Name: inverse z-transform using poles and residues
Uploaded: 2024-09-26T13:17:10.000Z
Duration: 1 min 20 s
Channel: JoVE

逆 Z 変換は、関数を周波数領域表現から時間領域に戻すために使用される重要なツールです。
関数 X(z) について考えてみます。これは、時間領域表現に変換し直す必要があります。
X(z) を分解するには、関数の極が特定され、これらの極で表されます。
各極は、部分分数展開に項を寄与します。
展開の各項の係数は、z に特定の値を代入することによって決定されます。
すべての係数を決定した後、関数は分解された形式で再構成されます。
この新しい表現は、より管理しやすくなっています。
各小数部は既知の Z 変換ペアに対応しているため、逆変換が簡単になります。
偏分数法は、関数を異なる係数を持つ単純な分数に分解することにより、逆 Z 変換を見つけるための効果的な手法です。
逆 Z 変換は、各小数項に個別に適用され、デルタ関数、指数シーケンス、ステップ関数の組み合わせになり、元の時間領域シーケンスをまとめて表します。

部分分数展開による逆 z 変換

Inverse z-Transform by Partial Fraction Expansion

逆 z 変換は、関数を z 領域表現から時間領域に戻すための重要な手法です。逆 z 変換を見つけるための効果的な方法の 1 つは、部分分数法です。この方法では、関数を異なる係数を持つより単純な分数に分解します。これらの分数は既知の z 変換ペアに対応し、逆変換プロセスを容易にします。

プロセスを開始するには、関数の極を特定し、関数をこれらの極で表現します。各極は、部分分数分解に項を提供します。分解の各項の係数は、各極の留数を評価することによって決定されます。

係数が決定されると、関数は分解された形式で再構成され、操作が簡単になります。次に、逆 z 変換を各分数項に個別に適用します。結果は、デルタ関数、指数シーケンス、および元の時間領域シーケンスを表すステップ関数を組み合わせます。

部分分数法を使用すると、複雑な関数の逆 Z 変換がより扱いやすくなり、時間領域への正確な変換が可能になります。この方法により、分解された関数の各要素が正しく変換され、元のシーケンスが正確に再構築されます。

逆 Z 変換は、関数を周波数領域表現から時間領域に戻すために使用される重要なツールです。
考えてみてください...

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

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

z-Transform

z変換

KEY TERMS AND CONCEPTS

主要用語と概念

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 term are found by evaluating the residues at each pole. The decomposed terms are then matched with known z-transform pairs. The inverse z-transform is applied to each fractional term separately.

The final result combines delta functions, exponential sequences, and step functions. Together, these terms reconstruct the original time-domain sequence. The partial fraction method makes complex inverse z-transforms easier to handle and supports accurate conversion back to the time domain.

Inverse z-Transform Using Poles and Residues

inverse-z-transform-using-poles-and-residues

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

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

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...

key terms and concepts

Learn how the inverse z-transform uses poles, residues, and partial fraction decomposition to turn a z-domain function back into a time-domain sequence.

Inverse z-Transform Using Poles and Residues | JoVE K12

JoVE Core 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Inverse z-Transform Using Poles and Residues

Description

Transcript

Tags