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This study provides a general framework to analyze the effects on correlation radiometers of a generic quantization scheme and sampling process. It reviews, unifies and expands several previous works that focused on these effects separately. In addition, it provides a general theoretical background that allows analyzing any digitization scheme including any number of quantization levels, irregular quantization steps, gain compression, clipping, jitter and skew effects of the sampling period.

Microwave radiometry is today a mature technology that was first used in radio-astronomy in the 1930s [

Digitization effects can be separated into quantization and sampling effects. Mainly, the effects related with the quantization of the input signal (thermal noise) imply the loss of its statistical properties due to the non-linear quantization process. Consequently, it is not possible to apply the well-known Gaussian statistical relationships to the quantified signal. This effect has a large impact when limited quantization levels are considered, and it can be mitigated by increasing the number of quantization levels. Non-linear effects studies on Gaussian signals started with the early analysis of the spectrum of clipped noise by Van Vleck and Middleton [

Sampling has an impact on the correlation of the two input signals by creating spectral replicas. Additional noise can be added to the mean correlation value due to spectra replication depending on the ratio between the sampling frequency and the signal’s bandwidth. Moreover, the sampling period and its inaccuracies (skew and jitter in the sampling periods) of the Analog to Digital Converter (ADC) can affect and distort the sampled signal and so the cross-correlation value.

To compare different sampling and quantization schemes an equivalent integration time is defined as the one required to obtain the same resolution as in the ideal (analog) correlation. Hagen and Farley [

More recent works have extended the initial digital radiometer proposed by Weinreb in 1961 [

This work provides a general context to analyze the digitization effects on the cross-correlation of two Gaussian random signals in the most general case including any number of quantization levels, irregular quantization steps, gain compression, clipping, bandwidth, sampling frequency, and the skew and jitter inaccuracies of the sampling periods.

In Section 2 a general analysis of the correlation over non-linear functions applied to Gaussian noise is given. In Section 3 quantization effects are analyzed as a particular case of the results of the previous section. Section 4 analyses the sampling process. Section 5 evaluates the correlation variance due to the digitization. Finally, Section 6 presents the main conclusions of this study.

The input signal of a microwave radiometer comes from natural thermal emission. Let us consider here two independent Gaussian random processes with a certain cross-correlation coefficient characterized by their probability density function (pdf). Furthermore, both input signals are assumed to be statistically identical and stationary, so the properties of the signals at a given time (T_{0}) are independent of T_{0}. Furthermore, both signals fulfill that their statistical properties can be deduced from a single, sufficiently long sample of the process (realization), _{x}_{,}_{y}

σ_{x,y}

_{xy}

_{xy}

_{xy}

_{1}(·) and

_{2}(·)) to the random variables (

_{xy}_{1}(·) and _{2}(·) are applied to

_{xy}_{1}(·) and _{2}(·))

^{th}

From _{xy}_{1}(_{2}(_{xy}

Considering the correlation of two signals at a delay different form zero (

_{xy}

_{x}_{y}_{0}_{x}_{y}_{0}_{xy}_{xy}τ

Assuming that the

This section particularizes the previous analysis to a generic ADC function. The degree of non-linearity of an ADC function depends on the number of quantization levels and the ADC span window (V_{ADC} = X_{M}_{o}_{ADC}, which can be considered linear. As explained before, quantization does not take into account the sampling, so that this analysis considers an infinite frequency sampling (_{s}

_{m}_{m}_{+1} are two consecutive threshold values of the input signal in the X̄ space (see _{m}_{m}_{m}_{−1} ≠ X_{m}_{+1} − X_{m}

_{m}_{m}

_{m}_{−1}, X_{m}

the values X_{0} and X_{M}_{−1} define the lower and upper bounds of the quantization window. Again, in a general case, the lower bound can be different from the upper bound (X_{m}_{M}_{−1}).

The first derivative function of _{m}_{m}

Substituting the results obtained in

Thereafter, it is easy to compute the integrals over the _{xy}

From _{xy}

In some particular and simplified cases,

Otherwise, when the quantization scheme is more complex

_{xy}_{xy}_{x,y}_{ADC} = 5_{x,y}_{2}15 = 3.9, note that it is not necessary to have an integer number of bits), but changing the non-linear function. Several

The root mean square error (RMSE) coefficient between R_{xy}_{xy}_{RMSE}. If the relationship between the quantized and the ideal correlations is linear enough (RMSE < Max_{RMSE}), then the value of the quantized correlation matches with the ideal correlation and there is no need of using the ^{−1}[·] function. Otherwise, if it is not linear enough (RMSE ≥ Max_{RMSE}) then, the obtained correlation has to be modified by the ^{−1}[·] function in order to retrieve the real correlation.

