Convolution example problems

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# Convolution example problems

Convolution is a mathematical operation used to express the relation between input and output of an LTI system. It relates input, output and impulse response of an LTI system as. Periodic convolution is valid for discrete Fourier transform. To calculate periodic convolution all the samples must be real. Periodic or circular convolution is also called as fast convolution.

If two sequences of length m, n respectively are convoluted using circular convolution then resulting sequence having max [m,n] samples.

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Here x[n] contains 3 samples and h[n] also has 3 samples. Now to get periodic convolution result, 1st 3 samples [as the period is 3] of normal convolution is same next two samples are added to 1st samples as shown below:. Correlation is a measure of similarity between two signals.

The general formula for correlation is. It is defined as correlation of a signal with itself. Consider a signals x t. The auto correlation function of x t with its time delayed version is given by. Auto correlation function of energy signal at origin i.

Consider two signals x 1 t and x 2 t. Cross correlation function corresponds to the multiplication of spectrums of one signal to the complex conjugate of spectrum of another signal. Parseval's theorem for energy signals states that the total energy in a signal can be obtained by the spectrum of the signal as.

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Previous Page. Next Page.Mastering convolution integrals and sums comes through practice.

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Here are detailed analytical solutions to one convolution integral and two convolution sum problems, each followed by detailed numerical verifications, using PyLab from the IPython interactive shell the QT version in particular. Here is a convolution integral example employing semi-infinite extent signals. The figure provides a plot of the waveforms. The convolution integral output is.

For the numerical convolution, use ssd. You first write Python code in the command window to generate the signals x t and h t and then carry out the convolution:.

## Signals and systems practice problems list - Rhea

For the case of discrete-time convolution, here are two convolution sum examples. The first employs finite extent sequences signals and the second employs semi-infinite extent signals.

Consider the convolution sum of the two sequences x [ n ] and h [ n ], shown here, along with the convolution sum setup. When convolving finite duration sequences, you can do the analytical solution almost by inspection or perhaps by using a table even a spreadsheet to organize the sequence values for each value of n, which produces a nonzero overlap between h [ k ] and x [ n — k ]. The support interval for the output follows the rule given for the continuous-time domain.

The output y [ n ] starts at the sum of the two input sequence starting points and ends at the sum of input sequence ending points. You can set up a spreadsheet table to evaluate the six sum-of-products related to the output support interval. To verify these hand spreadsheet calculation values, use Python functions in ssd. A plot of the waveforms is given here. With the help of Figure b, you have three cases to consider in the evaluation of the convolution for all values of n. The support interval for the convolution is.

Using the finite geometric series sum formula, the convolution sum evaluates to. Again, using the finite geometric series sum formula, the convolution sum evaluates to. To compare the analytical solution with the numerical solution, you follow the steps of plotting the analytical function against a plot of the actual convolution sum:.

Write a Python function to evaluate y [ n ] as a piecewise function:. Here, you see that the piecewise analytical solution compares favorably to the direct convolution sum numerical calculation. He is a member of the IEEE and is doing real signals and systems problem solving as a consultant with local industry.This page goes through an example that describes how to evaluate the convolution integral for a piecewise function. The key idea is to split the integral up into distinct regions where the integral can be evaluated.

This is done in detail for the convolution of a rectangular pulse and exponential. This is followed by several examples that describe how to determine the limits of integrations that need to be used when convolving piecewise functions. We will use convolution to find the zero input response of this system to the input given by a rectangular pulse, which we define piecewise by three distinction sections, and shown as a blue line.

The next section reiterates the development of the page deriving the convolution integral. If you feel you know that material, you can skip ahead to the mechanics of using the convolution integral. Now let's discuss how we can find an exact solution to this problem, which is not always straightforward with functions that are defined piecewise. To find the output of the system with impulse response. The result for the first part of our solution is the integral of the yellow line which is always zero.

So the integral becomes, in effect:.

### Concept of Convolution

This problem is solved elsewhere using the Laplace Transform which is a much simpler technique, computationally. To develop your ability to do this several examples are given below, each with a different number of "regions" for the convolution integral.

