GATE 2022 IN Q.25: solution and answer

GATE 2022 · INQ.25 · 1 mark · NAT

An analog-to-digital converter with resolution 0.01 V converts analog signals between 0 V to +10 V to an unsigned binary output. The minimum number of bits (in integer) in the output is ________

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Answer and solution

ANSWER

The minimum number of output bits is 10.

Official answer: 10Source: Official answer key, GATE 2022 IN (IIT Kharagpur)

Full-scale span of the input: $V_{F S} = 10 V - 0 V = 10 V$ .
An $n$ -bit ADC divides this span into equal steps. The step size (resolution) is $Δ = \frac{V _{F S}}{2 ^{n}}$ , or $Δ = \frac{V _{F S}}{2 ^{n} - 1}$ under the other common convention. To resolve $0.01 V$ the ADC needs $Δ \leq 0.01 V$ .
With $Δ = \frac{V _{F S}}{2 ^{n}}$ : $\frac{10}{2 ^{n}} \leq 0.01 \Rightarrow 2^{n} \geq 1000$ .
With $Δ = \frac{V _{F S}}{2 ^{n} - 1}$ : $\frac{10}{2 ^{n} - 1} \leq 0.01 \Rightarrow 2^{n} - 1 \geq 1000 \Rightarrow 2^{n} \geq 1001$ .
Check powers of two. $2^{9} = 512$ is too small: the step is $\frac{10}{512} \approx 0.0195 V$ (or $\frac{10}{511} \approx 0.0196 V$ ), which is coarser than $0.01 V$ . $2^{10} = 1024$ is enough: the step is $\frac{10}{1024} \approx 0.00977 V$ (or $\frac{10}{1023} \approx 0.00978 V$ ), which is finer than $0.01 V$ .
Both conventions give the same smallest integer: $n = ⌈ lo g_{2} 1000 ⌉ = ⌈ 9.97 ⌉ = 10$ bits.

KEY IDEA

The resolution of an $n$ -bit ADC is its full-scale range divided into about $2^{n}$ equal steps, so you need the smallest $n$ with $\frac{10 V}{2 ^{n}} \leq 0.01 V$ , which means $2^{n} \geq 1000$ . Here, with $2^{9} = 512 < 1000 < 1024 = 2^{10}$ , it does not matter whether you divide by $2^{n}$ or $2^{n} - 1$ .

WATCH OUT FOR

Taking $n = lo g_{2} 1000 \approx 9.97$ and rounding it down to 9 bits. With 9 bits the step is about $0.0195 V$ , nearly twice the required $0.01 V$ . The number of bits has to be rounded up. A second slip is to divide 10 V by $0.01 V$ , get 1000 levels, and write 1000 as the answer. The question asks for the number of bits, not the number of levels.

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GATE 2022 IN Q.25: solution and answer

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