Statistical properties of first-order bang-bang Pll with nonzero loop delay

A method to solve the stationary state probability is presented for the first-order bang-bang phase-locked loop (BBPLL) with nonzero loop delay. This is based on a delayed Markov chain model and a state How diagram for tracing the state history due to the loop delay. As a result, an eigenequation is obtained, and its closed form solutions are derived for some cases. After obtaining the state probability, statistical characteristics such as mean gain of the binary phase detector and timing error variance are calculated and demonstrated.

The role of the community service order and the suspended sentence in Ireland: a judicial perspective

Imprisonment is the most severe penalty utilised by the criminal courts in Ireland. In recent decades the prison population has grown significantly despite expressions both official and public to reduce the use of the sanction. Two other sanctions are available to the Irish sentencer which may be used as a direct and comparable sentence in lieu of a term of imprisonment namely, the community service order and the suspended sentence. The community service order remains under-utilised as an alternative to the custodial sentence. The suspended sentence is used quite liberally but its function may be more closely related to the aim of deterrence rather than avoiding the use of the custodial sentence. Thus the aim of decarceration may not be optimal in practice when either sanction is utilised. The decarcerative effect of either sanction is largely dependent upon the specific purpose which judges invest in the sanction. Judges may also be inhibited in the use of either sanction if they lack confidence that the sentence will be appropriately monitored and executed. The purpose of this thesis is to examine the role of the community service order and the suspended sentence in Irish sentencing practice. Although community service and the suspended sentence present primarily as alternatives to the custodial sentence, the manner in which the judges utilise or fail to utilise the sanctions may differ significantly from this primary manifestation. Therefore the study proceeds to examine the judges' cognitions and expectations of both sanctions to explore their underlying purposes and to reveal the manner in which the judges use the sanctions in practice. To access this previously undisclosed information a number of methodologies were deployed. An extensive literature review was conducted to delineate the purpose and functionality of both sanctions. Quantitative data was gathered by way of sampling for the suspended sentence and the part-suspended sentence where deficiencies were apparent to show the actual frequency in use of that sanction. Qualitative methodologies were used by way of focus groups and semi-structured interviews of judges at all jurisdictional levels to elucidate the purposes of both sanctions. These methods allowed a deeper investigation of the factors which may promote or inhibit such usage. The relative under-utilisation of the community service order as an alternative to the custodial sentence may in part be explained by a reluctance by some judges to equate it with a real custodial sentence. For most judges who use the sanction, particularly at summary level, community service serves a decarcerative function. The suspended sentence continues to be used extensively. It operates partly as a decarcerative penalty but the purpose of deterrence may in practice overtake its theoretical purpose namely the avoidance of custody. Despite ongoing criticism of executive agencies such as the Probation Service and the Prosecution in the supervision of such penalties both sanctions continue to be used. Engagement between the Criminal Justice actors may facilitate better outcomes in the use of either sanction. The purposes for which both sanctions are deployed find their meaning essentially in the practices of the judges themselves as opposed to any statutory or theoretical claims upon their use or purpose.