Exercise I: Number representation

Given the following numbers:

$A=(3855{)}_{10}$ is represented in decimal; $B=(11000100{)}_{c2}$ is represented in two's complement $8-$bits format; $C=(1001100{)}_2$ is represented in binary; $D=(1011010\mathrm{.}1011{)}_2$ is represented in binary as a fractional number.

a$)$ Convert A into binary, octal, and hexadecimal representations.

b$)$ Convert B into decimal representation.

c$)$ Perform the addition operation $B+C$ in binary. Then, convert the result into decimal representation.

d$)$ Convert D into decimal and octal representations.

Exercise II: Combinatory logic $-$ Synthesis of a $5-$to$-32$ Decoder

We would like to design a $5-$to$-32$ decoder using $3-$to$-8$ decoders. The goal is to efficiently utilize the $3-$to$-8$ decoders to create a circuit that can decode a $5-$bit input into one of $32$ output lines.

Design Specifications:

You have access to $3-$to$-8$ decoders $($as seen in the course$),$ each capable of decoding $3$ input lines into $8$ output lines if its input $V=1.$ Each to $3-$ to$-8$ decoder has $3$ inputs $($A$,B,C)$ one valid input $(V)$ and $8$ outputs to $($:$S_7\}.$

You can use additional logic gates to complete the system.

The final decoder should have $5$ input lines $($A$,$ B$,$ C$,$ D$,$ E$)$ and $32$ output lines $($S $S_0$ through $S_{31}).$

The circuit should activate only one of the $32$ output lines based on the $5-$bit input combination.

Exercise III: combinatory logic $-$ Temperature Control System Design

We would like to design a temperature control system for a bioreactor used in biopharmaceutical production. The system should maintain optimal temperature conditions inside the bioreactor to ensure the success of the production process. The temperature control system operates based on four Boolean inputs representing various environmental and operational factors, with a single output controlling the reversible heating$/$cooling system.

Inputs :

Temperature Sensor $($T$)$: Indicates the current temperature inside the bioreactor. High temperature or fluctuation in temperature are represented by $T=1,$ while normal$/$low temperature is represented by $T=0.$

Agitation Status $($A$)$: Indicates the status of agitation inside the bioreactor. Active agitation is represented by $A=1,$ while inactive agitation is represented by $A=0.$

Oxygen Level $(O)$ : Indicates the level of dissolved oxygen in the bioreactor. Low oxygen level is represented by $O=1,$ while normal$/$high oxygen level is represented by $O=0.$

Production Phase $(P)$ : Indicates the current phase of biopharmaceutical production. Active production phase is represented by $P=1,$ while inactive production phase is represented by $P=0.$