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Integrate a Fourier Series 8 term

Hi,

I need your help I've constructed and determined the Fourier 8 term series using CurveFitToolbox and cftool , where I set the x and y values to plot the Fourier series.
This represents first derivative.


Now I have 2 objectives in constructive a matlab script on the 2 points below (first derivative was determined, I made the plot and determined the parameters:)

1. To integrate the 8term Fouries Series using matlab due to fact I know the parameters with those values:

f2 = fit(month,pressure,'fourier8')
plot(f2,month,pressure)

f2 = 

     General model Fourier8:
     f2(x) = 
               a0 + a1*cos(x*w) + b1*sin(x*w) + 
               a2*cos(2*x*w) + b2*sin(2*x*w) + a3*cos(3*x*w) + b3*sin(3*x*w) + 
               a4*cos(4*x*w) + b4*sin(4*x*w) + a5*cos(5*x*w) + b5*sin(5*x*w) + 
               a6*cos(6*x*w) + b6*sin(6*x*w) + a7*cos(7*x*w) + b7*sin(7*x*w) + 
               a8*cos(8*x*w) + b8*sin(8*x*w)

a0=3.441, a1=1.856, b1= -4.55, a2= -1.771, b2= -3.592, a3= -3.28, b3= -0.918, 
a4= -1.236, b4=1.435, a5=1.256, b5=1.298, a6=1.924, b6= -0.14, a7=0.7718, b7= -0.9577, a8= -0.2835, b8= - 0.6792, w=6.995.


2. Second point will be to derivate this function somehow to obtain the second derivative and to know for which values of x the first derivative and  derivative are 0.

My model is the one described here:
https://www.mathworks.com/help/curvefit/fourier.html

http://www.math.psu.edu/wysocki/M412/Notes412_8.pdf


I can provide further details if needed.
0
Sprinceana
12/18/2016 2:03:03 PM
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I forgot to mention on point2 I want to plot as well the second derivative.

The first derivative was obtained by settings x and y value and using cftool.
I can provide the sources if needed and attach here.

function [fitresult, gof] = createFits(x, y, x2)
%CREATEFITS(X,Y,X2)
%  Create fits.
%
%  Data for 'untitled fit 1' fit:
%      X Input : x
%      Y Output: y
%  Data for 'untitled fit 3' fit:
%      X Input : x2
%      Y Output: y
%  Output:
%      fitresult : a cell-array of fit objects representing the fits.
%      gof : structure array with goodness-of fit info.
%
%  See also FIT, CFIT, SFIT.

%  Auto-generated by MATLAB on 04-Dec-2016 10:35:11

%% Initialization.

% Initialize arrays to store fits and goodness-of-fit.
fitresult = cell( 3, 1 );
gof = struct( 'sse', cell( 3, 1 ), ...
    'rsquare', [], 'dfe', [], 'adjrsquare', [], 'rmse', [] );

%% Fit: 'untitled fit 1'.
[xData, yData] = prepareCurveData( x, y );

% Set up fittype and options.
ft = fittype( 'fourier8' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';
opts.StartPoint = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.942477796076938];

% Fit model to data.
[fitresult{1}, gof(1)] = fit( xData, yData, ft, opts );

% Plot fit with data.
figure( 'Name', 'untitled fit 1' );
h = plot( fitresult{1}, xData, yData );
legend( h, 'y vs. x', 'untitled fit 1', 'Location', 'NorthEast' );
% Label axes
xlabel x
ylabel y
grid on

%% Fit: 'untitled fit 2'.
% Cannot generate code for fit 'untitled fit 2' because the data selection is incomplete.

%% Fit: 'untitled fit 3'.
[xData, yData] = prepareCurveData( x2, y );

% Set up fittype and options.
ft = fittype( 'fourier8' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Algorithm = 'Levenberg-Marquardt';
opts.Display = 'Off';
opts.StartPoint = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6.89770842105543];

% Fit model to data.
[fitresult{3}, gof(3)] = fit( xData, yData, ft, opts );

% Plot fit with data.
figure( 'Name', 'untitled fit 3' );
h = plot( fitresult{3}, xData, yData );
legend( h, 'y vs. x2', 'untitled fit 3', 'Location', 'NorthEast' );
% Label axes
xlabel x2
ylabel y
grid on
0
Sprinceana
12/18/2016 2:10:03 PM
The set of values used for x and y were:

