2. (25 pts) Let's study the general form of the flow graph for an all-pass system....

   

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2. (25 pts) Let's study the general form of the flow graph for an all-pass system. First of all, you should realize that the cascade of all-pass systems is still an all-pass system. Consider the following system function of an all-pass system, H (2)= (a) The direct form I flow graph (zero first, pole second) for the system is shown below, where there are two delays and two multipliers (not counting multiplication with +1). Show the difference equation based on the flow graph and verify if it indeed implements the all-pass system. X[n] 3 H (2) 1-/m (b) Revise the signal flow graph in part (a) such that it only uses one mul- tiplier and two delays. (Hint: manipulate the difference equation such that only one multiplication need to be performed.) (c) Now consider another all-pass system whose system function is -)(- = -1-2 1-22-1 Draw a signal flow graph for the system with two multipliers and three delays. (Hint: This is a cascade of two all-pass systems. First, draw the flow graph of the first system which is the same as in part (b). Second, draw the flow graph of the second system in the same way as in part (b). Then, revise the flow graph of the second system in a way such that the 2nd delay of the first system and the first delay of the second system can be combined. (d) Draw a flow graph of the entire system in direct form II. How many delays and multipliers did you use? Why would we often go for the flow graph based on DF1 but not DF2 in this case? 2. (25 pts) Let's study the general form of the flow graph for an all-pass system. First of all, you should realize that the cascade of all-pass systems is still an all-pass system. Consider the following system function of an all-pass system, H (2)= (a) The direct form I flow graph (zero first, pole second) for the system is shown below, where there are two delays and two multipliers (not counting multiplication with +1). Show the difference equation based on the flow graph and verify if it indeed implements the all-pass system. X[n] 3 H (2) 1-/m (b) Revise the signal flow graph in part (a) such that it only uses one mul- tiplier and two delays. (Hint: manipulate the difference equation such that only one multiplication need to be performed.) (c) Now consider another all-pass system whose system function is -)(- = -1-2 1-22-1 Draw a signal flow graph for the system with two multipliers and three delays. (Hint: This is a cascade of two all-pass systems. First, draw the flow graph of the first system which is the same as in part (b). Second, draw the flow graph of the second system in the same way as in part (b). Then, revise the flow graph of the second system in a way such that the 2nd delay of the first system and the first delay of the second system can be combined. (d) Draw a flow graph of the entire system in direct form II. How many delays and multipliers did you use? Why would we often go for the flow graph based on DF1 but not DF2 in this case?