# Wideband FM bandwidth estimation

In the book Modern Digital and Analog Communication Systems by B. P Lathi, there is a section for the estimation of WBFM bandwidth. First, the staircase approximation of $$m(t)$$ is constructed by the pulses of width $$1/2B$$. $$\hat{m}(t) = \sum_{k=-\infty}^{+\infty}m(t_k)\Pi(2B(t-t_k))$$ It's stated that FM signal corresponding to each of this cells is a sinusoid with frequency $$\omega_c + k_fm(t_k)$$ with duration $$1/2B$$. Why the corresponding FM signal is of finite duration? After that it's stated that "The FM signal for $$\hat{m}(t)$$ consists of a sequence of such constant frequency sinusoidal pulses of duration $$T = 1/2B$$ corresponding to various cells of $$\hat{m}(t)$$". It seems that constructing the FM signal assumed to be linear here. Or maybe I'm missing something?

Edit: Actually, there are two main questions here:

1. Why the FM signal corresponding to $$m(t_k)\Pi(2B(t-t_k))$$ is a sinusoid with frequency $$\omega_c + k_fm(t_k)$$ with duration $$1/2B$$? (I think it should be infinite duration.)

2. Why the FM signal corresponding to $$\sum_{k=-\infty}^{+\infty}m(t_k)\Pi(2B(t-t_k))$$ is sum of the sinusoids with duration $$1/2B$$ and frequencies $$\omega_c + k_fm(t_k)$$? (Assuming that FM signal corresponding to each cell is is a sinusoid with duration $$1/2B$$, it's not true that FM signal corresponding to sum of these cells is equal to sum of the FM signals corresponding to each cell.)