What Does a Complex Correlation Coefficient Mean? [Lecture Notes

Summary form only given. An aspect of wireless communication that is of paramount interest is reliable connectivity unhampered by signal fading caused by scatterers such as walls, buildings and other obstacles. For power limited ad hoc networks in complex environments, various phenomenon including multipath, diffraction from sharp corners contribute to the fading and distortion of electromagnetic waves that severely limit the coverage and reliability. Because of the difference in path length among the various signal components and in the absence of a direct signal, the received electric field will have uneven spatial distribution and significant fluctuations. This phenomenon is called fast fading and results in intermittent signal drop-offs causing the communication channel to be unreliable. A viable approach to mitigate fast fading is the use of antenna diversity systems. Antenna Diversity Systems, when used in multipath environments such as indoor and urban environments, enable improvement in signal-to-noise-ratio (SNR) without increasing the transmit power. A thorough and accurate analysis of a diversity system requires three main components: the Tx and Rx antenna radiation patterns (phase and amplitude), the multipath coherent propagation model and calculation of figures of merit for the performance of the diversity system including complex and envelop correlation coefficients and diversity gain. Existing simulation based diversity analysis techniques essentially model the multipath channel using a stochastic model such as Rayleigh and Rician distributions. While these probabilistic models provide generalized approximations to the indoor channel, such models do not accurately capture all the propagation mechanisms such as angle of arrival and polarization, and hence diversity analysis techniques based on such models inherently lack the information needed to assess the true performance of a given diversity system. In this work, we discuss a new diversity system analysis approach that takes into account the complex radiation pattern of the Tx and Rx diversity antennas and make use of an accurate deterministic, coherent, and polarization preserving propagation model for a complex indoor scenario. The new physics-based pattern diversity analysis approach will be introduced. The propagation modeling with specific focus on near-ground antennas will also be discussed. Measurement results in indoor scenarios utilizing a compact, co-located radiation pattern diversity antenna system will be utilized to assess the performance of the diversity system via complex correlation coefficients and diversity gain.

OFDM is a promising digital communications technique for high data rate transmissions. In this paper, we have analyzed the performance of mobile OFDM systems in the presence of channel estimation error. A new channel estimation error model is presented. Based on the estimation error model, new and simple expressions for the average bit error probability of M-QAM OFDM systems are derived for zero-forcing, minimum mean square error and maximum likelihood receivers. A complex correlation coefficient is used to quantify the quality of the channel estimation scheme. We have shown that if the complex correlation coefficient is a function of the signal to noise ratio, then the inter-carrier interference caused by channel variations is the dominant source of performance degradation. On the other hand, if the channel estimation scheme produces a constant value for the complex correlation coefficient (estimation quality is not a function of the signal to noise ratio), then the channel estimation error will cause an error floor that is larger than the one caused by the inter-carrier interference, and is considered as one of the dominant causes of degradation. And channel estimation using pilot symbols are discussed in terms of different pilot symbol placement schemes.

Complex Evidence Theory (CET) is a generalization of D-S Evidence Theory that can represent uncertain information and express uncertainty effectively. One of the hot topics in CET is how to measure the conflict between different pieces of evidence. Recently, a complex correlation coefficient has been proposed to measure the degree of similarity between two sets of evidence, where the similarity reflects the difference between the evidence, i.e. the degree of conflict between the sets of evidence. However, there is still a need to improve the precision of conflict metrics. In this paper, a new complex correlation coefficient called CCOR is proposed to measure the similarity between two sets of evidence. Through some numerical examples, CCOR can be used to measure the conflict between the two sets of evidence well. Finally, the validity and applicability of the method proposed in this paper are further illustrated by comparative experiments on real datasets.

ABSTRACTIn this letter, a methodology is proposed to improve the scattering powers obtained from model-based decomposition using Polarimetric Synthetic Aperture Radar (PolSAR) data. The novelty of this approach lies in utilizing the intrinsic information in the off-diagonal elements of the 3 × 3 coherency matrix T represented in the form of complex correlation coefficients. Two complex correlation coefficients are computed between co-polarization and cross-polarization components of the Pauli scattering vector. The difference between modulus of complex correlation coefficients corresponding to (i.e., the degree of polarization (DOP) optimized coherency matrix), and T (original) matrices is obtained. Then a suitable scaling is performed using fractions i.e., obtained from the diagonal elements of the matrix. Thereafter, these new quantities are used in modifying the Yamaguchi 4-component scattering powers obtained from . To corroborate the fact that these quantities have physical relevance, a quantitative analysis of these for the L-band AIRSAR San Francisco and the L-band Kyoto images is illustrated. Finally, the scattering powers obtained from the proposed methodology are compared with the corresponding powers obtained from the Yamaguchi et. al., 4-component decomposition and the Yamaguchi et. al., 4-component Rotated decomposition for the same data sets. The proportion of negative power pixels is also computed. The results show an improvement on all these attributes by using the proposed methodology.

