2. [4 Points] A linear time-invariant system is realized by the flow graph shown

Question

2. [4 Points] A linear time-invariant system is realized by the flow graph shown below. x[n] y[n] 2 a) b) c) Write the difference equation relating x[n] and y[n] for this flow graph. What is the system function of the system? In the realization of the figure above, how many real multiplications and real additions are required to compute each sample of the output? (Assume that x[n] is real, and assume that multiplication by 1 does not count in the total.) d) The realization of the figure above requires four storage registers (delay elements). Is it possible to reduce the number of storage registers by using a different structure? If so, draw the flow graph; if not, explain why the number of storage registers cannot be reduced 1/3

Explanation / Answer

a] y[n]= y[n-1] + 2y[n-2] + x[n] + 3x[n-1] + x[n-2]

b] H(z) = Y(z) /X (z) = (1 + z^-1 + 2z^-2 ) / (1 + 3z^-1 + z^-2)

c] 2 real multiplications and 4 real additions

d] the number of delay elements can be reduced but storage elements cannot be reduced as there are 4 different terms that have their own multiplication factor.

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