Theoretical and experimental investigation of chaperone effects on soluble recombinant proteins in Escherichia coli: effect of free DnaK level on temperature-induced recombinant streptokinase production Abstract Modeling and analysis of genetic networks have become increasingly important in the investigation of cellular processes. The genetic networks involved in cellular stress response can have a critical effect on the productivity of recombinant proteins. In this work, it was found that the temperature-inducible expression system for the production of soluble recombinant streptokinase in Escherichia coli resulted in a lower productivity compared to the chemically-induced system. To investigate the effect of the induced cellular response due to temperature up-shift a model-based approach is adopted. The role played by the major molecular chaperone teams DnaK-DnaJ-GrpE and GroEL-GroES on the productivity of recombinant streptokinase was experimentally determined. Based on these investigations, a detailed mechanistic mathematical model was developed for the cellular response during the temperature-induced recombinant streptokinase production. The model simulations were found to have a good qualitative agreement with the experimental results. The mechanistic mathematical model was validated with the experiments conducted on a r 32 mutant strain. Detailed analysis of the parameter sensitivities of the model indicated that the level of free DnaK chaperone in the cell has the major effect on the productivity of recombinant streptokinase during temperature induction. Analysis of the model simulations also shows that down regulation or selective redirection of the heat shock proteins could be a better way of manipulating the cellular stress response than overexpression or deletion. In other words, manipulating the system properties resulting from the interaction of the components is better than manipulating the individual components. Although our results are specific to a recombinant protein (streptokinase) and the expression system (E. coli), we believe that such a systems-biological approach has several advantages over conventional experimental approaches and could be in principle extended to bigger genetic networks as well as other recombinant proteins and expression systems. Escherichia coli has been used as an expression host for the production of a wide range of recombinant proteins (Baneyx 1999; Baneyx and Mujacic 2004) . Several expression systems have been developed for the high-level expression of recombinant proteins in Escherichia coli (Makrides 1996) . Among the various expression systems used, the temperature-inducible expression system has been beneficial due to the ease of induction and higher induction strength (Seeger et al. 1995; Gupta et al. 1999 ).. However, the use of temperature up-shift as an induction mechanism influences several cellular processes, including the up-regulation of the heat shock proteins such as molecular chaperones and proteases Weber et al. 2002) .. The role of induction mechanism on recombinant protein production has been investigated for certain recombinant proteins in the literature. For an aggregation prone recombinant human basic fibroblast growth factor, the temperature-induced expression was found to result in increased productivity and higher yield (Seeger et al. 1995) . In the case of soluble Chloramphenicol Acetyl Transferase (CAT) the productivity of the temperatureinducible system was lower compared to chemicallyinduced system (Harcum and Haddadin 2006) . Further, the global transcriptome response of recombinant E. coli producing soluble recombinant CAT was analyzed during chemical induction, temperature induction and dual induction conditions (chemical and temperature induction). The transcriptome response of the classical heat shock genes was found to be similar in the wild-type and dual stressed cultures, even though many other genes were found to be differentially expressed in these cultures. This indicates that modeling the dynamics of the heat shock proteins and their mechanism of interaction with the recombinant protein could give useful insights.. The role played by the various heat shock proteins in assisting solubilization, folding and degradation of recombinant proteins in E. coli has been extensively reviewed by Hoffmann and Rinas (2004) . Co-expression of molecular chaperones in general is employed to improve the solubility and activity of recombinant proteins, which are either difficult to express or insoluble (Georgiou and Valax 1996; Venkatesh et al. 2004 ). However, the role played by these chaperones in enhancing the solubility and activity depends on the nature and properties of the recombinant protein. For example, the co-expression of DnaK chaperone team was found to result in reduced activity for a soluble recombinant protein glutamate racemase expressed in E. coli (Kohda et al. 2002) . But, the same chaperone team was found to enhance the solubility of several aggregation prone recombinant proteins in E. coli (Georgiou and Valax 1996) .. In the present work, the chemical and temperature inducible expression systems for the production of recombinant streptokinase were compared. A detailed mechanistic model for the bacterial heat shock response was developed and the dynamics of the chaperones and proteases were simulated for the mild heat shock condition, which had been used earlier for recombinant streptokinase production (Yazdani and Mukherjee 2002; Ramalingam et al. 2007 ). A mechanistic model for the interaction of the heat-shock proteins with the predominantly soluble recombinant streptokinase was proposed from the experimental investigations. This was coupled with the heat-shock response model to obtain the complete model for temperature-induced recombinant streptokinase production. Model equations were simulated to examine the effect of the heat-shock response on the recombinant protein production. A hypothetical condition predicted by the model simulations was validated by experiments using an E. coli-mutant for the heat-shockspecific sigma factor (r 32 ). Temperature induction has been used for the production of several recombinant proteins in E. coli. The dynamics of the cellular response during temperature-induced recombinant protein over expression has been referred as a dual stress on the host cell (Harcum and Haddadin 2006). However, the dual stress situation arises mainly due to the nature and properties of the recombinant protein. The aggregation prone recombinant protein like basic human fibroblast growth factor, when induced using a temperature up-shift mechanism, was found to result in a continuous heat shock response (Hoffmann and Rinas 2000) . The DnaK level was found to be higher during temperatureinduced recombinant protein over-expression compared to the condition where temperature up-shift was done for the non-recombinant strain. This is because of the competition of both the misfolded cellular proteins and the aggregating recombinant protein for DnaK chaperone team. Thus, the DnaK chaperone is not free to down-regulate the heatshock response by binding to and assisting the degradation of r 32 . Further, certain aggregation prone recombinant proteins when expressed in E. coli were observed to induce a heat shock-like response without any temperature upshift (Kanemori et al. 1994 ).. In the case of soluble recombinant proteins like recombinant streptokinase the major stress on the cell is due to temperature up shift and not due to the misfolding of the recombinant protein. The up-regulation of the heat shock proteins or the induction of the heat-shock like response was not observed when recombinant streptokinase alone was over-expressed by chemical induction. However, when temperature up-shift was used as an induction mechanism, the up-regulated heat shock chaperones were found to influence the productivity of the soluble recombinant streptokinase. There are several reports in literature citing the effect of chaperones on recombinant protein expression, solubilization and folding in E. coli based expression systems. Although there have been many experimental investigations on the effects of chaperone co-expression on aggregating recombinant proteins (Caspers et al. 1994; Mogk et al. 1999; Nishihara et al. 2000) , there are very few such studies on soluble recombinant proteins (Kohda et al. 2002) . Our experimental investigations, involving coexpression of major chaperone teams, found that the role played by the molecular chaperones was different for a soluble recombinant protein as compared to the literature reports on aggregation-prone recombinant proteins (which often form inclusion bodies). Kohda et al. (2002) observed that coexpression of DnaK chaperone team lead to a significantly reduced level of the active recombinant protein (glutamate racemase) and attributed this to the accumulation of partially folded intermediates due to the absence of large quantities of GroEL chaperone team. However, in the present case, the DnaK chaperone team seems to impair productivity of streptokinase even in the presence of GroEL chaperone team and this indicates that the unavailability of GroEL chaperone team suggested by