org.apache.commons.math3.transform

## Class FastHadamardTransformer

- java.lang.Object
- - org.apache.commons.math3.transform.FastHadamardTransformer

All Implemented Interfaces:
  Serializable, RealTransformer

    public class FastHadamardTransformer
    extends Object
    implements RealTransformer, Serializable

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:
  $Id: FastHadamardTransformer.java 1385310 2012-09-16 16:32:10Z tn $
See Also:
  Serialized Form

### Field Summary

Fields
`(package private) static long` `serialVersionUID` Serializable version identifier.

### Constructor Summary

Constructors
Constructor and Description
`FastHadamardTransformer()`

### Method Summary

Methods
`protected double[]` `fht(double[] x)` The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.
`protected int[]` `fht(int[] x)` Returns the forward transform of the specified integer data set.
`double[]` `transform(double[] f, TransformType type)` Returns the (forward, inverse) transform of the specified real data set.
`int[]` `transform(int[] f)` Returns the forward transform of the specified integer data set.The integer transform cannot be inverted directly, due to a scaling factor which may lead to double results.
`double[]` `transform(UnivariateFunction f, double min, double max, int n, TransformType type)` Returns the (forward, inverse) transform of the specified real function, sampled on the specified interval.

### Methods inherited from class java.lang.Object

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

### Field Detail

#### serialVersionUID

    static final long serialVersionUID

Serializable version identifier.

See Also:
  Constant Field Values

### Constructor Detail

#### FastHadamardTransformer

    public FastHadamardTransformer()

### Method Detail

#### transform

    public double[] transform(double[] f,
                     TransformType type)

Returns the (forward, inverse) transform of the specified real data set.

Specified by:
  `transform` in interface `RealTransformer`
Parameters:
  `f` - the real data array to be transformed (signal)
  `type` - the type of transform (forward, inverse) to be performed
Returns:
  the real transformed array (spectrum)
Throws:
  `MathIllegalArgumentException` - if the length of the data array is not a power of two

#### transform

    public double[] transform(UnivariateFunction f,
                     double min,
                     double max,
                     int n,
                     TransformType type)

Returns the (forward, inverse) transform of the specified real function, sampled on the specified interval.

Specified by:
  `transform` in interface `RealTransformer`
Parameters:
  `f` - the function to be sampled and transformed
  `min` - the (inclusive) lower bound for the interval
  `max` - the (exclusive) upper bound for the interval
  `n` - the number of sample points
  `type` - the type of transform (forward, inverse) to be performed
Returns:
  the real transformed array
Throws:
  `NonMonotonicSequenceException` - if the lower bound is greater than, or equal to the upper bound
  `NotStrictlyPositiveException` - if the number of sample points is negative
  `MathIllegalArgumentException` - if the number of sample points is not a power of two

#### transform

    public int[] transform(int[] f)

Returns the forward transform of the specified integer data set.The integer transform cannot be inverted directly, due to a scaling factor which may lead to double results.

Parameters:
  `f` - the integer data array to be transformed (signal)
Returns:
  the integer transformed array (spectrum)
Throws:
  `MathIllegalArgumentException` - if the length of the data array is not a power of two

#### fht

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

The FHT (Fast Hadamard Transformation) which uses only subtraction and addition. Requires `N * log2(N) = n * 2^n` additions.

### Short Table of manual calculation for N=8

1. x is the input vector to be transformed,
2. y is the output vector (Fast Hadamard transform of x),
3. a and b are 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][N])
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. The consputation of `D_top` and `D_bottom` are best understood with the above example (for `N = 8`).
6. The output vector `y` is now in the last column of `hadm`.
7. Algorithm from chipcenter.

### Visually

| 0 | 1 | 2 | 3 | … | n + 1
0 | x0 | ↑ ← Dtop → ↓
1 | x1
2 | x2
… | …
N / 2 - 1 | xN/2-1
N / 2 | xN/2 | ↑ ← Dbottom → ↓
N / 2 + 1 | xN/2+1
N / 2 + 2 | xN/2+2
… | …
N | xN

Parameters:
  `x` - the real data array to be transformed
Returns:
  the real transformed array, `y`
Throws:
  `MathIllegalArgumentException` - if the length of the data array is not a power of two

#### fht

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

Returns the forward transform of the specified integer data set. The FHT (Fast Hadamard Transform) uses only subtraction and addition.

Parameters:
  `x` - the integer data array to be transformed
Returns:
  the integer transformed array, `y`
Throws:
  `MathIllegalArgumentException` - if the length of the data array is not a power of two