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Suppose we have a dead-band quantizer with $\ N = 5 \ quantizing \ levels $ which uses $\ b = 3 \ bits $ for each produced symbol. The quantizer gets as input a uniformly distributed signal $\ X $ at the interval $\ [-10,10] $. How can I calculate the signal to noise ratio $\ SNR $ of the quantizer ?

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Time align the quantized signal with your input signal and subtract to create an error signal (match the amplitudes such as to minimize the error signal). The SNR can then be determined as the ratio of the standard deviation of the input signal (after amplitude matching) to the standard deviation of the error signal (20Log10 of this ratio to be in dB).

The top diagram that I have shows this for a complex signal example, but it is also equally applicable to real signals.

Error Waveform

quantization and clipping noise

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