Реферат: Quantization error analysis of the quadrature components of narrowband signals
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.
Probability distribution laws of the amplitude and phase errors have also been evaluated by the means of computer simulation. For this purpose a LFM signal with time-compression ratio 6 400 was used. Statistical distributions were estimated with usage of 9 600 samples for inphase and 9 600 samples for quadrature components. Thirteen points of these statistical distributions were chosen. The plot of the statistical distribution law 
of the phase error values is shown in fig. 5 for various numbers of the A/D converters bits. fig. 6 shows the amplitude error distribution 
computed for the same case
.
Fig. 5. Probability distribution laws of the phase error for different word length, 
Fig. 6. Probability distribution laws of the amplitude error for different word-length, 
Conclusion
narrowband signal error
The results of theoretical analysis and computer simulation of the amplitude and phase errors of the narrowband signal, caused by quantizing of the signal's inphase and quadrature components show that the mean of the amplitude of the distorted signals remains equal to the input amplitude, but the output amplitude becomes fluctuated with the variance, determined by the variance of D/A converter error. The phase error has zero mean, maximum deviation 53° and a variance which is inversely proportional to the number of quantization levels. The results achieved may be used in digital filters' design.
r eferences
1. Rabiner, L.R. Theory and Application of Digital Signal Processing / L.R. Rabiner, В. Gold // Englewood Cliffs, NJ. – Prentice-Hall, 2008.