Question 1

In basic digital design, what logic function is produced at the SUM output of a half-adder circuit?

Accepted Answer

Introduction / Context: The half-adder is one of the first combinational circuits studied in digital logic. It adds two single binary digits and produces a SUM bit and a CARRY bit. Knowing which Boolean function corresponds to the SUM output is essential for understanding adders and arithmetic logic units (ALUs). Given Data / Assumptions: Inputs: A and B, each a single bit. Outputs: SUM and CARRY. No carry-in is present in a half-adder (unlike a full adder). Concept / Approach: The SUM output of a half-adder is 1 when exactly one of the inputs is 1. This is precisely the truth table of the Exclusive-OR function. The CARRY output is 1 only when both inputs are 1, which corresponds to the AND function. Step-by-Step Solution: 1) Write the truth table for A, B → SUM:A=0, B=0 → SUM=0A=0, B=1 → SUM=1A=1, B=0 → SUM=1A=1, B=1 → SUM=02) Compare with standard logic functions.3) The pattern matches Exclusive-OR (XOR): SUM = A ⊕ B. Verification / Alternative check: Boolean expression: SUM = A*B' + A'B, which is the canonical XOR identity. The carry is CARRY = AB. Together, they implement binary addition for 1-bit inputs without carry-in. Why Other Options Are Wrong: AND: that is the CARRY output, not the SUM.XNOR: high when inputs are equal, which would give 1 at A=0,B=0 and A=1,B=1—opposite of SUM.NAND/OR: do not match the SUM truth table. Common Pitfalls: Confusing SUM with CARRY, or assuming SUM is a simple OR. Remember, SUM must be 0 when both inputs are 1 (since 1+1=10 in binary), hence XOR, not OR. Final Answer: Exclusive-OR (XOR)

Question 2

Compute the binary subtraction: 01110010 − 01001000. Express the result as an 8-bit binary number.

Accepted Answer

Introduction / Context: Binary arithmetic is a core digital skill. This problem requires subtracting two unsigned 8-bit binary numbers and writing the 8-bit result. Understanding subtraction via direct arithmetic or via two's-complement addition is useful for both hardware design and programming contexts. Given Data / Assumptions: Minuend: 01110010 (binary) Subtrahend: 01001000 (binary) Unsigned subtraction; produce an 8-bit result. Concept / Approach: Two methods exist: (1) subtract directly bit-by-bit with borrows, or (2) add the two's complement of the subtrahend to the minuend. Both yield the same result if no underflow occurs (and here 114 − 72 is positive). Step-by-Step Solution: 1) Convert to decimal for a quick check: 01110010 = 114, 01001000 = 72.2) Compute decimal difference: 114 − 72 = 42.3) Convert 42 to 8-bit binary: 42 = 32 + 8 + 2 = 00101010.4) Therefore, 01110010 − 01001000 = 00101010. Verification / Alternative check: Two's-complement method: take 01001000 → invert bits 10110111, add 1 → 10111000. Now add to 01110010: 01110010 + 10111000 = 00101010 with a carry out (ignored in 8-bit arithmetic). This confirms the result 00101010. Why Other Options Are Wrong: 00011010 (26): too small; would correspond to 98 − 72.01110010 (114): that is the minuend, not the difference.00111100 (60): incorrect difference.11101010: negative two's-complement value, not applicable for 114 − 72. Common Pitfalls: Losing track of borrows in bitwise subtraction, or mis-converting between decimal and binary. Always verify by a secondary method such as two's complement addition. Final Answer: 00101010

Question 3

Using two's-complement arithmetic with 8-bit representation, compute the sum of −11 and −2. Give the final two's-complement result.

