(1) overlap-add method: $$ \begin{array}{l}{x(n)=2,2,1,1,2} \\ {h(n)=1,3}\end{array} $$ determine output y(n) using overlap-add method of convolution.

$\left.\begin{array}{l}{x_{1}(n)=2,2,1,0} \\ {x_{2}(n)=1,1,2,0}\end{array},\right\}{2}$ sequence

$h(n)=1,3,0,0$

No . of overlap =No. of samples in $h(n)$ ----1

$y_{1}(n)=x_{1}(n)(N) h(n)$

$\left[\begin{array}{llll}{1} & {0} & {0} & {3} \\ {3} & {1} & {0} & {0} \\ {0} & {3} & {1} & {0} \\ {0} & {0} & {5} & {1}\end{array}\right]\left[\begin{array}{l}{2} \\ {2} \\ {1} \\ {0}\end{array}\right]=\left[\begin{array}{l}{2} \\ {8} \\ {7} \\ {3}\end{array}\right]$

$Y_{2}(m)=x_{2}(n)(N) h(n)$

$\left[\begin{array}{llll}{1} & {0} & {0} & {3} \\ {3} & {1} & {0} & {0} \\ {0} & {5} & {1} & {0} \\ {0} & {0} & {5} & {1}\end{array}\right]\left[\begin{array}{l}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]=\left[\begin{array}{l}{1} \\ {4} \\ {5} \\ {6}\end{array}\right]$

$y_{1}(n)=2,8,4,3$

$y_{2}(n)=1,4,5,6$

$y(n)=2,8,7,4,4,5,6$

Crosscheck :-

$y(n)=2,8,7,4,4,5,6$

$\left.\begin{array}{l}{x_{1}(n)=2,2,0} \\ {x_{2}(n)=1,1,0} \\ {x_{3}(n)=1,2,0} \\ {h(n)=1,3,0}\end{array}\right\}$ 3 sequences

$Y_{1}(n)=x_{1}(n)(N) h(n)$

$\left[\begin{array}{lll}{1} & {0} & {3} \\ {3} & {1} & {0} \\ {0} & {3} & {1}\end{array}\right]\left[\begin{array}{l}{2} \\ {2} \\ {0}\end{array}\right]=\left[\begin{array}{l}{2} \\ {8} \\ {6}\end{array}\right]$

$y_{2}(n)=x_{2}(n)$

$\left[\begin{array}{lll}{1} & {0} & {3} \\ {3} & {1} & {0} \\ {0} & {3} & {1}\end{array}\right]\left[\begin{array}{l}{1} \\ {1} \\ {0}\end{array}\right]=\left[\begin{array}{l}{1} \\ {4} \\ {3}\end{array}\right]$

$y_{3}(n)=x_{3}(n)(N) h(n)$

$\left[\begin{array}{lll}{1} & {0} & {3} \\ {3} & {1} & {0} \\ {0} & {3} & {1}\end{array}\right]\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right]=\left[\begin{array}{l}{1} \\ {{5}}\\{{6}}\end{array}\right]$

$y_{1}(n)=2,8,6$

$y_{2}(n)=1,4,3$

$y_{3}(n)=$

$y(n)=2,8,7,4,4,5,6$

Crosscheck:-

$ | 2 2 1 1 1 2$

$1 | 2 2 1 1 1 2$

$3 | 6 6 3 3 3 6$

$y(n)=2,8,7,4,4,5,6$