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Difference between convolution sum and convolution integral

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Welcome to GoldSim Help!

Getting Started with GoldSim

Building a Model in GoldSim

Using the GoldSim Elements

Simulating Discrete Events

Customizing the Interface

Running a Model in GoldSim

Displaying Results in GoldSim

Documenting and Presenting Your Model

Advanced Modeling Concepts

Using Vectors and Matrices

Understanding Locally Available Properties

Modeling Aging Chains

Solving Convolution Integrals

What is a Convolution Integral?

Using the Convolution Element

Examples of the Use of the Convolution Element

Generating Stochastic Time Histories

Using Resources

Script Elements

Using Conditional Containers

Using External Application Elements

Localizing Containers

Cloning Elements

Referencing an Output's Previous Value

Using Looping Containers

Viewing and Modifying the Causality Sequence

Using SubModels to Embed Models Within Models

Customized Importance Sampling Using User-Defined Realization Weights

Dynamically Revising Distributions Using Simulated Bayesian Updating

Tracking Model Changes

Linking Elements to a Database

Distributed Processing Module

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Financial Module

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Glossary of Terms

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Advanced Modeling Concepts / Solving Convolution Integrals / What is a Convolution Integral?

What is a Convolution Integral?

Before describing how to use a Convolution element, it is first important to understand exactly what the element does mathematically, and what this mathematical operation represents conceptually. 

Consider a dynamic system, in which an input signal, say x(t), enters a “black box”, S, and is output as y(t):



Mathematically, we represent the “black box” system as S:

     y(t) = S[x(0 to t)]

where y(t) represents the output signal at time t, and x(0 to t) represents the time history of the input signal.  If the “black box” represents a convolution, it means that S takes on the following mathematical form:



In this equation hτ (t - τ) is the transfer function (also referred to as the impulse response function).  t - τ is referred to as the lag, as it represents the time lag between the input and the output time.  The transfer function is given a subscript τ to indicate that the function itself may take on different forms at different points in time.  That is, the response to an input impulse at t = τ1 could have a different form than the response to an impulse at t = τ2.  

Convolution elements require that you supply the input signal, x(t), and the transfer function, h(t – τ). The element then computes the output signal, y(t).  Note that the convolution integral is a linear operation.  That is, for any two functions x1(t) and x2(t), and any constant a, the following holds:

     S[x1(t) + x2(t)] = S[x1(t)] + S[x2(t)]

     S[ax1(t)] = a S[x1(t)]

Having defined mathematically what a convolution integral does, let us now try to understand what it represents conceptually.  Perhaps the simplest way to think about the convolution integral is that it is simply a linear superposition of response functions, h(t – τi), each of which is multiplied by the impulse x(τi)δτ.

A more common way to interpret the convolution integral is that the output represents a weighted sum of the present and past input values. We can see this if we write the integral in terms of a sum (and assume here that the system is discretized by a single unit of time):

     y(t) = x(0)h(t) + x(1)h(t-1) + x(2)h(t-2) + …

The convolution operation may also be thought of as afiltering operation on the signal x(t), where the transfer function is acting as the filter. The shape of the transfer function determines which properties of the original signal x(t) are "filtered out."

Note that if the transfer function was a constant, the integral would collapse and the output signal would just be a scaled version of the input signal.  Integration is necessary when the transfer function is not constant. Conceptually, this indicates that the system has memory and does not respond instantaneously to a signal.  Rather, the response is delayed and spread out over time.