Accepted Answer

Introduction / Context: Two's-complement representation is the standard way to encode signed integers in digital systems. This question practices encoding negative numbers and performing signed addition, which is identical to unsigned addition at the bit level when using two's complement. Given Data / Assumptions: Word size: 8 bits. Operands: −11 and −2. We must return the final 8-bit two's-complement sum. Concept / Approach: In two's complement, to represent −N, write N in binary, invert all bits, then add 1. Signed addition is then performed just like unsigned addition, with any carry out of the MSB discarded. The result remains valid within range −128 to +127 for 8 bits. Step-by-Step Solution: 1) Encode +11: 0000 1011.2) Two's complement for −11: invert → 1111 0100, add 1 → 1111 0101? (check carefully)Correct computation: 0000 1011 → invert 1111 0100 → add 1 = 1111 0101 is −11.3) Encode −2: +2 is 0000 0010 → invert 1111 1101 → add 1 = 1111 1110.4) Add: 1111 0101 + 1111 1110 = 1 1110 10011? Let’s add carefully:1111 0101+1111 1110=1 1111 0011 (discard the leading carry)5) Result: 1111 0011, which is the two's-complement code for −13. Verification / Alternative check: Decimal check: −11 + (−2) = −13. Two's-complement of 13: +13 = 0000 1101 → invert 1111 0010 → add 1 → 1111 0011. This matches the computed sum. Why Other Options Are Wrong: 1110 1101: corresponds to −19 (two's complement of 0001 0011).1111 1001: corresponds roughly to −7.1110 1001: corresponds to −23.0000 1101: this is +13, not −13. Common Pitfalls: Mistakes occur when inverting and adding 1, or when mishandling the end-around carry; remember to discard the final carry in fixed 8-bit arithmetic. Final Answer: 1111 0011

Question 4

What is the primary advantage of using Binary-Coded Decimal (BCD) instead of straight binary representation for decimal numbers?

Accepted Answer

Introduction / Context: Binary-Coded Decimal (BCD) encodes each decimal digit using four bits, making it simple to map between human-readable decimal numbers and their digital representation. This question probes your understanding of why BCD is sometimes preferred despite its storage overhead compared to straight binary. Given Data / Assumptions: BCD uses 4 bits per decimal digit (0000–1001 for 0–9). Straight binary represents the entire number in base-2 compactly. We are comparing representational advantages, not arithmetic performance. Concept / Approach: The chief benefit of BCD is the direct digit-by-digit correspondence with decimal. Conversions to and from decimal display are straightforward and avoid complex base-2 to base-10 conversion routines. This is valuable in financial and UI-centric systems where decimal precision and display are frequent operations. Step-by-Step Solution: 1) Identify what the question asks: the main advantage, not every advantage.2) Evaluate options: storage efficiency vs. conversion ease.3) BCD is less storage-efficient than binary (usually more bits).4) Therefore, the correct advantage is ease of decimal ↔ BCD conversion for each digit. Verification / Alternative check: Example: Decimal 59 → BCD digits 5 (0101) and 9 (1001) → 0101 1001. Converting back to decimal is trivial by reading each nibble; straight binary 59 (0011 1011) requires full base conversion to display digit characters. Why Other Options Are Wrong: Fewer bits required: false; BCD typically uses more bits.BCD always more efficient: incorrect; the opposite is common.BCD eliminates all rounding: arithmetic precision depends on algorithms; BCD helps with decimal rounding but does not eliminate all issues. Common Pitfalls: Assuming BCD is “better” universally; it is better only when decimal presentation or exact decimal digit control dominates. For computation, straight binary is usually smaller and faster. Final Answer: The relative ease of converting to and from decimal

Question 5

Digital design fundamentals A half-adder vs. full-adder: Would you normally use a half-adder whenever a carry input (Cin) is required in an adder circuit?

Accepted Answer

Introduction / Context: This question checks understanding of the difference between a half-adder and a full-adder in digital electronics. Adders are fundamental building blocks for arithmetic logic units; knowing when to use each type ensures correct handling of carry propagation in multi-bit addition. Given Data / Assumptions: A half-adder has two inputs: A and B; it outputs Sum and Cout (carry out) but has no carry input. A full-adder has three inputs: A, B, and Cin; it outputs Sum and Cout. We consider standard binary addition with possible incoming carry from a lower-order stage. Concept / Approach: When cascading adders to form multi-bit adders, each bit position except the least significant bit must accept a carry input from the previous position. Therefore, a circuit that requires Cin must be implemented with a full-adder, not a half-adder. Step-by-Step Solution: Define requirement: We need an adder stage with a carry input (Cin).Half-adder capability: computes A + B only; there is no terminal for Cin.Full-adder capability: computes A + B + Cin; it explicitly supports cascading.Conclusion: If Cin is required, use a full-adder, not a half-adder. Verification / Alternative check: Observe a 4-bit ripple-carry adder schematic: the least significant bit may be a half-adder if Cin = 0, but all higher bits must be full-adders because Cin is needed. Why Other Options Are Wrong: True / Depends / Only when Cin = 0 / Only for LSB: A half-adder lacks Cin entirely; only a full-adder supports Cin regardless of bit position. Common Pitfalls: Confusing the presence of Cout in a half-adder with the ability to accept Cin. Cout is an output; Cin requires an extra input port that the half-adder does not have. Final Answer: False

Question 6

4-bit adders and carry logic What is the role of fast look-ahead carry circuits commonly used in 4-bit full-adder designs?

