impulse response of a system

[PG1995]

Hi

Could you please help me with **broken link removed** query? Thank you.

Regards
PG

 

[MrAl]

Hi,


This one is a simpler question without messy drawings

When we find the impulse response we are finding out something about the system that tells us a lot about that system. We may NEVER want to hit the glass ball with a steel head hammer, but we may want to hit it with a soft head hammer and know how much it will flex and how it moves. The glass ball is a linear system with some elastic properties, even though it appears to be perfectly hard and ridged. Finding the impulse response (however you do this) will tell you just how elastic it is, and how it bends when you apply a force to say one side. If you didnt have this information you could not dare to hit it with anything even a soft cloth because you wont know how it responds, so you'd never be able to accurately use that glass ball as part of your system.
You might not want to hit the glass ball with a hammer of any sort, but you may want to use that glass ball to hit something else like a glass bell which would make a charming sound. You'd be able to analyze how the ball responds when it hits something if you knew the impulse response, but without that info you dont know anything about it.

For an electrical network, we like to know the impulse response because then we can calculate the response for other types of inputs much easier. You might not see this yet but you will in the future. This doesnt mean that we have to apply a huge pulse, but knowing how it responds to a theoretical huge pulse tells us a lot about it. And it doesnt matter if all the transistors blow out and caps short out with that hugh input, because we'll never actually apply that high of a signal. We will know other information about the system such as capacitor voltage ratings that will tell us to keep our real life signals below a certain level. And furthermore we will also know what kinds of signals might drive the system into a non linear mode which we also avoid using our secondary knowledge.

We always have some underlying knowledge about a system which we might call the secondary knowledge. We have the primary information that tells us how the system responds under normal conditions, and we have the secondary information which tells us how much the system can take before the primary information breaks down. The impulse response could be considered to be part of the primary knowledge about the system.
A simple example was the capacitor, where we know we can use a 1uf, 20v capacitor up to 20v (really more practical at 15v or the like) and here the primary info is the 1uf (which tells us how our circuit will work under normal conditions) and the secondary info is the 20v rating which tells us when the system breaks down under non normal conditions.

So for the glass ball, the primary info would be the impulse response that tells us how the thing responds to normal excitations, and the secondary info would be how much force it will take before it shatters and thus becomes non linear. So this secondary into here can be viewed as the limits of the system, of which every system has. We can use the impulse response information within the limits and even though these limits reduce the usable range, they give us a huge advantage in being able to characterize the system. As i mentioned above this will become more and more clear in the future.

Right now you are learning about the primary info when you learn about the impulse response, and just because the glass ball breaks when we hit it hard with a steel head hammer doesnt mean that we cant know anything about the glass ball with excitation forces much less than that.

 

[Winterstone]

PG1995,

I have a very short answer to your question:

The impulse response is a description of a LINEAR system in the time domain. Such a system does not know any limitations caused by finite supply voltages.
Is there any linear system in reality - without any restrictions (in particular as far as the dynamic range is concerned, which should be infinite for impulse measurements) ?
Of course, the answer is NO.
Therefore, we are only in the position to calculate/simulate the impulse response - but we cannot measure it. The only possibility is an approximation with a pulse that does not drive thy system into saturation.

 

[steveB]

I can reiterate the same thoughts as above.

When you measure an impulse response to a continuous time system, it is understood that you will use a pulse as an approximation. The shape of the pulse does not matter, as long as the input signal energy of the pulse (i.e. its integrated area) is known. Let's assume you use a perfect square pulse, to keep things simple. In this case the two things you can control are the pulse amplitude and the pulse duration. You choose the amplitude to be as large as possible, while still keeping it below the limit of nonlinearity. The pulse duration you try to keep as long as possible, but very well below the time constants in your system. In this way you will drive the system with enough amplitude and energy to make good measurements, and you will approximate the response of the impulse even though you are using a pulse.

Beginning students often assume that the pulse has to be (1) very very narrow and (2) very tall and (3) the area has to be equal to one and (4) the shape has to be perfect. But, all four of these assumptions are false. The shape and total integrated area are irrelevant, as long as you know what the area (energy) is and as long as the pulse is large enough to "kick" the system hard enough to do accurate measurements. The amplitude must also be low enough to be in the linear range of the system. And, the pulse duration also needs to be very small compared to the system time constants.

