Hartley transform
Encyclopedia
In mathematics
, the Hartley transform is an integral transform closely related to the Fourier transform
, but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform
by R. V. L. Hartley
in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real
functions to real functions (as opposed to requiring complex number
s) and of being its own inverse.
The discrete version of the transform, the Discrete Hartley transform
, was introduced by R. N. Bracewell
in 1983.
The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform
, with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase (Villasenor, 1994). However, optical Hartley transforms do not seem to have seen widespread use.
f(t) is defined by:
where
can in applications be an angular frequency
and
is the cosine-and-sine or Hartley kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain).
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.
(indeed, orthogonal
).
There is also an analogue of the convolution theorem
for the Hartley transform. If two functions
and 
have Hartley transforms 
and 
, respectively, then their convolution
has the Hartley transform:
Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively.
, and its definition as a phase-shifted trigonometric function
. For example, it has an angle-addition identity of:
Additionally:
and its derivative is given by:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hartley transform is an integral transform closely related to the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
by R. V. L. Hartley
Ralph Hartley
Ralph Vinton Lyon Hartley was an electronics researcher. He invented the Hartley oscillator and the Hartley transform, and contributed to the foundations of information theory.-Biography:...
in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
functions to real functions (as opposed to requiring complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s) and of being its own inverse.
The discrete version of the transform, the Discrete Hartley transform
Discrete Hartley transform
A discrete Hartley transform is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform , with analogous applications in signal processing and related fields. Its main distinction from the DFT is that it transforms real inputs to real outputs, with no...
, was introduced by R. N. Bracewell
Ronald N. Bracewell
Ronald Newbold Bracewell AO was the Lewis M. Terman Professor of Electrical Engineering, Emeritus of the at Stanford University.- Education :...
in 1983.
The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform
Fourier optics
Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...
, with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase (Villasenor, 1994). However, optical Hartley transforms do not seem to have seen widespread use.
Definition
The Hartley transform of a functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f(t) is defined by:
where
can in applications be an angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
and
is the cosine-and-sine or Hartley kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain).
Inverse transform
The Hartley transform has the convenient property of being its own inverse (an involution):Conventions
The above is in accord with Hartley's original definition, but (as with the Fourier transform) various minor details are matters of convention and can be changed without altering the essential properties:- Instead of using the same transform for forward and inverse, one can remove the
from the forward transform and use 
for the inverse—or, indeed, any pair of normalizations whose product is 
. (Such asymmetrical normalizations are sometimes found in both purely mathematical and engineering contexts.) - One can also use
instead of 
(i.e., frequency instead of angular frequency), in which case the 
coefficient is omitted entirely. - One can use cos−sin instead of cos+sin as the kernel.
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
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
.
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
Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series...
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.Properties
The Hartley transform is a real linear operator, and is symmetric (and Hermitian). From the symmetric and self-inverse properties, it follows that the transform is a unitary operatorUnitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
(indeed, orthogonal
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
).
There is also an analogue of the convolution theorem
Convolution theorem
In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...
for the Hartley transform. If two functions
and have Hartley transforms and , respectively, then their convolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
has the Hartley transform:Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively.
cas
The properties of the cas function follow directly from trigonometryTrigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
, and its definition as a phase-shifted trigonometric function
. For example, it has an angle-addition identity of:Additionally:
and its derivative is given by: