Z transform examples

Find Z transform of F[K] =Cos(ak) u[k] :

F(z) = Z (cos(ak) u(k) )

= Z ( ½ (ejak+e-jak) u(K) )

= ½ (Z (ejak  u(k) ) + Z (e-jak) u(K) )

=½ ( Z/ Z-eja + Z/ Z-e-ja)

= ½ ( Z/ Z-eja * Z-e-ja/ Z-e-ja + Z/ Z-e-ja * Z-e-ja/ Z-e-ja )

=½ ( Z2- Ze-ja / Z2- Zeja – Ze-ja +1  + Z2- Zeja / Z2- Zeja – Ze-ja +1)

=½ (2Z2 – (Zeja – Ze-ja) / Z2- Zeja – Ze-ja +1)

= ½(2Z2 – 2Zcos(a)/ Z2-2Zcos(a) +1 

= Z (Z-cos(a)) / Z2-2Zcos(a) +1 

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