_{ADC}. Each curve has a minimum that corresponds to the optimum configuration of V_{ADC} with respect _{x,y}_{ADC}/_{x,y}_{ADC} < 2_{x,y}

On the other hand, plots also have a common trend in the region above the minimum, where the ratio V_{ADC}/_{x,y}_{ADC}/_{x,y}_{ADC}/_{x,y}_{ADC} is so spread over _{x,y}

Another conclusion that can be drawn from _{ADC}) with relation of the standard deviation of the input signal. This is a critical design parameter and it has been summarized on

_{ADC}/_{x,y}_{ADC}/_{x,y}

In both cases, _{ADC}/σ_{x,y}_{hot} − T_{cold} calibration.

In [_{xy}

Furthermore, _{xy}

The effect of the quantization in the cross-correlation spectrum is to decrease and distort it in the pass-band and spread it over the rejected band. _{ADC} = 5_{x,y}

_{ADC} = 5_{x,y}

In this section, it has been shown that the quantization can be successfully expressed and analyzed as a non-linear transformation. Despite the quantization can affect significantly the value of the cross-correlation, it is possible to recover the ideal correlation using the inverse function ^{−1}[·], which can be obtained numerically. Furthermore, for each quantization scheme there is an optimum configuration, in terms of linearity, of the V_{ADC}/_{x,y}

The main effect of sampling is that it creates replicas of the spectrum. Usually, in a standard digital radiometer topology there is an anti-aliasing filter before the digitization process. Even in this case, the digitization can introduce aliasing due to the spread of the quantized spectrum, which depends on the sampling frequency (_{s}_{xy}_{Rxy}

Moreover, the SNR term is related with the radiometric resolution [_{xy}

_{xy}_{xy}_{s}_{s}_{xy}

If the sampling frequency increases, then the first replicas (_{s}

_{ADC} = 5_{x,y}_{s}_{Nyquist}) has the largest deformation: it is wider, sharper than the rest of them, and the side lobes are higher.

For the exactly Nyquist sampling rate the fringe-washing function is still sharp and wide. For a higher sampling rate (_{s} > 2 F_{Nyquist}) the function is not affected by the sampling rate. On the other hand, the spectrum in

_{s}_{Nyquist}) has the largest distortion: it is wider, sharper than the rest of them, and the side lobes are higher. For a larger sampling rate (_{s} ≥ F_{Nyquist}) the function is not affected by the sampling rate. On the other hand, in the

In the conversion from analog to digital signals, the sampling frequency is normally assumed to be constant. Nevertheless, the sampling clock is prone to interferences, thermal drifts and other effects that can change the oscillating frequency and introduce sampling inaccuracies.

_{Skew}), which is the delay offset between the sampling time of _{Jitter}). The impact of the skew is changing the point where the FWF is evaluated and a phase rotation of the correlation coefficient if the working frequency is not exactly zero. Meanwhile, jitter creates a perturbation for each correlation sample around the mean point with zero mean.

A clock skew error not only has an effect on the modulus of the correlation coefficient, but it affects its phase by rotating the real and imaginary parts [

〈_{1})_{2})〉 = _{R}_{1}

_{r}_{s}_{s}

_{xy}_{1} − T_{2}) is the fringe-washing function evaluated at T_{1} − T_{2}, and

T_{1} and T_{2} are the sampling periods for the _{1} − T_{2}) = T_{Skew} + T_{Jitter}(

On the other hand, the jitter is a random variable effect that can be statistically modeled as a polynomial of standard deviation depending on time (

the aperture jitter, which is related to the random sampling time variations in the ADC caused by thermal noise in the sample&hold circuit. The aperture jitter is commonly modeled as an independent Gaussian variable with zero mean and a standard deviation of _{Jitter} = _{app}, and

the clock jitter, which is a parameter of the clock generator that drives the ADC with the clock signal. The clock jitter is modeled as a Wiener process,

Taking into account a wideband signal in jitter terms (fulfill the inequality 2_{Jitter} > 1). From here, _{Jn}

〈e^{j2πfTJ|m−n|}〉 is the expected value of the jitter and it depends on the jitter type [

T_{J}_{|m−n|} is the existing jitter between the ^{th}^{th}

_{s}).

_{clk}

_{xy}

_{xy}

* denotes the convolution operator,

From _{app}. Particularizing

As it can be seen, in the case of aperture jitter the coherence loss ratio (_{s}N

In the previous sections, quantization and sampling have been analyzed in terms of the impact on the retrieved value of the correlation. In this section, the variances are analyzed taking into account both effects (quantization and sampling). In a general case, the output of a correlator of two quantized and sampled signals after averaging _{q}

T_{s}_{1} − T_{2}) = T_{Skew} + T_{Jitter}(

_{1,2} are defined in

Obviously, the product of a pair of samples of
_{|}_{m}_{−}_{n}_{|} follows a Rayleigh pdf when the _{1,2} functions, the distribution of _{|}_{m}_{−}_{n}_{|} follows different trends depending on the quantization scheme. However, the average of many of these products, over a large number of pairs of samples, approaches a Gaussian pdf, as implied by the central limit theorem. The variance of the correlator output can then be calculated as in

Thence,

_{1}(_{2}(

|_{xy}

The second term in

_{xx}_{s}_{yy}_{s}_{1}(_{2}(_{q}

The first part of the variance of the correlation is obtained by combining the results obtained in

Finally, the value of the variance of the _{q}_{q}

Following _{xy}_{ADC} = 5_{x,y}.

