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

Название: Quantization error analysis of the quadrature components of narrowband signals
Раздел: Топики по английскому языку
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The implementation of filters with digital circuits having finite word-length introduces unavoidable quantization errors. These effects have been widely studied [1–7]. The three common sources of quantization error are: input quantization, coefficient quantization and quantization in arithmetic operations. In [2–4, 6] papers the statistical characteristics of the quantization errors of scalar signals have been studied. The influence of all three sources of quantization errors on performance of a Chebyshev digital third-order highpass filter was investigated in [5] also for the scalar input signals. The quantization errors of complex input signals, which were represented by its inphase and quadrature components were studied in [7] to evaluate the performance of coder/decoders with phase shift keying. However, only computer simulation results were presented in this paper.

Usually digital signal processing of narrowband radio signals (i.e. signals for which inequality Quantization error analysis of the quadrature components of narrowband signals is valid) is carried out after the demodulation of the input signal into the quadrature components. Hence, our attention in this paper will be on input quantization of the complex signals. We adopt stochastic methods to analyse quantization errors [1–6]. The block diagram of the input narrowband signals converter, which produces the quadrature components of the signals and then transforms them into digital form is shown in fig. 1 (the left part of the plot).

Quantization error analysis of the quadrature components of narrowband signals

Fig. 1. Block diagram of narrowband signals' converter

The converter contains two frequency mixtures, two low pass filters (LPF), two analog-to-digital converters (A/D) and a control unit. The quantizing (roundoff) errors of the inphase Xi and the quadrature Yi components are caused by limited bit representation of the code words of these components. To quantitatively evaluate these errors we will transform the quadrature components which have the roundoff errors into the narrowband signal again, and then we will estimate the amplitude and phase errors in this signal in comparison with the input one. For this purpose we will add in the block-diagram in fig. 1 the necessary blocks (the right part of the plot): digital-to-analogue converters (D/A), low pass filters (LPF) which restore the continuous analogue signal, frequency mixtures and adder. Assume all blocks work in ideal mode, don't introduce the delay, then the magnitude of the transfer function of the LPF is

Quantization error analysis of the quadrature components of narrowband signals

If the Nyquist constraint is valid the values of the restored analogue quadrature components Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals (Quantization error analysis of the quadrature components of narrowband signalsis the clock period) will be equal to the discrete values of quadrature components –Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals respectively.

Preliminaries

Let Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals be the inphase and quadrature components at the input of the A/D converters. At each sampling instant i, the quantized outputs Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals, the quantization (roundoff) errors Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals, and the input Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals are related by

Quantization error analysis of the quadrature components of narrowband signals, Quantization error analysis of the quadrature components of narrowband signals. (1)

Suppose roundoff errors are independent with zero mean, variance Quantization error analysis of the quadrature components of narrowband signals and uniform distribution in interval Quantization error analysis of the quadrature components of narrowband signals, cf. [6]. Quantization error analysis of the quadrature components of narrowband signalsis the step of quantizing.

If the input signalQuantization error analysis of the quadrature components of narrowband signals is a narrowband signal

Quantization error analysis of the quadrature components of narrowband signals,

then the output signal Quantization error analysis of the quadrature components of narrowband signalsis also a narrowband signal and can be written in the form

Quantization error analysis of the quadrature components of narrowband signals (2)

where the values of Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals are given by formula (1).

The vector representation of the Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals signals is given in fig. 2. Obviously, we have

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

Quantization error analysis of the quadrature components of narrowband signals

Fig. 2. Vector representation of input and output (distorted) signals

Under the assumption about independent random variables Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals the hypothesis about uniform distribution of the random angles Quantization error analysis of the quadrature components of narrowband signals may be accepted. It is clear from the fig. 2 and formula (2) that the signal Quantization error analysis of the quadrature components of narrowband signalshas a parathytic amplitude modulation as well as a phase modulation. The parathytic modulation is caused by the quantizing errors of the signal's quadrature components.

Amplitude error analysis of the quantized narrowband signals.

The variance of the magnitude Quantization error analysis of the quadrature components of narrowband signals is

Quantization error analysis of the quadrature components of narrowband signals

where smax is the maximum available amplitude of the input signals of the A/D converter, n – is the number of bits of the A/D converter.

