Consider the discrete-time time-invariant system with input $x[n]$ and output $y[n]$ satisfying

$$y[n] = \sum_{k=1}^5{x[n-k]}$$

Consider approximating the desired system with a second-order IIR system with system function

$$H^{'}(z)= \frac{1}{1+a_1z^{-1}+a_2z^{-2}}$$

Use the following error criterion:

where $h_d$ is the desired impulse response. $$E = \sum_{n=-\infty}^{\infty}\left\lvert h_d[n]+a_1h_d[n-1]+a_2h_d[n-2]\right\rvert^2$$

How can the particular error function be useful in solving the system of equations or the desired impulse response?

Do I need to take derivative w.r.t both $a_1$ and $a_2$ and make it 0 to get the system of equations?

Now this is what I am getting: $$0 = \sum_{n=-\infty}^{\infty}( h_d[n]h_d[n-1]+a_1h_d[n-1]h_d[n-1]+a_2h_d[n-2]h_d[n-1])$$ and
$$0 = \sum_{n=-\infty}^{\infty}( h_d[n]h_d[n-2]+a_1h_d[n-1]h_d[n-2]+a_2h_d[n-2]h_d[n-2])$$

Now how can I solve these system of two equations and how can I approximate the desired impulse response from that?

  • $\begingroup$ This question appears to be homework. Complete answers to homework are off-topic, but specific questions about homework are acceptable if they include enough detail. Please edit the question to include more background about what you don't understand. $\endgroup$ – Marcus Müller Oct 30 '20 at 10:38
  • $\begingroup$ How can the particular error function be useful in solving the question? $\endgroup$ – ranjana sengupta Oct 30 '20 at 12:04
  • $\begingroup$ it's not "useful", it's the thing you need to optimize. $\endgroup$ – Marcus Müller Oct 30 '20 at 12:15
  • $\begingroup$ How can I do this ? Can you give a little details? $\endgroup$ – ranjana sengupta Oct 30 '20 at 12:41
  • $\begingroup$ you mean I need to take derivative w.r.t both a1 and a2 and make it 0 to get the system of equations? $\endgroup$ – ranjana sengupta Oct 30 '20 at 12:42

I think it's instructive to try to understand how this error function is derived and why it makes sense. First, the desired impulse response $h_d[n]$ is given implicitly by the input-output relation of the desired system:


The given error function is the error function minimized by Prony's method for the design of IIR filters. We try to approximate a given transfer function $H_d(z)$ by an IIR filter $H(z)=B(z)/A(z)$:


For the given example we get (with $B(z)=1$)


For equation $(3)$ to be satisfied, we need all coefficients associated with negative powers of $z$ to vanish:

$$\begin{align}h_d[1]+h_d[0]a_1&\stackrel{!}{=}0\\ h_d[2]+h_d[1]a_1+h_d[0]a_2&\stackrel{!}{=}0\\ h_d[3]+h_d[2]a_1+h_d[1]a_2&\stackrel{!}{=}0\\\vdots\end{align}\tag{4}$$

In practice, we can solve $(4)$ in an approximate way by minimizing the sum of the squares of the left-hand side, leading to the given error function


Finally, the optimal coefficients $a_1$ and $a_2$ are obtained by taking the derivative of $(5)$ w.r.t. $a_1$ and $a_2$ and equating them to zero. This results in two linear equations with two unknowns.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.