The integrals are not actually performed, only the limits of integration for each region are given. Click on any of the examples below the text at the left side of the page to show or hide it. Each of the examples also has a link to an interactive demo which will allow you to vary t as well as to see the output of the convolution.

Since both functions are constant throughout the integration, the product is just a rectangle whose width varies as t varies.

Since both functions are constant throughout the integration, the product is just a rectangle whose width is 2. The area i. There are two integrals, one to the left of the apex at t-1 and one to the right. The product is zero elsewhere and so doesn't contribute to the integral. Since both functions are constant throughout the integration, the product is just a rectangle whose width is 1. As a challenge, figure this one out yourself: Interactive Demo.

Heres an even trickier one: Interactive Demo. For continuity with the page deriving the convolution integral we can approximate the input by a series of impulses Interactive Demo. Convolution of triangular pulse, f tand a unit step function, h t. As a challenge, figure this one out yourself: Interactive Demo Heres an even trickier one: Interactive Demo.

The integral is defined piecewise.Just as with discrete signals, the convolution of continuous signals can be viewed from the input signalor the output signal. The input side viewpoint is the best conceptual description of how convolution operates. In comparison, the output side viewpoint describes the mathematics that must be used. These descriptions are virtually identical to those presented in Chapter 6 for discrete signals. Figure shows how convolution is viewed from the input side. An input signal, x tis passed through a system characterized by an impulse response, h tto produce an output signal, y t.

The input signal is divided into narrow columns, each short enough to act as an impulse to the system. In other words, the input signal is decomposed into an infinite number of scaled and shifted delta functions. Each of these impulses produces a scaled and shifted version of the impulse response in the output signal. The final output signal is then equal to the combined effect, i.

For this scheme to work, the width of the columns must be much shorter than the response of the system. Of course, mathematicians take this to the extreme by making the input segments infinitesimally narrow, turning the situation into a calculus problem. In this manner, the input viewpoint describes how a single point or narrow region in the input signal affects a larger portion of output signal. In comparison, the output viewpoint examines how a single point in the output signal is determined by the various values from the input signal. Just as with discrete signals, each instantaneous value in the output signal is affected by a section of the input signal, weighted by the impulse response flipped left-for-right.

In the discrete case, the signals are multiplied and summed.

## How to Work and Verify Convolution Integral and Sum Problems

In the continuous case, the signals are multiplied and integrated. In equation form:. This equation is called the convolution integral, and is the twin of the convolution sum Eq. Figure shows how this equation can be understood.

The goal is to find an expression for calculating the value of the output signal at an arbitrary time, t. The first step is to change the independent variable used to move through the input signal and the impulse response. Greek tau.

This change of variable names is needed because t is already being used to represent the point in the output signal being calculated. The input signal is then weighted by the flipped and shifted impulse response by multiplying the two, i. The value of the output signal is then found by integrating this weighted input signal from negative to positive infinity, as described by Eq.

If you have trouble understanding how this works, go back and review the same concepts for discrete signals in Chapter 6. Figure is just another way of describing the convolution machine in Fig. The only difference is that integrals are being used instead of summations. Treat this as an extension of what you already know, not something new. An example will illustrate how continuous convolution is used in real world problems and the mathematics required.In mathematics in particular, functional analysis convolution is a mathematical operation on two functions f and g that produces a third function expressing how the shape of one is modified by the other.

The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted. And the integral is evaluated for all values of shift, producing the convolution function.

Convolution has applications that include probabilitystatisticscomputer visionnatural language processingimage and signal processingengineeringand differential equations. The convolution can be defined for functions on Euclidean spaceand other groups. A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebraand in the design and implementation of finite impulse response filters in signal processing.

Continuous time convolution example: Barker sequence

Computing the inverse of the convolution operation is known as deconvolution. As such, it is a particular kind of integral transform :.

