DISCRETE FOURIER TRANSFORM: PROPERTIES, APPLICATIONS, EXAMPLES - MATHS - 2024

The discrete Fourier transform is a numerical method used to define samples referring to the spectral frequencies that make up a signal. It studies periodic functions in closed parameters, yielding another discrete signal as a result.

In order to obtain the discrete Fourier transform of N points, on a discrete signal, the following 2 conditions must be met on a sequence x

TDF

The discrete Fourier transform can be defined as an N-point sampling of the Fourier transform.

Interpretation of the discrete Fourier transform

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There are 2 points of view from which the results obtained on a sequence x _s can be interpreted 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 x _s.

-The second deals with the spectrum of a discrete aperiodic signal, with samples corresponding to the sequence x _s.

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.

Properties

The algebraic foundations of structure make up the rationale for the following sections.

Linearity

C. S _n → C. F; If a sequence is multiplied by a scalar, its transform will also be.

T _n + V _n = F + F; The transform of a sum is equal to the sum of the transforms.

Duality

F → (1 / N) S _-k; If the discrete Fourier transform is recalculated to an already transformed expression, the same expression is obtained, scaled in N and inverted with respect to the vertical axis.

Convolution

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.

X _n * R _n → F.F; The transform of a convolution is equal to the product of the transforms.

X _n. R _n → F * F; The transform of a product is equal to the convolution of the transforms.

Displacement

X _n-m → F e ^{–i (2π / N) km}; If a sequence is delayed by m samples, its effect on the discrete transform will be a modification of the angle defined by (2π / N) km.

Symmetry

X _t = X * _t = X _t

Modulation

W ^-nm _N. x ↔ X _t

Product

xy ↔ (1 / N) X _t * Y _t

Symmetry

X ↔ X _t = X * _t

Conjugate

x * ↔ X * _t

Parseval equation

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 a real variable.

The discrete Fourier transform, unlike, receives a discrete signal and transforms it into another discrete signal, that is, a sequence.

What is the discrete Fourier transform for?

They serve primarily to greatly simplify equations, while transforming derived expressions into power elements. Denoting differential expressions in integrable polynomial forms.

In the optimization, modulation and modeling of results, it acts as a standardized expression, being a frequent resource for engineering after several generations.

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History

This mathematical concept was introduced by Joseph B. Fourier in 1811, while developing a treatise on the propagation of heat. 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.

Discrete Fourier transform and its inverse

The discrete transform is obtained through the expression:

After given a discrete sequence X

The inverse of the discrete Fourier transform is defined through the expression:

Reverse PTO

Once the discrete transform is achieved, it allows defining the sequence in the time domain X.

Winged

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, defining the behavior of the sequence in a controlled parameter.

X. V

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.

Applications

Calculation of the fundamental solution

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 core function of heat or Dirichlet core is generated, which applies to sampling values ​​in a defined parameter.

Signal theory

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.

The Fourier series

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.

Fourier series are even more general than Taylor series, because they develop periodic discontinuous functions that do not have Taylor series representation.

Other forms of the Fourier series

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.

-Fourier series on a function of period 2L:

The interval 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.

-Complex notation of the Fourier series

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 using its complex notation:

Examples

Regarding the calculation of the fundamental solution, the following examples are presented:

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

-Problem with the consistency of the output signal

-Problems with signal filtering

Exercises

Exercise 1

Calculate the discrete Fourier transform for the following sequence.

You can define the PTO of x as:

X _t = {4, -j2, 0, j2} for k = 0, 1, 2, 3

Exercise 2

We want to determine the spectral signal defined by the expression x (t) = e ^-t through a digital algorithm. Where the maximum frequency requesting coefficient is f _m = 1Hz. A harmonic corresponds to f = 0.3 Hz. The error is limited to less than 5%. Calculate f _s, D and N.

Taking into account the sampling theorem f _s = 2f _m = 2 Hz

A frequency resolution of f ₀ = 0.1 Hz is chosen, from which we obtain 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 f _s = N / D = 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:

f ₀ = 0.3 Hz

D = 1 / 0.3 = 3.33s

k = 1

N = 1 × 8 = 8

References

1. Mastering the Discrete Fourier Transform in One, Two or Several Dimensions: Pitfalls and Artifacts. Isaac Amidror. Springer Science & Business Media, Jul 19. 2013
2. The DFT: An Owners' Manual for the Discrete Fourier Transform. William L. Briggs, Van Emden Henson. SIAM, Jan 1. nineteen ninety five
3. Digital Signal Processing: Theory and Practice. D. Sundararajan. World Scientific, 2003
4. Transforms and Fast Algorithms for Signal Analysis and Representations. Guoan Bi, Yonghong Zeng. Springer Science & Business Media, Dec 6. 2012
5. Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms. Eleanor Chu. CRC Press, Mar 19. 2008