Accepted Answer

Introduction / Context: Carry computation limits the speed of binary adders. Ripple-carry adders propagate carry bit-by-bit, causing delay. Look-ahead carry logic accelerates this process, enabling faster arithmetic operations in CPUs and digital systems. Given Data / Assumptions: Target device: 4-bit full-adder block with look-ahead carry (often 74xx83 + 74xx182 or integrated CLA). Goal: understand the function of the fast look-ahead carry circuitry. Binary addition with generate (G) and propagate (P) signals is used. Concept / Approach: Carry look-ahead computes carries C1, C2, C3, … directly from inputs and the initial carry using formulas with propagate and generate terms, avoiding sequential rippling. This reduces the worst-case carry chain time. Step-by-Step Solution: Define ripple problem: carry must settle at each stage before the next starts, creating linear delay with bit-width.Introduce CLA: compute Ci = Gi + Pi * Ci-1, with logic that expands Ci in terms of C0 and input bits at all positions.Net effect: parallelized carry computation → significantly lower propagation delay than ripple-only designs.Therefore, fast look-ahead carry circuits reduce propagation delay. Verification / Alternative check: Compare gate-level delays of ripple vs. look-ahead for identical widths; timing reports show CLA paths are much shorter for the carry signals. Why Other Options Are Wrong: Determine sign/magnitude: unrelated to carry acceleration.Add a 1 to complemented inputs: this describes two’s-complement subtraction facilitation, not CLA function.Increase ripple delay: the opposite of the CLA’s purpose. Common Pitfalls: Assuming look-ahead only helps for wide adders; even 4-bit blocks benefit and can be cascaded hierarchically for 16- or 32-bit adders. Final Answer: reduce propagation delay

Question 7

Binary arithmetic – division practice Compute the integer result of the binary division 01000110 ÷ 00001010.

Accepted Answer

Introduction / Context: Binary division mirrors decimal long division but is applied to base-2 numbers. This exercise reinforces number-base conversions and binary arithmetic skills widely used in digital logic and computer architecture. Given Data / Assumptions: Dividend: 01000110 (binary). Divisor: 00001010 (binary). We seek the integer quotient; fractional remainder is ignored. No signed arithmetic (both are treated as unsigned). Concept / Approach: Convert to decimal to verify, or perform binary long division directly. 01000110 is 70 in decimal. 00001010 is 10 in decimal. Therefore, 70 / 10 = 7. In binary, 7 is 0111. Step-by-Step Solution: Check magnitudes: 01000110 (70) ≥ 00001010 (10).Determine quotient: 70 / 10 = 7.Convert 7 to binary: 7 = 4 + 2 + 1 → 0111.Confirm via binary long division if desired: successive subtract-shifts yield remainder 0 and quotient 0111. Verification / Alternative check: Back-multiply: 0111 (7) * 00001010 (10) = 01110010 (decimal 70) with remainder 0, matching the original dividend. Why Other Options Are Wrong: 10011 (19) and 1001 (9) are not equal to 7; they mismatch the decimal check.0011 (3) is too small; 3 * 10 = 30 ≠ 70.0000 (0) is incorrect since the dividend is larger than the divisor. Common Pitfalls: Dropping leading zeros and misreading values, or confusing binary 1010 (10) with decimal 1010. Always verify by converting both numbers to the same base for a quick sense check. Final Answer: 0111

Question 8

Using a 4-bit adder to perform subtraction What modification enables a standard 4-bit adder to subtract one operand from another (A – B) using two’s complement?

