Page - 51 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

2.7 Exercises 51 Exercise2.18: Fit sines tostraight line A lot of technology, especiallymost types of digital audio devices for processing sound, is basedonrepresentinga signalof timeasa sumofsine functions. Say the signal is some functionf.t/on the interval Œ ; (amoregeneral interval Œa;b can easily be treated, but leads to slightlymore complicated formulas). Instead of workingwithf.t/directly,weapproximatef by the sum SN.t/D NX nD1 bnsin.nt/; (2.1) where thecoefficientsbnmustbeadjustedsuch thatSN.t/ is agoodapproximation tof.t/.Weshall in this exerciseadjustbnbya trial-and-errorprocess. a) Make a functionsinesum(t, b) that returnsSN.t/, given the coefficientsbn in an arrayb and time coordinates in an arrayt. Note that ift is an array, the returnvalue is alsoanarray. b) Write a functiontest_sinesum() that callssinesum(t, b) in a) anddeter- mines if the function computes a test case correctly. As test case, let t be an arraywith values =2 and =4, chooseN D 2, and b1 D 4 and b2 D 3. ComputeSN.t/byhand toget referencevalues. c) Makeafunctionplot_compare(f, N, M) thatplotstheoriginalfunctionf.t/ togetherwith the sumof sinesSN.t/, so that the quality of the approximation SN.t/ can be examined visually. The argumentf is a Python function imple- mentingf.t/, N is the numberof terms in the sumSN.t/, andM is thenumber ofuniformlydistributed t coordinatesused toplotf andSN . d) Write a functionerror(b, f, M) that returns amathematicalmeasure of the error inSN.t/asanapproximation tof.t/: E D sX i .f.ti/ SN.ti//2; where the ti values areM uniformly distributed coordinates on Œ ; . The arraybholds thecoefficients inSN andf isaPythonfunctionimplementingthe mathematical functionf.t/. e) Make a function trial(f, N) for interactively giving bn values and getting a plot on the screen where the resulting SN.t/ is plotted together with f.t/. Theerror in theapproximationshouldalsobecomputedas indicated ind). The argumentf is aPythonfunction forf.t/ andN is thenumberof termsN in the sumSN.t/. Thetrialfunctioncanrunaloopwheretheuser isaskedforthebn values in eachpass of the loop and the correspondingplot is shown. Youmust find away to terminate the loopwhen the experiments are over. UseM=500 in thecalls toplot_compareanderror. Hint To make this part of your program work, you may have to insert from matplotlib.pylab import * at the top and also add show() after the plot command in the loop.