# how to identify an impulse response in a system?

Now I'm trying to identify the impulse response in a system. Basically, as I know, there are consisted with impulse response and gaussian response in a system.

But I don't know how to identify the impulse response. Would you let me know how to identify the impulse response ina system? Even if some hints please.

• Get a hammer and a chart recorder? In all seriousness, that would depend on what sort of system - do you want to do this analytically or are you trying to measure something? – Chris Stratton Mar 15 '16 at 18:04
• @ChrisStratton No joke- we have a calibrated and instrumented hammer. Like this one – Spehro Pefhany Mar 15 '16 at 18:53

If you are speaking about a real system, i would suggest making an impulse and measuring response. Also it would be easier (for most cases) to measure frequency response- by sweeping sine waves through the whole spectrum of interest and measuring the output. Then just calculate the pulse response.

By the way, this is a nice feature in control systems.

• Is this able to identify to just calculate the pulse response? How to mesure and determine? – sogood Mar 15 '16 at 18:16
• You measure the frequency response, then you can calculate anything. The rest depends on the system. – Gregory Kornblum Mar 15 '16 at 18:20
• Thanks can you let me know do you have any recommend example ? – sogood Mar 16 '16 at 1:41
• or any hint or article also good for me, – sogood Mar 16 '16 at 4:15
• Again, i first need to know, what is your system – Gregory Kornblum Mar 16 '16 at 4:20

If your trying to find the transfer function of a physical system, then you need system identification theory. In short there is an input $x(t)$ and an output $y(t)$ and the idea is to find a frequency model $H(s)$ to find out what the "black box" is doing to the input.

The general idea is that you have to excite all frequencies to come up with a good model. This can be done with a frequency sweep (chirp signal), an impulse, or noise (like white noise which is noise across all frequencies).

If you are trying to do this physically, certain methods work better than others. Airplane and satellite frames are physically hit with a hammer (input) and then monitored with accelerators (output). Motors are best characterized with a frequency sweep to find the transfer function. To find a low pass filter pole, random noise can be injected and then an fft can find the frequency cutoff.

If you are doing for academics sakes, its all math. The transfer function will be given. Just plug in.

$x(t) = \delta(t)$

$Y(s) = H(s)*\mathcal{L}\{\delta(t)\}$

• Thanks Good answer, BTW can you let me know a little more easily? I'm not doing with academics just want to know practically in the fields – sogood Mar 16 '16 at 1:38
• Oh, this smelled like a homework problem. Thats why I included both. However, you really need to understand lapalace if you want to understand transfer functions. – Voltage Spike Mar 16 '16 at 2:41
• No, this is not a homework. As I know, the impulse signal is a very short strong magnitude. so I can't imagine what if I put the impulse signal into a system. how can I identify. Also the reason of like this dummy question is i'm not familiar with math. – sogood Mar 16 '16 at 3:25
• A dirac delta function has infinite amplitude at DC (if that makes any sense.) If you added up every sine wave across all frequencies (from DC to frequencies that aren't physically possible), you would get a dirac delta function. Thats the maths part. The unmaths part is snapping your fingers, you will have a broad spectrum of frequencies summed up at one point in time. If you happened to snap your fingers next to a tuning fork, it would ring. But none of this makes sense if you don't understand how things translate from the time domain to the frequency domain, it just won't. – Voltage Spike Mar 16 '16 at 3:32
• Did you mean that we can find the impulse signal after transferring time domain to frequency domain? I know what if I use DCT(Discrete cosine transform or FFT) then can extract the frequency information from time domain. BTW what is the relationship between DC,AC and impulse signal)? did you mean that the dc imply impulse signal? – sogood Mar 16 '16 at 4:09

A good way of identifying the system is to inject a PRBS (pseudo random binary signal) sequence into a system and record the response. The PRBS is used because the autocorrelation of PRBS gives a neat peak signal at time zero, in other words with the use of correlation between input excitation PRBS signal and output system you are able to get impulse response.
1. Inject PRBS
2. Record system response (t)
3. Corellation PRBS vs Sytem response
4. DFT
Then you get system impulse response

• Is this the same as Laplace inversion conversion ? – sogood Mar 16 '16 at 14:17