_{s}_{Nyquist} curve converges faster, because its samples are uncorrelated among them. For a given number of averaged samples, the variance of the quantized correlation increases with the sampling rate, because successive samples are more and more correlated. Recall that, abscises axis of these plots are displayed in number of averaged samples, not in integration time units. Obviously, for a given integration time, the plot with the highest sample rate has the lowest

In _{s}_{Nyquist} and two quantization schemes has been analyzed (3 and 15 levels). The three-level curve converges more quickly to zero than the 15 levels curve. This is because the fringe washing function is shaper due to the higher non-linear distortion in three levels than in the 15 levels quantization. This effect is equivalent to have more uncorrelated samples. Despite the mean of the correlation is lower for the three-level that the 15-level quantization, the variance of the correlation is also lower in this case.

The impact of a general quantization scheme has been expressed and analyzed as a non-linear transformation. Despite the quantization can significantly affect the value of the ideal cross-correlation (_{xy}_{xy}_{xy}_{xy}^{−1}[_{xy}_{ADC}/_{x,y}

Sampling also has an impact in the spectrum of the cross-correlation function, and depending on the sampling rate, replicas of the spectrum can overlap the main spectrum. This can be due to a sampling rate below the Nyquist criterion or, even above the Nyquist criterion, due to the non-linear quantization process which spreads the spectrum. In both cases, there is an impact on the SNR, and so on the radiometric resolution. Other sampling effects have been analyzed such as the clock inaccuracies: skew and jitter.

Finally, the impact of quantization and sampling on the variance of the measurements has been studied. It has been found that it strongly depends on the relationship between the bandwidth of the cross-correlation and the sampling rate. As the sampling rate increases, the successive samples are more correlated, so they add less new information to the measurements and the variance decreases slowly.

This work has been conducted as part of the award “Passive Advanced Unit (PAU): A Hybrid L-band Radiometer, GNSS Reflectometer and IR-Radiometer for Passive Remote Sensing of the Ocean” made under the European Heads of Research Councils and European Science Foundation European Young Investigator (EURYI) awards scheme in 2004, it was supported in part by the Participating Organizations of EURYI, and the EC Sixth Framework Program. It has also been supported by the Spanish National Research and EU FEDER Project TEC2005-06863-C02-01 (FPI grant BES-2006-11578, 2006-2010), and by the Spanish National Research and EU FEDER Project AYA2008-05906-C02-01/ESP.

Generic analog-to-digital transfer function. The function plotted has compression gain and different steps in analog and digitized domains.

Relationship between the non-linear and the ideal correlation for different digitization schemes.

Root mean square error and ADC span window relationship for different equally spaced quantification levels.

Minimum root mean square errors for different quantization levels, it decreases following an exponential trend,

Root mean square error for different levels and different spaced quantification levels,

Effect of the quantization on the cross-correlation spectrum, _{xy}

Effect of the quantization on the cross-correlation spectrum, _{xy}

Effect of the sampling on the correlation spectrum, (

Impact on the quantized correlation of different sampling frequencies. For 2 quantization levels, a bandwidth 2B (assuming a band-pass signal), and V_{ADC} = 5_{x,y}

Impact on the quantized correlation of different sampling frequencies. For 31 quantization levels, a bandwidth 2B, and V_{ADC} = 5_{x,y}

Impact of the clock inaccuracies on the correlation value. T_{skew} is the sampling rate offset between _{Jitter} is its fluctuations due to clock inaccuracies.

Impact of the number of averaged samples on the

Summary of the critical design parameter VADC/σx,y. Relationship between its optimal configuration and the minimum root mean square error obtained for some quantization schemes.

_{ADC}/σ_{x,y} optimum |
||
---|---|---|

2 (1 bit) | no optimum | 14.2 |

3 (1.6 bits) | 0.7 | 4.7 |

7 (2.8 bits) | 1.8 | 1.4 |

15 (3.90 bits) | 2.2 | 0.3 |

31 (4.95 bits) | 3.0 | 0.02 |

63 (5.97 bits) | 3.5 | 0.0066 |

Summary of the spectrum distortion and spread for V_{ADC} = 5σ_{x,y}

2 (1 bit) | 80% | 13% |

7 (2.8 bits) | 87% | 5.0% |

15 (3.90 bits) | 97% | 1.3% |

31 (4.95 bits) | 99% | 1.0% |

Ideal | 100% | 0.0% |