Name: dft through the unit circle
Uploaded: 2024-09-26T13:17:11.000Z
Duration: 1 min 20 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

HS-PS4-2

Relation of DFT to z-Transform

Дискретное преобразование Фурье (DFT) анализирует частотный состав сигналов дискретного времени.
Он сопоставляет дискретную последовательность во временной области с дискретной последовательностью в частотной области, где k представляет индекс частоты от 0 до N минус один.
Для дискретной временной последовательности z-преобразование определяется путем суммирования членов последовательности, умноженных на степени обратной z.
Рассмотрим z-преобразование дискретной последовательности. Для причинно-следственной последовательности z-преобразование упрощается до конечного суммирования.
Путем выборки z-преобразования в равноудаленных точках единичной окружности, представленных комплексными экспоненциалами, эти значения подставляются в z-преобразование.
Результирующее выражение соответствует определению DFT, показывая, что DFT последовательности является выборочной версией ее z-преобразования на единичной окружности.
DFT — это выборочная версия z-преобразования, оцениваемая в определенных точках единичной окружности в комплексной плоскости, связывающая последовательности во временной области с их представлениями в частотной области.
Это показывает, что DFT — это частный случай z-преобразования, вычисленного на единичной окружности.

Связь ДПФ с z-преобразованием

Дискретное преобразование Фурье (ДПФ) является важнейшим инструментом для анализа частотного содержания дискретных по времени сигналов. Оно преобразует последовательность N выборок из временного домена в соответствующую ей последовательность в частотном домене, где каждая выборка представляет собой определенный частотный компонент.

Чтобы понять, как работает ДПФ, полезно рассмотреть z-преобразование, которое является методом представления дискретных последовательностей в комплексном частотном домене. Z-преобразование включает суммирование членов последовательности, каждый из которых умножается на степень комплексного числа. Для последовательностей, которые начинаются в определенной точке времени и распространяются вперед (каузальные последовательности), z-преобразование может быть выражено как бесконечная или конечная сумма, в зависимости от длины последовательности.

Оценивая z-преобразование в N равноотстоящих точках вокруг единичной окружности в комплексной плоскости, получают значения, соответствующие коэффициентам ДПФ. Эти точки являются корнями единицы, и оценка z-преобразования в этих точках эффективно дискретизирует частотное содержание сигнала на этих конкретных частотах.

Таким образом, ДПФ можно рассматривать как конкретное применение z-преобразования, сосредоточенное на оценке последовательности в этих точных местах на единичной окружности. Этот процесс переводит последовательности во временном домене в их аналоги в частотном домене, что позволяет анализировать различные частотные компоненты дискретных сигналов.

Способность ДПФ раскрывать частотное содержание сигналов подчеркивает его важность в цифровой обработке сигналов и других смежных областях. Оно широко используется в таких приложениях, как обработка аудиосигналов, анализ изображений и системы связи. Кроме того, быстрое преобразование Фурье (БПФ) является эффективным алгоритмом, обычно используемым для вычисления ДПФ, что позволяет обрабатывать сигналы в реальном времени.

Связь ДПФ с Z-преобразованием

Дискретное преобразование Фурье (DFT) анализирует частотный состав сигналов дискретного времени.
Он отображает N-дискретную последовательность во временной области с ее ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

z-Transform

z-преобразование

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

dft through the unit circle

The Discrete Fourier Transform (DFT) shows how a discrete-time signal breaks into frequency parts. It takes N samples from the time domain and gives N values in the frequency domain. Each value represents one frequency component.

The z-transform helps explain this process. A z-transform represents a discrete sequence in the complex frequency domain by summing sequence terms multiplied by powers of a complex number. For causal sequences, which start at a fixed time and move forward, the sum can be infinite or finite, depending on the sequence length.

The DFT comes from evaluating the z-transform at N equally spaced points on the unit circle in the complex plane. These points are the roots of unity. Sampling the z-transform at these locations gives the DFT coefficients, which capture the signal’s frequency content at specific frequencies.

This view makes the DFT a special case of the z-transform. It turns a time-domain sequence into a frequency-domain sequence by checking the signal at exact points on the unit circle. That is why the DFT is so useful in digital signal processing and related fields.

