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# Derive the quantization formula of a sine wave.

Derive the quantization formula based upon the variance of a sinewave and the variance of distortion due to quantization.

This is the signal to distortion (SDR) ratio for a quantized sinewave. A is the amplitude of the quantized sinewave and Delta is the quantization interval of the analog to digital converter, and the quantization performed is rounding.

Using the variance of a random variable based upon statistical samples. The ratio of these two variances is the signal to distortion ratio for the quantized sine wave.

See attached for formula

keywords: steplength

#### Solution Preview

The average power in the sinewave can be measured by averaging over the period,

P_S = (1/Period) * int_{period} [A sin(2pi*t/Period)]^2 dt = (1/2)A^2 (1)

The average power in the quantization distortion can be measured in the same way:

P_D = (1/StepLength) * int_{StepLength} (Distortion)^2 dt, (2)

where StepLength is the length of the ...

#### Solution Summary

The quantization formula for a sine wave is derived. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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