P = conv(wrev(w),w)*2, where P is the "Lagrange trous" filter for N = 4. Evaluate P and plot it. P is a symmetric filter and wdb4 is a minimum phase solution of the previous equation based on the roots of P.

P = conv(wrev(wdb4),wdb4)*2;
stem(P)
title('''Lagrange trous'' filter')

Generate wsym4, the order 4 symlet scaling filter and plot it. The Symlets are the "least asymmetric" Daubechies' wavelets obtained from another choice between the roots of P.

wsym4 = symaux(4);
stem(wsym4)
title('Order 4 Symlet Scaling Filter')

Compute conv(wrev(wsym4),wsym4)*2 and confirm that wsym4 is another solution of the equation P = conv(wrev(w),w)*2.

P_sym = conv(wrev(wsym4),wsym4)*2;
err = norm(P_sym-P)

err = 
1.2491e-15

Input Arguments

n — Order of symlet

positive integer

Order of the symlet, specified as a positive integer.

sumw — Sum of coefficients

1 (default) | positive real number

Sum of the scaling filter coefficients, specified as a positive real number. Set to sqrt(2) to generate vector of coefficients whose norm is 1.

Output Arguments

w — Filter coefficients

row vector

Vector of scaling filter coefficients of the order n symlet.

The scaling filter coefficients satisfy a number of properties. You can use these properties to check your results. See Unit Norm Scaling Filter Coefficients for additional information.

More About

Least Asymmetric Wavelet

The Haar wavelet, also known as the Daubechies wavelet of order 1, db1, is the only compactly supported orthogonal wavelet that is symmetric, or equivalently has linear phase. No other compactly supported orthogonal wavelet can be symmetric. However, it is possible to derive wavelets which are minimally asymmetric, meaning that their phase will be very nearly linear. For a given support, the orthogonal wavelet with a phase response that most closely resembles a linear phase filter is called least asymmetric.

Extremal Phase Wavelet

Constructing a compactly supported orthogonal wavelet basis involves choosing roots of a particular polynomial equation. Different choices of roots will result in wavelets whose phases are different. Choosing roots that lie within the unit circle in the complex plane results in a filter with highly nonlinear phase. Such a wavelet is said to have extremal phase, and has energy concentrated at small abscissas. Let {h_k} denote the set of scaling coefficients associated with an extremal phase wavelet, where k = 1,…,M. Then for any other set of scaling coefficients {g_k} resulting from a different choice of roots, the following inequality holds for all J = 1,…,M:

$${\displaystyle \sum _{k=1}^J{g}_k^2\le }{\displaystyle \sum _{k=1}^J{h}_k^2}.$$

The {h_k} are sometimes called a minimal delay filter [2].

The polynomial equation mentioned above depends on the desired number of vanishing moments N for the wavelet. To construct a wavelet basis involves choosing roots of the equation. In the case of least asymmetric wavelets and extremal phase wavelets for orders 1, 2, and 3, there are effectively no choices to make. For N = 1, 2, and 3, the dbN and symN filters are equal. The example Symlet and Daubechies Scaling Filters shows that two different scaling filters can satisfy the same polynomial equation. For additional information, see Daubechies [1].

References

[1] Daubechies, I. Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: SIAM Ed, 1992.

[2] Oppenheim, Alan V., and Ronald W. Schafer. Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1989.

Version History

Introduced before R2006a

R2026a: To be removed

symaux runs without warning but will be removed in a future release. Use daubfactors instead. The function unifies the functionality of symaux and dbaux, as well, enabling you to obtain the scaling filters of the three Daubechies families, "csym", "db", and "sym", through a single interface.