GATE 2023 EC Q.31: solution and answer

GATE 2023 · ECQ.31 · 1 mark · NAT

The signal-to-noise ratio (SNR) of an ADC with a full-scale sinusoidal input is given to be 61.96 dB. The resolution of the ADC is ________ bits (rounded off to the nearest integer).

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The ADC has a resolution of 10 bits.

Official answer: 10Source: Official answer key, GATE 2023 EC (IIT Kanpur)

Take an ideal $n$ -bit ADC whose full-scale range is $2 V_{m}$ (a sine of peak $V_{m}$ just fills it). The step size is $Δ = \frac{2 V _{m}}{2 ^{n}}$ . The quantisation error is uniformly distributed over $(- \frac{Δ}{2}, \frac{Δ}{2})$ , so the quantisation noise power is $\frac{Δ ^{2}}{12}$ .
A full-scale sinusoid has power $\frac{V _{m}^{2}}{2}$ , so $SNR = \frac{V _{m}^{2} /2}{Δ ^{2} /12} = \frac{V _{m}^{2}}{2} \cdot \frac{12 \cdot 2 ^{2 n}}{4 V _{m}^{2}} = \frac{3}{2} 2^{2 n}$
In decibels: $SNR_{dB} = 10 lo g_{10} (\frac{3}{2}) + 20 n lo g_{10} 2 = 1.76 + 6.02 n dB$
Set this equal to the given value: $6.02 n + 1.76 = 61.96$ , so $6.02 n = 60.20$ .
$n = \frac{60.20}{6.02} = 10$ . (With the unrounded constants $1.7609$ and $6.0206$ , $n = \frac{60.1991}{6.0206} \approx 9.999$ , which also rounds to 10.) The resolution of the ADC is $10$ bits.

KEY IDEA

For an ideal $n$ -bit ADC driven by a full-scale sine wave, the signal power is $V_{m}^{2} /2$ and the quantisation noise power is $Δ^{2} /12$ with $Δ = 2 V_{m} / 2^{n}$ . That gives $SNR_{dB} = 6.02 n + 1.76$ . Solve it for $n$ .

WATCH OUT FOR

Using the rough '6 dB per bit' rule and leaving out the 1.76 dB term: $61.96/6.02 \approx 10.3$ happens to round to the right answer here, but the method is wrong and fails on other values. Taking the signal power as $V_{m}^{2}$ instead of $V_{m}^{2} /2$ , or the full-scale range as $V_{m}$ instead of $2 V_{m}$ , moves the constant away from 1.76 dB.

From the GATE 2023 EC paper. Every question of this paper · Download the full paper · All 18 ADC & DAC questions, with notes

GATE 2023 EC Q.31: solution and answer

Answer and solution

More ADC & DAC questions