Consider a bank account as a discrete-time system where x[n] denotes the system input (i.e. amount...

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Consider a bank account as a discrete-time system where x[n] denotes the system input (i.e. amount of money deposited on the account on the nth month) and y[n] denotes the system output (i.e. total amount of money in the account at the end of the nth month). Assume that the interest rate is kept constant and does not dependent on the amount held on the account or any other parameter in the banking system (i.e. the system under consideration is a linear, time- invariant discrete-time system). i) Find the impulse response of this system if the constant interest rate is 0.5%. ii) Consider some arbitrary input to the system - deposit £1,000 on the first month, withdraw £500 on the second month, deposit of £200 on the fifth month and £50 on the tenth month. Find the state of the account at the end of the tenth month assuming the same interest rate as in part i) above. [Hint: use convolution equation to find the required value]. iii) Show that the same system can be described with the following equation: y[n] = 1.005y[n 1] + x[n] Draw the direct form implementation of the system. iv) Is this system IIR or FIR type system? Is it stable or not? Explain your answer. v) Consider the same type of system, but with negative interest rate of 25%. Find the impulse response of this system and evaluate the response of the system to step input x[n] = 1000u[n] (i.e. £1,000 is deposited on the account each month). Consider a bank account as a discrete-time system where x[n] denotes the system input (i.e. amount of money deposited on the account on the nth month) and y[n] denotes the system output (i.e. total amount of money in the account at the end of the nth month). Assume that the interest rate is kept constant and does not dependent on the amount held on the account or any other parameter in the banking system (i.e. the system under consideration is a linear, time- invariant discrete-time system). i) Find the impulse response of this system if the constant interest rate is 0.5%. ii) Consider some arbitrary input to the system - deposit £1,000 on the first month, withdraw £500 on the second month, deposit of £200 on the fifth month and £50 on the tenth month. Find the state of the account at the end of the tenth month assuming the same interest rate as in part i) above. [Hint: use convolution equation to find the required value]. iii) Show that the same system can be described with the following equation: y[n] = 1.005y[n 1] + x[n] Draw the direct form implementation of the system. iv) Is this system IIR or FIR type system? Is it stable or not? Explain your answer. v) Consider the same type of system, but with negative interest rate of 25%. Find the impulse response of this system and evaluate the response of the system to step input x[n] = 1000u[n] (i.e. £1,000 is deposited on the account each month).

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Solution Part i The impulse response of a system is the output of the system when the input is a uni... View the full answer