Integrated combinational blocks

Explain what a $n$-input to $2^n$-output decoder does: what binary combination it receives and what exactly happens at its outputs (only one is activated, corresponding to the input code). Give the example of a $2$-to-$4$ decoder with its complete truth table.

A $3$-to-$8$ decoder receives the code $101$ at its inputs (with the most significant bit on the left). a) State which output is activated, numbering the outputs from $0$ to $7$. b) Write the complete truth table of the decoder. c) Describe an application: selecting one of eight memory devices sharing the same data bus.

A BCD to 7-segment decoder ($7$ segments) receives the code $0101$. a) State which digit that BCD code represents. b) Write which segments of the display must light up to show it (name them using the standard convention: $a$ the top one, $b$ and $c$ the right-hand ones, $d$ the bottom one, $e$ and $f$ the left-hand ones, and $g$ the middle one). c) Explain what the circuit does if it receives the code $1100$.

Explain what a multiplexer does: how many data inputs and how many select inputs a $4$-channel multiplexer has, and what appears at its single output. Compare it to the source selector of an audio amplifier (CD, radio, auxiliary).

Explain what a demultiplexer does and how it differs from a multiplexer (a data input routed to one of several outputs, chosen by the select inputs). Provide an analogy with a railway switch that routes a train carriage to one of four tracks.

The figure shows a $4$-channel multiplexer with the values of its data inputs and its select inputs. a) Write its selection table (which channel passes to the output for each value of $S_1 S_0$). b) Determine the value of the output marked with the question mark. c) Explain what would need to be changed so that the value of channel $D_1$ is output.

Explain what a half adder adds (two bits) and what two outputs it produces (sum and carry). Explain why it is called a "half" adder: what input is missing for it to be cascaded in adding multi-bit numbers.

Explain what additional input a full adder has compared to a half adder (the carry-in, $C_{in}$) and why it is essential when adding multi-bit numbers: describe the role played by the carry that "travels" from one column to the next, just like in manual decimal addition.

The figure shows the circuit of a half adder with its truth table to be completed. a) Fill in the sum ($S$) and carry ($C$) columns in your notebook for the four input combinations. b) Identify which gate produces each output and write the two logic functions. c) Check with the table that the set $\{S, C\}$ represents the arithmetic sum of $a$ and $b$ in binary.

Explain what a medium-scale integration (MSI) integrated circuit is compared to assembling the same using individual logic gates: advantages in terms of size, reliability and cost. Name three typical MSI blocks and an application of each.

Explain what a digital comparator does: what two data inputs it receives and what three outputs it can provide ($A > B$, $A = B$, $A < B$). Give two applications: a digital thermostat that compares the measured temperature with the setpoint, and a system that detects whether a counter has reached its maximum value.

The figure shows a $4$-bit adder consisting of four cascaded full adders, with the two addends $A = 0101$ and $B = 0011$ applied. a) Solve the addition step by step, indicating the carry coming out of each stage. b) Write the result and its final carry, and check in decimal that it is correct. c) Explain what would happen if the result required $5$ bits and there were only outputs for $4$ (overflow).