# Find the unit impulse response of a system specified by the equation

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up. This year I'm having trouble with my Signals and Systems class.

This question was my previous exam question but I coudn't write anything about it as a solution. If somebody could at least show a way to solve this problem it would be great. From here, to find the output I think I will use convolution.?

I used Laplace transform to find the inverse fourier transform of the function H jw. What was I going to do if Laplace transform would not be suitable to situation? It looks like your transfer function is correct, but there's a small mistake in your partial fraction expansion:. A method which is generally ignored.

The procedure: Consider an LTI system which is causal with initial rest conditions. Remember that this system was defined to be under inital rest with complete zero inital conditions prior to the application of any excitation.

As we stated, this system was LTI and causal with initial rest condition. A stable system, isn't it? Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. How to obtain impulse response from the differential equation of a system? Ask Question. Asked 4 years, 3 months ago. Active 2 months ago. Viewed 17k times. Thanks in advance Isma 6 6 bronze badges. Jay M. Jay 53 1 1 gold badge 1 1 silver badge 5 5 bronze badges.

Jan 5 '16 at Not sure if it is correct or not but this is all i can come up with so far. Jay Jan 5 '16 at Was is because of the orientation of the picture or the solution?One of the most common test inputs used is the unit step function.

This page serves as a review of the method of finding the step response of first and second order system. It is assumed that the reader has studied this topic previously and merely needs a quick overview. This section serves as a review of the calculation of the step response of first order systems by the method of homogeneous and particular or natural and forced response.

Consider the systems shown below. In the circuit the input and output are e in and e outrespectively. In the mechanical system the input and output are x in and x outrespectively. Find the unit step response of the sytstems. Solution: First we need to find the differential equation representing the systems. We do this by summing currents at a node e out or forces at x out. Electrical Equations sum of currents at e out Mechanical Equations sum of forces at x out. As a second example, consider the systems shown below in which we have swapped the components from the fires example.

Find the unit step response of the systems. As you would expect, the response of a second order system is more complicated than that of a first order system. Whereas the step response of a first order system could be fully defined by a time constant and initial conditions, the step response of a second order system is, in general, much more complex. As a start, the generic form of a second differential equation that we might solve is iven by:.

### How do you find the Impulse Response, h(t)

We will proceed by examining a particular system. To find the homogeneous solution of a second order system, we proceed as before, i. We set the input in the differential to zero, and substitute this expression and solve for "s. This last equation is called the "characteristic equation" of the system. Since we get a second order polynomial, we get two values for s.

We now have two unknown coefficients, A 1 and A 2 to be determined from initial conditions. However we have three distinct cases that we will consider based upon the characteristics of s 1 and s 2. The simplest case occurs when s 1 and s 2 are real and negative and not equal to each other; we call this the overdamped case.

If s 1 and s 2 are real and negative and equal to each other we call this the critically damped case.Note: This page does not assume knowledge of the Laplace Transform. If you do understand the Laplace Transform, many of these results are easier and are given here. Before reading this section you must first become familiar with the unit impulse function and the unit step response.

To develop this relationship, consider first the unit step response of a system.

### Difference Equations and Impulse Responses

If we delay the step, we simply delay the response:. By linearity, if we apply the sum of two inputs, the output is simply the sum of the individual outputs:. It is important to keep in mind that the impulse response of a system is a zero state response i.

If the problem you are trying to solve also has initial conditions you need to include a zero input response i. Note: Though it is not yet apparent why the impulse response may be useful, we will see later with the convolution integral that the impulse response lets us solve for the system response for any arbitrary input. When we apply it to a system we multiply by an amplitude with units equal to those of the input to the system.

For example, if the input of the system shown below has units of volts, then the step function must implicitly be multiplied by a constant of 1V. In the same way we did with the step, if our system input has units of volts then we must implicitly multiply the unit impulse by its area, or 1V-s. The calculation of the impulse response of a system will proceed in two steps. First we find the unit step response as described elsewherewe then differentiate it.

The only non-obvious step is that we must represent the unit step response in a functional form. Some examples will clarify. Consider the systems shown below. In the circuit the input and output are e in and e outrespectively. In the mechanical system the input and output are x in and x outrespectively. Find the unit impulse response of the systems. Solution: Step 1 is to find the unit step response. Both systems have identical step responses with outputs e out or x out derived elsewhere.