Basically, the mathematics and the practice are very different things, but if you have good intuition (which takes experience) you can mentally jump between the theoretical and the practical domains effortlessly.

 

[Winterstone]

I totally agree with the above detailed and very clear description.

PG, my question: Why are you interested in the impulse response?
You should know that the most common parameters describing the time domain properties of a system are based on the STEP response (rise time, settling time, overshoot, ringing,)

 

[MrAl]

Hello,

The impulse response is one of the most important characterizations of a system. It allows immediate conversion to other excitation responses including but not limited to the step response. I could never see a serious study into electrical circuits and systems without ample consideration of the impulse response and what it means.

And yes there are other ways of measuring the impulse response using well behaved real life excitations (that dont even have to be impulses) that we havent talked about too much yet.

To sum up, once we know the impulse response we can find the response to any other excitation simply by the convolution of the excitation with the impulse response. It can't get any easier than that
That means we can quickly get the response to other types of signals, so it should be clear that the impulse response is a more basic property of a circuit or system than any other type of response.

 

[misterT]

You can actually simulate any acoustic place (concert hall, a church..) by recording a starter pistol shot in the room and using that impulse response to digitally mimic what an instrument or somebody singing would sound like in that room. Ever seen a person clapping hes hands to check the acoustics in a room.. he is listening to the impulse response. An impulse response is the same as full frequency response because an impulse is made of "all frequencies".

 

[PG1995]

Thank you very much, MrAl, WS, Steve, misterT.

All the replies were very helpful and straightforward. I would still need some time to understand this stuff. When I get free I will get back to this thread with some follow-on queries.

@misterT: I really liked your example from 'real' life.

Best wishes
PG

 

[PG1995]

Hi

I think before going any further I first need to understand what an impulse really is in practical terms.

Q1:

When you measure an impulse response to a continuous time system, it is understood that you will use a pulse as an approximation. The shape of the pulse does not matter, as long as the input signal energy of the pulse (i.e. its integrated area) is known. Let's assume you use a perfect square pulse, to keep things simple. In this case the two things you can control are the pulse amplitude and the pulse duration. You choose the amplitude to be as large as possible, while still keeping it below the limit of nonlinearity. The pulse duration you try to keep as long as possible, but very well below the time constants in your system. In this way you will drive the system with enough amplitude and energy to make good measurements, and you will approximate the response of the impulse even though you are using a pulse.

Why does it have to well below the time constants of a system? By time constant I presume you mean τ, tau, which could be RC or R/L, and it takes one τ to charge a capacitor to 63% of its full voltage from an initial zero voltage or it takes one τ for a capacitor to get discharged by amount of 63% from its maximum charged voltage and the same goes for an inductor.

Q2:

Beginning students often assume that the pulse has to be (1) very very narrow and (2) very tall and (3) the area has to be equal to one and (4) the shape has to be perfect. But, all four of these assumptions are false. The shape and total integrated area are irrelevant, as long as you know what the area (energy) is and as long as the pulse is large enough to "kick" the system hard enough to do accurate measurements. The amplitude must also be low enough to be in the linear range of the system. And, the pulse duration also needs to be very small compared to the system time constants.

Perhaps, I'm splitting hairs now. Anyway, let's do it. Is total integrated area the same as energy? At one you point you said total integrated area is irrelevant and then you say one must know what the area (energy) is. What am I missing here? Please let me know. Thanks.

Q3:
In a system made of of resistor (say, an incandescent light bulb) there is no time constant, then does that mean we can apply perfect square pulse of any duration? Of course, we should control the amplitude of the pulse under some maximum value so that the bulb doesn't burn out. By the way, what does impulse response of such a system tells us? To me, it will only tell that the bulb is functioning!

Thank you very much for your time and help.

Regards
PG

 

[steveB]

I think before going any further I first need to understand what an impulse really is in practical terms.

In impulse is a kick. It is a very brief excitation that sets the system into a response, but then immediately goes away. It is an injection of energy that happens before the system can begin to respond.

