Name: lti system tricks for easier convolution
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 20 s
Channel: JoVE

畳み込み計算は、その固有の特性を利用することで簡略化できます。
この可換特性は、LTI システムの入力とインパルス応答を出力に影響を与えることなく交換できることを示しています。
連想特性は、3 つの関数のマージされた畳み込みが、畳み込みが実行される順序に関係なく、変わらないことを示唆しています。
インパルス応答を持つ 2 つの LTI システムを直列に接続すると、それぞれの方程式を連想特性を使用して組み合わせて、個々のインパルス応答の畳み込みと同様の等価ジョイント インパルスを導き出すことができます。
分散特性により、複数の入力信号の合計に対する畳み込み演算が可能になり、複雑なインパルス応答をより単純なコンポーネントに分解することで簡略化されます。
タイムシフト プロパティは、時間不変システムの入力を遅らせるか、システムに遅延が組み込まれている場合は、出力が 2 つの遅延の合計だけ遅延することを意味します。
計算上、このプロパティを使用すると、信号の対称性や因果関係を活用するために信号を遅延または進行させることができるため、畳み込み演算が簡略化されます。

畳み込みの特性-i

畳み込みの計算は、その固有の特性を利用することで簡略化できます。

交換律は、LTI (線形時不変) システムの入力とインパルス応答は、出力に影響を与えずに交換できることを示しています。

<img alt="Equation1" src="/files/ftp_upload/16050/16050_PT%20EQUATION_1.svg" style="height: 16px;">

結合律は、3 つの関数の結合された畳み込みは、畳み込みの順序に関係なく変更されないことを示しています。たとえば、3 つの関数 x(t)、h_1(t)、および h_2(t) は次のように記述されます。


<img alt="Equation2" src="/files/ftp_upload/16050/16050_PT%20EQUATION_2.svg" style="height: 16px;">

インパルス応答を持つ 2 つの LTI システムを直列に接続すると、結合法則を使用してそれぞれの方程式を結合し、個々のインパルス応答の畳み込みである同等の接合インパルス応答を導出できます。

分配律により、複数の入力信号の合計に対する畳み込み演算が可能になり、複雑なインパルス応答をより単純な要素に分解できます。数学的には、次のように表されます。


<img alt="Equation3" src="/files/ftp_upload/16050/16050_PT%20EQUATION_3.svg" style="height: 16px;">

時間シフトの法則は、時間不変システムの入力を遅らせると、出力が同じ量だけ遅れることを意味します。同様に、システムに組み込みの遅延がある場合、出力は入力遅延とシステム遅延の合計だけ遅れます。時間シフト t_0 は次のように表されます:

<img alt="Equation4" src="/files/ftp_upload/16050/16050_PT%20EQUATION_4.svg" style="height: 16px;">

計算上、この法則により、信号の対称性または因果関係を利用して畳み込み演算を簡素化し、信号を遅らせたり進めたりすることができます。

これらの法則 (可換律、結合律、分配律、時間シフト) は、LTI システムでの畳み込み演算を簡素化するための基本的なツールであり、複雑な信号処理タスクをより管理しやすく効率的にします。

畳み込みの特性-I

畳み込み計算は、その固有の特性を利用することで簡略化できます。
可換特性は、入力とインパルス...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Linear Time- Invariant Systems

線形時不変システム

KEY TERMS AND CONCEPTS

主要用語と概念

Convolution Properties I

lti system tricks for easier convolution

LTI system properties can make convolution easier to work with. LTI stands for linear time-invariant. In these systems, the input signal and the impulse response can be handled in helpful ways that simplify the final output.

The commutative property means the input and the impulse response can be switched without changing the result. The associative property says that when three functions are convolved, the order of grouping does not change the answer. For two LTI systems in series, this lets you combine their equations into one equivalent impulse response.

The equivalent joint impulse response is found by convolving the two individual impulse responses. The distributive property also helps by allowing convolution to be applied to a sum of input signals. This makes it easier to break a complex impulse response into simpler parts.

The time-shift property shows what happens when a signal is delayed. If the input of a time-invariant system is delayed, the output is delayed by the same amount. If the system already has a delay, the total output delay is the input delay plus the system delay. This same property can also help when signals are shifted forward or backward, especially when symmetry or causality makes the convolution simpler.

These properties include commutative, associative, distributive, and time-shift rules. Together, they are basic tools for simplifying convolution in LTI systems. They make signal processing tasks more manageable and efficient.

LTI System Tricks for Easier Convolution

linear time-invariant systems

Impulse Response in LTI Systems

Linear time-invariant systems are defined by linear elements and constant parameters. A system is linear when it shows homogeneity and additivity. These two ideas together are called the superposition property. Superposition is a key rule for analyzing linear systems. For a linear time-invariant system, the response to an impulse at the input can describe the whole input-output behavior. An impulse is a very short, sudden input. Once the impulse response is known, the response to other inputs...

impulse response

RC Circuit Impulse Response

RC circuit impulse response describes how the capacitor voltage changes after a brief input signal. In this system, the voltage source is the input, and the capacitor voltage is the output. The state of the circuit before and after excitation is defined separately so the response can be tracked clearly. Kirchhoff’s law is used to write the input signal equation for the RC circuit. The capacitor current and voltage provide the output relation. After substituting the current expression and...

convolution: math, graphics, and discrete signals

convolution-math-graphics-and-discrete-signals

Convolution in LTI Systems and Signal Response

Convolution describes how a linear time-invariant, or LTI, system responds to an input signal. In an LTI system, the convolution of two signals is written with a convolution operator. The method assumes that all initial conditions are zero. The convolution integral can be split into two parts. These are the zero-input response, also called the natural response, and the zero-state response, also called the forced response. The initial time is marked as t0. To make the integral easier to work...

convolution properties i

LTI system properties can make convolution easier to work with. LTI stands for linear time-invariant. In these systems, the input signal and the impulse response can be handled in helpful ways that simplify the final output. The commutative property means the input and the impulse response can be switched without changing the result. The associative property says that when three functions are convolved, the order of grouping does not change the answer. For two LTI systems in series, this lets...

convolution properties ii

Convolution Rules for Width and Area

Convolution rules help describe how an output signal changes when two signals are combined in an LTI system. The main rules here are width, area, differentiation, and integration. These rules make convolution easier to analyze and use in signal processing. The width property says that if two input signals last for T1 and T2, the output response lasts for T1 plus T2. This is true no matter what shape the two functions have. For example, convolving two rectangular pulses with durations of 2...

deconvolution

Finding an Impulse Response from Signals

Deconvolution is a method for finding a system’s impulse response from known input and output signals. It is also called inverse filtering. This technique is useful when the system’s characteristics are not known directly and must be inferred from the signals that can be observed. One common way to do deconvolution is polynomial division. In this approach, the input and output sequences are treated as coefficients of polynomials written in descending order. Long division is then used to divide...

bibo stability of continuous and discrete -time systems

bibo-stability-of-continuous-and-discrete-time-systems

Testing System Stability with Convolution

System stability in signal processing is often tested with convolution. A system is BIBO stable, or bounded-input bounded-output stable, when every bounded input signal produces a bounded output signal. A bounded input signal has a modulus that never exceeds a set constant at any time. For a continuous-time Linear Time-Invariant, or LTI, system, BIBO stability is checked with the convolution integral. The input is bounded by a constant, and the convolution integral shows whether the output...

key terms and concepts

Learn how commutative, associative, distributive, and time-shift properties simplify convolution in LTI systems for high school signal processing students.

LTI System Tricks for Easier Convolution | 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

LTI System Tricks for Easier Convolution

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