Kohda et al. could not be the reason for the reduced productivity. Enhancement in the biological activity of recombinant proteins during the co-expression of folding accessories has been extensively reported (Georgiou and Valax 1996) . The enhancement in the activity of the recombinant protein is generally brought about by increasing the solubility of the recombinant protein (Caspers et al. 1994; Baneyx and Palumbo 2003) . However, the enhancement in the activity of highly soluble recombinant protein often results from better folding of the recombinant protein. Further, such a state of the protein cannot be generally quantified unless the recombinant protein possesses an enzymatic activity, as in the case of recombinant streptokinase or glutamate racemase (Kohda et al. 2002) .. Further, based on isolated experimental investigations of individual chaperones, it is very difficult to predict the combined effect of all the major chaperones on a particular recombinant protein, given that the outcome depends on the recombinant protein characteristics as well as the dynamics of the stress response network. In this context, it would be more appropriate to formulate mathematical models which can simulate the network dynamics well and give insights on key parameters affecting the productivity as well as the quality of the recombinant protein. However, given the large number of heat-shock and other host cell proteins which interact with the recombinant protein, modeling and simulation can become a complex and even intractable task. Therefore, it may be essential to isolate the important components of a network and its interactions with a recombinant protein in order to simplify the model and its corresponding simulations.. This work makes a modest beginning in this direction by formulating a deterministic model-based approach for investigating the effect of heat shock response on a soluble recombinant protein, induced by a temperature up-shift. A mechanistic network model is proposed for the interaction of the major heat shock proteins and the soluble recombinant streptokinase. The framework presented is generally applicable for the chaperone-mediated folding, aggregation, dis-aggregation and proteolysis of recombinant proteins. This model was coupled to a modified model of heat-shock response, in order to simulate the chaperone effects on temperature-induced recombinant protein production. The model simulations were found to have strong qualitative agreement with the experimental results. The sensitivity analysis for the model parameters, together with the model simulations for the r 32 mutant strain have indicated that down-regulation of r 32 (and hence DnaK) levels in the cell could result in enhanced productivity of recombinant streptokinase during temperature-induced expression. Further, the sensitivity analysis has also shown that selective down-regulation of the DnaK chaperone level or activity could lead to enhancement in the productivity.. Thus, our theoretical investigations have shown that it is the amount of free DnaK chaperone (and not the total DnaK level per se) that has the most significant effect on the productivity of the soluble recombinant streptokinase. Such insights may be difficult to capture by purely experimental investigations of chaperone co-expression, thus making it difficult to design rational strategies for enhancing cellular productivity. Therefore, a model-based approach may be more justified in certain circumstances, where protein characteristics and its interaction with host cell proteins are quantitatively understood. Careful examination of these model parameters could allow design of expression vectors or experimental strategies which are capable of operating at multiple values for these model parameters and would be very useful in enhancing the productivity of recombinant proteins.

O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Personalisation for user agents This paper is concerned with personalisation of user agents by symbolic, on-line machine learning techniques. The application of these ideas to an infotainment agent is discussed in detail. Also experimental results, which indicate that a high level of personalisation can be achieved by this approach, are presented. The goal of the research described in this paper is to apply machine learning techniques to building user agents that facilitate interaction between a user and the Internet. This paper concentrates on the topic of personalisation in which the agent adapts its behaviour according to the interests and preferences of the user. There are many practical applications of personalisation that could exploit the technology presented here.. The research is set in the context of an infotainment agent, which is a multi-agent system that contains a number of agents with functionalities for recommending movies, TV programs, music and the like, as well as information agents with functionalities for searching for information on the Internet. This paper concentrates on the TV recommender as a typical such agent and shows how a high degree of personalisation can be achieved by symbolic, on-line machine learning techniques. The techniques can be ported comparatively easily to other agents of the system, thus providing a fully personalised infotainment system.. There is an extensive literature on recommender systems; typical recent work is in [1] . Differently to that work and other work on recommender systems that we are aware of is the emphasis here on symbolic machine learning techniques to discover the user's interests and preferences. Instead of more typical machine learning techniques such as naive Bayes classifiers that are used in recommender systems, here we employ a decision-list learner that produces comprehensible hypotheses. This has two advantages: the agent can explain to the user why it has made a recommendation and users can use the hypothesis language to give explicit rules about what they want to be recommended. This approach comes from the field of inductive logic programming ( [5] , [8] ) in which (first-order) logic is used as a knowledge representation language and in which there is a strong emphasis on learning comprehensible hypotheses. Apparently the only area of learning in agents where symbolic techniques have been employed so far is in relational reinforcement learning [3] , [4] ; thus this paper serves to introduce symbolic learning techniques to another problem of interest, that of personalisation of user agents.. Section 2 gives an outline of the infotainment agent, concentrating on its TV recommender. Section 3 briefly describes the approach to adaptation taken here. Section 4 describes how the TV recommender is personalised to the user. Section 5 presents results of experiments that show the level of personalisation achieved. Section 6 contains some conclusions and future directions for research. In all the experiments the performance of the system was evaluated using 10-fold cross-validations. One could also use an independent test set to collect performance statistics. This was in fact done for each user and the results closely correspond. As an example we have included the test-set results for user 2 in Chart (c).. The first set of experiments suggest that it is possible for the TV recommender to personalise well to different users, given sufficient training data. This is due in part to the nature of the problem. Television programming tends to be rather repetitive. Programs with the same title are scheduled cyclically at daily, weekly and sometimes hourly periods. In these circumstances instance-based learning is expected to work well. This form of learning is captured implicitly by the use of predicates that test for equality of program titles in the decision lists. This works well in combination with more conventional rule-based learning that exploits more general conditions in the hypothesis language.. The second set of experiments suggest that real-world learning is more problematic. Collecting sufficient training data is difficult. Users in general prefer to give minimal feedback, mostly correcting mistakes as they occur. They also expect the TV recommender to make useful predictions on times of the day not previously trained on.. The results for user 3 show that even after a moderately large period of training the TV recommender can fail to perform as well as in the first set of experiments. This user chose two specialist movie channels from the TV guide and noted that the system performed poorly on movies compared with the more day-to-day TV programming of other channels. One reason for this is that there is in general an insufficient overlap between a person's preference for movies and TV programs for it to be possible to learn a theory that models both simultaneously. This suggests the need for a sep-arate movie recommender with a more appropriate representation of movie individuals. We also note that the meta-data for movies in the TV guide was not very rich. For example, the genre for many movies was often simply designated as 'movie'. Users 4 and 5 reported that subjectively the system performed well enough, and that further training to improve performance seemed unnecessary.

B O B I O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Tractability and learnability arising from algebras with few subpowers A k-edge operation ϕ on a finite set A is a k + 1-ary operation that satisfies the identities ϕ(x, x, y, . . . , y) ≈ ϕ(x, y, x, y, . . . , y) ≈ y, ϕ(y, y, y, x, y, . . . , y) ≈ ϕ(y, y, y, y, x, y, . . . , y) ≈ · · · · · · ≈ ϕ(y, y, y, . . . , y, x) ≈ y.We prove that any constraint language Γ that, for some k > 1, has a k-edge operation as a polymorphism is globally tractable. We also show that the set of relations definable over Γ using quantified generalized formulas is polynomially exactly learnable using improper equivalence queries. Special instances of k-edge operations are Mal'cev and near-unanimity operations and so this class of constraint languages includes many well known examples. A wide variety of combinatorial problems can be expressed within the framework of the Constraint Satisfaction Problem (CSP). Given an instance of the CSP the aim is to determine if there is an assignment of values to the variables of the instance that satisfies all of its constraints. While the class of all CSPs forms an NP-complete class of problems there are many naturally defined subclasses that are tractable (i.e., lie in the class P). A main research goal is to identify the subclasses of the CSP that are tractable.. One common way to define a subclass of the CSP is to restrict the relations that appear in the constraints of an instance to a specified set of relations over some fixed domain, called a constraint language. The Dichotomy Conjecture of Feder and Vardi [14] states that for any constraint language Γ, the corresponding subclass of the CSP, denoted CSP(Γ), is either NP-complete or tractable. The Dichotomy Conjecture has been verified in a number of special cases, most notably over domains of size 2 [20] and size 3 [4] . Over larger domains, a number of general results have been obtained [7, 3, 12] .. In this paper we introduce a condition that when satisfied by a constraint language Γ, guarantees that the subclass CSP(Γ) is tractable. The polynomial-time algorithm that we use to establish tractability is essentially that presented by Dalmau in [12] and our result generalizes his and also generalizes earlier results of Feder and Vardi, Bulatov, and Jeavons, Cohen and Cooper [14, 6, 17] . The class of constraint languages that satisfy our condition is quite large and includes some well known constraint languages such as those that have Mal'cev or near unanimity polymorphisms.. A polymorphism of a set of relations Γ over some set A is a finitary operation on A that preserves all of the relations in Γ. Work of Jeavons and his co-authors has shown that the tractability of a constraint language is determined by the set of its polymorphisms and many tractability results can be expressed in terms of the existence of certain kinds of polymorphisms [8] . A benefit of focusing on polymorphisms is that it allows the introduction of ideas and techniques from universal algebra into the study of tractability and in this paper we adopt this algebraic approach.. Our main result is that if a constraint language Γ has a kedge operation (see Definition 3.2) as a polymorphism then Dalmau's algorithm can be used to solve in polynomial time all instances in CSP(Γ). This algorithm can be regarded as a natural generalization of the familiar Gaussian elimination algorithm for solving systems of linear equations since it makes essential use of the fact that, in this case, all solution sets of instances of CSP(Γ) have small generating sets (akin to bases of vector spaces) when considered as universes of particular algebras.. On the other hand, we argue that if a constraint language Γ fails to have a k-edge operation as a polymorphism then no "Gaussian-like" algorithm can solve all instances of CSP(Γ) in polynomial-time. So, in some sense, our result sharply determines the scope of algorithms like Dalmau's to quickly settle instances of the CSP.. In Section 5 we establish that if Γ is a set of relations over a finite set that is invariant under a k-edge operation then the set of relations defined by quantified generalized formulas over Γ is exactly learnable, in polynomial time, via an algorithm that makes improper equivalence queries. This extends the result of Bulatov, Chen, and Dalmau in [5] and earlier results of Dalmau and Jeavons [13] . The study of algebras with few subpowers has led to a number of surprising algebraic results and has helped to further work on the Dichotomy Conjecture for constraint languages. As noted earlier, our results delimit the scope of Dalmau-like algorithms for solving instances of the CSP and so new algorithms will need to be devised to deal with finite algebras that support non-trivial term operations more general than k-edge terms.. Given a finite algebra (or constraint language) there is a fairly straightforward, but inefficient algorithm to determine if it has a Mal'cev term operation (or Mal'cev polymorphism). In [15] a polynomial time algorithm to test for a Mal'cev term is presented that works for idempotent algebras. It is apparently much more difficult to test for the presence of a near unanimity term, but at least we know from a recent result of Maróti [18] that this question is decidable.

O O B O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Preliminary Evaluation of Head and Facial Feature Tracking for Input on Mobile Devices This paper discusses the concept of using head and facial feature tracking as an input mechanism for mobile devices. We present our concept and ideas, along with preliminary findings from two prototype implementations. Suggestions for further work and implications are presented in the final sections of the paper.

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Statistical efficiency of curve fitting algorithms We study the problem of fitting parametrized curves to noisy data. Under certain assumptions (known as Cartesian and radial functional models), we derive asymptotic expressions for the bias and the covariance matrix of the parameter estimates. We also extend Kanatani's version of the Cramer-Rao lower bound, which he proved for unbiased estimates only, to more general estimates that include many popular algorithms (most notably, the orthogonal least squares and algebraic fits). We then show that the gradient-weighted algebraic fit is statistically efficient and describe all other statistically efficient algebraic fits. In many applications one fits a parametrized curve described by an implicit equation P (x, y; Θ) = 0 to experimental data (x i , y i ), i = 1, . . . , n. Here Θ denotes the vector of unknown parameters to be estimated. Typically, P is a polynomial in x and y, and its coefficients are unknown parameters (or functions of unknown parameters). For example, a number of recent publications [2, 10, 11, 16, 19] are devoted to the problem of fitting quadrics Ax 2 +Bxy +Cy 2 +Dx+Ey +F = 0, in which case Θ = (A, B, C, D, E, F ) is the parameter vector. The problem of fitting circles, given by equation (x−a) 2 +(y−b) 2 −R 2 = 0 with three parameters a, b, R, also attracted attention [8, 14, 15, 18] .. We consider here the problem of fitting general curves given by implicit equations P (x, y; Θ) = 0 with Θ = (θ 1 , . . . , θ k ) being the parameter vector. Our goal is to investigate statistical properties of various fitting algorithms. We are interested in their biasedness, covariance matrices, and the Cramer-Rao lower bound.. First, we specify our model. We denote byΘ the true value of Θ. Let (x i ,ȳ i ), i = 1, . . . , n, be some points lying on the true curve P (x, y;Θ) = 0. Experimentally observed data points (x i , y i ), i = 1, . . . , n, are perceived as random perturbations of the true points (x i ,ȳ i ). We use notation x i = (x i , y i ). T andx i = (x i ,ȳ i ) T , for brevity. The random vectors e i = x i −x i are assumed to be independent and have zero mean. Two specific assumptions on their probability distribution can be made, see [4] :. Cartesian model: Each e i is a two-dimensional normal vector with covariance matrix σ 2 i I, where I is the identity matrix.. Radial model: e i = ξ i n i where ξ i is a normal random variable N (0, σ 2 i ), and n i is a unit normal vector to the curve P (x, y;Θ) = 0 at the point x i .. Our analysis covers both models, Cartesian and radial. For simplicity, we assume that σ 2 i = σ 2 for all i, but note that our results can be easily generalized to arbitrary σ 2 i > 0. Concerning the true pointsx i , i = 1, . . . , n, two assumptions are possible. Many researchers [6, 13, 14] consider them as fixed, but unknown, points on the true curve. In this case their coordinates (x i ,ȳ i ) can be treated as additional parameters of the model (nuisance parameters). Chan [6] and others [3, 4] call this assumption a functional model. Alternatively, one can assume that the true pointsx i are sampled from the curve P (x, y;Θ) = 0 according to some probability distribution on it. This assumption is referred to as a structural model [3, 4] . We only consider the functional model here.. It is easy to verify that maximum likelihood estimation of the parameter Θ for the functional model is given by the orthogonal least squares fit (OLSF), which is based on minimization of the function. where d i (Θ) denotes the distance from the point x i to the curve P (x, y; Θ) = 0. The OLSF is the method of choice in practice, especially when one fits simple curves such as lines and circles. However, for more general curves