Accepted Answer

Introduction / Context: Many arithmetic logic designs reuse adders for subtraction by leveraging two’s complement. Understanding how to adapt hardware adders avoids extra circuitry and simplifies ALU design. Given Data / Assumptions: Target operation: A – B implemented with an adder. Adder: 4-bit binary adder with a carry-in (Cin). We can optionally complement inputs and set Cin as needed. Concept / Approach: Two’s complement method for A – B computes A + (two’s complement of B). The two’s complement of B equals B' + 1, where B' is the bitwise inversion of B. Hardware trick: invert each B input and set Cin = 1; the adder then performs A + B' + 1 = A – B. Step-by-Step Solution: Two’s complement: (-B) = B' + 1.Hardware: feed B through inverters to form B'.Set Cin = 1 to add the +1 term automatically.The adder computes A + B' + 1 = A – B. Verification / Alternative check: Example: A = 0110 (6), B = 0011 (3). Invert B → 1100, set Cin = 1: 0110 + 1100 + 0001 = 0011 with Cout ignored, which is decimal 3 = 6 – 3. Why Other Options Are Wrong: Inverting the output: does not add the +1 term and does not create two’s complement of B.Inverting the carry-in: Cin is a single bit; inverting it alone cannot implement subtraction.Grounding B inputs: forces B = 0, yielding A – 0 only.Tying carry-out to carry-in: risks oscillation and does not realize two’s complement subtraction. Common Pitfalls: Forgetting to set Cin = 1 in addition to inverting B; both steps are required in practical circuits (many MCUs or ALUs implement this via XOR gates on B and an add-with-carry instruction). Final Answer: inverting the B inputs

Question 9

Two’s-complement interpretation and addition For 8-bit two’s-complement numbers 11110010 and 11110011, determine each value in decimal and the sum.

Accepted Answer

Introduction / Context: Two’s-complement is the standard signed binary representation. Correctly decoding negatives and performing additions is essential for digital design and embedded programming. Given Data / Assumptions: Word size: 8 bits. Numbers: N1 = 11110010, N2 = 11110011. Two’s-complement interpretation is required. Concept / Approach: If the most significant bit is 1, the number is negative. Magnitude is found by taking the two’s complement: invert all bits and add 1. Sum in two’s-complement naturally handles negatives; overflow is ignored if the true mathematical result fits within range. Step-by-Step Solution: N1 = 11110010 → invert = 00001101, add 1 → 00001110 = 14 → N1 = –14.N2 = 11110011 → invert = 00001100, add 1 → 00001101 = 13 → N2 = –13.Sum: –14 + –13 = –27.Check range: 8-bit two’s-complement range is –128 to +127, so –27 is representable. Verification / Alternative check: Binary addition: 11110010 + 11110011 = 11100101 with carry out discarded; interpret 11100101 → invert 00011010, add 1 → 00011011 = 27 → negative → –27. Matches. Why Other Options Are Wrong: –113 and –114, 227: incorrect magnitudes and impossible sum for 8-bit signed range.–27 and –13, 40: first magnitude wrong; sum does not match.–11 and –16, –27: wrong individual values though the sum coincidentally matches. Common Pitfalls: Forgetting the add-1 step after inversion, or misinterpreting 8-bit wrap-around when checking sums. Always re-verify with both decimal reasoning and binary addition. Final Answer: –14 and –13, –27

Question 10

BCD addition rule Compute the BCD result of adding 0100 (BCD 4) and 0110 (BCD 6).

Accepted Answer

Introduction / Context: Binary-Coded Decimal (BCD) encodes each decimal digit in four bits. Arithmetic in BCD requires digit-wise operations and correction when a nibble exceeds 9. This question tests basic BCD addition and correction logic. Given Data / Assumptions: BCD digits: 0100 (4) and 0110 (6). Single-digit BCD addition; no incoming carry. We express the final answer as an 8-bit BCD (two nibbles) with the proper correction. Concept / Approach: Add the two nibbles as binary. If the sum is greater than 1001 (9) or a carry occurs, add 0110 (decimal 6) to correct, producing a carry into the next decimal digit. For 4 + 6, the raw sum is 10 (1010), which requires BCD correction. Step-by-Step Solution: Raw binary sum: 0100 + 0110 = 1010 (decimal 10).BCD correction: since 1010 > 1001, add 0110 → 1010 + 0110 = 1 0000 (carry into tens place).Final BCD: 0001 0000 (representing decimal 10). Verification / Alternative check: Decimal check: 4 + 6 = 10. In BCD, 10 is encoded as two digits: 1 and 0 → 0001 0000. Why Other Options Are Wrong: 00010111BCD (17): incorrect; 4 + 6 ≠ 17.00001011BCD (11): incorrect; not the sum of 4 and 6.00010011BCD (13): incorrect result.00000110BCD (6): this is just digit 6, not the sum. Common Pitfalls: Forgetting the 0110 correction or misplacing the carry into the tens place. Always check whether the nibble exceeds 9 or produced a carry. Final Answer: 00010000BCD

