Name: finding an impulse response from signals
Uploaded: 2024-09-26T13:16:56.000Z
Duration: 1 min 20 s
Channel: JoVE

システムのインパルス応答を利用して、入力信号とインパルス応答畳み込みによる出力応答を決定できます。
このインパルス応答を、入力信号と出力が与えられた場合、デコンボリューションまたは逆フィルタリングと呼ばれます。これは、畳み込み和の構成信号の 1 つを取得するプロセスです。
入力信号と出力応答が与えられると、多項式除算または再帰的アルゴリズム法を使用してデコンボリューションを実行し、インパルス応答を生成できます。
多項式除算アプローチでは、シーケンスは降順多項式の係数と見なされます。次に、インパルス応答を取得するために筆算が実行されます。
再帰的アルゴリズム手法では、出力応答は最初に畳み込み和として定義され、再帰的アルゴリズムとして定式化できます。この方程式は、変数nをゼロに設定することで簡略化され、nの正の値に対するインパルス応答が得られるようになります。
インパルス応答に必要な評価の数は、信号の長さを与えられた関係に代入することによって決定されます。最終的なインパルス応答値は、取得した数値に対して計算されます。

デコンボリューション

デコンボリューションまたは逆畳み込みは、既知の入力信号と出力信号からインパルス応答を抽出するプロセスです。この手法は、システムの特性が不明で、観測可能な信号から推測しなければならないシナリオで不可欠です。

デコンボリューションには、インパルス応答を導出するためのいくつかの数学的手法が含まれます。一般的なアプローチの 1 つは多項式除算です。この方法では、入力シーケンスと出力シーケンスは降順多項式の係数として扱われます。これらの多項式に対して長除算を実行すると、インパルス応答が得られます。この方法は簡単で、システムの入力と出力の関係が多項式形式で表現されている場合にインパルス応答を決定するための効率的な手段を提供します。

デコンボリューションのもう 1 つの効果的な手法は、再帰アルゴリズム法です。ここでは、出力応答は畳み込みの和として表され、再帰アルゴリズムに変換できます。この方法の再帰的な性質により、畳み込みの和を体系的に簡略化できます。変数 n をゼロに設定することで、方程式が簡略化され、n の正の値に対するインパルス応答を決定できます。この方法は、デコンボリューションに伴う計算の複雑さを軽減するため、長いシーケンスを扱う場合に特に便利です。

インパルス応答を決定するために必要な評価の数は、入力信号と出力信号の長さによって異なります。これは、信号の長さを特定の関係に代入することで計算できます。必要な評価の数が決まると、インパルス応答の最終値を正確に計算できます。この手順は、導出されたインパルス応答がさまざまな入力条件下でのシステムの動作を予測する上で正確で信頼できるものであることを保証するために重要です。

システムのインパルス応答を利用して、入力信号とインパルス応答畳み込みによる出力応答を決定できます。
これを取得すると...

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

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 the polynomials and recover the impulse response.

A second method uses a recursive algorithm. The output response is written as a convolution sum, and that sum is converted into a recursive form. By setting the variable n to zero, the equation is simplified, and the impulse response for positive values of n can be found.

This recursive method is especially helpful for long sequences because it lowers the amount of computation needed. The number of evaluations needed to find the impulse response depends on the lengths of the input and output signals. Those lengths are placed into a relation to calculate the required evaluations, and the final impulse response value can then be determined accurately. That result helps predict how the system will behave for different input conditions.

Finding an Impulse Response from Signals

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 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...

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

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 deconvolution uses polynomial division and a recursive algorithm to extract an impulse response from known input and output signals.

Finding an Impulse Response from Signals | 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

Finding an Impulse Response from Signals

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