CN102647190B - Method for setting optimal input power of A/D (Analog to Digital) convertor applicable to Gaussian distribution signal - Google Patents

Abstract

The invention belongs to the technical field of signal processing. For achieving equilibrium between a general quantization error and an overload error, and a quantization output signal noise ratio is optimized. In order to achieve the aims, the technical scheme adopted by the invention is as follows: a method for setting optimal input power of an A/D (Analog to Digital) convertor applicable to a Gaussian distribution signal comprises the following steps of: step I, according to table I, finding out a value z when a quantization bit-width of the A/D convertor is n; step II, obtaining a variance of a Gaussian distribution input signal, according to a formula (shown in the description) and known conditions of maximum quantization amplitude Vmax of the A/D convertor; and step III, calculating the optimal power Px of the Gaussian distribution input signal of the A/D convertor, according to a formula (shown in the description). The method for setting the optimal input power of the A/D convertor applicable to the Gaussian distribution signal is mainly applied to the signal processing.

Description

Be applicable to the best input power establishing method of Gaussian Profile signal A/D transducer

Technical field

The invention belongs to signal processing technology field, specifically, relate to and be applicable to the best input power establishing method of Gaussian Profile signal A/D transducer.

Background technology

In order to resist the fluctuation of channel transfer characteristic, modern broadband communication system, particularly wireless communication system all adopts OFDM modulation, and different from general single-carrier modulated signal, the amplitude of ofdm modulation signal presents Gaussian Profile.A/D converter is used for analog signal to be converted to digital signal, and quantitative graphs is the important indicator weighing A/D converting system performance, A/D converter is in the front end of whole digital information processing system, and it is most important on the impact of the performance of whole digital information processing system.General A/D converting system requires that the maximum level amplitude of input signal is no more than the maximum quantizating level of A/D converter, namely overload error (clipping distortion) is avoided, now quantitative graphs is determined by general quantization error, input signal amplitude be uniformly distributed time, optimal quantization signal to noise ratio can be obtained.

To Gaussian Profile signal, the higher probability that it occurs of range value is less, now, if the blasting of avoiding, input signal power must reduce, and generally quantizes error power by the quantization level of quantization digit and maximum permission and determine, the quantitative graphs outputed signal now certainly will be caused to worsen.Therefore certain overload error is allowed can to improve the quantitative graphs of A/D converter output signal.How to average out between general quantization error and overload error and there is important using value.

Summary of the invention

The present invention is intended to solution and overcomes the deficiencies in the prior art, averages out between general quantization error and overload error, makes A/D converter quantize output signal-to-noise ratio optimum.For achieving the above object, the technical scheme that the present invention takes is, is applicable to the best input power establishing method of Gaussian Profile signal A/D transducer, comprises the following steps:

Suppose to treat that the Gaussian Profile signal that A/D converter quantizes is x (n), its average is 0, and variance is then its probability density function is the quantification bit wide of A/D converter is n, and A/D converter maximum quantization amplitude is V _max:

Step one: the power P of the Gaussian Profile signal of setting input _xbe approximately equal to the variance of the Gaussian Profile signal of input

Step 2: be n position by Gaussian Profile signal quantization by A/D converter, then quantize progression M=2 ⁿ, maximum and the minimum value of the Gaussian Profile signal after A/D converter quantizes are respectively

x _q，max＝(2 ^n-1-1)Δ

x _q，min＝-2 ^n-1Δ

Wherein Δ is quantization step, and A/D converter maximum quantization amplitude V _max=x _{q, max}| ≈ | x _{q, min}|, then

$\mathrm{\Δ}\≈\frac{{2x}_{q,\max }}{M}=\frac{x_{q,\max }}{2^{n-1}}=\frac{V_{\max }}{2^{n-1}},$

Now, the power of amplitude limit and overload error can be obtained:

$N_s={\∫}_{\mathrm{xq},\max }^{\∞}{(x-x_{q,\max })}^{2}f\left(x\right)\mathrm{dx}+{\∫}_{-\∞}^{x_{q,\min }}{(x-x_{q,\min })}^{2}f\left(x\right)\mathrm{dx}$

$\≈\frac{2}{\sqrt{2\mathrm{\π}}{\mathrm{\σ}}_x}{\∫}_{V\max }^{\∞}{(x-V_{\max })}^2{e}^{-\frac{x^2}{2{\mathrm{\σ}}_x^2}}\mathrm{dx}$

$=\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}$

Wherein x represent A/D converter quantize before Gaussian Profile signal, it is complementary error function;

