The discrete Fourier transform is a numerical method used to define samples referring to the spectral frequencies that make up a signal. Study periodic functions in closed parameters, yielding another discrete signal.
In order to obtain the discrete Fourier transform of N points, on a discrete signal, the following 2 conditions must be fulfilled on a sequence x [n]
x [n] = 0 n < 0 ˄ n > N - 1
If these conditions are satisfied, the discrete Fourier transform can be defined as
The discrete Fourier transform can be defined as an N-point sampling of the Fourier transform.
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There are 2 points of view from which the results obtained on a sequence x can be interpreteds[n] through the discrete Fourier transform.
-The first corresponds to the spectral coefficients, already known from the Fourier series. It is observed in discrete periodic signals, with samples coinciding with the sequence xs[n].
-The second deals with the spectrum of a discrete aperiodic signal, with samples corresponding to the sequence xs[n].
The discrete transform is an approximation to the spectrum of the original analog signal. Its phase depends on the sampling instants, while its magnitude depends on the sampling interval..
The algebraic foundations of structure make up the logical bases of the following sections.
C. Sn → C. F[Sk]; If a sequence is multiplied by a scalar, its transform will be too.
Tn + Vn = F [Tk] + F [Vk]; The transform of a sum is equal to the sum of the transforms.
F [Sn] → (1 / N) S-k; If the discrete Fourier transform is recalculated to an expression already transformed, the same expression is obtained, scaled in N and inverted with respect to the vertical axis.
Pursuing similar objectives as in the Laplace transform, the convolution of functions refers to the product between their Fourier transforms. Convolution also applies to discrete times and is responsible for many modern procedures..
Xn * Rn → F [Xn] .F [Rn]; The transform of a convolution is equal to the product of the transforms.
Xn . Rn→ F [Xn] * F [Rn]; The transform of a product is equal to the convolution of the transforms.
Xn-m → F [Xk] e -i (2π / N) km ; If a sequence is delayed in m samples, its effect on the discrete transform will be a modification of the angle defined by (2π / N) km.
Xt [-k] = X *t[k] = Xt [N - K]
W-nmN . x [n] ↔ Xt[k - m]
x [n] y [n] ↔ (1 / N) Xt[k] * Yt[k]
X [-n] ↔ Xt[-k] = X *t[k]
x * [n] ↔ X *t[-k]
With respect to the conventional Fourier transform, it has several similarities and differences. The Fourier transform converts a sequence into a solid line. In this way it is said that the result of the Fourier variable is a complex function of real variable.
The discrete Fourier transform, unlike, receives a discrete signal and transforms it into another discrete signal, that is, a sequence.
They serve primarily to significantly simplify equations, while transforming derived expressions into power elements. Denoting Differential Expressions in Forms of Integrable Polynomials.
In the optimization, modulation and modeling of results, it acts as a standardized expression, being a frequent resource for engineering after several generations..
This mathematical concept was presented by Joseph B. Fourier in 1811, while developing a treatise on the heat spread. It was quickly adopted by various branches of science and engineering.
It was established as the main work tool in the study of equations with partial derivatives, even comparing it with the existing work relationship between the Laplace transform and ordinary differential equations.
Every function that can be worked with a Fourier transform must present null outside a defined parameter.
The discrete transform is obtained through the expression:
After given a discrete sequence X [n]
The inverse of the discrete Fourier transform is defined through the expression:
It allows, once the discrete transform has been achieved, to define the sequence in the time domain X [n].
The parametrization process corresponding to the discrete Fourier transform lies in the windowing. To work the transform we must limit the sequence in time. In many cases the signals in question do not have these limitations.
A sequence that does not meet the size criteria to apply to the discrete transform can be multiplied by a “window” function V [n], defining the behavior of the sequence in a controlled parameter.
X [n]. V [n]
The width of the spectrum will be dependent on the width of the window. As the width of the window increases, the calculated transform will be narrower.
The discrete Fourier transform is a powerful tool in the study of discrete sequences.
The discrete Fourier transform transforms a continuous variable function, into a discrete variable transform.
The Cauchy problem for the heat equation presents a frequent field of application of the discrete Fourier transform. Where the function is generated heat core or Dirichlet core, which applies to sampling of values in a defined parameter.
The general reason for the application of the discrete Fourier transform in this branch is mainly due to the characteristic decomposition of a signal as an infinite superposition of more easily treatable signals.
It can be a sound wave or an electromagnetic wave, the discrete Fourier transform expresses it in a superposition of simple waves. This representation is quite frequent in electrical engineering.
They are series defined in terms of Cosines and Sines. They serve to facilitate work with general periodic functions. When applied, they are part of the techniques for solving ordinary and partial differential equations..
The Fourier series are even more general than the Taylor series, because they develop periodic discontinuous functions that do not have a Taylor series representation..
To understand the Fourier transform analytically, it is important to review the other ways in which the Fourier series can be found, until we can define the Fourier series in its complex notation..
Many times it is necessary to adapt the structure of a Fourier series to periodic functions whose period is p = 2L> 0 in the interval [-L, L].
The interval [-π, π] is considered, which offers advantages when taking advantage of the symmetric characteristics of the functions.
If f is even, the Fourier series is established as a series of Cosines.
If f is odd, the Fourier series is established as a series of Sines.
If we have a function f (t), which meets all the requirements of the Fourier series, it is possible to denote it in the interval [-t, t] using its complex notation:
Regarding the calculation of the fundamental solution, the following examples are presented:
Laplace equation
Heat equation
Schrödinger equation
Wave equation
On the other hand, the following are examples of the application of the discrete Fourier transform in the field of signal theory:
-System identification problems. Established f and g
-Output signal consistency problem
-Problems with signal filtering
Calculate the discrete Fourier transform for the following sequence.
You can define the PTO of x [n] as:
Xt[k] = 4, -j2, 0, j2 for k = 0, 1, 2, 3
We want to determine through a digital algorithm the spectral signal defined by the expression x (t) = e-t. Where the maximum frequency requesting coefficient is fm= 1Hz. A harmonic corresponds to f = 0.3 Hz. The error is limited to less than 5%. Calculate Fs , D and N.
Taking into account the sampling theorem Fs = 2fm = 2 Hz
A frequency resolution of F0 = 0.1 Hz, from where you get D = 1 / 0.1 = 10s
0.3 Hz is the frequency corresponding to the index k = 3, where N = 3 × 8 = 24 samples. Indicating that Fs = N / A = 24/10 = 2.4> 2
Since the aim is to get the lowest possible value for N, the following values can be considered as a solution:
F0 = 0.3 Hz
D = 1 / 0.3 = 3.33s
k = 1
N = 1 × 8 = 8
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