# Conceptual Question from Signal Processing - Impulse Response and AR Coefficients

In continuation to the previous question Conceptual questions from signal processing I have a doubt which is: Consider an Autoregressive model (AR(2)): $$y(t) = ay(t-1) + by(t-2)$$

and a FIR (Moving Average, MA(2)) model

$$x(t) = a\epsilon(t-1) + b\epsilon(t-2).$$

According to the reply in the prev question, in time domain

$$y[n] = h[n]\star x[n]$$ $h$ is the impulse response.

• Is there any relation between impulse response and the coefficients of AR and MA model?
• What is the intuition of the coefficients and how do we get them?

Impulse Response is basically the FIR coefficients of the system.
Namely, a system $H$ with an impulse response given by $f [n]$ and a Filter $F$ with an FIR representation of ${f[0], f[1], \cdots, f[n]}$ are equivalent.

Now, systems with Feedback are equivalent of both FIR and IIR (AR) filters.
But given infinite length of FIR model any LTI system can be represented by FIR coefficients.

The relation could be easily displayed by the Laplace Transform of the system.

• Thank you. FIR coefficients = Moving Average model and IIR = Autoregressive model. So, impulse response is only applicable to MA models? Then what does the coefficients of AR model represent? Could you kindly elaborate on the last statement - how to display the relation by Laplace tranform – Ria George May 22 '14 at 4:45
• Also, why do we need to do Laplace transform instead of Fourier transform since the Fourier transform of impulse response gives the frequency response (a totally offbeat question, but I have asked as you mentioned about Laplace transform). And since I am new to this field, am very confused as to use Fourier or Laplace. Please help – Ria George May 22 '14 at 4:50
• Using Laplace transform you can analyze a system by looking at Polynomials. To be specific, Rational Polynomials, Where the IIR / AR part is the denominator and the FIR / MA part is the denominator. Yet you must remember that you can approximate any Rational Polynomial by infinite length FIR. This is why you can always look at the impulse response to evaluate the System (Yet, it might be infinite). – Royi May 22 '14 at 5:35
• @RiaGeorge, Nope, the MA is the Numerator and the AR is the Denominator. I'd go for: "Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing, 2nd Edition, Prentice Hall Signal Processing Series. ISBN 0-13-754920-2." – Royi May 22 '14 at 21:16
• @RiaGeorge, Well, AR model means the current output depends on previous output. Which means the output data is going back to the output, so the system has feedback about its state. Have a look here: fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html – Royi May 22 '14 at 21:27