While the symbol t is used above, it need not represent the time domain. As t changes, the weighting function emphasizes different parts of the input function. For the multi-dimensional formulation of convolution, see domain of definition below. A common engineering convention is: . Convolution describes the output in terms of the input of an important class of operations known as linear time-invariant LTI.

In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. In other words, the output transform is the pointwise product of the input transform with a third transform known as a transfer function. See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.

The term itself did not come into wide use until the s or 60s. Prior to that it was sometimes known as Faltung which means folding in Germancomposition productsuperposition integraland Carson's integral. The summation is called a periodic summation of the function f. And if the periodic summation above is replaced by f Tthe operation is called a periodic convolution of f T and g T. For complex-valued functions fg defined on the set Z of integers, the discrete convolution of f and g is given by: .

The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomialsthen the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences.

This is known as the Cauchy product of the coefficients of the sequences. The summation on k is called a periodic summation of the function f. Circular convolution arises most often in the context of fast convolution with a fast Fourier transform FFT algorithm. In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation.

That can be significantly reduced with any of several fast algorithms. Digital signal processing and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O N log N complexity.If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Donate Login Sign up Search for courses, skills, and videos. Math Differential equations Laplace transform The convolution integral. Introduction to the convolution. The convolution and the Laplace transform.

Using the convolution theorem to solve an initial value prob.

### The convolution and the Laplace transform

Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Now that we know a little bit about the convolution integral and how it applies to the Laplace transform, let's actually try to solve an actual differential equation using what we know.

So I have this equation here, this initial value problem, where it says that the second derivative of y plus 2 times the first derivative of y, plus 2 times y, is equal to sine of alpha t.

And they give us some initial conditions. They tell us that y of 0 is equal to 0, and that y prime of 0 is equal to 0. And that's nice and convenient that those initial conditions tend to make the problem pretty clean. But let's get to the problem. So the first thing we do is we take the Laplace transform of both sides of this equation. The Laplace transform of the second derivative of y is just s squared.

This should be a bit of second nature to you by now. It's s squared times the Laplace transform of Y, which I'll just write as capital Y of s, minus s-- so we start with the same degree as the number of derivatives we're taking, and then we decrement that every time-- minus s times y of you kind of think of this as the integral, and you take the derivative 1, so this isn't exactly the derivative of that-- minus, you decrement that 1, you just have a 1 there, y prime of o.

And that's the Laplace transform of the second derivative. Now, we have to do the Laplace transform of 2 times the first derivative. That's just going to be equal to plus 2, times sY of s-- s times the Laplace transform of Y; that's that there-- minus y of 0. And we just have one left. The Laplace transform of 2Y. That's just equal to plus 2 times the Laplace transform of Y. And then that's going to be equal to the Laplace transform of sine of alpha t. We've done that multiple times so far.

That's just alpha over s squared plus alpha squared.We would like a way to take the inverse transform of such a transform. We can use a convolution integral to do this. First note that we could use 11 from out table to do this one so that will be a nice check against our work here. Also note that using a convolution integral here is one way to derive that formula from our table. Now, since we are going to use a convolution integral here we will need to write it as a product whose terms are easy to find the inverse transforms of.

This is easy to do in this case. Note however, that this did require a massive use of trig formulas that many do not readily recall. One final note about the integral just to make a point clear. We can only do that when the variables do not, in any way, depend on the variable of integration. First, notice that the forcing function in this case has not been specified. Prior to this section we would not have been able to get a solution to this IVP. With convolution integrals we will be able to get a solution to this kind of IVP.

We factored out a 4 from the denominator in preparation for the inverse transform process. Now, the first two terms are easy to inverse transform.

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The two functions that we will be using are. We can shift either of the two functions in the convolution integral. Taking the inverse transform gives us. Notes Quick Nav Download. Go To Notes Practice and Assignment problems are not yet written. As time permits I am working on them, however I don't have the amount of free time that I used to so it will take a while before anything shows up here.

Assignment Problems Downloads Problems not yet written. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Example 1 Use a convolution integral to find the inverse transform of the following transform. Take the Laplace transform of all the terms and plug in the initial conditions. 