# Is an FIR filters 'transition width' frequency dependent?

I was recently brushing up on some FIR filter design and was browsing the Wikipedia page. A claim in the article caught my attention:

The main disadvantage of FIR filters is that considerably more computation power in a general purpose processor is required compared to an IIR filter with similar sharpness or selectivity, especially when low frequency (relative to the sample rate) cutoffs are needed. However many digital signal processors provide specialized hardware features to make FIR filters approximately as efficient as IIR for many applications.

Article

Now what caught my attention was the statement made about how more computation is required when a low cutoff frequency relative to the sample rate (i.e. low normalized cutoff frequency) is specified.

My question: is this true? Or is it just a poorly written statement that assumes you need sharp transitions when working with low cutoff frequencies?

I was always been under the impression that FIR filters become very difficult to implement when trying to achieve sharp transitions, not certain frequencies. Have I been wrong in this assumption?

Furthermore, this article seems to confirm the fact that the decay rate of the impulse response is independent of a chosen cutoff frequency. So assuming the impulse response has a constant decay rate, shouldn't an FIR filter of fixed length being equally good (or bad) regardless of cutoff frequency?

• The transition width of a FIR filter is on the order of proportional to the width of the FIR, not to the cut-off frequency, thus get wider relative to cutoff the lower the cutoff, perhaps enough that the stop band no longer meet the definition of "low" for some given criteria. – hotpaw2 Sep 19 '16 at 16:41

You're right: the required filter order is approximately inversely proportional to the desired transition bandwidth $\Delta\omega$, regardless of the cut-off frequency. This is reflected in the empirical formulas for estimating the required filter orders for the Kaiser window design method as well as for the Parks McClellan equiripple design of low pass filters:
\begin{align}M&=\frac{-20\log_{10}\delta-8}{2.285\Delta\omega}&\qquad\text{(Kaiser Window)}\tag{7.63}\\{}\\ M&=\frac{-10\log_{10}(\delta_1\delta_2)-13}{2.324\Delta\omega}&\qquad\text{(Parks McClellan)}\tag{7.104}\end{align}
where $\delta$ is the maximum allowable approximation error (Kaiser), and $\delta_1$ and $\delta_2$ are the maximum pass band and stop band errors, respectively (Parks McClellan).