Name: z-transform convergence and stability
Uploaded: 2024-09-26T13:17:10.000Z
Duration: 1 min 17 s
Channel: JoVE

z 変換は、z の特定の値に対してのみ収束します。この値の範囲は収束領域(ROC)と呼ばれ、システムまたは信号の動作と安定性を決定するために不可欠です。
これは、z 変換が収束する複素平面内の領域を指定します。
中華民国は、円の内側、円の外側、または円環内など、さまざまな形をとることができます。
指数離散時間信号 x[n] について考えます。Z 変換は等比級数であり、ROC は半径 a を原点とする円の外側の領域に対応します。
半径の値にもよりますが、ROC に単位円が含まれている場合はシステムは安定しており、単位円の外側にある場合は不安定です。
ROCが単位円と一致すると、システムはわずかに安定しています。
DTFT は、ROC に単位円が含まれている場合にのみ存在します。
Z 変換は極で収束しないため、ROC には極は含まれません。
また、元の時間領域信号を取得する逆 z 変換にも影響します。

収束領域

Z 変換は、離散時間信号およびシステムの解析で使用される強力な数学ツールです。離散時間システムの解析に不可欠なツールですが、その収束は複素変数 z の特定の値に制限されます。この値の範囲は収束領域 (ROC) と呼ばれ、システムまたは信号の動作と安定性を決定する上で非常に重要です。ROC は、複素平面内で Z 変換が収束する領域を定義します。この領域は、円の内側、円の外側、または環状空間の内側など、さまざまな形を取ることができます。

たとえば、指数関数的な離散時間信号 x[n] を考えてみましょう。この信号の z 変換は幾何級数を形成し、その ROC は原点を中心とする半径 a の円の外側の領域に対応します。単位円に関する ROC の位置は、システムの安定性を評価する上で重要です。ROC に単位円が含まれている場合、システムは安定しています。逆に、ROC が単位円の外側にある場合、システムは不安定です。ROC が単位円と正確に一致する場合、システムは限界的に安定していると見なされます。

信号の離散時間フーリエ変換 (DTFT) は、z 変換の ROC に単位円が含まれている場合にのみ存在します。ROC の重要性は、z 変換から元の時間領域信号を取得するために使用される逆 z 変換にも及びます。z 変換は ROC から除外される極では収束しないため、このプロセスでは ROC を慎重に考慮する必要があります。

ROC を理解することは、z 変換の収束を確実にするためだけでなく、離散時間システムの安定性と応答を分析および予測するためにも不可欠です。z 変換が収束する特定の領域を描写することにより、ROC は安定し、予測どおりに動作するシステムを設計するのに役立ちます。 ROC が逆 Z 変換に与える影響は、信号処理におけるその重要性を強調しており、離散時間信号やシステムを扱うすべての人にとって重要な概念となっています。

z 変換は、z の特定の値に対してのみ収束します。この値の範囲は、収束領域(ROC)として知られており、これは...

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

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 exponential discrete-time signal x[n], the z-transform becomes a geometric series. In that case, the ROC is the region outside a circle with radius a and centered at the origin.

The ROC also helps show whether a system is stable. If the ROC includes the unit circle, the system is stable. If the ROC is outside the unit circle, the system is unstable. If the ROC matches the unit circle exactly, the system is marginally stable.

The ROC matters for the Discrete-Time Fourier Transform, or DTFT, as well. The DTFT exists only when the z-transform ROC includes the unit circle. The ROC is also important in the inverse z-transform, which is used to recover the original time-domain signal from its z-transform.

Poles are values where the z-transform does not converge, so they are not part of the ROC. Because of this, the ROC must be chosen with care when working backward from the z-transform. Understanding the ROC helps explain convergence, stability, and the response of discrete-time systems.

Z-Transform Convergence and Stability

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

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

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 ROC defines where convergence happens, how it affects the unit circle, and why it matters for DTFT, inverse z-transform, and stability.

Z-Transform Convergence and Stability | 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

Z-Transform Convergence and Stability

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