Continuous Mathematics The complex form of the Fourier series is:…

Question Answered step-by-step Continuous Mathematics The complex form of the Fourier series is:… Continuous MathematicsThe complex form of the Fourier series is:f(x) =+?Xk=??ckei2?kxwhere ck is a complex number and ck = c??k.(a) Prove that the complex coeffiffifficient, ck, encodes the amplitude and phasecoeffiffifficients, Ak and ?k, in the alternative form:f(x) =+?Xk=0Ak cos(2?kx ? ?k)[10 marks](b) What is special about the case k = 0? [2 marks](c) Explain how the coeffiffifficients, ck, of the Fourier series of the periodic function,f(x):f(x) = f(x + T), ?xcan be obtained from the Fourier transform, FL(?), of the related function,fL(x):fL(x) = f(x), ?T2 6 x < T20,otherwise[8 marks]2 Concurrent SystemsAn interprocess communication environment is based on synchronous messagepassing. A server is to be designed to support a moderate number of simultaneousclient requests.Clients send a request message to the server, continue in parallel with serveroperation, then wait for the server's reply message.Discuss the design of the server's interaction with the clients. Include any problemsyou foresee and discuss alternative solutions to them. [20 marks]2CST.2001.4.33 Further Java(a) Describe how mutual-exclusion locks provided by the synchronized keywordcan be used to control access to shared data structures. In particular youshould be clear about the behaviour of concurrent invocations of difffferentsynchronized methods on the same object, or of the same synchronized methodon difffferent objects. [6 marks](b) Consider the following class defifinition:class Example implements Runnable {public static Object o = new Object();int count = 0;public void run() {while (true) {synchronized (o) {count ++;}}}}Show how to start two threads, each executing this run method. [2 marks](c) When this program is executed, only one of the count fifields is found toincrement, even though threads are scheduled preemptively. Why might thisbe? [2 marks](d) Defifine a new class FairLock. Each instance should support two methods, lockand unlock, which acquire and release a mutual exclusion lock such that callsto unlock never block the caller, but will allow the longest-waiting blockedthread to acquire the lock. The lock should be recursive, meaning that thethread holding the lock may make multiple calls to lock without blocking.The lock is released only when a matched number of unlock operations havebeen made.You may wish to make use of the fact the Thread.currentThread() returnsthe instance of Thread that is currently executing. [10 marks]3[TURN OVERCST.2001.4.44 Compiler ConstructionConsider the following grammar giving the concrete syntax of a language:E ? idC ? E = E;C ? {B}C ? C repeatwhile EC ? if E then CC ? if E then C else CB ? B CB ? CS ? C eofwhere C repeatwhile E has the same meaning as do C while E in C or Java.(a) List the terminals and non-terminals of this grammar and explain thesignifificance of S. [3 marks](b) Identify any ambiguities in the above grammar and rewrite it to remove them,ensuring that your new grammar generates exactly the same set of strings.[4 marks](c) Specify a suitable abstract syntax, for example by giving a type declarationin a programming language of your choice, which might be used to hold parsetrees for this language. [3 marks](d) Give either a recursive descent parser or a characteristic fifinite state machine(e.g. for SLR(1)) with associated parser for your grammar. Your parser neednot return a parse tree—it suffiffiffices for your parser either to accept or to rejectthe input string. [10 marks]4CST.2001.4.55 Data Structures and Algorithms(a) Outline how you would determine whether the next line segment turns left orright during the Graham scan phase of the standard method of computing theconvex hull of a set of points in a plane. [5 marks](b) Describe in detail an effiffifficient algorithm to determine how often the substringABRACADABRA occurs in a vector of 106 characters. Your algorithm should beas effiffifficient as possible. [10 marks](c) Roughly estimate how many character comparisons would be made when youralgorithm for part (b) is applied to a vector containing 106 characters uniformlydistributed from the 26 letters A to Z. [5 marks]6 ECAD(a) When designing clocked circuits there are times when asynchronous inputshave to be sampled which may result in metastable behaviour in state holdingelements. How might metastability be avoided when sampling asynchronousinputs? [5 marks](b) An optical shaft encoder (e.g. used on the internal rollers of a mechanicalmouse) consists of a disk with an evenly spaced alternating transparent andopaque grating around the circumference. Two optical sensors are positionedsuch that when one sensor is at the middle of an opaque region, the otheris at the edge. Consequently, the following Gray code sequence is produced,depending upon the direction of rotation:positive rotationnegative rotation A shaft decoder module is required to convert the Gray code into an 8-bitposition. The 8-bit position should be incremented every time the inputchanges from one state to another in a positive direction (e.g. from 00 to01, or from 10 to 00). Similarly, the 8-bit position should be decrementedevery time the input changes from one state to another in a negative direction(e.g. from 00 to 10, or from 01 to 00).Image transcription textA vapor stream that is 52.5 mole%styrene and 47.5 moles toluene is inequilibrium with a liquid m... Show more... Show moreImage transcription textQuantitative methods: Emphasizeobjective measurements and thestatistical, mathematical, ... Show more... Show more  Engineering & Technology Computer Science CPSC 202 Share QuestionEmailCopy link Comments (0)