It is interesting to note that quantizing errors exist only when the input signals exists, nevertheless these errors are additive but not multiplicative because the values of these errors depend on the quantizing step Quantization error analysis of the quadrature components of narrowband signals, but do not depend on the amplitude of the input signal Quantization error analysis of the quadrature components of narrowband signals. (See formula (5)). We are interested in the amplitude and phase of the output signal Quantization error analysis of the quadrature components of narrowband signals. Let us find the statistical characteristics of the amplitude and phase.

The length Quantization error analysis of the quadrature components of narrowband signals of the vector Quantization error analysis of the quadrature components of narrowband signals can easily be found from the triangle OAB (see fig. 2)

Quantization error analysis of the quadrature components of narrowband signals, (6)

where Quantization error analysis of the quadrature components of narrowband signals.

As the amplitude Quantization error analysis of the quadrature components of narrowband signals is the random variable, let us find the mean of this amplitude

Quantization error analysis of the quadrature components of narrowband signals.(7)

Since for many practical interesting cases Quantization error analysis of the quadrature components of narrowband signals, we shall use the decomposition Quantization error analysis of the quadrature components of narrowband signals, hence

Quantization error analysis of the quadrature components of narrowband signals. (8)

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

Quantization error analysis of the quadrature components of narrowband signals, (9)

Quantization error analysis of the quadrature components of narrowband signals. (10)

The angle Quantization error analysis of the quadrature components of narrowband signals is (see fig. 2)

Quantization error analysis of the quadrature components of narrowband signals, hence

Quantization error analysis of the quadrature components of narrowband signals, (11)

because Quantization error analysis of the quadrature components of narrowband signals is a random variable with uniform distribution in interval Quantization error analysis of the quadrature components of narrowband signals.

By inserting the values given by formulas (9)–(11) into the formula (8) we get the mean of the amplitude Quantization error analysis of the quadrature components of narrowband signals

Quantization error analysis of the quadrature components of narrowband signals.(12)

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

Quantization error analysis of the quadrature components of narrowband signals (12a)

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

Analysis of formula (12) shows that if Quantization error analysis of the quadrature components of narrowband signals and if the number of bits of the A/D converter Quantization error analysis of the quadrature components of narrowband signals then the mean Quantization error analysis of the quadrature components of narrowband signals is equal to the Quantization error analysis of the quadrature components of narrowband signals 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 Quantization error analysis of the quadrature components of narrowband signals can be found considering formula (6) and the fact, that Quantization error analysis of the quadrature components of narrowband signals

Quantization error analysis of the quadrature components of narrowband signals


Supposing that Quantization error analysis of the quadrature components of narrowband signals and using the decomposition Quantization error analysis of the quadrature components of narrowband signals, the formula (13) can be written

Quantization error analysis of the quadrature components of narrowband signals

Where

Quantization error analysis of the quadrature components of narrowband signals.

If we have identical A/D converters, then

Quantization error analysis of the quadrature components of narrowband signals, (15)

Where

Quantization error analysis of the quadrature components of narrowband signals.

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

Quantization error analysis of the quadrature components of narrowband signals

Quantization error analysis of the quadrature components of narrowband signals

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

Quantization error analysis of the quadrature components of narrowband signals.

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 Quantization error analysis of the quadrature components of narrowband signalsi 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

Quantization error analysis of the quadrature components of narrowband signals hence

Quantization error analysis of the quadrature components of narrowband signals.(17)

Let us define the limits of the angle Quantization error analysis of the quadrature components of narrowband signals variation. From the triangle OBF we get

Quantization error analysis of the quadrature components of narrowband signals, (18)

and from the triangle OAG we get

Quantization error analysis of the quadrature components of narrowband signals. (19)

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

Quantization error analysis of the quadrature components of narrowband signals. (20)

It is obvious from formula (20) what the maximum phase error Quantization error analysis of the quadrature components of narrowband signals will be, provided the value of the inphase component is minimum and the quantization error Quantization error analysis of the quadrature components of narrowband signals is maximum, i.e. provided

Quantization error analysis of the quadrature components of narrowband signals. (21)

Inserting these values into formula (20), we get

Quantization error analysis of the quadrature components of narrowband signals. (22)

