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The Hadamard Transform

The document discusses the Hadamard transform. It is well-suited for digital signal processing as its basis vectors only take values of ±1. The Hadamard transform matrices Hn are N x N and can be generated recursively using a core matrix and Kronecker products. As an example, the matrix for n=3 is given. The basis vectors can also be generated from Walsh functions, which form an orthonormal basis. The transform of a vector u is defined as v = Hu and the inverse as u = Hv. Properties include the transform being real, symmetric, orthogonal and fast to compute in O(N log2 N) time using only additions and subtractions.

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Krishanu Modak
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894 views4 pages

The Hadamard Transform

The document discusses the Hadamard transform. It is well-suited for digital signal processing as its basis vectors only take values of ±1. The Hadamard transform matrices Hn are N x N and can be generated recursively using a core matrix and Kronecker products. As an example, the matrix for n=3 is given. The basis vectors can also be generated from Walsh functions, which form an orthonormal basis. The transform of a vector u is defined as v = Hu and the inverse as u = Hv. Properties include the transform being real, symmetric, orthogonal and fast to compute in O(N log2 N) time using only additions and subtractions.

Uploaded by

Krishanu Modak
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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THE HADAMARD TRANSFORM

1. The elements of the basis vectors of the Hadamard transform take only the binary
values ± 1 and are, therefore, well suited for digital signal processing.

2. The Hadamard transform matrices, Hn , are N x N matrices, where

These can be easily generated by the core matrix

and the Kronecker product recursion

-------(1)

3. As an example for n=3, the Hadamard matrix becomes

--------(2)

which gives
4. The basis vectors of the Hadamard transform can also be generated by sampling a
class of functions called the Walsh functions. These functions also take only the
binary values ± 1 and form a complete orthonormal basis for square integrable
functions. For this reason the Hadamard transform just defined is also called the
Walsh-Hadamard transform.

5. The number of zero crossings of a Walsh function or the number of transitions in


a basis vector of the Hadamard transform is called its sequency. In the Hadamard
matrix generated via (2), the row vectors are not sequency ordered.

6. The existing sequency order of these vectors is called the Hadamard order. The
Hadamard transform of an N x 1 vector u is written as

v = Hu ----------------------------(3)

and the inverse transform is given by

u = Hv ----------------------------(4)

where

7. In series form, the transform pair becomes

where
and {ki},{mi} are the binary representations of k and m, respectively.

8. The basis vectors of the Hadamard transform is given by


9. Properties of the Hadamard Transform

(1)The Hadamard transform H is real, symmetric, and orthogonal, that is,

(2) The Hadamard transform is a fast transform. The one-dimensional transformation of


eqn (3) can be implemented in O(N log2 N) additions and subtractions.

Since the Hadamard transform contains only ±1 values, no multiplications are


required in the transform calculations.

(3) The natural order of the Hadamard transform coefficients turns out to be equal to
the bit reversed gray code representation of its sequency. Table below shows the
conversion of sequency s to natural order h, and vice versa, for N = 8.

(4) The Hadamard transform has good to very good energy compaction for highly
correlated images.

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