Реферат: Quantization error analysis of the quadrature components of narrowband signals

.(7)

Since for many practical interesting cases , we shall use the decomposition , hence

. (8)

Considering the formulas (4) and (5) we will find the mean of values in formula (8)

, (9)

. (10)

The angle is (see fig. 2)

, hence

, (11)

because is a random variable with uniform distribution in interval .

By inserting the values given by formulas (9)–(11) into the formula (8) we get the mean of the amplitude

.(12)

Notice that the value of s₀ in the formula (12) has to satisfy

(12a)

as the amplitude of the input signal must exceed the quantization step.

Analysis of formula (12) shows that if and if the number of bits of the A/D converter then the mean is equal to the with the error less than 0,5 %. This means that the mean amplitude of output signal is practically equal to the amplitude of the input signal.

The variance of amplitude can be found considering formula (6) and the fact, that

Supposing that and using the decomposition , the formula (13) can be written

Where

.

If we have identical A/D converters, then

, (15)

Where

.

Finally we get, considering formula (11) and the fact that