engineering recuitment GATE ECE 2023-24 Test Series Signals and Systems Linear Time Invariant Systems Convolution of Signals
A cascade system having the impulse response \({{\rm{h}}_1}\left( {\rm{n}} \right) = \left\{ {\begin{array}{*{20}{l}} {1, - 1}\\ \uparrow \end{array}} \right\}\) and \({{\rm{h}}_2}\left( {\rm{n}} \right) = \left\{ {\begin{array}{*{20}{l}} {1,{\rm{\;}}1}\\ \uparrow \end{array}} \right\}\) is shown in the figure below, where symbol ↑ denotes the time origin.
The input sequence x(n) for which the cascade system produces an output sequence \({\rm{y}}\left( {\rm{n}} \right) = \left\{ {\begin{array}{*{20}{l}} {1,{\rm{\;}}2,{\rm{\;}}1,{\rm{\;}} - 1,{\rm{\;}} - 2,{\rm{\;}} - 1}\\ \uparrow \end{array}} \right\}\) is
1
\({\rm{x}}\left( {\rm{n}} \right) = \left\{ {\begin{array}{*{20}{l}} {1,{\rm{\;}}2,{\rm{\;}}1,{\rm{\;}}1}\\ \uparrow \end{array}} \right\}\)
2
\({\rm{x}}\left( {\rm{n}} \right) = \left\{ {\begin{array}{*{20}{l}} {1,{\rm{\;}}1,{\rm{\;}}2,{\rm{\;}}2}\\ \uparrow \end{array}} \right\}\)
3
\({\rm{x}}\left( {\rm{n}} \right) = \left\{ {\begin{array}{*{20}{l}} {1,{\rm{\;}}1,{\rm{\;}}1,{\rm{\;}}1}\\ \uparrow \end{array}} \right\}\)
4
\({\rm{x}}\left( {\rm{n}} \right) = \left\{ {\begin{array}{*{20}{l}} {1,{\rm{\;}}2,{\rm{\;}}2,{\rm{\;}}1}\\ \uparrow \end{array}} \right\}\)