0.000000	-0.026590	3.977996E-5
0.000488	-0.132251	0.000271
0.000977	-0.338744	0.000937
0.001465	-0.634508	0.002275
0.001953	-1.006813	0.004471
0.002441	-1.450356	0.007664
0.002930	-1.968410	0.011956
0.003418	-2.566635	0.017408
0.003906	-3.244713	0.024051
0.004395	-3.995591	0.031886
0.004883	-4.810065	0.040893
0.005371	-5.672310	0.051026
0.005859	-6.557590	0.062226
0.006348	-7.441490	0.074420
0.006836	-8.305276	0.087524
0.007324	-9.136336	0.101445
0.007812	-9.929480	0.116086
0.008301	-10.681959	0.131346
0.008789	-11.384348	0.147118
0.009277	-12.021010	0.163298
0.009766	-12.580998	0.179780
0.010254	-13.064760	0.196458
0.010742	-13.481637	0.213231
0.011230	-13.840761	0.230001
0.011719	-14.138680	0.246674
0.012207	-14.359522	0.263160
0.012695	-14.491962	0.279375
0.013184	-14.539982	0.295242
0.013672	-14.516250	0.310689
0.014160	-14.435629	0.325653
0.014648	-14.315235	0.340076
0.015137	-14.171688	0.353907
0.015625	-14.016977	0.367100
0.016113	-13.855920	0.379616
0.016602	-13.688030	0.391425
0.017090	-13.514479	0.402503
0.017578	-13.340102	0.412833
0.018066	-13.172151	0.422405
0.018555	-13.021352	0.431218
0.019043	-12.899040	0.439276
0.019531	-12.810367	0.446591
0.020020	-12.752616	0.453180
0.020508	-12.720998	0.459064
0.020996	-12.709994	0.464271
0.021484	-12.710983	0.468829
0.021973	-12.716122	0.472766
0.022461	-12.720961	0.476114
0.022949	-12.722392	0.478906
0.023437	-12.716874	0.481175
0.023926	-12.702243	0.482957
0.024414	-12.682480	0.484288
0.024902	-12.667333	0.485201
0.025391	-12.664768	0.485731
0.025879	-12.674999	0.485909
0.026367	-12.692799	0.485765
0.026855	-12.713666	0.485330
0.027344	-12.738110	0.484631
0.027832	-12.769148	0.483698
0.028320	-12.807419	0.482557
0.028809	-12.852370	0.481236
0.029297	-12.905762	0.479760
0.029785	-12.971606	0.478152
0.030273	-13.053582	0.476438
0.030762	-13.150236	0.474639
0.031250	-13.249551	0.472779
0.031738	-13.332601	0.470878
0.032227	-13.384014	0.468957
0.032715	-13.399421	0.467033
0.033203	-13.386378	0.465120
0.033691	-13.360775	0.463231
0.034180	-13.340246	0.461378
0.034668	-13.340474	0.459573
0.035156	-13.378248	0.457830
0.035645	-13.466207	0.456160
0.036133	-13.606969	0.454574
0.036621	-13.794873	0.453081
0.037109	-14.015224	0.451685
0.037598	-14.248097	0.450393
0.038086	-14.471345	0.449207
0.038574	-14.663096	0.448131
0.039062	-14.810485	0.447164
0.039551	-14.910600	0.446307
0.040039	-14.964357	0.445558
0.040527	-14.970966	0.444917
0.041016	-14.924931	0.444381
0.041504	-14.821480	0.443950
0.041992	-14.660058	0.443620
0.042480	-14.441205	0.443388
0.042969	-14.169630	0.443252
0.043457	-13.860399	0.443208
0.043945	-13.536220	0.443256
0.044434	-13.217602	0.443391
0.044922	-12.918670	0.443610
0.045410	-12.648538	0.443908
0.045898	-12.408524	0.444279
0.046387	-12.191808	0.444713
0.046875	-11.990070	0.445200
0.047363	-11.795989	0.445726
0.047852	-11.607297	0.446278
0.048340	-11.432151	0.446843
0.048828	-11.284795	0.447410
0.049316	-11.178649	0.447967
0.049805	-11.121722	0.448505
0.050293	-11.114535	0.449014
0.050781	-11.153167	0.449488
0.051270	-11.236794	0.449923
0.051758	-11.366695	0.450315
0.052246	-11.538416	0.450660
0.052734	-11.742720	0.450957
0.053223	-11.967818	0.451206
0.053711	-12.199537	0.451406