the OLSF becomes intractable, because the precise distance d i is hard to compute. For example, when P is a generic quadric (ellipse or hyperbola), the computation of d i is equivalent to solving a polynomial equation of degree four, and its direct solution is known to be numerically unstable, see [2, 11] for more detail. Then one resorts to various approximations. It is often convenient to minimize. instead of (1.1). This method is referred to as a (simple) algebraic fit (AF), in this case one calls |P (x i , y i ; Θ)| the algebraic distance [2, 10, 11 ] from the point (x i , y i ) to the curve. The AF is computationally cheaper than the OLSF, but its accuracy is often unacceptable, see below.. The simple AF (1.2) can be generalized to a weighted algebraic fit, which is based on minimization of. where w i = w(x i , y i ; Θ) are some weights, which may balance (1.2) and improve its performance. One way to define weights w i results from a linear approximation to d i :. where ∇ x P = (∂P/∂x, ∂P/∂y) is the gradient vector, see [20] . Then one minimizes the function. This method is called the gradient weighted algebraic fit (GRAF). It is a particular case of (1.3) with. The GRAF is known since at least 1974 [21] and recently became standard for polynomial curve fitting [20, 16, 10] . The computational cost of GRAF depends on the function P (x, y; Θ), but, generally, the GRAF is much faster than the OLSF. It is also known from practice that the accuracy of GRAF is almost as good as that of the OLSF, and our analysis below confirms this fact. The GRAF is often claimed to be a statistically optimal weighted algebraic fit, and we will prove this fact as well.. Not much has been published on statistical properties of the OLSF and algebraic fits, apart from the simplest case of fitting lines and hyperplanes [12] . Chan [6] , Berman and Culpin [4] investigated circle fitting by the OLSF and the simple algebraic fit (1.2) assuming the structural model. Kanatani [13, 14] used the Cartesian functional model and considered a general curve fitting problem. He established an analogue of the RaoCramer lower bound for unbiased estimates of Θ, which we call here Kanatani-CramerRao (KCR) lower bound. He also showed that the covariance matrices of the OLSF and the GRAF attain, to the leading order in σ, his lower bound. We note, however, that in most cases the OLSF and algebraic fits are biased [4, 5] , hence the KCR lower bound, as it is derived in [13, 14] , does not immediately apply to these methods.. In this paper we extend the KCR lower bound to biased estimates, which include the OLSF and all weighted algebraic fits. We prove the KCR bound for estimates satisfying the following mild assumption:. Precision assumption. For precise observations (when x i =x i for all 1 ≤ i ≤ n), the estimateΘ is precise, i.e.Θ (x 1 , . . . ,x n ) =Θ (1.5). It is easy to check that the OLSF and algebraic fits (1.3) satisfy this assumption. We will also show that all unbiased estimates ofΘ satisfy (1.5).. We then prove that the GRAF is, indeed, a statistically efficient fit, in the sense that its covariance matrix attains, to the leading order in σ, the KCR lower bound. On the other hand, rather surprisingly, we find that GRAF is not the only statistically efficient algebraic fit, and we describe all statistically efficient algebraic fits. Finally, we show that Kanatani's theory and our extension to it remain valid for the radial functional model. Our conclusions are illustrated by numerical experiments on circle fitting algorithms.

O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O


Pattern recognition after image processing of low-contrast images, the case of the Shroud of Turin Abstract. We discuss the potentially misleading effect of software techniques for elaborating low-contrast images. In particular, we present the example of the stains embedded into one of the most studied archaeological objects in history, the Shroud of Turin. We show for the first time that image processing of both old and recent photographs of the Shroud may lead some researchers to perceive inscriptions and patterns that do not actually exist, confirming that there is a narrow boundary between image enhancement and manipulation. Human visual perception is a complex process which involves the eyes and the brain. The eyes behave as electronic devices that collect the data, while the brain performs real-time information processing. The result of this mind-related process is called "perception" and it may transform the objective image into a subjective sensation [1] . In fact, both shape and color of objects can be modified by our brain, depending on our past experience, state of mind and the context where objects are located.. In this paper, starting from the basic concepts of "dynamic" and "static" contrast, we show how our visual perception can be deceived after image processing of low-contrast images, like the case of invisible letters and patterns perceived on the Shroud of Turin. When observing an object, we trigger a complex process involving our eye-brain system. The result of this process is what we call "perception". Our perception may be subjective in that it is filtered by our experience [1] , depends on the relations among the framed objects [4, 5] , and it is influenced by our mood and state of mind. Even when the answer to a stimulus is the same for almost all, we cannot be sure that our perception is correct, like in the cases illustrated in Figs. 1, 3, 4a, 8 and 9. In fact, our eye-brain system is very efficient in supplying the missing information in order to make sense out of any pattern we can see [4, 5] , thus enhancing pareidolia effects [16] . It should be pointed out that many aspects of perception, including pareidolia, are "automatic", in that they are especially rapid, non-conscious, mandatory and capacity-free [17, 23] . In other words, usually we are not aware when pareidolia deceives our perception.. The limited static contrast of our eyes may render problematic the perception of low-contrast images, as in the case of the various stains on the Shroud. Several works have proposed interpretations of patterns on the Shroud that become visible only after the use of image processing tools, like presumed inscriptions [13] , coins [12, 25] and flowers [26] "seen" on the frontal side of the Shroud, as well as a face-like image perceived on the reverse side of the Shroud [20] . The results summarized in Sections 4 and 5 suggest that these "discoveries" and interpretations should be considered interesting tracks to stimulate further studies, but they cannot be considered selfconsistent proofs. In fact, we have shown that image processing of recent, high-resolution photographs of the Shroud may lead some to perceive inscriptions and patterns that do not actually exist, see Fig. 3 . This is because our ability to retrieve incomplete information acts in synergy with the pareidolia effect to interpret false image pixels emerging after image processing, to finally perceive patterns that "make sense" but do not exist in the original image. In addition, we have shown for the first time that the "discovery" of a hidden image of a face on the reverse side of the Shroud reported in [20] has a very poor spatial correlation with the corresponding face image on the frontal side of the Shroud, see Figs. 4 and 6. As a consequence, it is likely the face-like pattern we perceive on the reverse side of the Shroud after a deep image processing is due to a combined effect of the brain's ability to supply the missing contours [4] , Gestalt [5] , pareidolia [16, 17] and to our innate propensity to interpret stimuli as faces based on minimal cues [19, 23, 24] , see Figs. 8 and 9.. The validity of the above findings is not limited to the images of the Shroud, as very similar arguments can be used in a number of low-contrast images, like those analyzed e.g., in radiology [27] and forensic science [28] . As an example, studies of expert perception of forensic pattern recognition [29] have revealed that the perception process can be influenced by emotional context, external events, expectation and personal beliefs linked to the observed pattern, in addition to the previously discussed pareidolia, Gestalt and "filling information" ability of our brain.. A method to reduce bias which is used in forensic science is the blind test process. According to the American Bar Association Standard for criminal justice [30] "testing should follow procedures designed to minimize bias when interpreting test results" and this can be achieved if the examiner is "blind" to the case's circumstances and other evidences.. A blind investigation could be useful in the case of letters on the Shroud (see figure 2) , but it cannot circumvent our innate propensity to interpret stimuli as faces based on minimal cues ( figures 4, 8 and 9 ). The use of clustering [31] in the case of letters and of facial recognition algorithms [32] in the case of face-like patterns may help to obtain objective data for decisions. Table 1 summarizes causes and possible methods to prevent the incorrect perception of the low-contrast patterns discussed in this paper. In conclusion, we should consider the subjectivity risk and cognitive bias when using image processing tools, because people tend to see what they expect to see but what may not be really embedded in the original, low contrast image. In general, the results illustrated in this paper confirm there is a narrow line between enhancement and manipulation of low-contrast images.