The power of truncated error or rounding error is:

$N_q\≈\frac{{\mathrm{\Δ}}^2}{12}=\frac{1}{12}\·{\left(\frac{V_{\max }}{2^{n-1}}\right)}^2=\frac{V_{\max }^2}{{3\·4}^n}$

Total quantization error power is:

$N=N_q+N_s$

$=\frac{V_{\max }^2}{{3\·4}^n}+\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}$

Then quantitative graphs is

$\mathrm{snr}=10\lg \left(\frac{P_x}{N}\right)=10\lg \left(\frac{{\mathrm{\σ}}_x^2}{\frac{V_{\max }^2}{{3\·4}^n}+\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}}\right)$

Order to quantitative graphs about differentiate be equivalent to y about differentiate

$\frac{\∂y}{\∂{\mathrm{\σ}}_x^2}=\frac{V_{\max }^2}{{3\·4}^n}+V_{\max }^2\·\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{2{\mathrm{\σ}}_x^2}}=0$

Order then above formula can be reduced to

$\frac{z}{{3\·4}^n}+z\·\mathrm{erfc}\left(z\right)-\frac{1}{\sqrt{\mathrm{\π}}}{e}^{-z^2}=0$

The z value that different A/D converter quantizes under bit wide n correspondence is as shown in table 1,

Z value under the different n value of table 1.

n	z	n	z	n	z
						1	0.8767	11	3.3653	21	4.9105
2	1.2096	12	3.5449	22	5.0421
						3	1.5214	13	3.7173	23	5.1707
4	1.8096	14	3.8834	24	5.2965
						5	2.0762	15	4.0438	25	5.4196
6	2.3242	16	4.1990	26	5.5402
						7	2.5563	17	4.3494	27	5.6585
8	2.7747	18	4.4954	28	5.7746
						9	2.9814	19	4.6374	29	5.8886
10	3.1779	20	4.7757	30	6.0005

Step 3: by the z value that obtains and known conditions A/D converter maximum quantization amplitude V _maxsubstitute into namely the variance of the Gaussian Profile input signal making A/D converter quantitative graphs maximum is obtained value;

Step 4: utilize formula to Gaussian Profile signal variance carry out revising the exact value P of the best power obtaining A/D converter Gaussian Profile input signal _x, R is the input impedance of A/D converter.

Described step is reduced to further:

Step one: according to table 1, finds A/D converter and quantizes the z value of bit wide when being n;

Step 2: according to formula with known conditions A/D converter maximum quantization amplitude V _maxobtain the variance of Gaussian Profile input signal

Step 3: according to calculate the best power P of A/D converter Gaussian Profile input signal _x.

Technical characterstic of the present invention and effect:

By the enforcement of scheme, the power of the best Gaussian Profile input signal under specific A/D converter quantification bit wide and maximum quantization amplitude can be obtained, make A/D converter quantize output signal-to-noise ratio optimum.Such as quantizing bit wide at A/D converter is 16, and when A/D converter maximum quantization amplitude is 2.2V, the Gaussian Profile input signal power of the best is 2.74mW, and its A/D converter quantizes output signal-to-noise ratio can reach 85dB.

Accompanying drawing explanation

Fig. 1. be applicable to the A/D converter input power initialization system of Gaussian Profile signal.

Embodiment

Suppose to treat that the Gaussian Profile signal that A/D converter quantizes is x (n), its average is 0, and variance is then its probability density function is the quantification bit wide of A/D converter is n, and A/D converter maximum quantization amplitude is V _max, its system as shown in drawings;

Step one: the power P of the Gaussian Profile signal of input _xbe approximately equal to the variance of the Gaussian Profile signal of input

Step 2: be n position by Gaussian Profile signal quantization by A/D converter, then quantize progression M=2 ⁿ.Maximum and the minimum value of the Gaussian Profile signal after A/D converter quantizes are respectively

x _q，max＝(2 ^n-1-1)Δ

x _q，min＝-2 ^n-1Δ

Wherein Δ is quantization step.And A/D converter maximum quantization amplitude V _max=| x _{q, max}| ≈ | x _{q, min}|, then

$\mathrm{\Δ}\≈\frac{{2x}_{q,\max }}{M}=\frac{x_{q,\max }}{2^{n-1}}=\frac{V_{\max }}{2^{n-1}}.$

Now, the power of amplitude limit (overload) error can be obtained:

