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Chapter 15: Problem 8

Show that, by using 4-bit 2's-complement words, the correct final sum is obtained when the numbers \(+5,+5\), and \(-7\) are added together while any overflow effects are ignored.

Short Answer

Expert verified
The correct final sum is +3 with no overflow affecting the result.

Step by step solution

01

Converting to 2's-Complement Binary

First, represent each number using 4-bit 2's-complement binary. For \(+5\), the binary is \(0101\). For \(-7\), find the binary of \(+7\), \(0111\), invert the digits to get \(1000\) and add 1 to get \(1001\) as the 2's-complement of \(-7\). So, the numbers are \(+5 = 0101\), \(+5 = 0101\), \(-7 = 1001\).
02

Add the First Two Numbers

Add the first two numbers \(+5\) and \(+5\) in binary: \[ \begin{array}{c} 0101 \ +0101 \ ------ \ 1010 \end{array} \] Therefore, the sum is \(1010\), which is \(+10\) in decimal, ignoring overflow as specified.
03

Add the Third Number

Add the result from Step 2 to \(-7\). Using \(1010\) for \(+10\) and \(1001\) for \(-7\): \[ \begin{array}{c} 1010 \ +1001 \ ------ \ 0011 \end{array} \] The result is \(0011\), which is \(+3\) in decimal.
04

Verify No Overflow

In 2's-complement addition, overflow occurs if the carry into the most significant bit (MSB) is different from the carry from the MSB. Here, the MSB's carry is the same with no disparity, so it confirms the absence of overflow affecting the result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binary Addition
Binary addition works like regular decimal addition but is limited to two digits: 0 and 1. The carryover, when needed, is 1 instead of 10 in decimal format. When adding in binary:
  • 0 + 0 = 0 (no carry)
  • 0 + 1 = 1 (no carry)
  • 1 + 0 = 1 (no carry)
  • 1 + 1 = 0 (with a carryover of 1)
  • 1 + 1 + 1 = 1 (with a carryover of 1)
In this exercise, the problem hinges on the addition of 4-bit numbers such as adding two 5s represented as 0101. By following binary rules, the sum is 1010, which equates to 10 in decimal without any issues of overflow in this step. Understanding these basics of binary arithmetic is crucial for interpreting results in computer logic and digital electronics.
Overflow Detection
Overflow detection in binary arithmetic, especially in 2's-complement systems, occurs when the calculated sum exceeds the representational limits of the given bit-length. This typically happens in signed number systems like 2's-complement, where a fixed number of bits is used. When adding, overflow is detected when:
  • The sign of the resulting sum is different from the sign of the two numbers being added (when adding two positive numbers, the result should be positive).
  • There's a discrepancy between the carry into the MSB and the carry out.
In the provided solution, there is no overflow effect because the discrepancy condition didn't occur. The carry into the MSB matched the carry out, thereby confirming the result is accurate and within the representational limits.
4-bit Representation
In computing, a 4-bit representation is a way of expressing numbers using four binary digits. In 2's-complement notation, these 4 bits can represent numbers from -8 to +7. This method is widely used because it simplifies both the hardware design and the mathematics involved in processing these numbers.
  • Positive numbers remain the same, but their range is halved compared to unsigned notation.
  • Negative numbers are represented by inverting all bits of their absolute magnitude and adding 1.
In this exercise, the numbers +5, +5, and -7 were encoded using 4-bit 2's-complement: 0101, 0101, and 1001 respectively. Given the limited range, carefully inspecting for overflow is crucial. However, by ignoring overflow effects as in this exercise, 4-bit arithmetic can simplify calculations for limited-range numerical problems.

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Most popular questions from this chapter

An ideal quantizer is to cover the range from 0 to \(1 \mathrm{~V}\) with less than \(10 \mu \mathrm{V}_{\mathrm{rms}}\) quantization noise power. How many bits of resolution does the converter require?

Consider the following measured voltage values for a 2-bit D/A with a reference voltage of \(4 \mathrm{~V}\) : $$ \\{00 \leftrightarrow 0.01 \mathrm{~V}\\} \quad\\{01 \leftrightarrow 1.02 \mathrm{~V}\\} \quad\\{10 \leftrightarrow 1.97 \mathrm{~V}\\} \quad\\{11 \leftrightarrow 3.02 \mathrm{~V}\\} $$ In units of LSB, find the offset error, gain error, worst absolute and relative accuracies, and worst differential nonlinearity. Restate the relative accuracy in terms of an N-bit accuracy.

An input signal arrives at a quantizer already corrupted with some noise and having a SNR of \(35 \mathrm{~dB}\). How many bits of resolution does the quantizer require to ensure that the quantization noise is at least \(3 \mathrm{~dB}\) smaller than the input noise?

What is the SNR for an ideal 12-bit unipolar \(\mathrm{A} / \mathrm{D}\) converter with \(\mathrm{V}_{\text {ref }}=3 \mathrm{~V}\), when a sinusoidal input of \(1 \mathrm{~V}_{\mathrm{pp}}\) is applied? What size input would result in an SNR of \(0 \mathrm{~dB}\) ?

A 10-bit A/D converter has a reference voltage, \(\mathrm{V}_{\text {ref }}\), tuned to \(10.24 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\). Find the maximum allowable temperature coefficient in terms of \((\mu \mathrm{V}) /{ }^{\circ} \mathrm{C}\) for the reference voltage if the reference voltage is allowed to cause a maximum error of \((\pm 1 / 2)\) LSB over the temperature range of 0 to \(50{ }^{\circ} \mathrm{C}\).
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