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Improving the performance of multi-hop wireless networks using frame aggregation and broadcast for TCP ACKs As data rates supported by the physical layer increase, PHY and especially MAC overheads increasingly dominate the throughput achievable by wireless networks. A promising approach for reducing these overheads is to aggregate a number of frames together into a single transmission. The 802.11n standard uses such an approach for unicast frames. We present the design of a system that can aggregate both unicast and broadcast frames. Further, the system can classify TCP ACK segments so that they can be aggregated with TCP data flowing in the opposite direction. A novel aspect of our work is that we implement and validate our design not through simulation, but rather using our wireless node prototype, Hydra, which supports a high performance PHY based on 802.11n. Our validation shows significant improvements in throughput for each kind of aggregation we support. As the demand for high data rates increases, wireless networking systems are deploying broadband communication high-throughput technologies such as orthogonal frequency division multiplexing (OFDM) and multiinput multi-output (MIMO). These technologies allow the data portions of frames to be transmitted at high data rates, which decreases the time spent transmitting data, but does not generally decrease the time spent on a variety of overheads. These include the time spent waiting to gain access to the transmission floor, exchanging control frames required by the MAC protocol, and physical layer (PHY) headers. The result is that these overheads begin to dominate performance even when the PHY is capable of high data rates. In general, this problem becomes more severe as rate increases because the time to transmit the data decreases, but the transmit time for most of the overheads does not. Similarly, the effect of these overheads is more dominant for short frames, such as those typically used for control, than for longer ones.. One approach to reducing these overheads and thus achieving the potential performance gains offered by modern PHYs is to group (or aggregate) several frames together into one transmission. This has two benefits: one, it reduces the total number of transmissions, resulting in less time waiting for the floor and transmitting control frames, and two, it reduces header overhead by allowing several frames to share headers. As an example of this approach, the IEEE 802.11n standard includes several frame aggregation schemes to support high data rates as part of its high throughput MAC design [8] .. Most frame aggregation schemes require that frames that are aggregated all be destined to the same receiver. This approach neglects the fact that transmissions are broadcast and a single transmission will potentially be received by many receivers. A simple way of taking advantage of this is to aggregate broadcast frames along with a group of unicast frames all destined for one receiver. Because broadcast frames do not require acknowledgement, this can be done while still having a single ACK for the unicast frames.. We expect TCP traffic to be important for wireless networks and it can also benefit from frame aggregation. The key observation is that TCP ACKs are small packets that flow in the opposite direction from the (typically) larger TCP data packets. Since TCP ACKs are cumulative and thus carry redundant information, they have lower reliability requirements than the data packets. We take advantage of this by treating TCP ACKs as if they were broadcast frames and do not require link-level acknowledgements. This allows the ACKs to be aggregated with TCP data traveling in the opposite direction, significantly reducing the cost of the TCP ACKs. By allowing the MAC to investigate TCP headers, our design breaks layering abstractions and thus is a cross-layer algorithm.. We present a design, implementation, and performance evaluation of three aggregation techniques: unicast aggregation, broadcast aggregation, and TCP ACK aggregation. Although our design is a general modification of 802.11, our implementation is specifically for our Hydra wireless node prototype [6] , which supports an 802.11n-based PHY, with OFDM and MIMO. Thus our performance evaluation is based on a real operational wireless network, rather than simulation.. Our paper is organized as follows. Section 2 outlines existing aggregation schemes. Section 3 details the design of our protocol, while Section 4 describes its implementation on the Hydra prototype. Sections 5 and 6, presents our experimental setup and performance evaluation of our approach. We discuss our conclusions and future enhancements in Section 7. We presented a design for unicast aggregation, broadcast aggregation, and for treating TCP ACKs as broadcasts so as to enable them to be aggregated. We used our wireless networking prototype, Hydra, to implement our aggregation schemes and to perform a series of experiments to validate our design. The performance results are promising for all approaches.. Possible future work falls broadly into two categories, experimental work and protocol development and validation. Experimentally, we plan to perform further experiments when both TCP and standard broadcast frames are generated. In addition, to verify the impact of treating TCP ACKs as broadcasts on performance, we will extend our experiments to the large-scale networks using simulation.. In terms of protocols, we plan to extend our system to make better use of frame aggregation by implementing a block ACK scheme. As an extension to our existing rate adaptation protocol, we plan to design and implement a rate-adaptive frame aggregation scheme. By adapting rates for broadcasts and unicasts, we anticipate further performance improvements. In addition, we will implement a delayed ACK scheme that allows the MAC to aggregate unicast subframes which have different destination addresses. Despite the wide variety of physical layer approaches to increasing wireless network throughput, MAC overhead ultimately limits maximum achievable throughput [21] . Further, these limits have more impact for frames with small data payloads and the impact increases as the rate used for data increases. Frame aggregation can help to address both the problem of short frames and of overhead becoming dominant at high data rates. Here we describe previous efforts to improve throughput using frame aggregation.. To improve throughput, the IEEE 802.11n [8] high throughput standard adopts two approaches to frame aggregation: the aggregated MAC service data unit (A-MSDU), and the aggregated MAC protocol data unit (A-MPDU) [1] . A-MSDU aggregates packets from the upper layer and adds a single MAC header and checksum. This scheme is effective when the MAC aggregates many small user packets such as TCP ACKs or other control oriented data. A-MPDU concatenates normal 802.11 MAC frames each having its own MAC header and checksum. Each of these subframes is separated by a MAC delimiter, which includes a length, checksum, and delimiter signature. The MAC delimiter allows a receiver to robustly separate each subframe, even in the case where some errors occur in the individual subframe. This approach has more overhead than the first, but supports a block ACK scheme. Block ACKs allow each subframe to be acknowledged separately, thus allowing retransmission of only the subframes in error. This approach will have an advantage with high error rates. Kim et al. [12] evaluate the throughput of an early variant of 802.11n frame aggregation as a function of payload size and physical data rate.. The 802.11n MAC also specifies a bi-directional data transfer method that can reduce floor acquisition overhead [1] . It is particularly useful for reducing the overhead of a bi-directional stream of TCP data and ACKs. When a node transmits a frame, instead of relinquishing the floor when the transmission completes, the node can grant the receiver permission for a reverse direction transmission destined for the original transmitter. This approach allows both TCP data and ACKs to be transmitted in turn. This saves a floor acquisition time and the time to exchange a request-to-send (RTS) and clear-to-send (CTS) if they are being used. However, this method does not reduce the MAC and PHY header overheads or the cost of link-level ACKs.. Skordoulis et al. [18] proposed a two-level frame aggregation scheme that mixes 802.11n's two aggregation methods. In the first stage, the MAC aggregates user packets from the upper layer into an A-MSDU with a MAC header and checksum. Then, a series of these AMSDUs are concatenated into an A-MPDU with each A-MSDU separated by a MAC delimiter. This scheme increases the maximum aggregation size compared to using A-MSDUs and reduces MAC header overheads compared to using A-MPDUs. It allows the block ACK scheme to be applied to the A-MSDUs.. Kim et al. [11] proposed a multi-layer scheme that provides aggregation at both the MAC