Digital Word Representation

A digital word is a binary representation of data. The number of bits in a digital word determines the total number of unique values it can represent. In this question, we are given a 4-bit digital word. A 4-bit word can represent \(2^4\) different unique values.

• Total possible values for a 4-bit word = \(2^4 = 16\) values.
• These values typically range from 0 to 15 in decimal.
• In binary, these values range from 0000 (decimal 0) to 1111 (decimal 15).

Analog Signal to Digital Mapping

The analog signal varies from 0 V to 15 V. This range perfectly aligns with the 16 unique values that a 4-bit digital word can represent (0 to 15). This indicates a direct, linear mapping where each volt corresponds to one decimal increment in the digital word.

Therefore:

• 0 V corresponds to the digital value 0 (binary 0000).
• 1 V corresponds to the digital value 1 (binary 0001).
• ...and so on.
• 15 V corresponds to the digital value 15 (binary 1111).

Converting 7V to its Digital Word

Based on the direct mapping established above, to find the digital word corresponding to 7 V, we simply need to convert the decimal number 7 into its 4-bit binary equivalent.

To convert decimal 7 to binary:

Alternatively, using positional notation:

• We need to find a combination of powers of 2 (starting from \(2^0\)) that sum up to 7.
• \(2^3 = 8\)
• \(2^2 = 4\)
• \(2^1 = 2\)
• \(2^0 = 1\)

So, \(7 = 4 + 2 + 1\). This means we need the \(2^2\), \(2^1\), and \(2^0\) bits to be '1', and the \(2^3\) bit to be '0'.

Therefore, the 4-bit binary representation of decimal 7 is 0111.

Final Digital Word for 7V

The digital word D corresponding to 7 V for an analog signal varying from 0 V to 15 V is 0111.