Name: convolution in lti systems and signal response
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 24 s
Channel: JoVE

どの LTI システムでも、2 つの信号の畳み込みは、すべての初期条件が 0 であると仮定して、畳み込み演算子を使用して示されます。
畳み込み積分は、ゼロ入力応答または自然応答と、システムのゼロ状態または強制応答の 2 つの部分に分割され、t0 は初期時間を示します。
畳み込み積分は、負の時間値に対して入力信号とインパルス応答の両方がゼロであると仮定することで簡略化できます。
グラフィカル畳み込みには、フォールディング、シフト、乗算、積分の 4 つのステップが含まれます。入力パルス信号と出力応答が指定されたRC回路を考えてみましょう。最初に、フォールディングは、入力信号の鏡像を y 軸に沿って作成することによって実行されます。これに続いて、結果の信号のわずかなシフト、乗算、および積分が続きます。結果として生じる畳み込みはグラフィカルに描かれています。
離散時間畳み込みでは、離散時間システムに入力を適用し、インパルス応答と畳み込み積分を使用して応答を決定します。
離散入力信号とインパルス応答の畳み込みは、システム応答の畳み込み和を形成します。

数学、グラフ、および離散信号

どの LTI (線形時不変) システムでも、2 つの信号の畳み込みは、すべての初期条件がゼロであると仮定して、畳み込み演算子を使用して表されます。畳み込み積分は、ゼロ入力または自然応答とゼロ状態または強制応答の二部分に分けることができ、t_0 は初期時間を示します。

畳み込み積分を簡略化するために、負の時間値に対しては入力信号とインパルス応答の両方がゼロであると仮定します。グラフの畳み込みプロセスには、折りたたみ、シフト、乗算、および積分の 4 つのステップが含まれます。

指定された入力パルス信号と出力応答を持つ RC 回路を考えてみましょう。最初に、y 軸に沿って入力信号の鏡像を作成することによって折りたたみが実行されます。次に、折りたたみ信号を時間軸に沿ってスライドさせるシフトが行われます。次に、折りたたみ信号とシフト信号の乗算が点ごとに実行されます。最後に、結果として得られた信号を時間とともに積分すると、畳み込みの結果が得られます。このプロセスは、グラフで表すことができます。

離散時間畳み込みでは、離散時間システムに入力を適用し、インパルス応答と畳み込みの合計を使用して、システムの応答が決定されます。離散入力信号 x[n] とインパルス応答 h[n] の畳み込みにより、システム応答の畳み込みの和が形成されます。

<img alt="Equation1" src="/files/ftp_upload/16049/16049_PT%20EQUATION_1.svg" style="height: 28px;">

この和により、各離散時間ステップ n での出力信号 y[n] が計算されます。連続畳み込みと離散畳み込みの両方を理解することは、LTI システムを分析し、さまざまな入力に対する動作を予測するために不可欠です。

どの LTI システムでも、2 つの信号の畳み込みは、すべての初期条件が 0 であると仮定して、畳み込み演算子を使用して示されます。
畳み込み...

convolution-math-graphics-and-discrete-signals

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: 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 with, both the input signal and the impulse response are taken to be zero for negative time values.

Graphical convolution follows four steps: folding, shifting, multiplication, and integration. First, folding makes a mirror image of the input signal along the y-axis. Then the folded signal is shifted, or slid, along the time axis. After that, the folded and shifted signals are multiplied point by point.

The last step is integration over time, which gives the convolution result. This process can be shown graphically for an RC circuit with a given input pulse signal and output response. The same idea also appears in discrete-time systems. There, the input signal x[n] and the impulse response h[n] are combined with the convolution sum to find the output y[n] at each discrete time step n. Knowing both continuous and discrete convolution helps describe LTI system behavior for different inputs.

Convolution in LTI Systems and Signal Response

convolution-in-lti-systems-and-signal-response

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 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

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 convolution predicts LTI system response using folding, shifting, multiplication, and integration, plus the discrete convolution sum for x[n] and h[n].

Convolution in LTI Systems and Signal Response | 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

Convolution in LTI Systems and Signal Response

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