Name: convolution rules for width and area
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 17 s
Channel: JoVE

重要な畳み込みプロパティは、幅、面積、微分、および積分のプロパティです。
width プロパティは、入力信号の持続時間がそれぞれ T1 と T2 の場合、出力応答の幅は、2 つの関数の形状に関係なく、両方の持続時間の合計に等しくなることを示します。たとえば、持続時間が 2s と 1s の 2 つの直交パルスを畳み込むと、幅が 3s の関数になります。
area プロパティは、2 つの関数の畳み込みの下の面積が各関数の下の面積の積と等しいことをアサートします。
微分プロパティは、畳み込みの導関数が入力信号の導関数とインパルス応答の畳み込みに等しい、またはその逆であることを示します。
方程式を組み合わせて高次導関数を一般化すると、一般微分関係が得られます。
特定の畳み込みの積分特性により、インパルス応答と入力応答の畳み込みが得られます。これは、逆の順序で述べることもできます。
ステップ関数を使用して畳み込みが実行されると、LTI システムは理想的な積分器のように動作します。

畳み込み特性-ii

重要な畳み込み特性には、幅、面積、微分、積分特性があります。

幅特性は、入力信号の持続時間が T_1 と T_2 の場合、出力応答の幅は、2 つの関数の形状に関係なく、両方の持続時間の和に等しいことを示します。たとえば、持続時間が 2 秒と 1 秒の 2 つの矩形パルスを畳み込むと、幅が 3 秒の関数が生成されます。

面積特性は、2 つの関数の畳み込みの下の面積が、各関数の下の面積の積に等しいことを示します。数学的には、x(t) と h(t) が 2 つの関数である場合、面積特性の式は次のように表せます。

<img alt="Equation1" src="/files/ftp_upload/16051/16051_PT%20EQUATION_1.svg" style="height: 16px;">

微分特性は、畳み込みの導関数が入力信号の導関数とインパルス応答の畳み込みに等しいこと、またはその逆であることを示します。これは次のように表せます。


<img alt="Equation2" src="/files/ftp_upload/16051/16051_PT%20EQUATION_2.svg" style="height: 34px;">

これらの式を組み合わせて高次の導関数に一般化すると、一般的な微分関係が得られます。

積分特性は、畳み込みの積分がインパルス応答の畳み込みと入力信号の積分をもたらすこと、またはその逆であることを示します。これは数学的に次のように表されます:


<img alt="Equation3" src="/files/ftp_upload/16051/16051_PT%20EQUATION_3.svg" style="height: 42px;">

畳み込みがステップ関数で実行される場合、LTI システムは理想的な積分器のように動作します。

これらの特性 (幅、面積、微分、積分) は、LTI システムでの畳み込み演算を簡素化して理解するのに重要であり、複雑な信号処理タスクの分析と設計を容易にします。

畳み込み特性-II

重要な畳み込みプロパティは、幅、面積、微分、および積分のプロパティです。
width プロパティは、...

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 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 seconds and 1 second gives an output with a width of 3 seconds.

The area property states that the area under the convolution of two functions equals the product of their individual areas. In other words, if x(t) and h(t) are convolved, the total area of the result depends on the area of each input. The differentiation property says that the derivative of a convolution is the convolution of the derivative of the input signal and the impulse response, or the other way around.

The integration property gives the matching rule for integrals. It says that integrating a convolution produces the convolution of the impulse response and the integral of the input signal, or vice versa. If a convolution is performed with a step function, the LTI system acts like an ideal integrator. Together, these width, area, differentiation, and integration rules help simplify convolution in LTI systems and support the analysis and design of more complex signal processing tasks.

Convolution Rules for Width and Area

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 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 the key convolution rules for width, area, differentiation, and integration in LTI systems, with a simple rectangular pulse example and step-function behavior.

Convolution Rules for Width and Area | 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 Rules for Width and Area

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