Q$2[40$ Points$].$ You are designing a system to control the brightness of an LED using timer$-$based Pulse Width Modulation

$($PWM$)$ on a microcontroller.

$($Part $1)$ Calculate the period of a PWM signal with a frequency of $1$ kHz in microseconds. The microcontroller$$s clock

frequency is $16$ MHz$,$ and the timer uses a prescaler of $64.$ Calculate the value to be loaded into the timer$$s register

$($TA$0$CCRx value$)$ to achieve a $1$ kHz PWM frequency.

$($Part $2)$ Given the calculated TA$0$CCRx value from part $(1),$ compute the TA$0$CCRx for the $50\%$ and $75\%$ duty cycle.

$($Part $3)$ Assume the timer is in $8-$bit mode $(255$ max$).$ Calculate the minimum and maximum achievable PWM fre$-$

quencies using the $16$ MHz clock and a prescaler of $64.$ Determine the resolution of the duty cycle as a percentage.

$($Part $4)$ Suppose you switch to a $16-$bit timer with the same $16$ MHz clock and a prescaler of $64.$ Calculate the

maximum TA$0$CCRx value. Calculate the minimum achievable PWM frequency.

Q$3[20$ Points$].$ A push button is used to control an LED based on these rules: $(1)$ Every time the push button is pressed,

the LED turns ON for $5$ seconds; $(2)$ If the button is pressed again within the $5-$second window, the timer resets, and the

LED stays ON for another $5$ seconds from the latest press. A user interacts with the button as follows:

$$ Press $1$: Time $=0$ seconds

$$ Press $2$: Time $=3$ seconds

$$ Press $3$: Time $=8$ seconds

$$ Press $4$: Time $=12$ seconds

$($Part $1)$ Calculate the total time the LED stays ON based on the button press sequence. Provide a timeline showing

the ON and OFF states of the LED.

$($Part $2)$ Explain if there are any periods where pressing the button has no effect and why that happens.

$($Part $3)$ Consider a scenario where the button is pressed every $4$ seconds starting at Time $=0$ seconds. How long

would the LED remain ON by the $20$th press?