Given that CNDR of 256 QAM signal operating at 12.5 M Symbols/s is 100 dBHz. The value of E_b/N₀ (Energy-per-bit to Noise Density ratio) is:

Explanation:

Numerical Solution for Eb/N0 Calculation

Given Data:

• CNDR (Carrier-to-Noise Density Ratio): 100 dBHz
• Modulation Scheme: 256-QAM
• Symbol Rate: 12.5 M Symbols/s (or 12.5 × 10⁶ Symbols/s)

Step-by-Step Solution:

To calculate the energy-per-bit to noise density ratio (Eb/N0), we can use the relationship between CNDR, symbol rate, and the modulation scheme. The formula is as follows:

Formula:

Eb/N0 = CNDR - 10 × log₁₀(R × log₂(M))

Where:

• CNDR: Carrier-to-Noise Density Ratio (in dBHz)
• R: Symbol rate (in Symbols/s)
• M: Modulation order (number of distinct symbols in the modulation, for 256-QAM, M = 256)

Step 1: Convert CNDR to dB Scale

In this question, CNDR is already provided in dBHz (100 dBHz), so no additional conversion is necessary.

Step 2: Calculate the Logarithmic Term for M

The modulation order (M) for 256-QAM is 256. The logarithmic term log₂(M) is calculated as:

log₂(256) = log₂(2⁸) = 8

Step 3: Calculate the Logarithmic Term for Symbol Rate

The symbol rate (R) is given as 12.5 M Symbols/s, which is equivalent to:

R = 12.5 × 10⁶ Symbols/s

The logarithmic term 10 × log₁₀(R × log₂(M)) is calculated as:

10 × log₁₀(12.5 × 10⁶ × 8)

First, calculate the product:

12.5 × 10⁶ × 8 = 100 × 10⁶ = 10⁸

Now, calculate the logarithm:

log₁₀(10⁸) = 8

Finally, multiply by 10:

10 × log₁₀(R × log₂(M)) = 10 × 8 = 80 dB

Step 4: Subtract the Logarithmic Term from CNDR

Eb/N0 = CNDR - 10 × log₁₀(R × log₂(M))

Substitute the values:

Eb/N0 = 100 dB - 80 dB

Eb/N0 = 20 dB

The value of Eb/N0 is 20 dB.