The simulation diagram for the α – β filter is given in Figure. This filter is second orde...

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.

(a) Find a state model for the α-filter described by the equation

y[n + 1] – (1 – α)y[n] = αx[n + 1],

where x[n] is the input signal. Hint: Draw a simulation diagram first.

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(b) Verify the results of Part (a) by (i) finding the transfer function from the describing equation and (ii) finding the transfer function from the state equations.