Parseval's theorem
Encyclopedia
In mathematics
, Parseval's theorem usually refers to the result that the Fourier transform
is unitary
; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series
by Marc-Antoine Parseval
, which was later applied to the Fourier series
. It is also known as Rayleigh's energy theorem, or Rayleigh's Identity, after John William Strutt, Lord Rayleigh.
Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics
and engineering
, the most general form of this property is more properly called the Plancherel theorem
.
and
respectively. Then
where i is the imaginary unit
and horizontal bars indicate complex conjugation.
Parseval, who apparently had confined himself to real-valued functions, actually presented the theorem without proof, considering it to be self-evident. There are various important special cases of the theorem. First, if A = B one immediately obtains:
from which the unitarity of the Fourier series follows.
Second, one often considers only the Fourier series for real-valued functions A and B, which corresponds to the special case:
real, 
, 
real, and 
. In this case:
where
denotes the real part. (In the notation of the Fourier series
article, replace
and 
by 
.)
and engineering
, Parseval's theorem is often written as:
where
represents the continuous Fourier transform
(in normalized, unitary form) of x(t) and f represents the frequency component (not angular frequency
) of x.
The interpretation of this form of the theorem is that the total energy contained in a waveform x(t) summed across all of time t is equal to the total energy of the waveform's Fourier Transform X(f) summed across all of its frequency components f.
For discrete time
signals, the theorem becomes:
where X is the discrete-time Fourier transform
(DTFT) of x and Φ represents the angular frequency
(in radians per sample) of x.
Alternatively, for the discrete Fourier transform
(DFT), the relation becomes:
where X[k] is the DFT of x[n], both of length N.
as the inner product form, and to
as the norm
form. It is not difficult to show that they are (pointwise
) equivalent. One can use the polarization identity
which is true for all complex numbers a and b, and the linearity of both integration and the Fourier transform.
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...
, Parseval's theorem usually refers to the result that 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...
is unitary
Unitary 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...
; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
by Marc-Antoine Parseval
Marc-Antoine Parseval
Marc-Antoine Parseval des Chênes was a French mathematician, most famous for what is now known as Parseval's theorem, which presaged the unitarity of the Fourier transform....
, which was later applied to the Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
. It is also known as Rayleigh's energy theorem, or Rayleigh's Identity, after John William Strutt, Lord Rayleigh.
Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, the most general form of this property is more properly called the Plancherel theorem
Plancherel theorem
In mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
.
Statement of Parseval's theorem
Suppose that A(x) and B(x) are two Riemann integrable, complex-valued functions on R of period 2π with (formal) Fourier seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
and
respectively. Then
where i is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
and horizontal bars indicate complex conjugation.
Parseval, who apparently had confined himself to real-valued functions, actually presented the theorem without proof, considering it to be self-evident. There are various important special cases of the theorem. First, if A = B one immediately obtains:
from which the unitarity of the Fourier series follows.
Second, one often considers only the Fourier series for real-valued functions A and B, which corresponds to the special case:
real, , real, and . In this case:where
denotes the real part. (In the notation of the Fourier seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
article, replace
and by .)Applications
In physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, Parseval's theorem is often written as:
where
represents the continuous Fourier transformContinuous Fourier transform
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...
(in normalized, unitary form) of x(t) and f represents the frequency component (not angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
) of x.
The interpretation of this form of the theorem is that the total energy contained in a waveform x(t) summed across all of time t is equal to the total energy of the waveform's Fourier Transform X(f) summed across all of its frequency components f.
For discrete time
Discrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...
signals, the theorem becomes:
where X is the discrete-time Fourier transform
Discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function . But the DTFT requires an input function...
(DTFT) of x and Φ represents the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
(in radians per sample) of x.
Alternatively, for the discrete Fourier transform
Discrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
(DFT), the relation becomes:
where X[k] is the DFT of x[n], both of length N.
Equivalence of the norm and inner product forms
We shall refer toas the inner product form, and to
as the norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
form. It is not difficult to show that they are (pointwise
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f of some function f. An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values...
) equivalent. One can use the polarization identity
Polarization identity
In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \|x\| \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...
which is true for all complex numbers a and b, and the linearity of both integration and the Fourier transform.
See also
- Bessel's inequalityBessel's inequalityIn mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence....
- Parseval's identityParseval's identityIn mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is thePythagorean theorem for inner-product spaces....
- Plancherel's Theorem
- Parseval–Gutzmer formula
External links
- Parseval's Theorem on Mathworld