0.054199	-12.426862	0.451559
0.054687	-12.643092	0.451669
0.055176	-12.842325	0.451740
0.055664	-13.019409	0.451777
0.056152	-13.169648	0.451785
0.056641	-13.288508	0.451771
0.057129	-13.372544	0.451739
0.057617	-13.419688	0.451694
0.058105	-13.430669	0.451639
0.058594	-13.412934	0.451574
0.059082	-13.382451	0.451501
0.059570	-13.362438	0.451420
0.060059	-13.378490	0.451332
0.060547	-13.450389	0.451236
0.061035	-13.585473	0.451134
0.061523	-13.778759	0.451027
0.062012	-14.022201	0.450917
0.062500	-14.306146	0.450805
0.062988	-14.612823	0.450688
0.063477	-14.926102	0.450565
0.063965	-15.240031	0.450434
0.064453	-15.547668	0.450293
0.064941	-15.838157	0.450142
0.065430	-16.105303	0.449982
0.065918	-16.349179	0.449814
0.066406	-16.573497	0.449639
0.066895	-16.779941	0.449460
0.067383	-16.967062	0.449279
0.067871	-17.133060	0.449097
0.068359	-17.275307	0.448912
0.068848	-17.394216	0.448726
0.069336	-17.494909	0.448539
0.069824	-17.579765	0.448350
0.070312	-17.646322	0.448161
0.070801	-17.693018	0.447974
0.071289	-17.721284	0.447789
0.071777	-17.732668	0.447606
0.072266	-17.726223	0.447428
0.072754	-17.701415	0.447253
0.073242	-17.659650	0.447085
0.073730	-17.603025	0.446924
0.074219	-17.536406	0.446769
0.074707	-17.462028	0.446622
0.075195	-17.372691	0.446480
0.075684	-17.254953	0.446344
0.076172	-17.090840	0.446215
0.076660	-16.864528	0.446091
0.077148	-16.571577	0.445972
0.077637	-16.218014	0.445857
0.078125	-15.815916	0.445746
0.078613	-15.377910	0.445638
0.079102	-14.913577	0.445534
0.079590	-14.430265	0.445431
0.080078	-13.935120	0.445329
0.080566	-13.433273	0.445228
0.081055	-12.925531	0.445126
0.081543	-12.412066	0.445023
0.082031	-11.893884	0.444916
0.082520	-11.374210	0.444808
0.083008	-10.865126	0.444700
0.083496	-10.383487	0.444594
0.083984	-9.938051	0.444493
0.084473	-9.525494	0.444401
0.084961	-9.138844	0.444321
0.085449	-8.777169	0.444254
0.085937	-8.442978	0.444204
0.086426	-8.135075	0.444170
0.086914	-7.847891	0.444156
0.087402	-7.573802	0.444163
0.087891	-7.304596	0.444192
0.088379	-7.031100	0.444246
0.088867	-6.742415	0.444325
0.089355	-6.431847	0.444433
0.089844	-6.105211	0.444569
0.090332	-5.777917	0.444735
0.090820	-5.467409	0.444927
0.091309	-5.191711	0.445145
0.091797	-4.962417	0.445388
0.092285	-4.772544	0.445653
0.092773	-4.598264	0.445940
0.093262	-4.416651	0.446245
0.093750	-4.221174	0.446566
0.094238	-4.013666	0.446897
0.094727	-3.789135	0.447233
0.095215	-3.539346	0.447570
0.095703	-3.259813	0.447903
0.096191	-2.946607	0.448230
0.096680	-2.596625	0.448549
0.097168	-2.213229	0.448863
0.097656	-1.805604	0.449170
0.098145	-1.382136	0.449473
0.098633	-0.943432	0.449772
0.099121	-0.482905	0.450065
0.099609	0.005696	0.450352
0.100098	0.525262	0.450631
0.100586	1.073850	0.450901
0.101074	1.646006	0.451156
0.101562	2.240365	0.451390
0.102051	2.860296	0.451598
0.102539	3.505375	0.451774
0.103027	4.168385	0.451917
0.103516	4.842812	0.452025
0.104004	5.526579	0.452097
0.104492	6.220059	0.452134
0.104980	6.924644	0.452137
0.105469	7.641579	0.452107
0.105957	8.377542	0.452044
0.106445	9.146859	0.451951
0.106934	9.961724	0.451828
0.107422	10.828216	0.451675
0.107910	11.745548	0.451495
0.108398	12.704397	0.451291
