Confusion calculating length of Kaiser window I am reading Digital Signal Processing Using Matlab by Ingle and Proakis (3rd ed.)
In Chapter 7, Section 7.3,
I am confused why he is adding 1 to main lobe width for calculating length of window.
I have attached snapshots below.
First/above snap is of matlab code of example 7.8 and 2nd snap is of code of example, both snaps contain the confusing line as highlighted.
This is just an empirical formula found by Kaiser for determining the necessary filter length for a given transition width. That formula is given as Equation $(7.30)$ on page $332$:
$$M=\frac{A_s-7.95}{2.285\,\Delta\omega}+1\tag{1}$$
I think that Kaiser came up with a formula for determining the filter order (hence without the $+1$ in the equation), and the authors of your book preferred to have a formula for the filter length, so they took the original formula and added $1$ to it.
Judging from some of your previous questions, you seem to be confused when it comes to the terms filter order and filter length. For FIR filters, filter length is the number of coefficients (taps). Filter order is the (minimum) number of delay elements necessary to implement the filter. It's just like with polynomials: their order is one less than their number of coefficients. E.g., a second-order polynomial has $3$ coefficients:
$$P_2(x)=a_2x^2+a_1x+a_0\tag{2}$$
Coming back to FIR filters, you always have
$$\textrm{filter length}=\textrm{filter order}+1\tag{3}$$