The DFT is widely used in audio signal processing, image analysis, and communication systems. The Fast Fourier Transform, or FFT, is the efficient algorithm often used to compute the DFT. The FFT makes real-time signal processing possible in many applications.

DFT Through the Unit Circle

definition of z-transform

z-Transform for Discrete-Time Signals

The z-transform is a key math tool for analyzing discrete-time signals and systems. It is the discrete-time counterpart of the Laplace transform used for continuous-time systems. In signal and systems analysis, it works alongside the discrete-time Fourier transform. The z-transform changes a discrete-time signal into a power series with a complex variable, z. Each term in the series represents one sampling instant of the signal. This form makes it easier to study the signal in detail and solve...

region of convergence

Z-Transform Convergence and Stability

The z-transform and its Region of Convergence (ROC) are key ideas in discrete-time signals and systems. The z-transform is a mathematical tool for analyzing signals that change at separate time steps. Its values do not always converge, so the ROC identifies the range of complex numbers z where the transform is valid. The ROC can appear in different shapes in the complex plane. It may lie inside a circle, outside a circle, or within an annulus, which is the space between two circles. For an...

properties of the z-transform i

Z-Transform Rules for Signals

The z-transform rules for signals help students analyze discrete-time systems in digital signal processing. These rules turn time-domain changes into simpler expressions in the z-domain, which makes signal work easier to follow. Linearity is one key rule. It means the z-transform of a linear combination of two discrete-time signals equals the same linear combination of their individual z-transforms. This is useful when signals are added together or superimposed. Time-shifting is another...

properties of the z-transform ii

Z-Transform Rules for Sum and Convolution

The z-transform rules for sum and convolution help analyze discrete-time signals in signal processing. These rules show how changes in the time domain connect to the z-domain, which is the transform domain used for discrete-time systems. The accumulation property comes from the accumulated sum of a discrete-time signal. By applying the time-shifting property, the z-transform of the summed signal can be found from the z-transform of the original signal. The result is a simple multiplicative...

inverse z-transform by partial fraction expansion

inverse-z-transform-by-partial-fraction-expansion

Inverse z-Transform Using Poles and Residues

The inverse z-transform converts a function from the z-domain back to the time domain. One useful approach is the partial fraction method. It breaks a complex function into simpler fractions with distinct coefficients. The process starts by identifying the poles of the function. A pole is a value that makes the function grow without a finite limit. The function is then written in terms of these poles, and each pole adds one term to the partial fraction form. Next, the coefficients for each...

difference equation solution using z-transform

difference-equation-solution-using-z-transform

Solving Discrete-Time Responses with z-Transform

The z-transform helps solve discrete-time systems that are written as linear difference equations. These equations are common in digital signal processing and control systems. For a higher-order difference equation, you need the input signal and the initial conditions up to one term less than the order of the equation. The method also makes delayed signals easier to handle. A delay in time becomes a shift in the z-domain. An advance in time is shifted in the opposite direction. This link...

relation of dft to z-transform

The Discrete Fourier Transform (DFT) shows how a discrete-time signal breaks into frequency parts. It takes N samples from the time domain and gives N values in the frequency domain. Each value represents one frequency component. The z-transform helps explain this process. A z-transform represents a discrete sequence in the complex frequency domain by summing sequence terms multiplied by powers of a complex number. For causal sequences, which start at a fixed time and move forward, the sum can...

key terms and concepts

Learn how the DFT connects to the z-transform by sampling points on the unit circle, and see why this method matters in signal processing and the FFT.

DFT Through the Unit Circle | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

HS-PS4-5

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HS-PS4-1

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HS-PS4-3

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Evaluate the validity and reliability of claims in published materials of the effects that different frequencies of electromagnetic radiation have when absorbed by matter. [Clarification Statement: Emphasis is on the idea that photons associated with different frequencies of light have different energies, and the damage to living tissue from electromagnetic radiation depends on the energy of the radiation. Examples of published materials could include trade books, magazines, web resources, videos, and other passages that may reflect bias.] [Assessment Boundary: Assessment is limited to qualitative descriptions.]

HS-PS4-4

DFT Through the Unit Circle

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