Determine the resolution and the maximum quantization error of an 8-bit linear sign magnitude PCM code for a maximum decoded voltage of 1.27 V. With explanation

The given data -

Linear Sign Magnitude PCM Code = 8 bits

Maximum Decoded Voltage = $V_{max}$ = 1.27 V

To find -

Resolution = ?

Maximum Quantization Error = $Q_e$ = ?

Formulae -

$$ Resolution = \frac {V_{max}}{2^n - 1} $$

$$Maximum\ Quantization\ Error = Q_e = \frac {Resolution}{2} $$

Solution -

First, find out the Resolution using the formula

$$ Resolution = \frac {V_{max}}{2^n - 1} $$

Where,

n = Number of bits in the PCM code but, excluding the Sign bit.

Therefore,

$$n = 8 - 1 = 7$$

$$ Resolution = \frac {V_{max}}{2^n - 1} = \frac {1.27}{2^7 - 1} = \frac {1.27}{127} = 0.01\ V$$

Now, calculate the maximum quantization error

$$Maximum\ Quantization\ Error = Q_e = \frac {Resolution}{2} = \frac {0.01}{2} = 0.005\ V $$

Answer:

Resolution = 0.01 V

Maximum Quantization Error = $ Q_e $ = 0.005 V