Question 11

Two’s-complement conversion practice Convert the following 8-bit two’s-complement numbers to signed decimal: 00000101, 11111100, 11111000.

Accepted Answer

Introduction / Context: Interpreting two’s-complement values is a core skill for embedded systems and computer architecture. This exercise reinforces the conversion of binary to signed decimal for positive and negative values. Given Data / Assumptions: Word size: 8 bits; range is –128 to +127. Values: 00000101, 11111100, 11111000. Two’s-complement rules apply. Concept / Approach: If MSB = 0, the number is positive; convert directly. If MSB = 1, the number is negative; magnitude is found by inverting all bits and adding 1, then applying a minus sign. Step-by-Step Solution: 00000101: MSB = 0 → positive. Value = 5 → +5.11111100: MSB = 1 → negative. Invert = 00000011; add 1 → 00000100 = 4 → –4.11111000: MSB = 1 → negative. Invert = 00000111; add 1 → 00001000 = 8 → –8. Verification / Alternative check: Re-encode: +5 = 00000101; –4 → two’s complement of 00000100 = 11111100; –8 → two’s complement of 00001000 = 11111000. All match originals. Why Other Options Are Wrong: –5 +4 +8, –5 +252 +248, +5 –252 –248: These misinterpret two’s-complement and bit width.+5 +4 +8: Treats negatives as positives; incorrect signs. Common Pitfalls: Forgetting the add-1 step after inversion for negatives, or misreading the 8-bit range. Always check MSB and then apply the two-step conversion for negative numbers. Final Answer: +5 –4 –8

Question 12

Multi-nibble binary subtraction (treating the numbers as continuous) Compute the result of subtracting the 24-bit binary value 0010 0011 0011 1000 0101 0111 from 0101 1000 1010 0011 1101 1110.

Accepted Answer

Introduction / Context: Large binary subtractions are common when working with addresses, checksums, and digital signal processing. Grouping bits into nibbles (4-bit hex digits) simplifies long subtractions via hexadecimal arithmetic. Given Data / Assumptions: Minuend: 0101 1000 1010 0011 1101 1110. Subtrahend: 0010 0011 0011 1000 0101 0111. Interpret numbers as continuous 24-bit values, not independent nibbles. Concept / Approach: Convert grouped nibbles to hexadecimal, subtract in hex, then convert back to binary. Hex is base-16 and aligns exactly with 4-bit groups, streamlining arithmetic while preserving accuracy. Step-by-Step Solution: Convert minuend to hex: 0101 1000 1010 0011 1101 1110 → 0x58A3DE.Convert subtrahend to hex: 0010 0011 0011 1000 0101 0111 → 0x233857.Subtract: 0x58A3DE − 0x233857 = 0x356B87.Back to binary: 0x35 6B 87 → 0011 0101 0110 1011 1000 0111. Verification / Alternative check: Decimal cross-check: 0x58A3DE = 5800158 (decimal); 0x233857 = 2301527 (decimal); difference = 3498631 (decimal) = 0x356B87, consistent. Why Other Options Are Wrong: Each incorrect option differs by one or more nibbles from 0011 0101 0110 1011 1000 0111; these reflect common borrow/hex conversion mistakes. Common Pitfalls: Attempting nibble-wise subtraction with independent borrows per nibble instead of continuous subtraction across the entire word, and mis-converting between binary and hex digits. Final Answer: 0011 0101 0110 1011 1000 0111

Question 13

Two's-complement arithmetic (8-bit signed): Interpret the signed binary numbers 11110010 and 11110011 in decimal (sign and magnitude), and then compute their sum.

