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

In any LTI system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero.
The convolution integral is divided into two parts: the zero-input or natural response and the zero-state or forced response of the system, with t0 indicating the initial time.
The convolution integral can be simplified by presuming that both the input signal and impulse response are zero for negative time values.
Graphical convolution involves four steps: Folding, Shifting, Multiplication, and Integration. Consider an RC circuit with a specified input pulse signal and output response. Initially, folding is performed by creating a mirror image of the input signal along the y-axis. This is followed by a slight shift, multiplication, and integration of the resulting signals. The resulting convolution is depicted graphically.
In discrete-time convolution, the response is determined by applying an input to a discrete-time system and using the impulse response and convolution integral.
The convolution of the discrete input signal and impulse response forms the convolution sum for the system response.

convolution: math, graphics, and discrete signals

In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process involves four steps: folding, shifting, multiplication, and integration.
Consider an RC circuit with a specified input pulse signal and output response. Initially, folding is performed by creating a mirror image of the input signal along the y-axis. This is followed by shifting, where the folded signal is slid along the time axis. Next, the multiplication of the folded and shifted signals is done point-by-point. Finally, the integration of the resulting signal over time provides the convolution result. This process can be depicted graphically.
In discrete-time convolution, the system's response is determined by applying an input to a discrete-time system and using the impulse response and convolution sum. The convolution of the discrete input signal x[n] and the impulse response h[n] forms the convolution sum for the system response:
<img alt="Discrete convolution formula, \(y[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k]\), mathematical concept." src="/files/ftp_upload/16049/16049_PT%20EQUATION_1.svg" style="text-align: center; height: 28px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
This sum computes the output signal y[n] at each discrete time step n. Understanding both continuous and discrete convolution is essential for analyzing LTI systems and predicting their behavior in response to various inputs.

Convolution: Math, Graphics, and Discrete Signals

In any LTI system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero.
The convolution ...

convolution-math-graphics-and-discrete-signals

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Electrical Engineering

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Linear Time- Invariant Systems

KEY TERMS AND CONCEPTS

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 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 Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

Convolution in LTI Systems and Signal Response

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