and PHY layers. The MAC aggregates multiple MAC frames into an A-MPDU, and then the PHY aggregates a series of A-MPDUs into a single physical frame. Within the physical frame, an additional physical delimiter precedes each of the A-MPDUs. The physical delimiter contains modulation and coding scheme information for each A-MPDU, and thus allows each A-MPDU to be transmitted at a different rate. Unlike the other existing approaches, this scheme also supports multi-destination aggregation because each A-MPDU can be addressed to a different destination. To facilitate this, the protocol employs a polling scheme so that frames sent to different destinations can be acknowledged.. Sadeghi et al. [16] proposed the opportunistic autorate (OAR) method, which uses frame aggregation to take advantage of favorable channel conditions. When the underlaying rate adaptation algorithm shows that a frame can be sent at higher than base-rate, the MAC attempts to aggregate frames so that the time spent sending the frame at the higher rate equals the time that would be the same as the time to send a single frame at base-rate. This preserves the basic fairness capabilities of the 802.11 MAC while taking advantage of higher rates and the overhead reduction of frame aggregation.. There have been cross-layer approaches to improve TCP performance by piggybacking small TCP ACKs with link-level frames [14, 19, 20, 17] . Parsa et al. [14] proposed the transport unaware link improvement protocol (TULIP) that provides a piggyback method which transfers a TCP ACK with a link-level frame. Tourrilhes [19] proposed PiggyData of which idea is to transmit PiggyData ACK, a link-level acknowledgement with a flag indicating status of transmit queue, as a response of TCP data. Setting the flag to one means that a TCP ACK is followed after SIFS interval. Xiao [20] suggested a piggyback mechanism which allows the MAC to piggyback a link-level acknowledgement with a TCP ACK in a single frame immediately after the node receives TCP data. Our scheme differs from these approaches in that these do not reduce the MAC and PHY header overheads or the cost of link-level ACKs.. Scalia et al. [17] suggested PiggyCode which allows the MAC to combine TCP data with TCP ACK by using network coding technology. This approach can reduce the MAC and PHY header overheads for TCP ACKs, similar to our approach. However, the PiggyCode requires additional header to encode and decode packets, and it allows only packets of different types to be coded together, which restricts transmitting a single TCP data and TCP ACK at a time. We implemented this protocol using Hydra, our wireless node/network prototype. In this section, we overview Hydra and present the details of our implementation. For our experiments, we equipped our USRP frontends to operate in the 2.4 GHz frequency band and we used a transmission power of 7.7 mW which ensured a signalto-noise ratio (SNR) level of 25 dB for the node spacing we used. This SNR did not allow reliable operation of the rates that required 64-QAM. We used the cyclicdiversity MIMO mode and thus transmitted a single data stream from our two antenna systems. We ultimately did experiments at 0.65, 1.30, 1.95, and 2.60 Mbps.. Higher rates would have shown greater improvements Figure 7 : Throughput vs. Aggregation size for our techniques, but the rates used serve to validate our concept. We used three different topologies: 2-and 3-hop linear topologies as shown in Figure 5 ; and a star topology as shown in Figure 6 . Spacing between the nodes is roughly 2.5 meters. Although Click provides several ad-hoc routing protocols, because all of our nodes are within transmission range of the others, they would have not discovered any multi-hop routes. Instead we used static routing to force the topologies in the figures.. To create UDP traffic we used an application that simply sent UDP packets at a controllable rate. For our experiments, we sent UDP packets that resulted in 1140B MAC frames. For TCP, we used a one-way file transfer to send a 0.2 Mbyte file. The maximum segment size for TCP was set to 1357 bytes, resulting in a MAC frame of 1464 bytes. TCP ACKs had MAC frame sizes of 160 bytes.

O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Relations Between Redundancy Patterns of the Shannon Code and Wave Diffraction Patterns of Partially Disordered Media The average redundancy of the Shannon code, , as a function of the block length , is known to exhibit two very different types of behavior, depending on the rationality or irrationality of certain parameters of the source: It either converges to as grows without bound, or it may have a nonvanishing, oscillatory, (quasi-) periodic pattern around the value for all large . In this paper, we make an attempt to shed some insight into this erratic behavior of , by drawing an analogy with the realm of physics of wave propagation, in particular, the elementary theory of scattering and diffraction. It turns out that there are two types of behavior of wave diffraction patterns formed by crystals, which are correspondingly analogous to the two types of patterns of . When the crystal is perfect, the diffraction intensity spectrum exhibits very sharp peaks, a.k.a. Bragg peaks, at wavelengths of full constructive interference. These wavelengths correspond to the frequencies of the harmonic waves of the oscillatory mode of . On the other hand, when the crystal is imperfect and there is a considerable degree of disorder in its structure, the Bragg peaks disappear, and the behavior of this mode is analogous to the one where is convergent. T HE analysis of the average redundancy of lossless codes for data compression schemes is a topic that attracted the attention of considerably many researchers throughout the history of Information Theory (cf. e.g., [1] , [3] , [6] - [12] , [14] and many references therein).. In [14] Szpankowski has derived the asymptotic behavior of the average redundancy , as a function of the block length , for the Shannon code, the Huffman code, and other codes, focusing primarily on the binary memoryless source, parametrized by -the probability of zero. His analysis revealed a rather interesting behavior of , especially in the cases of the Shannon code and the representation as the ratio between two integers whose greatest common divisor is 1, and designates the fractional part of a real number . In several places in his paper, Szpankowski describes this behavior of as "erratic" and this qualifier is, of course, understandable.. Our purpose in this paper is to make an attempt to give some insight into this erratic behavior of by drawing an analogy with the physics of wave diffraction. From the theory of X-ray scattering (see, e.g., [2, Ch. 2] , [15] ), it is known that if the object that causes the diffraction of an incident wave is a perfect crystal, then the intensity profile of the scattered wave (as a function of the wavelength or the wave number) exhibits very sharp peaks, known as Bragg peaks, at wavelengths that correspond to full coherence, where the optical distance differences to all scattering elements (layers of the crystal) are exactly integer multiples of the wavelength. This continues to be the case as long as there is enough order in the medium such that all these distances are commensurable and therefore have a common divisor (common unit of length), which can serve as the fundamental wavelength. In the realm of the average redundancy analysis, this corresponds to the case where is rational and the fundamental frequency of the oscillatory term of is intimately related to the fundamental wavelength at which there is a Bragg peak. On the other hand, when the distances are incommensurable, perfect coherence between all scattered waves is not achieved at any wavelength and therefore no Bragg peaks are observed. This is the case of strong disorder, which in the lossless source coding problem, corresponds to the case of irrational, where is convergent. More concretely, the analysis of the scattered wave intensity function is based on a very simple model of disorder, which is due to Hendricks and Teller [5] (see also [4] ). According to the Hendricks-Teller (HT) model, the distances between every two consecutive layers in the solid are selected independently at random from a finite set of two or more distances. In the simplest case, where there are only two possible distances and , with probabilities and , this random selection process is analogous to the memoryless binary source of the data compression problem and the parameter of this source plays a role analogous to that of the ratio . Thus, irrational means that and are incommensurable, which is the case of strong disorder with no Bragg peaks and no oscillations in . On the other hand when is rational, we are in the (partially) ordered mode, as described earlier.. From the pure mathematical point of view, the analogy between the average redundancy problem and the diffraction problem is rooted in that at the heart of the analyzes of 0018-9448/$31.00 © 2012 