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In order for the elements in the spectrum
to represent different sequency components contained in the
signal in a low-to-high order, we can re-order the rows (or columns) of
the Hadamard matrix
according to their sequencies.
The conversion of a given sequency
into the corresponding index number
in Hadamard order is a three-step process:
- represent
in binary form:
- convert this binary form to Gray code:
where
represents exclusive or and
by definition.
- bit-reverse
's to get
's:
Now
can be found as
where
or equivalently
.
For example,
, we have
Now the sequency-ordered or Walsh-ordered Walsh-Hadamard matrix can be
obtained as
The first column on the right of the matrix is for the sequency
of
the corresponding row, which is the index for the sequency-ordered
matrix, and the second column is the index
of the Hadamard ordered.
We see that this matrix is still symmetric:
Next: Fast Walsh-Hadamard Transform (Sequency
Up: wht
Previous: The Walsh-Hadamard Transform (Hadamard
Ruye Wang
2013-10-22