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Step 2 is to differentiate the unit step response. However, there is a slight difficulty here because we have a piecewise description of the step response i. We need a functional description of the system if we are to differentiate it for all values of time.

Since the function is zero for negative times, we used the unit step function to represent the signal. We can take the derivative of the first term and use the fact that the derivative of the step function is the impulse function to rewrite the second.

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The rightmost term can be simplified. Such an impulse has no effect on system behavior, so we can drop this term and we get:.

As a second example, consider the systems shown below in which we have swapped the components from the fires example. Find the unit impulse response of the sytstems. Solution: Step 1 is to find the unit step response of the system derived elsewhere. The impulse function in the result is easily understood. Because the step response has a discontinuity in it i. If the step response of a system has no discontinuities, the impulse response has no impulse functions.

If the step response of a system has a discontinuity, the impulse response will have an impulse function as a part of it at the same time as the discontinuity. As expected, since the step response has a discontinuity, the impulse response has an impulse function as part of it.

Solution: Put in functional form and differentiate note, in the interest of brevity not all steps are in the differentiation are given :. As expected, the result has no impulse function in it because there was no discontinuity in the step response. The impulse response of a system is important because the response of a system to any arbitrary input can calculated from the system impulse response using a convolution integral.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.

I don't know where to start from, I'm not familiar with math. I'm trying to understand this question "how to identify impulse response of a system? In this case, if I put impulse function signal in to above system, then I can get the impulse response. This can be a difficult task in general. You can try some exercices in Exercises in Signals, Systems, and Transformsfor instance 1.

You can also check the applet in the joy of convolution. Since, in practice, it is impossible to generate a discrete pulse, there are other practical identification techniques for real-life systems, using random sequences or sine waves. It is called the impulse response because if you feed in the impulse signal you get the convolution filter as the output, e. So once you know the impulse response, you can calculate the response to any signal using the first equation.

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Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 4 years, 1 month ago. Active 4 years, 1 month ago. Viewed times. Question 1: What is the meaning of "how to identify impulse response of a system"?

Is this asking about convolution or auocorrelation? I can't understand what it does mean. Question 2: I want to know how to identify impulse response of a system? I'm very confused. If you have any hint, please let me know. Update I found a question in my case as below, 1. Active Oldest Votes.

Fat32 Laurent Duval Laurent Duval 21k 3 3 gold badges 21 21 silver badges 78 78 bronze badges. Would you help me any hint please?

I think the same both mechanisms except for time reversal. However, a convolution is just a multiplication in the Fourier domain, where as correlation is multiplication by the conjugate. I guess alot of filtering is done in the Fourier domain so the former is easier.

Interestingly, the wavelet literature uses correlation, even for wavelets defined in the Fourier domain. Have a look to this examples in Matlab: mathworks. Also can you tell me how to identify the impulse response of a system? Sign up or log in Sign up using Google.

Sign up using Facebook.Documentation Help Center. For state-space models, impulse assumes initial state values are zero. The impulse response of multi-input systems is the collection of impulse responses for each input channel. The duration of simulation is determined automatically to display the transient behavior of the response. Express Tfinal in the system time units, specified in the TimeUnit property of sys. Express t in the system time units, specified in the TimeUnit property of sys.

For discrete-time models, t should be of the form Ti:Ts:Tfwhere Ts is the sample time. For continuous-time models, t should be of the form Ti:dt:Tfwhere dt becomes the sample time of a discrete approximation to the continuous system see Algorithms.

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To plot the impulse responses of several models sys1See "Plotting and Comparing Multiple Systems" and the bode entry in this section for more details. No plot is drawn on the screen. For single-input systems, y has as many rows as time samples length of tand as many columns as outputs. In the multi-input case, the impulse responses of each input channel are stacked up along the third dimension of y. The dimensions of y are then. Similarly, the dimensions of x are. The left plot shows the impulse response of the first input channel, and the right plot shows the impulse response of the second input channel.

Because this system has two inputs, y is a 3-D array with dimensions. The impulse response of the first input channel is then accessed by. Fetch the impulse response and the corresponding 1 std uncertainty of an identified linear system. You can change the properties of your plot, for example the units. For information on the ways to change properties of your plots, see Ways to Customize Plots. Continuous-time models are first converted to state space. The impulse response of a single-input state-space model. To simulate this response, the system is discretized using zero-order hold on the inputs. Linear System Analyzer initial lsim step. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location.