Accepted Answer

Introduction / Context: This question checks your ability to read 8-bit two's-complement numbers, determine their decimal values (including sign), and perform signed addition. Two's-complement is the standard way computers represent signed integers and perform addition with the same hardware as for unsigned numbers. Given Data / Assumptions: Word size: 8 bits. Numbers: 11110010 and 11110011 in two's-complement. Task: find each decimal value and then the sum. Concept / Approach: For two's-complement, if the most significant bit (MSB) is 1, the number is negative. To find magnitude: invert all bits, add 1, then attach a negative sign. Addition of two's-complement values uses ordinary binary addition; overflow is ignored for the magnitude unless you exceed representable range (–128 to +127 for 8 bits). Step-by-Step Solution: Interpret 11110010: MSB = 1 ⇒ negative. Invert to get 00001101, add 1 ⇒ 00001110.00001110 is 14 decimal ⇒ value is –14.Interpret 11110011: MSB = 1 ⇒ negative. Invert to get 00001100, add 1 ⇒ 00001101.00001101 is 13 decimal ⇒ value is –13.Sum in decimal: –14 + (–13) = –27. Verification / Alternative check: Add in binary: 11110010 + 11110011 = 11100101 (with a carry out dropped). Convert 11100101: MSB 1 ⇒ negative; invert 00011010, add 1 ⇒ 00011011 = 27 ⇒ –27. Matches the decimal sum. Why Other Options Are Wrong: –113/–114, –227: Misinterprets 8-bit range and magnitude.–11 and –16, –27: Incorrect magnitudes for the given bit patterns.–27 and –13, –40: First operand misread; sum becomes inconsistent.–112 and –111, –223: Same magnitude error pattern as option A. Common Pitfalls: Forgetting to add 1 after inversion, treating the carry-out as part of the sum for 8-bit results, or confusing sign-magnitude with two's-complement. Final Answer: –14 and –13, –27

Question 14

Two's-complement representation: Find the 2's complement (binary) needed to represent –(110110₂).

Accepted Answer

Introduction / Context: Here we practice forming a two's-complement pattern for a negative value. When asked for the 2's complement of a positive binary pattern, the result is the bit pattern that represents its negative in two's-complement arithmetic. Given Data / Assumptions: Unsigned magnitude given: 110110₂. Goal: produce the two's-complement bit pattern that corresponds to –110110₂. Assume a fixed width just large enough to hold the bits shown (6 bits here). Concept / Approach: To form –N in two's-complement: take N, invert all bits (one's complement), then add 1. Ensure you keep the same bit width while performing the operation. Step-by-Step Solution: Start with N = 110110.Invert bits ⇒ 001001.Add 1 ⇒ 001001 + 1 = 001010.Therefore, two's-complement pattern for –110110₂ is 001010₂. Verification / Alternative check: Check by addition: 110110 + 001010 = 1000000 (carry out of the 6-bit range). Dropping the carry in fixed width yields 000000, confirming that 001010 is the additive inverse of 110110 in 6-bit two's-complement. Why Other Options Are Wrong: 110100₂, 101010₂, 001001₂, 110111₂: Each fails either the invert-plus-one rule or the additive-inverse check (sum within 6 bits must be 000000). Common Pitfalls: Forgetting the final +1, changing the bit width mid-calculation, or misreading the order of bits during inversion. Final Answer: 001010₂

Question 15

Full-adder chains and performance: What is true about carry propagation delay in cascaded full-adder (FA) stages?

Accepted Answer

Introduction / Context: Ripple-carry adders are built by cascading full adders, with each stage waiting for the previous carry. This architecture is simple but introduces carry propagation delay, a key metric that determines how fast multi-bit addition can be completed. Given Data / Assumptions: Adder type: ripple-carry (cascaded FAs). Carry out of stage i must settle before stage i+1 can settle. Each FA has a finite gate delay. Concept / Approach: Propagation delay through each full adder adds to the total delay because carry must ripple through all stages from least to most significant bit. Thus total delay roughly scales with the number of bits, directly impacting maximum clock or throughput. Step-by-Step Solution: Let t_FA be the worst-case carry delay of one full adder.For N bits in ripple form, worst-case delay ≈ N * t_FA (carry must traverse all stages).Therefore, the carry propagation delay is cumulative and limits speed. Verification / Alternative check: Compare with look-ahead adders: they reduce dependency chain length by computing carries in parallel, achieving much lower effective delay than N * t_FA. This contrast supports that ripple carry delay is indeed limiting. Why Other Options Are Wrong: Negligible: Not for wide adders; it dominates speed.Decreases with stages: Opposite of reality.Increases but not limiting: In practice it is often the limiting factor.Eliminated by sum-only logic: Sums still depend on carries. Common Pitfalls: Ignoring worst-case paths, or assuming average delay sets the clock limit; synchronous systems must accommodate worst case. Final Answer: It is cumulative per stage and limits arithmetic speed.