IEEE both problems, there is one very simple mathematical fact in common: Given a vector of nonnegative reals summing to unity (probabilities) and a vector , the complex number (1) has a modulus that obviously never exceeds unity, and (i.e., full coherence between all phasors) is attained for some integer values of if and only if are all rational. To see why this true, first observe that the "if" part is obvious: If are all rational, namely, , , for some positive integers and (where and are relatively prime for all ), let be their common denominator, i.e., the smallest common multiple of all . Then clearly, for all values of which are integer multiples of , since is integer for all at the same time. 1 The "only if" part follows from the simple fact that for to be equal to 1, must be integers for all . In other words, , which means that must all be rational. The analogy between the Shannon code redundancy analysis and the diffraction patterns under the HT model will center around (1) and its two types of behavior depending on the rationality or irrationality of . The remaining part of this short paper consists of two more main sections. In Section II, we briefly summarize the main ideas of the derivations in [14] . To gain some insight concerning the derivations in [14] , we present them in a somewhat sketchy way, emphasizing the use of the simple mathematical fact described in the previous paragraph. For reasons of simplicity, we focus on the Shannon code and the derivation specializes on the memoryless case. In Section III, we bring the derivation of the diffraction patterns of the HT model, with a focus on the analogy with Section II. We then describe in detail the mapping between the two problems under discussion. Finally, in Section IV we summarize and conclude, with a few speculations on a possible extension to the Markov case. In this short paper, we have made an attempt to provide some insight into the erratic behavior of the redundancy pattern of the Shannon code for lossless data compression. The insight we propose is rooted in the physical point of view, where the two modes of the behavior of the redundancy patterns are respectively analogous to partial order and complete disorder of a wave diffraction medium, which dictates the existence or nonexistence of Bragg peaks pertaining to perfectly constructive interference. It is hoped that this physical insight contributes to the intuitive understanding of the redundancy of the Shannon code and perhaps other codes as well.. Finally, as a suggestion for possible future work, it would be interesting to generalize the above analysis to the case of finite-state Markov sources (and indeed, Markov models have been proposed in the diffraction setting too [5] , [15] ). While we have not studied that Markov case thoroughly, it is speculated that in this case, both in the data compression problem and in the HT model, the role played by high powers of would be essentially replaced by high powers of state transition probability matrix whose entries are weighted by the appropriate complex exponentials (which depend on ). It is expected that what should matter then are the eigenvalues of this matrix. More concretely, it is not difficult to see that the spectral radius, in both settings, never exceeds unity. In the data compression problem, the critical behavior is believed to be dictated by the existence or nonexistence of integer values for which the spectral radius is exactly 1. When such values of exist, then should have an oscillatory behavior. In the diffraction problem, the distinction between the two types of behavior should be dictated by the existence of values of for which one of the eigenvalues is exactly equal to one.

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Some subjective validation methods using graphical displays of data Subjective methods for operational validity are presented that use graphical displays of histograms, box plots, and behavior graphs. These methods allow the data to be correlated, have any statistical distribution, and be limited in the number of observations. Model data are used for the reference distribution (instead of a theoretical distribution such as the t or F) and for reference to compare the system data against. These methods are very general and can be used in validating different types of models. Determining that a model is valid for its intended purpose is critically important in developing a model and doing this is called the validation process. (See, e.g., Sargent (1996) for a discussion of the validation process). One step of the validation process is performing operational validity. Operational validity (Sargent 1994 ) is defined as "determining that a model's output behavior has sufficient accuracy for the model's intended purpose over the domain of the model's intended applicability.". Various methods and techniques are used in conducting operational validity and they are applied either subjectively or objectively.. The major attribute determining which methods and techniques to use in performing operational validity is the observability of the problem entity being modeled. (A problem entity is some system (real or proposed), idea, situation, policy or phenomena.) A problem entity is (i) observable if it is possible to collect a reasonable amount of data on its operational behavior, (ii) partially observable if the amount of data that can be collected on its operational behavior is limited, and (iii) unobservable if no data can be collected on its operational behavior. The reason that observability is so important is because it is usually necessary to compare the model and problem entity behaviors (outputa) for different experimental conditions from the model's domain of intended application in order to establish a high degree of confidence that a model is valid.. It is preferable to make the comparisons c~fthe behavior data between the problem entity and the model in objective ways, which generally is through the use of statistical tests. However, it is often not possible to use statistical tests because (i) the assumpticms of the tests cannot (or only with extreme difficulty) be satisfied, (ii) the number of observations from the problem entity is insufficient, or (iii) the behavior of the problem entity is changing, e.g, is highly nonstationary. In these situations, other approaches for comparing the data are desirable.. We present some subjective approaches for performing operational validity that uses graphical displays for comparison of system and problem entity behavior data. We believe that the use of graphical displays is the most effective way of presenting the data for subjective comparison. (Furthermore, we also believe that the use of tables for comparing datab is highly ineffective.) While the approaches we present are applicable to different types of models and problem entities, we orient the presentation of our approaches and the discussions about them towards the vahdation of stochastic discrete event simulation models. (This assumes that the problem entities are also stochastic.). The remainder of this paper is organized as follows: Section 2 contains some different ways of clisplaying data graphically, Section 3 discusses some approaches for performing operational validity that use graphical displays of data, and Section 4 gives the summary.

O O B O O O O O B O O O O O O O O B O O O O O O O B O O O O O B O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O B O O O O O B O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O B O O O O O O O B O O O O O O O B O O O O O O B O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B B O O O O O O B O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B B O O O O B O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O B O O O O O O


Idle regulation in non-clairvoyant scheduling of parallel jobs The optimization of parallel applications is difficult to achieve by classical optimization techniques because of their diversity and the variety of actual parallel and distributed platforms and/or environments. Adaptive algorithmic schemes, capable of dynamically changing the allocation of jobs during the execution to optimize global system behavior, are the best alternatives for solving this problem. In this paper, we focus on non-clairvoyant scheduling of parallel jobs with known resource requirements but unknown running times, with emphasis on the regulation of idle periods in the context of general list policies. We consider a new family of scheduling strategies based on two phases which successively combine sequential and parallel execution of jobs. We generalize known worst-case performance bounds by considering two extra parameters, in addition to the number of processors and maximum processor requirements considered in the literature, namely, job parallelization penalty and idle regulation factor. Furthermore, we prove that under certain conditions of idle regulation, the performance guarantee of parallel job scheduling in space-sharing mode can be improved.