0.108887	13.698184	0.451068
0.109375	14.729744	0.450830
0.109863	15.800903	0.450582
0.110352	16.902733	0.450330
0.110840	18.018873	0.450077
0.111328	19.137332	0.449830
0.111816	20.250214	0.449593
0.112305	21.347591	0.449370
0.112793	22.421862	0.449168
0.113281	23.468740	0.448989
0.113770	24.482395	0.448837
0.114258	25.456376	0.448714
0.114746	26.387782	0.448621
0.115234	27.280586	0.448557
0.115723	28.144452	0.448518
0.116211	28.988518	0.448504
0.116699	29.818249	0.448514
0.117187	30.636753	0.448546
0.117676	31.444855	0.448598
0.118164	32.241260	0.448673
0.118652	33.024229	0.448769
0.119141	33.788112	0.448889
0.119629	34.521155	0.449030
0.120117	35.211205	0.449191
0.120605	35.849559	0.449370
0.121094	36.428393	0.449567
0.121582	36.940473	0.449779
0.122070	37.382874	0.450006
0.122559	37.757812	0.450248
0.123047	38.073334	0.450505
0.123535	38.344081	0.450775
0.124023	38.586910	0.451056
0.124512	38.819550	0.451344
0.125000	39.061523	0.451635
0.125488	39.327482	0.451928
0.125977	39.624555	0.452218
0.126465	39.954117	0.452501
0.126953	40.310341	0.452772
0.127441	40.684508	0.453025
0.127930	41.069705	0.453256
0.128418	41.457770	0.453457
0.128906	41.838611	0.453623
0.129395	42.204803	0.453750
0.129883	42.551194	0.453833
0.130371	42.872888	0.453869
0.130859	43.169843	0.453856
0.131348	43.447101	0.453791
0.131836	43.706095	0.453673
0.132324	43.941607	0.453504
0.132812	44.149623	0.453286
0.133301	44.332702	0.453023
0.133789	44.494164	0.452720
0.134277	44.630566	0.452386
0.134766	44.732109	0.452031
0.135254	44.790100	0.451662
0.135742	44.802556	0.451290
0.136230	44.773755	0.450921
0.136719	44.714588	0.450561
0.137207	44.640079	0.450214
0.137695	44.561648	0.449887
0.138184	44.486804	0.449583
0.138672	44.424193	0.449305
0.139160	44.379219	0.449057
0.139648	44.347909	0.448842
0.140137	44.324505	0.448662
0.140625	44.313378	0.448519
0.141113	44.326795	0.448410
0.141602	44.372993	0.448333
0.142090	44.453917	0.448286
0.142578	44.567247	0.448263
0.143066	44.705371	0.448261
0.143555	44.860599	0.448276
0.144043	45.031692	0.448304
0.144531	45.221779	0.448341
0.145020	45.430073	0.448384
0.145508	45.647594	0.448428
0.145996	45.860933	0.448471
0.146484	46.058288	0.448511
0.146973	46.236010	0.448547
0.147461	46.399109	0.448577
0.147949	46.551323	0.448603
0.148437	46.691742	0.448625
0.148926	46.817955	0.448645
0.149414	46.925923	0.448659
0.149902	47.012360	0.448669
0.150391	47.075446	0.448672
0.150879	47.112240	0.448668
0.151367	47.119545	0.448658
0.151855	47.093678	0.448646
0.152344	47.028066	0.448636
0.152832	46.915171	0.448633
0.153320	46.747808	0.448640
0.153809	46.522806	0.448658
0.154297	46.248284	0.448688
0.154785	45.938064	0.448730
0.155273	45.600602	0.448780
0.155762	45.235634	0.448837
0.156250	44.836145	0.448899
0.156738	44.394884	0.448965
0.157227	43.909570	0.449034
0.157715	43.383776	0.449104
0.158203	42.824951	0.449173
0.158691	42.241878	0.449239
0.159180	41.644277	0.449299
0.159668	41.040964	0.449350
0.160156	40.438609	0.449386
0.160645	39.846940	0.449403
0.161133	39.280970	0.449397
0.161621	38.755833	0.449364
0.162109	38.279004	0.449301
0.162598	37.845723	0.449205
0.163086	37.443492	0.449072
0.163574	37.059632	0.448900
0.164062	36.686666	0.448686