Question 16

Fast look-ahead carry in the 7483 (4-bit adder): Why is a look-ahead carry circuit used inside the 7483 full-adder?

Accepted Answer

Introduction / Context: The TTL 7483 is a classic 4-bit binary full-adder. Basic ripple-carry adders are simple but slow because each stage must wait for the incoming carry. The 7483 includes look-ahead carry circuitry to improve performance. Given Data / Assumptions: Adder word length: 4 bits. Built-in carry look-ahead logic. Goal: improve addition speed. Concept / Approach: Carry look-ahead uses generate (G) and propagate (P) signals from each bit to compute carries in parallel rather than sequentially. This reduces the carry chain dependency and therefore reduces total delay. Step-by-Step Solution: For each bit, compute propagate P = A ⊕ B and generate G = A * B.Use combinational equations to derive C1, C2, C3, C4 from C0 in parallel, e.g., C1 = G0 + P0C0, C2 = G1 + P1G0 + P1P0C0, etc.Parallel computation of carries significantly reduces the time to valid sum outputs. Verification / Alternative check: Compare critical paths: ripple delay ≈ N * t_FA; look-ahead delay ≈ t_logic for P/G plus a few gate levels for look-ahead equations, independent of N for small N, thus faster. Why Other Options Are Wrong: Decrease cost/make smaller: Look-ahead adds logic, often increasing complexity.Slow down the circuit: The opposite of the purpose.Reduce power only: Power may change, but the key benefit is speed. Common Pitfalls: Confusing look-ahead with pipelining or assuming it removes all delay; it reduces, not eliminates, carry latency. Final Answer: To speed up the circuit

Question 17

Two's-complement encoding (8-bit): Convert the signed decimal numbers +7, –3, and –12 into 8-bit two's-complement binary.

Accepted Answer

Introduction / Context: Two's-complement is the standard signed integer format for 8-bit microcontrollers and processors. This problem reinforces encoding both positive and negative decimal integers into fixed-width binary. Given Data / Assumptions: Word size: 8 bits for each value. Numbers: +7, –3, –12 (decimal). Format: two's-complement (MSB is sign bit). Concept / Approach: For a positive number, simply write its 8-bit binary with leading zeros. For a negative number –N: write N in binary, invert all bits, then add 1. Keep the width at 8 bits throughout. Step-by-Step Solution: +7: binary 7 = 0000 0111 (8 bits).–3: magnitude 3 = 0000 0011 → invert 1111 1100 → add 1 ⇒ 1111 1101.–12: magnitude 12 = 0000 1100 → invert 1111 0011 → add 1 ⇒ 1111 0100.Thus the triplet is 0000 0111, 1111 1101, 1111 0100. Verification / Alternative check: Add the two's-complement negatives to their magnitudes: 1111 1101 + 0000 0011 = 1 0000 0000 (discard carry) → 0000 0000, confirming additive inverse. Similarly, 1111 0100 + 0000 1100 also wraps to 0000 0000 with a carry out. Why Other Options Are Wrong: Options with 1000 xxxx for small positives misuse the sign bit.Options showing raw magnitude for negatives lack inversion and +1.Mixed patterns do not pass the additive-inverse check. Common Pitfalls: Dropping leading zeros, forgetting the +1 step, or accidentally expanding beyond 8 bits. Final Answer: 0000 0111 1111 1101 1111 0100

Question 18

VHDL coding basics: In VHDL, what are constants and how are they used?

