Document filter coefficients. [CI SKIP]

doc/filters/daubechies.qbk

@@ -40,20 +40,37 @@ The first ambiguity is whether they should be indexed by the number of vanishing
 /Boost.Math indexes the family by the number of vanishing moments./
 This indexing convention is the same as PyWavelets and Mathematica, but differs from Numerical Recipes.
 
-Next, we need to ask whether the coefficients 
+The next ambiguity is the overall scale.
+We (and PyWavelets) normalize the filters so that their L[sub 2] norm is 1.
+Others normalize so that the L[sub 2] norm is [sqrt]2.
 
-Here are some printing of the coefficents to help recognize the ambiguities:
+Finally, we can use a convolutional representation of the filters, or a dot product representation.
+The dot product has all elements reversed relative to the convolutional representation.
 
 
-WaveletFilterCoefficients[DaubechiesWavelet[2], "PrimalLowpass"]
-{{0, 0.341506}, {1, 0.591506}, {2, 0.158494}, {3, -0.0915064}}
 
->>> print(pywt.Wavelet('db2').dec_lo)
-[-0.12940952255126037, 0.2241438680420134, 0.8365163037378079, 0.48296291314453416]
->>> np.linalg.norm(pywt.Wavelet('db2').dec_lo)
-1.0
+The Boost convention:
+
+    // ... 
+    auto h = boost::math::filters::daubechies_scaling_filter<double, 2>();
+    std::cout << "h = {" << h[0] << ", " << h[1] << ", " << h[2] << ", " << h[3] << "}\n";
+    // output:
+    h = {0.48296291314453416, 0.83651630373780794, 0.22414386804201339, -0.12940952255126037}
 
 
+Mathematica conventions:
+
+    WaveletFilterCoefficients[DaubechiesWavelet[2], "PrimalLowpass"]
+    {{0, 0.341506}, {1, 0.591506}, {2, 0.158494}, {3, -0.0915064}}
+
+
+PyWavelet conventions:
+
+    >>> import pywt
+    >>> print(pywt.Wavelet('db2').dec_lo)
+    [-0.12940952255126037, 0.2241438680420134, 0.8365163037378079, 0.48296291314453416]
+    >>> np.linalg.norm(pywt.Wavelet('db2').dec_lo)
+    1.0
 
 
 

doc/math.qbk

@@ -730,6 +730,10 @@ and as a CD ISBN 0-9504833-2-X  978-0-9504833-2-0, Classification 519.2-dc22.
 [include differentiation/lanczos_smoothing.qbk]
 [endmathpart]
 
+[mathpart filters Filters]
+[include filters/daubechies.qbk]
+[endmathpart]
+
 [include complex/complex-tr1.qbk]
 [include quaternion/math-quaternion.qbk]
 [include octonion/math-octonion.qbk]