Watering on porous pavement for improvement of environmental human thermal comfort in an ecological community in arid area: A case study in Lanzhou, China

The article investigates the possibility of measuring the strength of a linear correlation relationship between nominal data and numerical data. Correlation coefficients for variables coded with real numbers as well as for variables coded with complex numbers were studied. For variables coded with real numbers, unambiguous measures of real linear correlation were obtained. In the case of complex coding, it has been observed that the obtained complex correlation coefficients change with the permutation of the phases in the complex numbers used to code classes of elements with equal cardinalities. It was found that a necessary condition for linear correlation is the possibility of linear ordering of a set with data. Since linear order is not possible in the set of complex numbers, complex correlation coefficients cannot be used as a measure of linear correlation. In the event of such a situation, a substitute action was suggested that would prevent equal cardinality of classes of identical elements contained in the set with nominal data. This action would consist in the correction of data, analogous to the correction during preprocessing or cleaning of data containing missing or outlier values.

In wireless communication systems, multipath interference has a significant impact on system design and performance. Fast fading variations are caused by the coherent summation of multiple echoes from many reflection points reaching the receive antenna. Antenna diversity is one technique that can be used to overcome multipath fading. A test system used to measure the diversity performance of an antenna pair was used to experimentally determine the complex correlation coefficient between the two antenna branches. A local mean estimation algorithm based on the channel mean square error equalization was implemented. Thus, the two parameters that determine the expected diversity gain, i.e., the complex correlation coefficient and the mean level signal difference, were estimated. The test system was used to evaluate the polarization diversity performance of different antenna pairs in Rayleigh and Rician environments, both in the absence and in the presence of a human head phantom.

Polarimetric and interferometric SAR data are frequently multilook processed for speckle reduction and data compression. The statistical characteristics of multilook data are quite different from those of single-look data. The authors investigate the statistics of their intensity and phase. Probability density function (PDF's) of the multilook phase difference, magnitude of complex product, and intensity and amplitude ratios between two components of the scattering matrix are derived, and expressed in closed forms. The PDF's depend on the complex correlation coefficient and the number of looks. Comparisons of these theoretically derived PDF's are made to measurements from NASA/JPL AIRSAR data. The results of this paper can be applied to feature classification using polarimetric SAR and to the estimation of decorrelation effects of the interferometric SAR. >

In this paper first results of a sensitivity study using dual polarimetric TerraSAR-X data for ice thickness estimation are presented. The sea ice thickness reference data set was measured, coincident to the SAR data take, by means of a helicopter-borne EM sounding device on April 28, 2008 in the Russian Arctic. For some of the signatures, namely the complex correlation coefficient, a relation to ice thickness could be found that is theoretically predicted for L-band SAR. The first results show, that the new generation of polarimetric space borne SAR sensors like TerraSAR-X may open a new opportunity for thin sea ice thickness monitoring from space.

The probability density function (pdf) of the multi-look interferometric phase between two complex synthetic aperture radar (SAR) images is parameterized by the number of looks and the complex correlation coefficient. In practice, adjacent pixels in a real SAR interferogram, are statistically dependent due to filtering, and hence, the number of looks is usually smaller than the number of samples averaged. It has been shown that compensation with an effective number of looks, rather than an intractable rederivation of the pdf, can account for the statistical dependence. This paper addresses the challenge of how to determine a suitable value for the effective number of looks. It is shown that an optimum value can be found via a maximum-likelihood estimator (MLE) based on the interferometric phase pdf. However, since such an MLE is computationally intensive and numerically unstable, an estimator based on the method of moments (MoM) possessing similar fidelity is proposed. MoM is fast and robust and can be used in operational applications, such as determining constant false alarm rate (CFAR) detection thresholds for moving-target detection in SAR along-track interferometry.

