Consider the α – β filter of Figure. Parts of this problem are repeated from Problem. Use...

Consider the α – β filter of Figure. Parts of this problem are repeated from Problem. Use those results if available.

(a) Write the state equations of the filter, with the state variables equal to the delay outputs and the output equal to y[n].

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(b) Use the results of Part (a) to find the filter-transfer function H(z) = Y(z)/U(z).

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(c) Show that, with β = 0, the transfer function is that of the α-filter in Problem.

The simulation diagram for the α – β filter is given in Figure. This filter is second order and is used in radar-signal processing. The input u[n] is the unfiltered target- position data, the output y[n] is the filtered position data, and the output v[n] is an estimate of the target velocity. The parameter T is the sample period. The parameters α and β are constants and depend on the design specifications for the filter.

(a) Write the state equations for the filter, with the state variables equal to the outputs of the delays and the system outputs equal to y[n] and v[n].

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(b) Let β = 0 in Part (a). Show that the resulting equations are equivalent to those of the α-filter of Problem.

Consider the α-filter of Problem. Parts of this problem are repeated from Problem. Use those results if available.

(b) Use the results of Part (a) to find the filter transfer function.