Fourier Transform
functionh:\mathbb{R}\to\mathbb{C}h:R→CAbout\hat{h} =
F\left[h\right]:\mathbb{R}\to\mathbb{C}h^=F[h]:R→Ca function called
\displaystyle \hat{h}(t) = F\left[h\right]\left(t\right)
\equiv \int_{-\infty} ^{\infty} e^{-2 \pi itx}
h\left(x\right) \mathrm{d}x \quad \left(i = \sqrt{1}\right)h^(t)=F[h](t)≡∫− ∞∞And−2πitxh(x)dx(i=− 1)
, and transform the above\hat{h}, F[h]h^,F[h]functionhhIt is defined as the Fourier
transform of[One][2] It can be defined similarly in higher dimensions. Fourier transform\int e^{itx} h\left(x\right) \mathrm{d}x∫And−itxh(x)dxSome mathematicians define it as[3] In
this case, the relationship between the derivative of the function and the Fourier transform is
neat (F\left[h'\right](t)=itF\left[h\right](t)F[h'](t)=i t F[h](t)) is mainly used in partial
differential equations. In physics, normal timettThe angular frequency is the Fourier domain
of\omegaωuse , locationxxwave number into the Fourier domain oftotouse
The above integral is an arbitrary functionhhis not well defined
(e.g.h(x)=1h(x)=Oneif ). However, in most cases, the integral expression is interpreted as
improper Riemann integral,h(x)e^{-\epsilon x^2}h(x)e−ϵx2After calculating the Fourier
transform of\ epsilon \ to 0ϵ→0It is solved by sending the extreme. In any case, it is
necessary to strictly define the Fourier transform in order to solve such a small problem.
Please refer to the following.
1.1. in Schwarz space[edit]
schwarz space\mathcal{S}Sis a space of functions that are infinitely differentiable and
whose derivatives are rapidly decreasing. In this case, the Fourier transform as an integral
expression is well defined. especially,F:\mathcal{S}\to \mathcal{S}F:S→Sis a one-toone correspondence, and allh,g\in
\mathcal{S}h,g∈SAbout
\displaystyle \int_{-\infty}^{\infty} h(x)\bar{g}(x) \mathrm{d}x = \int_{\infty}^{\infty} F[h](t) \overline{F[g]}(t) \mathrm{d}t∫− ∞∞h(x)gˉ
(x)dx=∫− ∞∞F[h](t)F[g](t)dt
satisfies Plancherel's theorem.
1.2. in L 1 space[edit]
First, you need to know about the Lebesgue space. (see comments)[4] L^{1}IOneThe
Fourier transform as an integral expression for a function is well defined, but the transformed
function isL^{1}IOnemay not go into, in factF:L^{1} \to L^{\infty}F:IOne→I∞Only Im
can know. Unlike in Schwarz space,F:L^{1} \to L^{\infty}F:IOne→I∞is not a
transcription function.[5]
1.3. in L 2 space[edit]
with this nature\mathcal{S}\subset
L^{2}S⊂I2by the compactness ofF:\mathcal{S}\to
\mathcal{S}F:S→SIsF:L^{2}\to L^{2}F:I2→I2It has a unique extension to the causal
operator. because of thisF:L^{2}\to L^{2}F:I2→I2is well defined
1.4. in a controlled distribution space.[edit]
\mathcal{S}'S'space is schwarz space\mathcal{S}SAs a dual space of , it has a weak-*
phase.\mathcal{S}'S'The elements of is called the control distribution. control distributionh
\in \mathcal{S}'h∈S'Fourier transform ofF[h] \in \mathcal{S}'F[h]∈S'is naturally
defined as follows by Plancherel's theorem.