B I O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O B O O O O O O


Resampling and requantization of band-limited Gaussian stochastic signals with flat power spectrum A theoretical analysis, aimed at characterizing the degradation induced by the resampling and requantization processes applied to band-limited Gaussian signals with flat power spectrum, available through their digitized samples, is presented. The analysis provides an efficient algorithm for computing the complete joint bivariate discrete probability distribution associated to the true quantized version of the Gaussian signal and to the quantity estimated after resampling and requantization of the input digitized sequence. The use of Fourier transform techniques allows deriving approximate analytical expressions for the quantities of interest, as well as implementing their efficient computation. Numerical experiments are found to be in good agreement with the theoretical results, and confirm the validity of the whole approach. Modern signal processing consists of algorithms applied to sequences of numbers, obtained by analogue to digital (A/D) conversion of analogue signals. The A/D conversion implies sampling in time domain and amplitude quantization, the second step being mandatory due to the finite length of the registers used for storing the samples amplitude in the processing machine. If the effect of quantization is disregarded, the exact reconstruction of the analogue signal from its samples is guaranteed by the sampling theorem, under the assumption that the signal itself is band-limited. Conversely, when quantization is applied, the exact reconstruction of the signal from the quantized samples is no longer possible. An important signal processing task is the rate conversion applied to a sequence of numbers representing a digitized signal. This task consists in obtaining samples of a signal taken at a certain rate, say 1/T 2 , based on the samples of the same signal available at a different rate, say 1/T 1 . This problem was extensively studied in the past years, for both rational and irrational values of the ratio T 2 /T 1 , assumed to be either larger (interpolation problem) or smaller (decimation problem) than unity [1, 2, 3, 4] . The above cited papers derive powerful techniques ensuring that the rate conversion is performed without degradation in all treated cases, under the assumption that the signals are not quantized.. In a non ideal condition, the input sequence available at rate 1/T 1 and the output sequence obtained at rate 1/T 2 as result of the rate conversion process are both quantized, in general (but not necessarily) according to the same quantization scheme. Requantization associated to rate conversion is applied in different contexts, like for example to signals received from radio sources in many applications of radio astronomy [5, 6] , or to coded video data in image processing [7] . In such cases it is of interest to establish theoretical bounds for the degradation occurring due to the quantization process, affecting both the input and the output sequences of numbers.. The inclusion of quantization effects within the context of rate conversion was studied by the authors, in the specific case of extreme clipping, when only the sign of the analogue signal is recorded, i.e. when only one bit of information is associated to the amplitude of each sample [8] . Under this hypothesis, and assuming that the input analogue signal is a realization x(t) of a band-limited Gaussian process X with flat power spectral density within the supporting bandwidth, results in closed form could be obtained about the degradation effect, in that context identified with the probability of error between the quantized version of x(t) at any instant of time estimated through the available quantized samples, and the true quantized value of x(t) (the "target").. The present paper is devoted to extending the results of Ref. [8] to the case of arbitrary quantization scheme, including multiple output levels, with the unique constraints of antisymmetry of the non-linear quantization function. The above mentioned probability of error, which was the metric used for quantifying the degradation effect in the binary case, is replaced by a complete bivariate discrete probability distribution or, in the case of large number of outputs, by the crosscorrelation coefficient between the estimated quantized value of x(t) and the target. In the above decomposition, the variable Z C retains the coefficients φ k for which k is close to the interval of interest of λ, while the variable Z I corresponds to the tails of the sum in Eq. (25), with coefficients φ k such that |φ k (λ)| is slowly varying with respect to k, being. which, for large values of |k|, goes like 1/|k|. Following Ref. [14] , we assume that Z I retains approximately a normal distribution, so that. where the variance σ. with. where the fact that k φ 2 k = 1 has been used. In Sec. 5, in order to minimize the computational effort, we have used the above decomposition with the minimum possible value of h, i.e., h = 1 and N = {0, 1}. As we will see, such a choice provides quite satisfactory results. As far as the Z C is concerned, this variable is now defined as a sum of a finite number of terms, which makes not prohibitive the computation of the quantity exp(i2πZ C ) , as required by Eq. (35). In particular, in Appendix B it is shown that. where erf(·) denotes the error function [18] , j = 1, ..., M , s ∈ {−1, +1}, and q ∈ {0, 1}.. where Accordingly, on recalling Eq. (30), through Eqs. (31)-(33) we eventually found. It is not difficult to show that, once Eqs. (36) and (19) are substituted into Eq. (21), the mixed moments take the following form (see Appendix C):. for even n and m and. for odd n and m, where the functions I (o) (·, ·), I (e,1) (·), I (e,2) (·), and I (e,3) (·, ·), together with the symbolsâ j andâ ′ j+1 are defined in Appendix C. The purpose of this paper was to extend the studies about the rate conversion applied to signal sequences, by taking into account the degradation effect associated to the quantization process.. In particular, we addressed the problem of computing the degradation induced by the resampling and re-quantization of a BL stationary and ergodic signal with Gaussian statistics and flat power spectrum within the supporting bandwidth, available through its quantized samples.. The analysis provides the algorithm for quantitatively characterizing the degradation effect induced by the resampling and re-quantization processes in terms of the knowledge of the complete bivariate discrete probability distribution associated to the target and the estimated quantized signals, or in terms of the correlation coefficient between the two quantities. The analysis makes use of FT representation of the quantization function and of its powers, to allow the application of linear analysis techniques. Numerical experiments have also been implemented in order to validate the theoretical methodology and results. The comparison showed an excellent agreement between theory and simulations. Finally, we provided an example of application of our theoretical results to an important area of Digital Signal Processing, the sampling rate conversion.. The class of stochastic processes considered in the present paper represents a fundamental model with important applications in radio astronomy, where the received noise-like signal originated from radio sources can be modeled, after filtering, with good accuracy by the ideal process here analysed. In such applications coarse quantization (1 or 2 bits) is commonly applied, and the obtained sequences are often subject to resampling and requantization. Of course, the results obtained in the present paper must be interpreted as a first step toward the extension of the methodology originally developed in Ref. [8] to other important classes of stochastic signals. Within the same perspective another important topic, which will be the subject of forthcoming studies, concerns the development of an asymptotic analysis dealing with the case of dense and noclipping quantization schemes, aimed at deriving closed-form limit expressions describing the degradation effect.. Finally, we wish to suggest a possible, future extension of our work which also could be of interest for the radio-astronomy community. In particular, we refer at the classical problem in very large baseline interferometry (VLBI), of different antennas observing the same object and producing digitized representations of the received signal which are then cross-correlated to extract observables of interest. The decorrelation induced by the quantization process has been studied in various works in the past, for example in Ref. [20] . We now think to the case, presently of practical interest, where the received downlink signal is digitized at the different antennas according to different sampling and quantization schemes. In such a case the data streams have to be brought to a common quantization scheme and sampling rate prior to cross-correlation. We believe that the additional de-correlation induced by the re-sampling and re-quantization can be studied by use of the methods exposed in our paper.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O