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Two LTI systems in cascaded have impulse response h1[n] and h2[n]

h1[n]=(0.9)nu[n]0.5(0.9)n1u[n1] h2[n]=(0.5)nu[n](0.5)n1u[n1]

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 10

Year : MAY 2014

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By using convolution property of z-transform,

h[n]=h1[n]h2[n](z)h(z)=h1(z).h2(z)h1[n]=(0.9)nu[n]0.5(0.9)n1u[n1]

By z-transform,

h1(z)=zz0.90.5z1zz0.9 (shifting property)

=zz0.90.5z0.9=z0.5z0.9Now,h2[n]=(0.5)nu[n](0.5)n1u[n1]

By z-transform,

h2(z)=zz0.5z1zz0.5 ….. (shifting property)

=zz0.51z0.5=z0.1z0.5h(z)=h1(z)h2(z)=z0.5z0.9×z0.1z0.5=z1z0.9=zz0.90.5z0.9=zz0.9z10.5zz0.9

By Inverse z-transform,

h[n]=(0.9)nu(n)(0.9)(n1)u(n1).

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