Consider a complex pulse amplitude modulated digital communication system. Suppose that a squareroot raised cosine is used for the transmit pulse shape and received matched filter with fifty percent excess bandwidth with a symbol rate of one mega symbols per second. a$)$ Find the expression of discretized root raised cosine filter assuming sampling interval T$.$ How will the expression change, if the sampling interval is Tx $=$ T $/$L$.$ b$)$ For given $,$ determine the smallest value of L in Z $+$ such that Tx $=$ T $/$L will still result in perfect reconstruction. $(2)(2$ points$)$

Consider a complex pulse amplitude modulated digital communication system. Suppose that a squareroot raised cosine is used for the transmit pulse shape and received matched filter with fifty percent excess bandwidth with a symbol rate of one mega symbols per second.

a$)$ Find the expression of discretized root raised cosine filter assuming sampling interval $\backslash ($ T $\backslash ).$ How will the expression change, if the sampling interval is $\backslash ($ T$\_\{$x$\}=$T $/$ L $\backslash ).$

b$)$ For given $\backslash (\backslash$alpha $\backslash ),$ determine the smallest value of $\backslash ($ L $\backslash$in $\backslash$mathbf$\{$Z$\}^\{+\}\backslash )$such that $\backslash ($ T$\_\{$x$\}=$T $/$ L $\backslash )$ will still result in perfect reconstruction.