Why does it have to well below the time constants of a system? By time constant I presume you mean τ, tau, which could be RC or R/L, and it takes one τ to charge a capacitor to 63% of its full voltage from an initial zero voltage or it takes one τ for a capacitor to get discharged by amount of 63% from its maximum charged voltage and the same goes for an inductor.

If the pulse time is much shorter than the time constants, then the system does not begin to respond until the pulse is over. In this way the energy goes in, and the response is not dependent on the actual pulse time and the response scales with integrated area (energy).

Perhaps, I'm splitting hairs now. Anyway, let's do it. Is total integrated area the same as energy? At one you point you said total integrated area is irrelevant and then you say one must know what the area (energy) is. What am I missing here?

The area is irrelevant from the point of view that you don't need an area of one (in some unit system) to say you have an impulse. However, you need to know what the area is in order to interpret the system response. If area is 10, then your measured response needs to be divided by 10 before you can call it a unit impulse response.

In a system made of of resistor (say, an incandescent light bulb) there is no time constant, then does that mean we can apply perfect square pulse of any duration? Of course, we should control the amplitude of the pulse under some maximum value so that the bulb doesn't burn out. By the way, what does impulse response of such a system tells us? To me, it will only tell that the bulb is functioning!

In the real world, there is always a time constant. The light bulb system has inductance from the wire loop that forms the circuit. However, if you could make a system with no time constant, then you would have an infinite bandwidth amplifier or attenuator. H(s) would equal a constant value, and h(t) would be an impulse multiplied by the constant value. In this case you would need to apply a real impulse to the system, because the lack of time constant implies the value of the time constant is zero. Only an impulse with zero width can be small compared to a time constant of zero. Of course, this is all nonsense from a practical point of view. Real impulses do not exist and infinite bandwidth circuits do not exist.

 

[misterT]

In a system made of of resistor (say, an incandescent light bulb) there is no time constant, then does that mean we can apply perfect square pulse of any duration? Of course, we should control the amplitude of the pulse under some maximum value so that the bulb doesn't burn out. By the way, what does impulse response of such a system tells us? To me, it will only tell that the bulb is functioning!

Incandescent bulb is a nonlinear device. When it heats its resistance increases. Simply put: The goal of an impulse response is to measure the frequency response of a linear system.

Take a look at the Fourier transform of an impulse (bottom of the picture). It is constant over all frequencies for perfect impulse. It contains all frequencies with same amplitude and therefore is perfect for measuring the frequency response of a system. Now take a look at the Fourier transform of a square pulse (top pictures) Its frequency spectrum is not flat at all.. its more like a Sinc function.
View attachment 68976

 

[misterT]

This might interest you:
https://www.dspguide.com/ch13/1.htm

 

[PG1995]

Thank you very much, Steve, misterT.

Q1:

In impulse is a kick. It is a very brief excitation that sets the system into a response, but then immediately goes away. It is an injection of energy that happens before the system can begin to respond.

Now this is strange. Let's look at mechanical analogue of an electric impulse. When a ball is hit by a bat, the force applied can be approximated as an impulse - a lot of force applied over a very short span of time. But as soon as the ball comes into contact with the bat, the ball starts its motion in the direction the force (or, impulse) is applied. Informally speaking, as soon as you inject energy into the ball by hitting it with a bat, it starts responding.

Q2:

If the pulse time is much shorter than the time constants, then the system does not begin to respond until the pulse is over. In this way the energy goes in, and the response is not dependent on the actual pulse time and the response scales with integrated area (energy).

Suppose we have a RC circuit. The capacitor's voltage is zero. An impulse of a voltage is applied by turning on the switch for an extremely brief time - the time much shorter than the time constant, RC. It takes one tau, τ, to get the cap charged up by 63% of its maximum allowed voltage. So, if the switch is turned on for a period much shorter than the "τ", then the cap would, probably, get charged up by only, say, 0.0001% value. Then, what kind of response is expected from such a low value? I think the reply to this question can be combined with the Q1.

