Simple question about Sinc()

Question

Suppose I use the ideal lowpass filter eq to attempt creating a IIR filter using it : $$ h[n] = \frac{\sin(\omega_c \, n)}{\pi \, n}, \qquad -\infty < n < \infty $$

Obviously, I can't really (unless I'm wrong) realize a filter with this impulse response because I don't have the finite difference equation to make the recursion happen.

But I'm told that it's also an unstable system (textbook solutions). My question is.. why?

The stability condition states that a system is stable if its impulse response satisfies: $$ \sum_{n=-\infty}^{\infty} \Big|h[n] \Big| < \infty $$

From what I see, if I look at the condition blindly, it does satisfy the stability criterion. Doesn't it? Then why it's unstable? Is it because of something like a discontinuity at $n=0$ (even if technically its equal to one)?

Thanks!

Answer 1

No, the signal $$ h[n] = \frac{\sin(\omega_c \, n)}{\pi \, n}, \qquad -\infty < n < \infty $$

is not absolutely summable, as the sum

$$ \sum_{n=-\infty}^{\infty} \Big|h[n]\Big| = \sum_{n=-\infty}^{\infty} \left |\frac{\sin(\omega_c \, n)}{\pi \, n} \right| $$

diverges to infinity. Hence the absolute sum does not converge and therefore it's considered as an unstable system's impulse response. Yet, its square is absolutely summable:

$$ \sum_{n=-\infty}^{\infty} \Big|h[n]\Big|^2 = \sum_{n=-\infty}^{\infty} \left |\frac{\sin(\omega_c \, n)}{\pi \, n} \right|^2 $$ does not diverge. This kind of condition is stated as mean square convergence; i.e., a typical condition for discontinuous frequency responses that has finite jumps in its frequency response which should be continuous for all $\omega$ according to the Fourier theorem that would require uniform convergence at all frequency points $\omega$.