Toggle Main Navigation. Search Support Support MathWorks. Search MathWorks. Off-Canvas Navigation Menu Toggle.One of the most common test inputs used is the unit step function. If the problem you are trying to solve also has initial conditions you need to include a zero input response in order to obtain the complete response.

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If you don't know about Laplace Transforms, there are time domain methods to calculate the step response. We can easily find the step input of a system from its transfer function.

Difference Equation to Impulse Response

Given a system with input x toutput y t and transfer function H s. Immediately we can determine two characteristics of the unit step response, the initial and final values, of the step response by invoking the initial and final value theorems.

First we will consider a generic first order system, then we will proceed with several examples. Consider a generic first order transfer function given by.

This last equation is important. Likewise if we experimentally determine the initial value, final value and time constant, then we know the transfer function.

The time constant of first order systems is often easy to find. The time constants of some typical first order systems are given in the table below:. If the input force of the following system is a unit step, find v t. Also shown is a free body diagram. Solution: The differential equation describing the system is.

If the input force of the following system is a step of amplitude X 0 meters, find y t. Note the input is not a unit step, but has a magnitude of X 0. Therefore all system outputs must also be scaled by X 0. Note: input and output are in different directions because they were defined that way in system drawing.

If the input voltage, e in tof the following system is a unit step, find e out t. Solution: First we find the transfer function. We note that the circuit is a voltage divider with two impedances. Clearly that was much simpler than the previous solution using partial fraction expansion. Therefore, as before. As you would expect, the response of a second order system is more complicated than that of a first order system.

Whereas the step response of a first order system could be fully defined by a time constant determined by pole of transfer function and initial and final values, the step response of a second order system is, in general, much more complex. As a start, the generic form of a second order transfer function is given by:.

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Then we can rewrite the transfer function as. The choice of these constants may seem arbitrary, but we will soon show that the choice simplifies the mathematics, and that all three constants have a physical interpretation that helps give insights into a system.

We call this the prototype second order lowpass system because the frequency response of this system is "lowpass," don't worry if you don't know what that means yet. The first three cases are most important, and the last two will be discussed only briefly in what follows.

In the overdamped case we have two real poles of the transfer function or zeros of the characteristic equation at. We can look this form up as the "asymptotic double exponential" in the Laplace transform table or do an inverse Laplace transform using partial fraction expansion to get:.

You can experiment with how the pole locations affect the step resonse with an interactive demo. To find the response of the critically damped case we proceed as with the overdamped case. This is the "asymptotic critically damped" form in the Laplace transform tableso. You can experiment with how the pole location affects the step resonse with an interactive demo.

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For a better experience, please enable JavaScript in your browser before proceeding. How do you find the Impulse Response, h t. Thread starter martnll2 Start date Apr 13, Homework Statement How do i find the impulse response, h tgiven the following? Do you know about the Laplace transform? What is the relationship between input, output, and the impulse response in that domain?

We didn't cover Laplace yet, that comes next week. Is there an alternate way to solve this? RoshanBBQ said:. I don't think so. Insights Author. Gold Member. Last edited: Apr 14, I already tried dividing both sides by 8t, and taking the derivative, but it didn't seem to work. Here is my work. I tried this but got unreal coefficients for the impulse term. The reason I'm showing you this is in an intro course to systems, this should be a plain fact stated in the book It was in mine, and rude man's comment got my memory jogging.

I doubt the problem expected you to derive this, and the real culprit is you most likely not reading your book. Since you will be encountering this, I'll go ahead and remind you that you have to use product rule with u t being one function and the other stuff being another to find the right answer. Theorem: If y t is the reponse to a ramp, then y' t is the response to a step input, and y'' t is the response to an impulse input Dirac delta function.

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I can't give you the proof of this since you haven't had the Laplace transform. The problem is easy after that. You can do it in the time domain. I'm still confused. I don't know how you could use these integrals not knowing h t since the problem is asking to find h t. I don't know how to use these facts to solve h t. Do I just take the derivative with respect to tau on both sides and solve h t that way? Last edited: Apr 15, I do read my text book. I guess this is one of the main reasons I am changing my major; I can't comprehend anything!