Q3:

In the real world, there is always a time constant. The light bulb system has inductance from the wire loop that forms the circuit. However, if you could make a system with no time constant, then you would have an infinite bandwidth amplifier or attenuator. H(s) would equal a constant value, and h(t) would be an impulse multiplied by the constant value. In this case you would need to apply a real impulse to the system, because the lack of time constant implies the value of the time constant is zero. Only an impulse with zero width can be small compared to a time constant of zero. Of course, this is all nonsense from a practical point of view. Real impulses do not exist and infinite bandwidth circuits do not exist.

But I don't think an incandescent bulb without any time constant can work as an amplifier or attenuator! Or, can it? In other words, what are you saying? If it's complex then please just leave it. Remember, I have studied op-amp.

Q4:
I remember once you told me that the rising edge of a square wave contains high frequencies and the falling edge contains low frequencies, or something similar. At that time, I didn't want to stir up a hornet's nest for myself so I just skipped that detail. But it seems now it's the time to enter deep waters.

Frequency simply means number of cycles per second. So, in simplest terms, I don't see any cycles in the rising or falling edge of a square wave. So, what does that mean?

Another related detail, which is quite mystery to me, is that an impulse contains all the possible frequencies with the same amplitude (misterT also mentioned it twice in this thread). Now, an impulse is nothing more than a vertical straight line and I can't see any cycles there! If I have a graph of sine or cosine wave I can clearly see the cycles and hence frequency. This proves I can trust my eyesight but not my mind!

Thanks a lot for the help and your time.

Best regards
PG

 

[steveB]

Now this is strange. Let's look at mechanical analogue of an electric impulse. When a ball is hit by a bat, the force applied can be approximated as an impulse - a lot of force applied over a very short span of time. But as soon as the ball comes into contact with the bat, the ball starts its motion in the direction the force (or, impulse) is applied. Informally speaking, as soon as you inject energy into the ball by hitting it with a bat, it starts responding.

You should conclude that the act of hitting a ball with a bat maybe is not actually a good example of an impulse response. There are very complicated dynamics going on in this particular action. To understand this you need to see a slow motion video of what happens to the ball when the bat is in contact with it. It does not immediately start it's motion in the direction of force. The ball compresses and decelerates. Then it stops for an instant and starts moving in the opposite direction. Then the ball uncompresses as it leaves the bat with velocity greater than the bat speed and greater than the original ball speed, but less than the addition of both speeds.

A truly good approximation of an impulse against a ball would be from a short pulse that ended before the ball began to compress appreciably. Although, we need to be careful of what we say until we clearly define exactly what the system is and what the input and output are from the system.

Suppose we have a RC circuit. The capacitor's voltage is zero. An impulse of a voltage is applied by turning on the switch for an extremely brief time - the time much shorter than the time constant, RC. It takes one tau, τ, to get the cap charged up by 63% of its maximum allowed voltage. So, if the switch is turned on for a period much shorter than the "τ", then the cap would, probably, get charged up by only, say, 0.0001% value. Then, what kind of response is expected from such a low value? I think the reply to this question can be combined with the Q1.

I've attached an impulse response for a first order low pass filter, which is essentially a series RC circuit with the voltage output taken across the capacitor. Don't forget that an impulse has infinite amplitude. It can do a lot even though its duration is infinitely brief.

But I don't think an incandescent bulb without any time constant can work as an amplifier or attenuator! Or, can it? In other words, what are you saying? If it's complex then please just leave it. Remember, I have studied op-amp.

It's not complex at all, but I probably should not say attenuator/amplifier because this implies that the input and output have the same units, such as voltage to voltage, or current to current. Rather, the transfer function is a constant for all frequencies when you have infinite bandwidth. It might be relevant to talk about the current response to an applied input voltage to a light bulb. However, the point is that such a simple model is not valid, unless we are restricting ourselves to a limited range of operation/study. A correct model would include the nonlinear resistance of the bulb and the line inductance, and even radiation effects are possible if the frequency (EDIT: actually wavelength converted from frequency) gets large (EDIT: comparable or small) enough compared to the circuit dimensions.

I remember once you told me that the rising edge of a square wave contains high frequencies and the falling edge contains low frequencies, or something similar. At that time, I didn't want to stir up a hornet's nest for myself so I just skipped that detail. But it seems now it's the time to enter deep waters.

I hope i did not say that because it is not correct. If i did say it, then I misspoke. Basically fast edges (both rising and falling) imply high frequency content.

