Consider a system with impulse responseh[n] = (1)cos u[n]Determine the system transfer function H(Q).

To determine the system transfer function H(Q), we need to take the Z-transform of the impulse response h[n].

The Z-transform of cos u[n] can be found using the property of the Z-transform:

Z{cos u[n]} = (1/2)(Z{e^(ju[n])} + Z{e^(-ju[n])})

Using the Z-transform property for the unit step function u[n]:

Z{u[n]} = 1/(1 - Q^(-1))

where Q^(-1) represents the inverse of the Z-transform variable Q.

Applying this property to the Z-transform of cos u[n], we get:

Z{cos u[n]} = (1/2)(Z{e^(ju[n])} + Z{e^(-ju[n])}) = (1/2)(1/(1 - Q^(-1)) + 1/(1 - Q^(-1)))

Simplifying this expression, we have:

Z{cos u[n]} = 1/(1 - Q^(-1))

Now, since the impulse response h[n] is given as (1)cos u[n], we can write the Z-transform of h[n] as:

Z{h[n]} = Z{(1)cos u[n]} = Z{1} * Z{cos u[n]} = 1 * 1/(1 - Q^(-1)) = 1/(1 - Q^(-1))

Therefore, the system transfer function H(Q) is equal to 1/(1 - Q^(-1)).