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Given: $ y[n]=x[n]*h[n]=\sum_{k=-\infty}^{\infty}(x[k]h[n-k]) $

  1. $ x[n]*(h_1[n]*h_2[n])=x[n]*(h_1[n]*h_2[n]) $
  2. $ x[n]*(h_1[n]*h_2[n])=x[n]*(h_2[n]*h_1[n]) $ Commutative property of discrete time
  3. $ x[n]*(h_1[n]*h_2[n])=x[n]*(\sum_{k=-infty}^{infty}h_2[k]h_1[n-k]) $
  4. $ x[n]*(h_1[n]*h_2[n])=\sum_{j=-infty}^{infty}x[j](\sum_{k=-infty}^{infty}h_2[k]h_1[n-k-j]) $
  5. $ x[n]*(h_1[n]*h_2[n])=\sum_{j=-infty}^{infty}\sum_{k=-infty}^{infty}x[j](h_2[k]h_1[n-k-j]) $
  6. $ x[n]*(h_1[n]*h_2[n])=\sum_{j=-infty}^{infty}\sum_{k=-infty}^{infty}h_2[k]x[j]h_1[n-k-j] $
  7. $ x[n]*(h_1[n]*h_2[n])=\sum_{k=-infty}^{infty}h_2[k]\sum_{j=-infty}^{infty}x[j]h_1[n-k-j] $
  8. $ x[n]*(h_1[n]*h_2[n])=h_2[n]*\sum_{j=-infty}^{infty}x[j]h_1[n-j] $
  9. $ x[n]*(h_1[n]*h_2[n])=h_2[n]*(x[n]*h_1[n]) $
  10. $ x[n]*(h_1[n]*h_2[n])=(x[n]*h_1[n])*h_2[n] $ Commutative property of discrete time

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