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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?

Source: http://www.cs.csub.edu/~vvakilian/CourseECE423/LectureNotes/Lecture9.pdf

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.)

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