What does "how to identify impulse response of a system?" mean?

Question

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?".

As you can see as below figure, as I know, the input delta function and output is $h[n]$.

In this case, if I put impulse (function) signal in to above system, then I can get the impulse response. Then I think the output $h[n]$ and impulse response are the same, doesn't it?

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?
Question 3: Why do we get the output of $y[n]$ when if we know that input $x[n]$, impulse response $h[n]$. I mean the output $y[n]$ is the same $h[n]$ isn't it? I'm very confused.

If you have any hint, please let me know.

Update

I found a question in my case as below,

1.2.8 Consider a discrete-time LTI system described by the rule $$y(n)=x(n-5)+\frac 12 x(n-7).$$ What is the impulse response $h(n)$ of the system ?

How to identify $h[n]$?

the answer :

$h[5]$ = 1+ 0 = 1

$h[7]$ = 0 + 1/2 = 1/2

. . .

@Laurent Duval Can you help me please ? – sogood Mar 17 '16 at 08:00 — sogood, Mar 17 '16 at 08:00

score 1 · Accepted Answer · edited Mar 18 '16 at 01:12

1

Given a system, one often wants to be able to predict the output $y$, given an input $x$. This amounts to identifying a mathematical relation between all inputs and outputs, optimaly as $y = S(x)$. This can be a difficult task in general.

When the system is linear, and time-invariant (its properties do not change over time), then it is proven that it suffices to know the system's response to a discrete unit pulse $\delta$, to be able to compute the output for any other input $x$, in the form of a convolution equation.

So suppose that your system outputs $h[n]$ when you input $\delta[n]$, then for any $x[n]$, the output will be: $$y[n] = \sum_k x[k]h[n-k]\,.$$

To identify the impulse response of the system, you ought to provide the numbers, or a generic formula, that give the value for each $h[n]$.

You can try some exercices in Exercises in Signals, Systems, and Transforms, for instance 1.2.4 and 1.2.7. 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.

edited Mar 18 '16 at 01:12

Fat32

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answered Mar 17 '16 at 08:22

Laurent Duval

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I have updated with one question. Would you help me any hint please ? – sogood Mar 17 '16 at 10:09
Can you provide us with the first steps of your reasoning, based on the information you have already received – Laurent Duval Mar 17 '16 at 10:18
Basically, I know to get the output of y[n]. most textbook show that how to get y[n] (how to convolution). but this question is a quite difficult to me. – sogood Mar 17 '16 at 10:30
So input $\delta$ in your system, as it were the $x$ in your new equation, and evaluate the $y$ – Laurent Duval Mar 17 '16 at 11:35
Sorry I don't understand what you mean – sogood Mar 17 '16 at 11:48
Please write the equation from 1.2.8 replacing $x$ by $\delta$, and take for instance $n=5$. What is the value of $y[5]$ then (forget about convolution for the moment, do simple things)? – Laurent Duval Mar 17 '16 at 11:52
Sir, I think I found the answer. then what next? and Can we determine whether IIR or FIR from the results? – sogood Mar 17 '16 at 13:45
Indeed! Do you have a finite or a infinite number of no-zero coeffcients in the impulse response? – Laurent Duval Mar 17 '16 at 14:01
Sir I think the equation is FIR. because the finite number of non-zero coefficients in the impulse response. am i right? is there any specific name or calling this to find what i did? – sogood Mar 17 '16 at 14:04
FIR, perfect. I do not understand your second question – Laurent Duval Mar 17 '16 at 14:14
i wonder that what is this calling what i did (to find FIR), is this having any specific name likely " finding FIR" or anything? I want to know this because when I meet expert, I want to tell this as expert words – sogood Mar 17 '16 at 14:29
I am not fully sure there is a specific term for it, this is a subset of filter identification. Some may call it "filter class identification" or "filter type identification". – Laurent Duval Mar 17 '16 at 15:03
By the way, you can improve your skills with the related question Is this system an FIR filter or an IIR filter? – Laurent Duval Mar 17 '16 at 15:04
Thanks Sir. it's very good related question. Thanks a lot! – sogood Mar 17 '16 at 15:07

score 0 · Answer 2 · edited Mar 18 '16 at 01:10

0

The operation of linear time invariant system on a signal $f[n]$ can be described as a convolution, e.g.

$$(f \ast h)[n] = g[n]$$

where $g[n]$ is the output and $h[n]$ is convolution filter representing the system.

It is called the impulse response because if you feed in the impulse signal you get the convolution filter as the output, e.g.

$$(\delta \ast h)[n] = h[n]$$

So once you know the impulse response, you can calculate the response to any signal using the first equation.

edited Mar 18 '16 at 01:10

Fat32

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answered Mar 17 '16 at 03:39

geometrikal

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Thanks can you tell me why not use the correlation ? I think the same both mechanisms except for time reversal. – sogood Mar 17 '16 at 03:51
@sogood They are very similar (see http://dsp.stackexchange.com/questions/2654/what-is-the-difference-between-convolution-and-cross-correlation). 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. – geometrikal Mar 17 '16 at 03:57
I almost got it what you mean. you mean that xn * h[n](impulse response) = yn. then how to identify impulse response of a system ? is there any way ? – sogood Mar 17 '16 at 04:22
if you know y[n], you can then go to the frequency domain and do Y[w]/DFT(impulse) which is 1 and get H(w), then you go again in to the discrete time domain and get h[n]. Have a look to this examples in Matlab: http://www.mathworks.com/matlabcentral/fileexchange/55885-dft-h--n- and http://www.mathworks.com/matlabcentral/fileexchange/55822-low-pass-filtering – Behind The Sciences Mar 17 '16 at 07:28
@BehindTheSciences Thanks but I don't understand it, how to go from y[n] to frequency domain? Also can you tell me how to identify the impulse response of a system? – sogood Mar 17 '16 at 07:57
using the frequency domain is one way that I thought it would be useful if you know about the Fourier Transform: https://en.wikipedia.org/wiki/Fast_Fourier_transform if not, just convolve your input, x[n] with the impulse delta[n]: have a look to the discrete time domain here: https://en.wikipedia.org/wiki/LTI_system_theory – Behind The Sciences Mar 17 '16 at 18:55

What does "how to identify impulse response of a system?" mean?

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