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What is the relationship between the Error Vector Magnitude and the Signal to Noise Ratio for a 64-QAM signal?

The relationship between EVM(Vector Magnitude) and SNR(Signal to Noise Ratio), also known as MER (Modulation Error Ratio), is relatively straightforward as EVM is a voltage ratio of the error signal to the peak state voltage, and SNR is a ratio of the signal power to the error power. The equation is as follows.

SNR = - [3.7 + 20 * log10 (EVM / 100%)]

This equation is for 64 QAM only, and is accurate only when averaging is disabled and 64 states are equally probable in the measured signal. When averaging is enabled, the measured SNR and the SNR that is calculated from EVM will be different as the SNR number is biased because of the averaging of a log quantity.

As EVM is a ratio of voltages, we use 20 log10 to get to a power ratio. The minus sign is necessary because SNR is a ratio of signal to noise whereas EVM is a ratio of noise to signal.

The 3.7 represents the peak-to-average power ratio (in dB) of the 64 QAM signal. This number can be easily computed from the constellation diagram by computing the ratio of the average power for all 64 states (which is the same as the average power in one quadrant) and dividing by the power of a corner state (the corner state has the highest power).