Frequency simply means number of cycles per second. So, in simplest terms, I don't see any cycles in the rising or falling edge of a square wave. So, what does that mean?

You should know this from Fourier analysis. What are the Fourier coefficients for a square wave? Don't they extend up to an infinite number of harmonics of the fundamental frequency?

Another related detail, which is quite mystery to me, is that an impulse contains all the possible frequencies with the same amplitude (misterT also mentioned it twice in this thread). Now, an impulse is nothing more than a vertical straight line and I can't see any cycles there! If I have a graph of sine or cosine wave I can clearly see the cycles and hence frequency. This proves I can trust my eyesight but not my mind!

You can't see the cycles because when you add up an infinite number of sine waves with all frequencies, you get a narrow pulse. It's just a result of the math. If you want to see the cycles, write a computer program to try and do the addition of the sine waves for both a square wave and a Dirac impulse function. Since you can't really add up an infinite number of things with a computer program, you will see functions that only approximate these functions, and you will see ripples with higher frequencies. (EDIT: I found a picture with the square wave and attached it)

 

[PG1995]

Thanks a lot, Steve.

1:

I hope i did not say that because it is not correct. If i did say it, then I misspoke. Basically fast edges (both rising and falling) imply high frequency content.

I'm sorry for this. I shouldn't have ascribed something to you without double checking. What do you mean by the part in bold? Kindly also have a look on **broken link removed**. By the way, I only started learning fourier series yesterday, so please don't mind if my queries are just way too silly!

2:
You said that this is the impulse of a RC low pass circuit. A pulse with a very high amplitude and very short duration (assume area of the pulse is unity then this approximates to a unit impulse) is applied to the terminals Vin. Doesn't the response you attached show the capacitor discharging? I think the capacitor should get charged up when a pulse is applied and the response should look like this. Please let me know where I'm having it wrong. Thanks.

3:

You can't see the cycles because when you add up an infinite number of sine waves with all frequencies, you get a narrow pulse. It's just a result of the math.

I couldn't find any diagram which shows that as you add up more and more number of sine waves with increasing frequencies, the result gets closer to an ideal impulse. Could you please help? I have also searched for an applet of fourier transform but couldn't find it.

This PDF shows fourier series approximation of a square wave.

Best wishes
PG

 

[steveB]

What do you mean by the part in bold? Kindly also have a look on **broken link removed**. By the way, I only started learning fourier series yesterday, so please don't mind if my queries are just way too silly!

The bold part means what it says, or to expand the sentence for clarity. The presence of fast edges (such as with a square wave) or sharp corners (such as with a triangle wave) implies that the signal can be resolved into frequency components that include high frequency (even infinitely high frequency). In other words, we use the idea of superposition and claim that the signal we have can be represented as a sum (even an infinite sum is allowed) of sinusoids made from the fundamental frequency and any or all harmonics of that frequency.

Also, when talking about non-periodic functions, then fast changes and sharp edges and corners imply high frequency components. However for non-periodic signals, we are not restricted to only the fundamental frequency and harmonics of that frequency. But, any and sometimes all frequencies are present. Again, the idea is that superposition applies and the full signal can be represented as a sum of high frequency sine waves. When viewed in totality, we can't see the actual cycles, however, if we were to place a narrow band filter on the signal, we would be able to see the cycles of the part of the signal we allow to pass through the filter.

I guess you need to trust the math and work with it enough to get comfortable with it.

You said that this is the impulse of a RC low pass circuit. A pulse with a very high amplitude and very short duration (assume area of the pulse is unity then this approximates to a unit impulse) is applied to the terminals Vin. Doesn't the response you attached show the capacitor discharging? I think the capacitor should get charged up when a pulse is applied and the response should look like this. Please let me know where I'm having it wrong. Thanks.

What you showed is a step response, not an impulse response. An impulse will instantaneously charge up the capacitor, and then the response is just the natural decay of the system.

I couldn't find any diagram which shows that as you add up more and more number of sine waves with increasing frequencies, the result gets closer to an ideal impulse. Could you please help? I have also searched for an applet of fourier transform but couldn't find it.