This Fourier transform is very similar to the Laplace transform . right abovetttoisistry
putting The derivative of a function is multiplied with a variable by Fourier transform, and the
product is transferred to convolution. Therefore, the Laplace transform solution of the
differential equation can be corrected with the Fourier transform solution as it is. However,
what is even better than the Laplace transform is that the inverse transform is very easy. No,
it's just an inverse transformation! strictly speakingF^{2} h\left(t\right) =
F\left[F\left[h\right]\right]\left(t\right) = h\left(t\right)F2h(t)=F[F[h]](t)=h(−t)is established.[6]
3. Deepen[edit]
3.1. inverse transform[edit]
Inverse Transform of Fourier TransformF^{-1}
\left[g\right]\left(x\right) = \int e^{2\pi
itx} g\left(t\right) dtF− 1[g](x)=∫And2πitxg(t)d tatg = F\left[h\right]g=F[h]if put
toh\left(x\right) = \int e^{2\pi itx} F\left[h\right] dth(x)=∫And2πitxF[h]d tbecomes,
which ish\left(x\right)h(x)is an exponential functione ^ {2 \ pi itx}And2πitxIt means that
they can be expressed as 'continuous primary bonds'. For this purpose, the Fourier series and
the Fourier transform can be combined together to be called a Fourier analysis.[7]
3.2. Discrete Fourier Transform[edit]
There is a method of applying the above-described Fourier transform within a discrete
range. This is called a Discrete Fourier Transform (DFT). Discrete Fourier Transform is a
Fourier Transform of a Discrete Data Set While the basic Fourier Transform is used for a
function whose domain is continuous, the Discrete Fourier Transform does not completely
transform any function, but rather the Fourier Transform for specific discrete data. Since it is
used to calculate values, it is mainly a function that is not an electrolytic function.It is used
when you want to apply a Fourier transform to [8] .
The discrete Fourier transform is defined by the following equation.
Discrete Fourier Transform (DFT)
\displaystyle X[k]= \sum_{n=0}^{N-1} x[n] e^{-i\frac{2 \pi k n}{N}}
\quad \quadX[k]=n=0∑N−1x [ n ] e−iN2πkn
Inverse Discrete Fourier Transform (IDFT)
\displaystyle x[n]=\dfrac{1}{N} \sum_{k=0}^{N-1} X[k] e^{i\frac{2 \pi
kn}{N}} \quad \quadx[n]=NOnek = 0∑N−1X [ k ] eiN2πkn
The discrete Fourier transform seeks the transformation of specific discrete data, not the
perfect transformation of the electrolytic function, so the purpose of the transformation is the
integral symbol.\int∫Instead, the sum sign\sum∑write In addition, since the range of
conversion from zero is finite, it is\infty∞Instead, set the range to a finite arbitrary integer N1.
3.2.1. Linearity Definition[edit]
The Fourier analysis is an analysis in which a linear combination between the exponential
function and the function space is defined. Therefore, if Fourier analysis is applied to the set
of discrete parts as well, the inner orthogonalization process makes the sum of products
calculated at each point to be zero, so that the eigenfunction is orthogonalized in the same
discrete set as the spatial point of the orthogonalization interval. features exist.
\displaystyle \langle f, g\rangle = \int_{x_0}^{x_0+P} f(x) \overline{g(x)}
dx⟨ f,g⟩ =∫x0x0+Pf(x)g(x)dx
for the set of natural numbers Nx_kxtocast\displaystyle
x_k=\dfrac{2\pi k}{N}xto
=N2πk [9] is defined as an arbitrary integer n andx_kxtoa function defined as\phi_n(x)ϕn
(x)and exponential functione^{inx}AndinxAssume that is equal to Therefore, if the Fourier
analysis defined as a linear combination is applied, the Fourier transform in the discrete set is
also defined as follows.
