Please use the red id $827748525$In this question, you will need to design a sequential counter that counts based on your own Red ID number $($Be careful, you will need to use your own Red ID number in this question. Using others' solutions will result in a complete false answer, which will in turn lead to a complete failure in this exam$).$

$3\mathrm{.}1$ We plan to display your counted number using seven segment LED display that contains $7$ LEDs. Each LED is called a segment and they are identified as $(a,b,c,d,e,f,g)$ segments as shown below:

To display a single decimal digit represented by $4$ binary numbers $A,B,C,D,$ we use a $7447$ decoder as shown below. For example, If ABCD$=0000,$ the output of the decoder should be abcdef $g=1111110$ so that a $"0"$ pattern is displayed in the LED segments; If ABCD$=0001,$ the output of the decoder should be abcdef $g=0110000$ so that a $"1"$ pattern is displayed in the LED segments. We are only interested in displaying decimal numbers ranging from $0$ to $9$ in this device.

Figure $2,744/$ vecouer useu ivi dev urapidy

Page $\frac{3}{8}$

Complete the following truth table of the $7447$ decoder in the circuit above by writing the output values in each row $(20$ points$).$

$\backslash$table$[[ABCD,$abcdefg$],[0000,],[0001,],[0010,],[0011,],[0100,],[0101,],[0110,],[0111,],[1000,],[1001,],[1010,],[1011,],[1100,],[1101,],[1110,],[1111,]]$

$3\mathrm{.}2$ Now we will design your own Red ID divided$-$by$-8$ remainder counter using a Finite State Machine $($FSM$).$ We aim at designing a Moore FSM$.$ The number of states in this FSM will be the total number of digits in your Red ID$.$ The binary outputs in each state correspond to the remainder value of a division process which takes your current decimal Red ID digit as the dividend, and the decimal value $8$ as the divisor. For example, if your current Red ID digit is $6,$ then $68=0r6,$ so your binary outputs should be $0110$ which reflect your remainder $6$ in the division process. We use three binary variables $A,B,C$ to represent the binary outputs. We use one external input in this counter, namely for upward counting and $x=0$ for downward counting$).$

Here is an example of the sequential Red ID counter: Suppose that your Red ID number is $"876543210".$ Since there are $9$ numbers in your Red ID sequence, there should be $9$ states in your FSM$.$ By applying the aforementioned divided$-$by$-8$ process, the expected decimal output sequence will be $"{076543210}^{\text{'}\text{'}}.$ Using $x$ as an input, a typical state graph of the counter for the Red ID $"876543210"$ is shown Figure $3.$

Figure $3.$ Example of a sequential Red ID counter.

Indicate your Red ID number and the corresponding binary output values for each digit of your Red ID number. $(5$ points$)$

Fill up the next state table of your counter. Please use $Q_1{Q}_2{Q}_3{Q}_4$ as your binary state variables, $A,B,C$ as your binary outputs, and $x$ as your binary input. $(15$ points$)$

$3\mathrm{.}3$ Express each of the next state variables $Q_1^{+},Q_2^{+},Q_3^{+},Q_4^{+}$as a minterm expansion $($no need to simplify the logic expressions$)$ with $Q_1,Q_2,Q_3,Q_4,x$ as inputs. $(20$ points$)$

$3\mathrm{.}4$ Using Karnaugh maps or any logic simplification methods of your choice, obtain the simplified expressions for each of the three output variables $A,B,C$ as a function of $Q_1,Q_2,Q_3,Q_4.(20$ points$)$