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

Quantization error analysis of the quadrature components of narrowband signals. (23)

Solving the equation (23) with respect to Quantization error analysis of the quadrature components of narrowband signals we get

Quantization error analysis of the quadrature components of narrowband signals. (24)

It is clear that maximum value of the angle Quantization error analysis of the quadrature components of narrowband signals will be, if Quantization error analysis of the quadrature components of narrowband signals, hence

Quantization error analysis of the quadrature components of narrowband signals.(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 %)

Quantization error analysis of the quadrature components of narrowband signals. (26)

The mean of the phase error Quantization error analysis of the quadrature components of narrowband signals is

Quantization error analysis of the quadrature components of narrowband signals, (27)

where Quantization error analysis of the quadrature components of narrowband signals.Quantization error analysis of the quadrature components of narrowband signals

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

Quantization error analysis of the quadrature components of narrowband signals

Inserting the value of Quantization error analysis of the quadrature components of narrowband signals , given by formula (5) into formula (28), we finally get the phase variance

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')

Quantization error analysis of the quadrature components of narrowband signals.

Fig. 3 shows a plot of phase variance a against number of A/D converter bits for various values of ratio Quantization error analysis of the quadrature components of narrowband signals (solid curves). The computation was carried out in accordance with formula (29).

Quantization error analysis of the quadrature components of narrowband signals

Fig. 3. Standard deviation of the phase quantization error for different rations Quantization error analysis of the quadrature components of narrowband signals as a function of code word length

Quantization error analysis of the quadrature components of narrowband signals

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

Quantization error analysis of the quadrature components of narrowband signals

where Quantization error analysis of the quadrature components of narrowband signals – operator of quantization.

Quantization error analysis of the quadrature components of narrowband signalsis 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

Quantization error analysis of the quadrature components of narrowband signals, Quantization error analysis of the quadrature components of narrowband signals. (31)

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

Quantization error analysis of the quadrature components of narrowband signals.

The phase error was then founded as the difference between Quantization error analysis of the quadrature components of narrowband signals and Quantization error analysis of the quadrature components of narrowband signals. 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 ± 0,1Quantization error analysis of the quadrature components of narrowband signals). 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 Quantization error analysis of the quadrature components of narrowband signals 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 Quantization error analysis of the quadrature components of narrowband signals computed for the same case

.

Quantization error analysis of the quadrature components of narrowband signals

Fig. 5. Probability distribution laws of the phase error for different word length, Quantization error analysis of the quadrature components of narrowband signals

Quantization error analysis of the quadrature components of narrowband signals

Fig. 6. Probability distribution laws of the amplitude error for different word-length, Quantization error analysis of the quadrature components of narrowband signals

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.

2. Агеев, Р.В. Логарифмическая дискретизация сигналов с заданной абсолютной погрешностью / Р.В. Агеев, Ю.Н. Овчаров // Автометрия. – 2008. – № 6. – С. 23–27.

3. Лифшиц, Н.А.Численные характеристики ошибок квантования амплитуды / Н.А. Лифшиц, В.Е. Фарбер // Автоматика и телемеханика. – 2008. – т. 39, № 12. – с. 176–179.

4. Домрачеев, В.Г. Критерий оценки точности цифровых преобразователей угла / В.Г. Домрачеев, Б.С. Мейко // Измерительная техника. – 2008. – т. 18, № 11. – С. 22–25.

5. Koffler, H. Quantization and roundoff errors in a digital MTI filter, Siemens Forsch. and Entwicklungsber / H. Koffler. – Germany, 2010 – Vol. 2, № 2. – p. 73–78.

6. Snipad, A.B. A necessary and sufficient condition for quantization errors to be uniform and white / A.B. Snipad, D.L. Snyder // IEEE Trans, on Acoust. Speech and Sign. Proc.Vol. – ASSP-25. – 2007. – № 5 (Oct.)–p. 442–448.

7. O'Neal, Iz. Digital encoding of phase shift keying voiceband data signals / Iz. O 'Neal, R.R. Koneru, I.P. Agrawal // Conf. record of Int. Conf. on Acoustics Speech and Signal processing, ICASSP-80. Denver. Co. – 2010. – April 9–11. – P. 315–318.

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