A microprocessor$-$based real$-$time control loop has a single ADC with a double buffer, a control microcontroller, and a single DAC. The ADC rate at which the measured feedback signal is sampled into the input buffer is $10$ k words$/$s$.$ The microprocessor can read data from the input buffer at a rate of $1$ M words$/$s$.$ It can write data into the output register at the same rate. A typical control computation cycle involves reading the contents of the input buffer, performing a $$system identification$($i$.$e$.,$ computation of a dynamic model$)$ using this data, computation of the new control signal, and loading the control signal into the output register to be picked up by the DAC of the control channel. The computations alone require a processing time of $1$ ms during which time the microprocessor is not available for reading data from the input buffer or for sending data to the DAC.

In the following questions, we will

$($a$)$ Estimate the required minimum size of the input buffer

and

$($b$)$ determine the best control frequency and bandwidth that can be provided by this controller.