Accepted Answer

Introduction / Context: Understanding objects in VHDL (constants, signals, variables, types) is foundational for writing readable, maintainable RTL. Constants capture literal values with meaningful names and prevent magic numbers from scattering across code. Given Data / Assumptions: Language: VHDL for digital hardware description. Focus: the role and definition of constants. Concept / Approach: A constant in VHDL binds a name to an immutable value of a given type (e.g., integer, std_logic_vector, time). It is elaborated at compile/elaboration time and cannot be changed during simulation. Syntax example: constant DATA_WIDTH : integer := 8; Using constants improves clarity and reduces errors when values are reused. Step-by-Step Solution: Identify that constants are named, typed, and assigned once.They can be declared in declarative regions (package, architecture) and referenced throughout.They are not variables (which can change) nor types (which define sets of values). Verification / Alternative check: Attempting to assign to a constant in a process or concurrent statement results in a compile error, reaffirming immutability. Simulation logs will also show constants retaining the same value for the entire run. Why Other Options Are Wrong: Fixed variables: Variables are mutable within processes; constants are not.Fixed number types: Types are categories, not instances of values.Constants do not exist: They do; they are heavily used for generics, timing, and widths.Signals that cannot be assigned: Signals are drivers on nets; constants are not signals. Common Pitfalls: Confusing constants with generics (which are parameterizable at instantiation) or with signals (which can have drivers and resolution). Final Answer: Fixed numbers represented by a name

Question 19

Adder architectures: What distinguishes a look-ahead-carry adder compared to a ripple-carry adder?

Accepted Answer

Introduction / Context: Adder speed is often limited by how quickly carry information propagates. Ripple-carry adders propagate carries serially, while look-ahead-carry adders compute carries in parallel logic to accelerate addition. Given Data / Assumptions: We compare ripple-carry vs. look-ahead-carry. Goal: identify the key performance difference. Concept / Approach: Look-ahead-carry uses generate (G) and propagate (P) signals to derive all intermediate carries from the inputs and the initial carry in only a few gate levels. This reduces total delay from roughly proportional to word length (N) down to a small, mostly fixed number of levels for modest N. Step-by-Step Solution: Define P_i = A_i ⊕ B_i and G_i = A_i * B_i.Compute carries with equations like C1 = G0 + P0C0, C2 = G1 + P1G0 + P1P0C0, etc.Since carries are produced in parallel, sums stabilize sooner than in ripple designs. Verification / Alternative check: Critical-path comparison confirms lower logic depth for carry computation in look-ahead designs, yielding higher maximum frequency. Why Other Options Are Wrong: Slower: Contrary to the design goal.Easier to implement: It is actually more complex than ripple-carry.Requires advance knowledge: Carries are computed from inputs, not from the result.Eliminates carry bits: Carries are still computed, just faster. Common Pitfalls: Assuming look-ahead scales perfectly for very large N without hierarchical structures; practical adders use multi-level look-ahead or carry-select. Final Answer: It is faster than a ripple-carry adder.

Question 20

BCD adder count: How many BCD adder stages are needed to add the decimal numbers 973₁₀ and 39₁₀?

Accepted Answer

Introduction / Context: Binary-coded decimal (BCD) adders operate on digits (0–9) encoded in 4 bits each. Multi-digit decimal addition uses one BCD adder per decimal digit, plus carry correction logic for sums ≥ 10. Given Data / Assumptions: Numbers: 973 and 39 (both base-10). BCD addition aligns digits by place value. Each BCD adder handles one decimal digit. Concept / Approach: Write the smaller addend with leading zeros so both numbers have equal digits. The number of digits equals the number of required BCD adder stages, connected least significant to most significant with carry propagation. Step-by-Step Solution: Rewrite 39 as 039 to align with 973.Digits: hundreds (9 vs 0), tens (7 vs 3), ones (3 vs 9).Therefore, three BCD digit positions ⇒ three BCD adder stages. Verification / Alternative check: Perform decimal addition: 973 + 39 = 1,012. Even though the result has four digits, the hardware still uses three cascaded BCD adders for the three input columns; a carry out from the most significant digit produces the extra thousand. Why Other Options Are Wrong: 2: Would handle only two least significant digits, losing the hundreds column.4, 5, 6: Exceed the number of aligned input digits. Common Pitfalls: Confusing the number of digits in the result with the number of digit adders required; hardware sizing follows input width, not the potential carry out. Final Answer: 3

Digital Arithmetic Operations and Circuits Questions