\displaystyle \langle \phi_n, \phi_m\rangle = \sum_{k=0}^{N-1}
\overline{\phi_n(x_k)} \phi_m(x_k)Φ ϕn,ϕm⟩ =k = 0∑N−1ϕn(xto)ϕm(xto)
here\displaystyle
x_k=\dfrac{2\pi k}{N}xto=N2πkAgain, to remind you that
\ displaystyle \ langle \ phi_n, \ phi_m \ rangle = \ sum_ {k = 0} ^ {N-1} e
^ {\ frac {2 \ pi ik (nm)} {N}} = \ sum_ {k = 0 } ^ {N-1} r ^ kΦ ϕn,ϕm
⟩ =k = 0∑N−1AndN2πik(n−m)=k = 0∑N−1rto
When r = 1, the above expression has a value of N, or\sum∑finite sum formula of\frac{1-
r^{N}}{1-r}1−r1−rNis a finite geometric series. But,r^{N} = e^{2 \ pi i
(nm)}rN=And2πi(n−m), and since n and m are limited integer values, when n and m are equal
to or different from each other,r^N=1rN=OneThis is defined Therefore , using the definition
of Kronecker delta ,
\displaystyle \langle \phi_n, \phi_m\rangle =N \sum_{n=-\infty}^{\infty}
\delta_{m-n},nNΦ ϕn,ϕm⟩ =Nn=−∞∑∞dm−n,n N
Infinite sum using Kronecker delta is not zero, but if mn is not a multiple of N, all infinite sums
in the above equation are zero. but the function\phi_nϕnSince is defined at N points, N is
uniquely linearly independent, so the above equation is rather complicated. therefore,
\phi_{n+N}(x_k)=e^{\frac{2\pi
i(n+N)k}{N}}=e^{\frac{2\pi
ink}{N}}=\ phi_n(x_k)ϕn+N(xto)=AndN2πi(n+N)k=AndN2πink=ϕn(xto)
You can limit the values of n and m in the range between 0 and N-1. Therefore, the
orthogonalization relationship is defined as follows.
\displaystyle
⟩ =Nδm n
\langle
\phi_n,
\phi_m\rangle
=N\delta_{mn}Φ ϕn,ϕm
3.2.2. Discrete Time Fourier Transform[edit]
Discrete Time Fourier Transform (DTFT)
\displaystyle X(\omega) = \sum_{n=-\infty}^{\infty} x\left[n\right] \,e^{-i
\omega n}X ( ω )=n=−∞∑∞x[n]And- i ω n
DTFT is the application of a Fourier transform to a set of discrete data such
as digital . here\omegaωAlso called digital angular frequency, the unit is radian/sample. The
relationship between the Fourier transform and the DTFT can be seen as similar to the
relationship between the Laplace transform and the Z-transform.
DTFT is characterized in that the transformation result is periodic and continuous. That is, it is
discrete in the time domain but continuous in the frequency domain.
However, it is not used well on computers, because the transformation is a function type, it is
necessary to interpret an infinite series, and the inverse transformation is an integral in normal
programming.B
3.2.3. Fast Fourier Transform[edit]
Fast Fourier Transform (FFT)
이산적인 nnWhen given data of\mathcal{O}(n
\log n)O ( nlogn)It is an algorithm for fast
DFT with only the amount of computation of . It was developed by Cooley and Thukey in 1965
and is widely used under the name of Cooley-Tucky Algorithm, but it has already been
independently discovered and forgotten by other mathematicians several times before . . In
addition, various other algorithms have been discovered by various mathematicians.
4. Application[edit]
Of course, it is also covered in mathematics, but its importance grows in physics and electronic
engineering. It is almost always used in fields that contain
electromagnetic waves or frequencies . The fundamental reason for using the Fourier
transform is that signals that are difficult to interpret in the time domain can be easily
interpreted in the frequency domain (think Laplace transform when solving
ODE). Modulation techniques such as AM and FM all deal with a certain signal in the
frequency domain, which is a Fourier transform of time. For example, in AM, a message is
multiplied by a cosine function with a carrier frequency, but the shape of the graph is quite
dirty to be interpreted directly in the time domain. However, if it is Fourier transform, the Fourier
transform of the cosine function is a delta function, so it can be easily analyzed by analyzing
it in the frequency domain.
The principle of MRI image composition is based on physical knowledge of hydrogen atoms
and magnetic fields, and a virtual space called k-space is used in the process of processing
the obtained data. The Fourier transform is used in the information processing process. The
most commonly used transformation for making MRI images is that the sinc function is a
transformed square wave.[11]