A PCM system has step resolution of 2 V. Sinusoidal signal amplitude 10 V. SNR=? And total number of bits=?

1. What is PCM?

Pulse Code Modulation (PCM) converts an analog signal (like a sine wave) into digital bits in three steps:

1. Sampling: Measure the signal's value at regular time intervals.
2. Quantization: Round each sample to the nearest fixed level.
3. Encoding: Represent each level with a binary code of n bits.

Problems about bits and SNR come from the quantization and encoding steps.

2. Finding the number of bits

Step size (Δ) is the gap between two neighboring quantization levels, also called the step resolution.

A sinusoid with amplitude Aₘ swings from −Aₘ to +Aₘ, so the total range it covers is:

Range = 2Aₘ (peak-to-peak)

The number of levels needed to cover this range is:

L = 2Aₘ / Δ

Each level gets a unique binary code. With n bits you can make 2ⁿ codes, so you need:

2ⁿ ≥ L

Bits must be a whole number, so you always round up.

Solving

• Range = 2 × 10 = 20 V
• L = 20 / 2 = 10 levels
• 2³ = 8 (not enough), 2⁴ = 16 (enough)

n = 4 bits

3. Finding the SNR

Where does the noise come from?

When a sample is rounded to the nearest level, a small error is created. This is called quantization error (or quantization noise).

Since a sample is rounded to the nearest level, the error is never more than half a step:

Error range: −Δ/2 to +Δ/2

The error is equally likely to be anywhere in this range (a uniform distribution). The average power of such an error works out to:

Noise power, Nq = Δ² / 12

Signal power

For a sine wave of amplitude Aₘ, the average power (across a 1 Ω resistor) is:

Signal power, S = Aₘ² / 2

SNR

The Signal-to-Noise Ratio compares how strong the signal is relative to the noise:

SNR = S / Nq = (Aₘ² / 2) ÷ (Δ² / 12)

SNR (dB) = 10 log₁₀(SNR)

A higher SNR means a cleaner, more accurate digital signal.

Solving

• S = 10² / 2 = 100 / 2 = 50 W
• Nq = 2² / 12 = 4 / 12 = 0.333 W
• SNR = 50 / 0.333 = 150
• SNR (dB) = 10 log₁₀(150) ≈ 21.76 dB

4. Final answer

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