Problem $4(20$ points$)$

We want to design a PAM system using $M$ waveforms, where one of the waveforms has Fourier ts

At which values of $R_s($symbols per second$)$ and $R_b($bits per second$)$ are we able to transmit without ISI if the channel does not introduce any distortion? Justify your answer.

Define and sketch $M$ waveforms that can be utilized to avoid ISI at the rates established in $1)$ and provide the constellation of the $M-$PAM system.

If we still use an $M-$PAM system but choose different waveforms, can we avoid ISI and achieve the same $R_s$ as in $1)$ but using a bandwidth of just i$)16$ KHz $,$ and ii$)9$ KHz $?$ If possible, sketch the Fourier transforms of the $M$ waveforms, defining all the points of interest in one of the Fourier transforms.

Can we increase $R_b$ while leaving $R_s$ constant?