org.apache.commons.math.transform
Class FastHadamardTransformer

java.lang.Object
  org.apache.commons.math.transform.FastHadamardTransformer

All Implemented Interfaces:

RealTransformer

public class FastHadamardTransformer
extends Object
implements RealTransformer

Implements the Fast Hadamard Transform (FHT). Transformation of an input vector x to the output vector y.

In addition to transformation of real vectors, the Hadamard transform can transform integer vectors into integer vectors. However, this integer transform cannot be inverted directly. Due to a scaling factor it may lead to rational results. As an example, the inverse transform of integer vector (0, 1, 0, 1) is rational vector (1/2, -1/2, 0, 0).

Since:

2.0

Version:

$Revision: 1070725 $ $Date: 2011-02-15 02:31:12 +0100 (mar. 15 févr. 2011) $

Constructor Summary

FastHadamardTransformer()

Method Summary

protected double[]	fht(double[] x) The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.
protected int[]	fht(int[] x) The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.
double[]	inversetransform(double[] f) Inversely transform the given real data set.
double[]	inversetransform(UnivariateRealFunction f, double min, double max, int n) Inversely transform the given real function, sampled on the given interval.
double[]	transform(double[] f) Transform the given real data set.
int[]	transform(int[] f) Transform the given real data set.
double[]	transform(UnivariateRealFunction f, double min, double max, int n) Transform the given real function, sampled on the given interval.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

FastHadamardTransformer

public FastHadamardTransformer()

Method Detail

transform

public double[] transform(double[] f)
                   throws IllegalArgumentException

Transform the given real data set.

Specified by:

transform in interface RealTransformer

Parameters:

f - the real data array to be transformed (signal)

Returns:

the real transformed array (spectrum)

Throws:

IllegalArgumentException - if any parameters are invalid

transform

public double[] transform(UnivariateRealFunction f,
                          double min,
                          double max,
                          int n)
                   throws FunctionEvaluationException,
                          IllegalArgumentException

Transform the given real function, sampled on the given interval.

Specified by:

transform in interface RealTransformer

Parameters:

f - the function to be sampled and transformed

min - the lower bound for the interval

max - the upper bound for the interval

n - the number of sample points

Returns:

the real transformed array

Throws:

FunctionEvaluationException - if function cannot be evaluated at some point

IllegalArgumentException - if any parameters are invalid

inversetransform

public double[] inversetransform(double[] f)
                          throws IllegalArgumentException

Inversely transform the given real data set.

Specified by:

inversetransform in interface RealTransformer

Parameters:

f - the real data array to be inversely transformed (spectrum)

Returns:

the real inversely transformed array (signal)

Throws:

IllegalArgumentException - if any parameters are invalid

inversetransform

public double[] inversetransform(UnivariateRealFunction f,
                                 double min,
                                 double max,
                                 int n)
                          throws FunctionEvaluationException,
                                 IllegalArgumentException

Inversely transform the given real function, sampled on the given interval.

Specified by:

inversetransform in interface RealTransformer

Parameters:

f - the function to be sampled and inversely transformed

min - the lower bound for the interval

max - the upper bound for the interval

n - the number of sample points

Returns:

the real inversely transformed array

Throws:

FunctionEvaluationException - if function cannot be evaluated at some point

IllegalArgumentException - if any parameters are invalid

transform

public int[] transform(int[] f)
                throws IllegalArgumentException

Transform the given real data set.

The integer transform cannot be inverted directly, due to a scaling factor it may lead to double results.

Parameters:

f - the integer data array to be transformed (signal)

Returns:

the integer transformed array (spectrum)

Throws:

IllegalArgumentException - if any parameters are invalid

fht

protected double[] fht(double[] x)
                throws IllegalArgumentException

The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.
Requires Nlog2N = n2ⁿ additions.

Short Table of manual calculation for N=8:

1. x is the input vector we want to transform
2. y is the output vector which is our desired result
3. a and b are just helper rows

 
 +----+----------+---------+----------+
 | x  |    a     |    b    |    y     |
 +----+----------+---------+----------+
 | x0 | a0=x0+x1 | b0=a0+a1 | y0=b0+b1 |
 +----+----------+---------+----------+
 | x1 | a1=x2+x3 | b0=a2+a3 | y0=b2+b3 |
 +----+----------+---------+----------+
 | x2 | a2=x4+x5 | b0=a4+a5 | y0=b4+b5 |
 +----+----------+---------+----------+
 | x3 | a3=x6+x7 | b0=a6+a7 | y0=b6+b7 |
 +----+----------+---------+----------+
 | x4 | a0=x0-x1 | b0=a0-a1 | y0=b0-b1 |
 +----+----------+---------+----------+
 | x5 | a1=x2-x3 | b0=a2-a3 | y0=b2-b3 |
 +----+----------+---------+----------+
 | x6 | a2=x4-x5 | b0=a4-a5 | y0=b4-b5 |
 +----+----------+---------+----------+
 | x7 | a3=x6-x7 | b0=a6-a7 | y0=b6-b7 |
 +----+----------+---------+----------+
 
 

How it works

1. Construct a matrix with N rows and n+1 columns
   hadm[n+1][N]
   (If I use [x][y] it always means [row-offset][column-offset] of a Matrix with n rows and m columns. Its entries go from M[0][0] to M[n][m])
2. Place the input vector x[N] in the first column of the matrix hadm
3. The entries of the submatrix D_top are calculated as follows.
   D_top goes from entry [0][1] to [N/2-1][n+1].
   The columns of D_top are the pairwise mutually exclusive sums of the previous column
4. The entries of the submatrix D_bottom are calculated as follows.
   D_bottom goes from entry [N/2][1] to [N][n+1].
   The columns of D_bottom are the pairwise differences of the previous column
5. How D_top and D_bottom you can understand best with the example for N=8 above.
6. The output vector y is now in the last column of hadm
7. Algorithm from: http://www.archive.chipcenter.com/dsp/DSP000517F1.html

Visually

        +--------+---+---+---+-----+---+
        |   0    | 1 | 2 | 3 | ... |n+1|
 +------+--------+---+---+---+-----+---+
 |0     | x0     |       /\            |
 |1     | x1     |       ||            |
 |2     | x2     |   <= Dtop  =>       |
 |...   | ...    |       ||            |
 |N/2-1 | xN/2-1  |       \/            |
 +------+--------+---+---+---+-----+---+
 |N/2   | xN/2   |       /\            |
 |N/2+1 | xN/2+1  |       ||            |
 |N/2+2 | xN/2+2  |  <= Dbottom  =>      | which is in the last column of the matrix
 |...   | ...    |       ||            |
 |N     | xN/2   |        \/           |
 +------+--------+---+---+---+-----+---+
 

Parameters:

x - input vector

Returns:

y output vector

Throws:

IllegalArgumentException - if input array is not a power of 2

fht

protected int[] fht(int[] x)
             throws IllegalArgumentException

The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.

Parameters:

x - input vector

Returns:

y output vector

Throws:

IllegalArgumentException - if input array is not a power of 2