This example shows how to determine the Daubechies extremal phase scaling filter with a specified sum. The two most common values for the sum are $$\sqrt{2}$$ and 1.

Construct two versions of the db4 scaling filter. One scaling filter sums to $$\sqrt{2}$$ and the other version sums to 1.

NumVanishingMoments = 4;
h = dbaux(NumVanishingMoments,sqrt(2));
m0 = dbaux(NumVanishingMoments,1);

The filter with sum equal to $$\sqrt{2}$$ is the synthesis (reconstruction) filter returned by wfilters and used in the discrete wavelet transform.

[LoD,HiD,LoR,HiR] = wfilters('db4');
max(abs(LoR-h))

ans = 4.2590e-13

For orthogonal wavelets, the analysis (decomposition) filter is the time-reverse of the synthesis filter.

max(abs(LoD-fliplr(h)))

ans = 4.2590e-13

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)

wdb4 = 1×8

    0.1629    0.5055    0.4461   -0.0198   -0.1323    0.0218    0.0233   -0.0075

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)

wsym4 = 1×8

    0.0228   -0.0089   -0.0702    0.2106    0.5683    0.3519   -0.0210   -0.0536

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 = 1.8677e-15

This example demonstrates that for a given support, the cumulative sum of the squared coefficients of a scaling filter increase more rapidly for an extremal phase wavelet than other wavelets.

Generate the scaling filter coefficients for the db15 and sym15 wavelets. Both wavelets have support of width $2\times 15-1=29$.

[~,~,LoR_db,~] = wfilters('db15');
[~,~,LoR_sym,~] = wfilters('sym15');

Next, generate the scaling filter coefficients for the coif5 wavelet. This wavelet also has support of width $6\times 5-1=29$.

[~,~,LoR_coif,~] = wfilters('coif5');

Confirm the sum of the coefficients for all three wavelets equals $\sqrt{2}$.

sqrt(2)-sum(LoR_db)

ans = 2.2204e-16

sqrt(2)-sum(LoR_sym)

ans = 0

sqrt(2)-sum(LoR_coif)

ans = 2.2204e-16

Plot the cumulative sums of the squared coefficients. Note how rapidly the Daubechies sum increases. This is because its energy is concentrated at small abscissas. Since the Daubechies wavelet has extremal phase, the cumulative sum of its squared coefficients increases more rapidly than the other two wavelets.

plot(cumsum(LoR_db.^2),'rx-')
hold on
plot(cumsum(LoR_sym.^2),'mo-')
plot(cumsum(LoR_coif.^2),'b*-')
legend('Daubechies','Symlet','Coiflet')
title('Cumulative Sum')