Name: solving discrete-time responses with z-transform
Uploaded: 2024-09-26T13:17:11.000Z
Duration: 1 min 24 s
Channel: JoVE

ほとんどの実用的な離散時間システムは線形差分方程式で表すことができるため、z変換は特に便利なツールになります。
入力信号とNの初期条件を知ることは、N次差分方程式を解くために必要です。
遅延信号または高度な信号の場合、z 変換は、z または z の逆数をそれぞれ乗算することで、z 領域の信号をシフトします。
特定の係数と初期条件によって特徴付けられる 2 次差分方程式について考えてみます。入力は単位ステップ関数です。
各項の z 変換を取ると、方程式は入力信号と出力信号の z 領域表現を含む代数式に変換されます。
この代数方程式を z 領域の出力信号に対して解くと、偏分数分解を使用して簡略化できる式が得られます。
係数が計算され、システムの時間領域応答は、部分分数式の逆 z 変換によって与えられます。
このプロセスは、離散時間線形システムの解析と解法を簡素化するz変換の力を実証しており、さまざまなデジタル信号処理および制御システム分野で不可欠なツールとなっています。

z 変換を用いた差分方程式の解法

z 変換は、多くの場合線形差分方程式で表される実際の離散時間システムを解析するための強力なツールです。高次の差分方程式を解くには、方程式の次数より 1 項少ない項までの入力信号と初期条件の知識が必要です。

z 変換は、信号を z 領域でシフトすることで遅延信号の処理を容易にします。これは時間領域で信号を遅延させることに相当し、同様に反対方向にシフトすることで時間を進めることで信号を進めます。

特定の係数と初期条件を持つ二次差分方程式を考えます。ここで、入力は単位ステップ関数です。各項に z 変換を適用すると、差分方程式が z 領域の代数式に変換されます。この式には、入力信号と出力信号の両方の z 領域表現が含まれます。

z 領域の出力信号を解くには、この代数方程式を簡略化できます。多くの場合、部分分数分解を使用します。部分分数の係数を決定することで、逆 z 変換を使用して時間領域に戻すことができる扱いやすい形式が得られます。結果として得られる時間領域応答は、離散時間線形システムの解析を簡素化する z 変換の有効性を示しています。

このプロセスは、デジタル信号処理および制御システムにおける z 変換の有用性を強調しています。これは、時間領域と z 領域間の遷移、複雑な方程式の解決、および正確なシステム応答の取得を行える簡単な方法を提供します。正確で意味のある結果を確実に得るには、z 変換を適用するときに初期条件と収束領域の役割を考慮することが重要です。

ほとんどの実用的な離散時間システムは線形差分方程式で表すことができるため、z変換は特に便利なツールになります。
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difference-equation-solution-using-z-transform

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

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 between time shifts and z-domain shifts makes the algebra easier to work with.

A second-order difference equation with given coefficients and initial conditions can be converted step by step. In the example, the input is a unit step function. Applying the z-transform to each term turns the difference equation into an algebraic expression in the z-domain. That expression includes the z-domain forms of both the input and the output signals.

The algebraic equation for the output can then be simplified. Partial fraction decomposition is often used to break the result into smaller terms. After the coefficients are found, the expression can be converted back with the inverse z-transform. The final time-domain response shows how the z-transform simplifies analysis of discrete-time linear systems. The region of convergence must also be considered to make sure the result is accurate and meaningful.

Solving Discrete-Time Responses with z-Transform

solving-discrete-time-responses-with-z-transform

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

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 z-transform turns a difference equation into algebra, handles time shifts, and helps find the discrete-time response with initial conditions.

Solving Discrete-Time Responses with z-Transform | JoVE K12

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electrical engineering

Solving Discrete-Time Responses with z-Transform

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