Relation To Fourier Transform
This transform differs from the classic Fourier transform in the choice of the kernel. In the Fourier transform, we have the exponential kernel: where i is the imaginary unit.
The two transforms are closely related, however, and the Fourier transform (assuming it uses the same normalization convention) can be computed from the Hartley transform via:
That is, the real and imaginary parts of the Fourier transform are simply given by the even and odd parts of the Hartley transform, respectively.
Conversely, for real-valued functions f(t), the Hartley transform is given from the Fourier transform's real and imaginary parts:
where and denote the real and imaginary parts of the complex Fourier transform.
Read more about this topic: Hartley Transform
Famous quotes containing the words relation to, relation and/or transform:
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“Our sympathy is cold to the relation of distant misery.”
—Edward Gibbon (1737–1794)
“Bees plunder the flowers here and there, but afterward they make of them honey, which is all theirs; it is no longer thyme or marjoram. Even so with the pieces borrowed from others; one will transform and blend them to make a work that is all one’s own, that is, one’s judgement. Education, work, and study aim only at forming this.”
—Michel de Montaigne (1533–1592)