The cross power spectrum function is used to realize the operational modal analysis and identification of the dry gas seal device system through the multi-reference point least squares complex frequency domain method. The steady state diagram and mathematical indicators MAC, MPD, MPC, MOV and MIF are used to verify the modal results. At the same time, based on the response surface method, with two different operating conditions of medium pressure and rotating speed, modal direction and modal order as the response surface variables, a time-varying modal recognition model is established. Through the Full Factorial experiment design, Box-Behnken experiment design and Central Composite experiment design, the suitable variable sample points are formed. A complete quadratic polynomial response surface model of the system operational modal parameters is established. The complex correlation coefficient, the modified complex correlation coefficient and the root mean square error are used to verify the effectiveness of the response surface model. It provides new method and technical support for realizing time-varying modal identification in this paper.

Summary Megalopae of the common shore crab, Carcinus maenas (L.), were sampled in the Ria de Aveiro, during spring and summer of 1991. At weekly intervals, plankton samples were collected below the surface, at mid-water and above the bottom during night and day flood tides, with the use of plankton nets. In order to evaluate the relationship between wind speed and direction and input of megalopae to the estuary, a complex correlation coefficient between megalopae concentration and average daily wind stress was calculated. Maximum values of the correlation coefficient were obtained when the concentration of megalopae was lagged after wind stress by 3 d. The direction of the correlation was 2°, meaning that input of megalopae to the estuary increased as southward wind stress decreased. High densities of megalopae were not associated with strong southward wind stress. Correlation between input of megalopae to the Ria de Aveiro and wind stress improved when a cumulative wind stress vector was used, which was constructed as the sum of the wind stresses for time lags of 2, 3 and 4 days before the input of megalopae. The results are consistent with onshore transport of megalopae over the shelf following relaxation of southward, upwelling favourable, wind events.

The balances of momentum and heat were studied for an 18‐day period in October–November 1983 during the Mixed‐Layer Dynamics Experiment at a site 650 km off central California. Data were collected from a drifting platform from which surface winds, air temperature, and ocean currents (from a string of vector‐measuring current meters) were measured. The position of the unattended drifter was determined by LORAN‐C on the platform. Currents in the surface layers were partially wind driven and partially coherent with low‐frequency currents at deeper levels. The latter currents, though coherent, were sheared, turned significantly with depth, and were presumably nearly geostrophically balanced. Vertically differencing velocity in the mixed layer improved the balance between wind and current by removing a portion of the geostrophic flow. The complex correlation coefficient between wind stress and current integrated to 38 m was 0.76 for current relative to 38 m yet only 0.36 for the undifferenced case. Surface heat flux magnitudes were not clearly reflected in the heat content of the near‐surface layers although variability was. Over the first 13 days of the experiment, the correlation coefficient between daily‐averaged heat flux and heat content change was 0.89, but average net surface heating was 60 W m−2, while the average heat content change in the top 38 m was only 10 W m−2. Heat balance must have been restored by advection. Vertical advection was rejected as an explanation because heat content changes in layers defined by isotherm depths showed similar imbalances with regard to surface heating. Horizontal advection was estimated by two methods: (1) by using sparse thermistor chain tows around the current meter drifter to construct a history of horizontal temperature gradients and (2) by using current profiles at moderate depths to estimate pressure gradients (essentially through geostrophy), hence buoyancy gradients (by the hydrostatic relation), hence temperature gradients. The two methods gave very different estimates, though both tended to redress the imbalance between surface heat flux and local heat storage.

The authors studied the applicability of coherence information which is obtained from interferometric SAR data pair for the discrimination of forest types. Two test sites, around Mt. Fuji in Japan and Phuket Island in Thailand, were tested with two interferometric JERS-1/SAR data pairs for each site. The coherence was computed by complex correlation coefficient in a small corresponding patch between single-look SAR complex data pair. The temporal changes of SAR intensity and coherence were investigated among various forest types and the results indicated that the coherence information is much superior than the intensity to discriminate different forest types. Especially in Phuket test site, Mangrove forest was easily discriminated from other tropical forest due to its relatively higher coherence, while that forest type is hardly discriminated by using SAR intensity.

In this study, observations from a high-frequency radar (HFR) deployment are combined with numerical model simulations to investigate the relative contributions of surface current forcing. A tidal decomposition of the relevant datasets was applied by fitting the flow profile to a finite set of sinusoids at specific frequencies related to astronomical parameters. The resultant time-series pair comprised tidal harmonic constituents and residuals composed of primarily wind-driven surface flows. Complex correlation coefficients between these data and both numerical simulations and winds were used to further investigate both bay dynamics and numerical model performance. Results of analysis demonstrate good agreement between HFR and model data, particularly for tidal harmonics. Analysis of wind driven surface currents illustrate the complex nature of flow dynamics and the many factors requiring consideration.