$N_s={\∫}_{\mathrm{xq},\max }^{\∞}{(x-x_{q,\max })}^{2}f\left(x\right)\mathrm{dx}+{\∫}_{-\∞}^{x_{q,\min }}{(x-x_{q,\min })}^{2}f\left(x\right)\mathrm{dx}$

$\≈\frac{2}{\sqrt{2\mathrm{\π}}{\mathrm{\σ}}_x}{\∫}_{V\max }^{\∞}{(x-V_{\max })}^2{e}^{-\frac{x^2}{2{\mathrm{\σ}}_x^2}}\mathrm{dx}$

$=\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}$

Wherein x represent A/D converter quantize before Gaussian Profile signal, it is complementary error function;

The power of truncated error or rounding error is:

$N_q\≈\frac{{\mathrm{\Δ}}^2}{12}=\frac{1}{12}\·{\left(\frac{V_{\max }}{2^{n-1}}\right)}^2=\frac{V_{\max }^2}{{3\·4}^n}$

Total quantization error power is:

$N=N_q+N_s$

$=\frac{V_{\max }^2}{{3\·4}^n}+\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}$

Then quantitative graphs is

$\mathrm{snr}=10\lg \left(\frac{P_x}{N}\right)=10\lg \left(\frac{{\mathrm{\σ}}_x^2}{\frac{V_{\max }^2}{{3\·4}^n}+\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}}\right)$

Order to quantitative graphs about differentiate be equivalent to y about differentiate

$\frac{\∂y}{\∂{\mathrm{\σ}}_x^2}=\frac{V_{\max }^2}{{3\·4}^n}+V_{\max }^2\·\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{2{\mathrm{\σ}}_x^2}}=0$

Order then above formula can be reduced to

$\frac{z}{{3\·4}^n}+z\·\mathrm{erfc}\left(z\right)-\frac{1}{\sqrt{\mathrm{\π}}}{e}^{-z^2}=0$

The z value that different A/D converter quantizes under bit wide n correspondence is as shown in table 1,

Z value under the different n value of table 1.

n	z	n	z	n	z
						1	0.8767	11	3.3653	21	4.9105
2	1.2096	12	3.5449	22	5.0421
						3	1.5214	13	3.7173	23	5.1707
4	1.8096	14	3.8834	24	5.2965
						5	2.0762	15	4.0438	25	5.4196
6	2.3242	16	4.1990	26	5.5402
						7	2.5563	17	4.3494	27	5.6585
8	2.7747	18	4.4954	28	5.7746
						9	2.9814	19	4.6374	29	5.8886
10	3.1779	20	4.7757	30	6.0005

Step 3: by the z value that obtains and known conditions A/D converter maximum quantization amplitude V _maxsubstitute into namely the variance of the Gaussian Profile input signal making A/D converter quantitative graphs maximum is obtained value;

Step 4: utilize formula to Gaussian Profile signal variance carry out revising the exact value P of the best power obtaining A/D converter Gaussian Profile input signal _x, R is the input impedance of A/D converter.

Now citing illustrates the step of the best input power of the A/D converter obtaining being applicable to Gaussian Profile signal.Suppose that the quantification bit wide of A/D converter is n=16, A/D converter maximum quantization amplitude is V _max=± 2.2V, the input impedance of A/D converter is 50 Ω.

Step one: according to table 1, when finding A/D converter quantification bit wide n=16, z=4.1990.

Step 2: according to formula with A/D converter maximum quantization amplitude V _maxobtain the variance of Gaussian Profile input signal as follows:

${\mathrm{\σ}}_x^2=V_{\max }^2/{2y}^2=\frac{{2.2}^2}{2\×{4.1990}^2}=0.137$

Step 3: according to calculate the best power Px of A/D converter Gaussian Profile input signal:

$P_x={\mathrm{\σ}}_x^2/R=2.74\mathrm{mW}.$

Claims (2)

1. be applicable to the best input power establishing method of Gaussian Profile signal A/D transducer, it is characterized in that, comprise the steps:

Suppose to treat that the Gaussian Profile signal that A/D converter quantizes is x (n), its average is 0, and variance is then its probability density function is the quantification bit wide of A/D converter is n, and A/D converter maximum quantization amplitude is V _max:

Step one: the power P x of the Gaussian Profile signal of setting input is approximately equal to the variance of the Gaussian Profile signal of input

