Quantization error analysis of the quadrature components of narrowband signals (стр. 2 из 3)

.(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

Under the constraint given by formula (12') we get

.

The last expression means that the variance of the amplitude error of the signal caused by quantization errors of its quadrature components is practically equal to the variance of the quantization error of the A/D converter.

Phase error analysis of the quantized narrowband signals

The phase error

_i of the distorted signal (we measure the phase error by comparing the input phase with the output phase) can be found from fig. 2. Actually, from the triangle OBE we get

hence

.(17)

Let us define the limits of the angle

variation. From the triangle OBF we get

, (18)

and from the triangle OAG we get

. (19)

Transforming formula (18) considering the formula (19) we obtain

. (20)

It is obvious from formula (20) what the maximum phase error

will be, provided the value of the inphase component is minimum and the quantization error is maximum, i.e. provided

. (21)

Inserting these values into formula (20), we get

. (22)

Transforming in the formula (22) the sum of angles [8] we get

. (23)

Solving the equation (23) with respect to

we get

. (24)

It is clear that maximum value of the angle

will be, if , hence

.(25)

We have found that maximum phase error does not exceed 53°. Therefore we can replace sin in the formula (17) by its argument (with the error less than 10 %)

. (26)

The mean of the phase error

is

, (27)

where

.

The variance of the phase error can be found from formulas (6) and (9)

Inserting the value of

, given by formula (5) into formula (28), we finally get the phase variance

The maximum value of the phase variance will occur if the input signal has the minimum, given by formula (12')

.

Fig. 3 shows a plot of phase variance a against number of A/D converter bits for various values of ratio

(solid curves). The computation was carried out in accordance with formula (29).

Fig. 3. Standard deviation of the phase quantization error for different rations

as a function of code word length

Fig. 4. Standard deviation of the amplitude quantization error as a function of code word length

Сomputer simulation of the roundoff errors of the quadrature components. The computer simulation of the quantizing errors of the quadrature components of the narrowband signal was carried out with the intention to check the validity of the obtained formulas (16) and (29).

The LFM signal with time-compression ratio 100 was chosen as a narrowband signal. Quantization of the inphase and quadrature components was made in accordance with formulas

where

– operator of quantization.

is an integer part of variable u, n is a number of A/D converter bits.

For each sample of the input signal the quantizing values of inphase and quadrature components were defined and then amplitude and phase of the distorted signal were determined according to formulas

, . (31)

At the same time the phase of the input signal was computed

.

The phase error was then founded as the difference between

and . These operations were made for 150 samples of the input signal. Then mean and variance of the amplitude error were defined as well as the same parameters of the phase error. The achieved results show that the mean of the amplitude is very close to the amplitude of the input signal (within 3 %), the mean of phase error is close to zero (in all cases the mean was less than ± 0,1 ). The plots of the phase standard deviation against the number of bits of the A/D converter are shown in fig. 3 for different rations s0/smax by points. The plots of the amplitude standard deviation against number of bits n are shown in fig. The coincidence between theoretical and simulation results are rather good, which shows the validity of our assumptions.