# Performance bounds for low-pass filters

Precisely how does the performance of an FIR filter depend on its design criteria?

To illustrate, here is a specific question that probably does not have a convenient answer:

The Parks-McClellan algorithm converges to a FIR filter optimally aligned to design parameters. With it one can design a low-pass filter with minimum pass- and stop-band ripple:

Take $$\mathcal{S}_n$$ to be the collection of symmetric FIR filters of order $$n$$: \begin{align} \mathcal{S}_n := \left\{(h_1,\dots,h_n)\in \mathbb{R}^n: h_k=h_{n-k+1} \text{ for each }k\right\}. \end{align} Write the frequency response of $$h\in \mathcal{S}_n$$ as $$\hat{h}(\omega)=\sum_k h_ke^{-j2\pi \omega k}.$$

Specify a "stopband weight" $$\alpha > 0$$, cutoff $$\omega>0$$ and transition width $$\varepsilon>0$$. Define filter mismatch as follows: \begin{align} \gamma_n(h; \alpha,\omega,\epsilon)\! := \min_{f\in [0,\omega]} \left|1-|\hat{h}(f)|\right| + \alpha\cdot \max_{f\in[\omega+\epsilon, 2\pi]} |\hat{h}(f)|. \end{align} The Parks-McClellan algorithm with parameters $$(\alpha, \omega,\varepsilon)$$ converges to an $$h_{best}$$ which minimizes $$\gamma_n(h; \alpha, \omega, \varepsilon)$$ over all $$\mathcal{S}^n$$.

What estimating expressions depending on $$(n, \alpha,\omega, \varepsilon)$$ provide tight upper and lower bounds for $$\min_h \gamma_n(h;\alpha, \omega,\varepsilon)$$ ?

Alternatively, fixing $$n,\ (\alpha,\omega, \varepsilon)$$ and $$\Delta > 0$$, are there bounds for $$\min_h \max_{f\in[\omega+\epsilon, 2\pi]} |\hat{h}(f)|$$ among those $$h\in \mathcal{S}_n$$ that have $$\min_{f\in [0,\omega]} \left|1-|\hat{h}(f)|\right| \leq \delta$$

• The question is pretty cryptic. Could you expand? Maybe refer to the source material that's sparking it? And if the filter has outright zeros on the unit circle, then $\min_{f \in |0,\omega|} \left | \hat h (f) \right | = 0$ -- so that's probably not what you want for a meaningful expression for $\gamma$. Commented Dec 23, 2020 at 0:10
• No source material, unfortunately. I don't know of any strong references for this sort of thing. Added question details. Commented Dec 23, 2020 at 1:37
• I doubt you’ll find tight bounds. Most of the examples I know of are empirical. The Nehari (anti causal) approximant gives a lower bound, but I doubt its tight.
– Peter K.
Commented Dec 23, 2020 at 2:41

As far as I know there are no known tight upper and lower bounds, but there's a quite accurate empirical formula for estimating the required filter order $$M$$ given maximum passband and stopband deviations $$\delta_1$$ and $$\delta_2$$, and the width of the transition band $$\Delta\omega$$:
$$M=\frac{-10\log_{10}(\delta_1\delta_2)-13}{2.324\;\Delta\omega}\tag{1}$$
Formula $$(1)$$ can be found as Eq. $$(7.117)$$ in Oppenheim and Schafer's textbook Discrete-Time Signal Processing (3rd ed, p.570). According to the authors, this formula is due to Kaiser, who developed it based on experimental results published by Herrmann, Rabiner, and Chan.
Also Matlab's firpmord.m uses a similar but more complex formula to estimate the required filter order.