Step 2: be n position by Gaussian Profile signal quantization by A/D converter, then quantize progression M=2 ⁿ, maximum and the minimum value of the Gaussian Profile signal after A/D converter quantizes are respectively

x _q,max＝(2 ^n-1-1)△

x _q,min＝-2 ^n-1△

Wherein △ is quantization step, and A/D converter maximum quantization amplitude V _max=| x _{q, max}| ≈ | x _{q, min}|, then $\mathrm{\Δ}\≈\frac{{2x}_{q,\max }}{M}=\frac{x_{q,\max }}{2^{n-1}}=\frac{V_{\max }}{2^{n-1}},$

Now, the power of amplitude limit and overload error can be obtained:

$\begin{array}{c}{N}_s={\∫}_{x_{q,\max }}^{\∞}{(x-x_{q,\max })}^{2}f\left(x\right)\mathrm{dx}+{\∫}_{-\∞}^{x_{q,\max }}{(x-x_{q,\min })}^{2}f\left(x\right)\mathrm{dx}\\ \≈\frac{2}{\sqrt{2\mathrm{\π}}{\mathrm{\σ}}_x}{\∫}_{V_{\max }}^{\∞}{(x-V_{\max })}^2{e}^{-\frac{x^2}{{2\mathrm{\σ}}_x^2}}\mathrm{dx}\\ =\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)[(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x{V}_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{{V_{\max }}^2}{{2\mathrm{\σ}}_x^2}}\end{array}$

Wherein x represent A/D converter quantize before Gaussian Profile signal, it is complementary error function;

The power of truncated error or rounding error is:

$N_q\≈\frac{{\mathrm{\Δ}}^2}{12}=\frac{1}{12}\·{\left(\frac{V_{\max }}{2^{n-1}}\right)}^2=\frac{V_{\max }^2}{3\·4^n}$

Total quantization error power is:

$\begin{array}{c}N=N_q+N_s\\ =\frac{V_{\max }^2}{3\·4^n}+\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}\end{array}$

Then quantitative graphs is

$\mathrm{sur}=101g\left(\frac{P_x}{N}\right)=101g\left(\frac{{\mathrm{\σ}}_x^2}{\frac{V_{\max }^2}{3\·4^n}+\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)(V_{\max }^2+{\mathrm{\σ}}_x^2)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}}\right)$

Order to quantitative graphs about differentiate be equivalent to y about differentiate

$\frac{\∂y}{{\∂\mathrm{\σ}}_x^2}=\frac{V_{\max }^2}{3\·4^n}+V_{\max }^2\·\mathrm{erfc}\left(\frac{V_{\max }}{\sqrt{2}{\mathrm{\σ}}_x}\right)-\frac{{2\mathrm{\σ}}_x\·V_{\max }}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{V_{\max }^2}{{2\mathrm{\σ}}_x^2}}=0$

Order $z=V_{\max }/\sqrt{2}{\mathrm{\σ}}_x,$ Then above formula can be reduced to

$\frac{z}{3\·4^n}+z\·\mathrm{erfc}\left(z\right)-\frac{1}{\sqrt{\mathrm{\π}}}{e}^{-z^2}=0$

The z value that different A/D converter quantizes under bit wide n correspondence is as shown in table 1,

Z value under the different n value of table 1.

n z n z n z 1 0.8767 11 3.3653 21 4.9105 2 1.2096 12 3.5449 22 5.0421 3 1.5214 13 3.7173 23 5.1707 4 1.8096 14 3.8834 24 5.2965 5 2.0762 15 4.0438 25 5.4196 6 2.3242 16 4.1990 26 5.5402 7 2.5563 17 4.3494 27 5.6585 8 2.7747 18 4.4954 28 5.7746 9 2.9814 19 4.6374 29 5.8886 10 3.1779 20 4.7757 30 6.0005

Step 3: the z value obtained and known conditions A/D converter maximum quantization amplitude Vmax are substituted into namely the variance of the Gaussian Profile input signal making A/D converter quantitative graphs maximum is obtained value;

Step 4: utilize formula to Gaussian Profile signal variance carry out revising the exact value P of the best power obtaining A/D converter Gaussian Profile input signal _x, R is the input impedance of A/D converter.

2. be applicable to the best input power establishing method of Gaussian Profile signal A/D transducer as claimed in claim 1, it is characterized in that, described step is reduced to further:

Step one: according to table 1, finds A/D converter and quantizes the z value of bit wide when being n;

Step 2: according to formula with known conditions A/D converter maximum quantization amplitude V _maxobtain the variance of Gaussian Profile input signal

Step 3: according to calculate the best power P of A/D converter Gaussian Profile input signal _x.