What you can try is to write a computer program that adds up lots and lots of sine waves over a wide band with very narrow spacing between the frequencies. You won't be able to make an impulse, but you should be able to at least make a narrow pulse. This won't be a square pulse, but remember that an impulse does not really have a shape to it, so what you are looking for is a function that gets narrower and narrower as you add up more and more frequencies.

 

[steveB]

What you can try is to write a computer program that adds up lots and lots of sine waves over a wide band with very narrow spacing between the frequencies. You won't be able to make an impulse, but you should be able to at least make a narrow pulse. This won't be a square pulse, but remember that an impulse does not really have a shape to it, so what you are looking for is a function that gets narrower and narrower as you add up more and more frequencies.

Here I can give an example. Let's try summing up first 1 cosine wave, then 10, then 100 and then 200. I've attached the plot. Notice how a narrow pulse is formed and you can still see the ripples at high frequency. If the summation is extended up to infinity and if the frequency spacing is decreased to zero, you will get a prefect impulse function.

[latex] \cos(0.02\;t)[/latex] blue
[latex] \frac{\sum_1^{10}\cos(0.02\;n\;t)}{10}[/latex] red
[latex] \frac{\sum_1^{100}\cos(0.02\;n\;t)}{100}[/latex] green
[latex] \frac{\sum_1^{200}\cos(0.02\;n\;t)}{200}[/latex] amber

Note that i normalized the functions so that they have a peak at value of 1 at t=0, but the real function get larger and larger at t=0.

 

[steveB]

There is another way to calculate it too, using integrals instead of summations. Here is a plots of the following integrals.

[latex]\int_0^{100} \cos{(wt)}\;dw[/latex]

[latex]\int_0^{1000} \cos{(wt)}\;dw[/latex]

By letting the upper limit in the integration go to infinity, you get the impulse funciton.

 

[PG1995]

Thank you very much, especially for the graphs. They were very helpful.

Before proceeding with any other query I would like to clarify something you said in post #16.

The bold part means what it says, or to expand the sentence for clarity. The presence of fast edges (such as with a square wave) or sharp corners (such as with a triangle wave) implies that the signal can be resolved into frequency components that include high frequency (even infinitely high frequency). In other words, we use the idea of superposition and claim that the signal we have can be represented as a sum (even an infinite sum is allowed) of sinusoids made from the fundamental frequency and any or all harmonics of that frequency.

But whenever a periodic function is resolved into harmonics using fourier series each successive sinusoid (or, harmonic) has higher frequency than the previous one but lesser amplitude. This means that there will come a time when frequency will get infinite for the components of any function. Then, what's so special about functions such as square or triangle waves about having high frequency components? I think any function can have high frequency harmonics. I hope my query is clear. If not, then please let me. I will try to rephrase it. Thanks a lot.

Best wishes
PG

 

[steveB]

Thank you very much, especially for the graphs. They were very helpful.

Before proceeding with any other query I would like to clarify something you said post #16.


But whenever a periodic function is resolved into harmonics using fourier series each successive sinusoid (or, harmonic) has higher frequency than the previous one but lesser amplitude. This means that there will come a time when frequency will get infinite for the components of any function. Then, what's so special about functions such as square or triangle waves about having high frequency components? I think any function can have high frequency harmonics. I hope my query is clear. If not, then please let me. I will try to rephrase it. Thanks a lot.

Best wishes
PG

The thing that is special about square and triangle functions is that they are periodic functions that have discontinuities in the function and the derivative, respectively. (edit: i removed incorrect sentence here) I think we can extend this notion and say that the harmonics will usually not extend to infinity if the function and all derivatives of any order (i.e. first, second, third derivative ... etc.) are continuous. Also, the harmonics usually must extend to infinity when you have kinks in the function or any derivatives. I'm tempted to remove the work "usually" but that would require a lot of thinking to make sure there are no special cases/exceptions, since we know that most rules have exceptions.

Basically a finite frequency sine wave can't make a kink because it does not contain any kinks itself. However, a sine wave with infinite frequency (combined with an infinite sum) has the ability to change infinitely fast (in a crude sense, -please mathematicians don't throw a spear at me). Although, this is not in any way a formal proof, hopefully you can get a feel for it by thinking in these terms.