0.164551	36.321469	0.448428
0.165039	35.960707	0.448128
0.165527	35.594088	0.447789
0.166016	35.199432	0.447414
0.166504	34.750039	0.447008
0.166992	34.225394	0.446576
0.167480	33.615424	0.446127
0.167969	32.919927	0.445668
0.168457	32.144495	0.445206
0.168945	31.295412	0.444751
0.169434	30.380764	0.444311
0.169922	29.412540	0.443894
0.170410	28.402553	0.443506
0.170898	27.362052	0.443151
0.171387	26.302716	0.442831
0.171875	25.236600	0.442548
0.172363	24.176564	0.442303
0.172852	23.134140	0.442096
0.173340	22.118654	0.441930
0.173828	21.135773	0.441807
0.174316	20.185413	0.441730
0.174805	19.263721	0.441699
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1.041504	11.941026	0.451215
1.041992	11.084130	0.451348
1.042480	10.212999	0.451472
1.042969	9.329624	0.451587
1.043457	8.436612	0.451693
1.043945	7.533298	0.451790
1.044434	6.619248	0.451877
1.044922	5.695425	0.451953
1.045410	4.760555	0.452017
1.045898	3.811414	0.452071
1.046387	2.848954	0.452112
1.046875	1.875902	0.452141
1.047363	0.887458	0.452156
1.047852	-0.126718	0.452154
1.048340	-1.172501	0.452134
1.048828	-2.248433	0.452094
1.049316	-3.351145	0.452036
1.049805	-4.480697	0.451960
1.050293	-5.643864	0.451870
1.050781	-6.854120	0.451770
1.051270	-8.125231	0.451663
1.051758	-9.463960	0.451553
1.052246	-10.867747	0.451440
1.052734	-12.324487	0.451327
1.053223	-13.817682	0.451217
1.053711	-15.329836	0.451111
1.054199	-16.837152	0.451010
1.054687	-18.313182	0.450912
1.055176	-19.737423	0.450815
1.055664	-21.094028	0.450717
1.056152	-22.370797	0.450617
1.056641	-23.560091	0.450514
1.057129	-24.655015	0.450406
1.057617	-25.646036	0.450292
1.058105	-26.519515	0.450176
1.058594	-27.261759	0.450058
1.059082	-27.863911	0.449945
1.059570	-28.317743	0.449840
1.060059	-28.614812	0.449746
1.060547	-28.752335	0.449667
1.061035	-28.736397	0.449604
1.061523	-28.580843	0.449560
1.062012	-28.304482	0.449534
1.062500	-27.928225	0.449527
1.062988	-27.471995	0.449537
1.063477	-26.953504	0.449560
1.063965	-26.386366	0.449595
1.064453	-25.777296	0.449641
1.064941	-25.125138	0.449697
1.065430	-24.426554	0.449761
1.065918	-23.685565	0.449834
1.066406	-22.910927	0.449913
1.066895	-22.110080	0.449997
1.067383	-21.290855	0.450084
1.067871	-20.459851	0.450171
1.068359	-19.619956	0.450255
1.068848	-18.771657	0.450336
1.069336	-17.915302	0.450412
1.069824	-17.054150	0.450480
1.070312	-16.195189	0.450543
1.070801	-15.349893	0.450599
1.071289	-14.533212	0.450649
1.071777	-13.756836	0.450693
1.072266	-13.025961	0.450731
1.072754	-12.339738	0.450762
1.073242	-11.692044	0.450787
1.073730	-11.074715	0.450804
1.074219	-10.480771	0.450814
1.074707	-9.903993	0.450815
1.075195	-9.339210	0.450805
1.075684	-8.784933	0.450782
1.076172	-8.239461	0.450744
1.076660	-7.700625	0.450690
1.077148	-7.167661	0.450620
1.077637	-6.639366	0.450536
1.078125	-6.114688	0.450442
1.078613	-5.590566	0.450342
1.079102	-5.064821	0.450240
1.079590	-4.541129	0.450138
1.080078	-4.027576	0.450040
1.080566	-3.533592	0.449944
1.081055	-3.066438	0.449848
1.081543	-2.631339	0.449752
1.082031	-2.232618	0.449654
1.082520	-1.875240	0.449555
1.083008	-1.568133	0.449453
1.083496	-1.322828	0.449347
1.083984	-1.149971	0.449237
1.084473	-1.057601	0.449121
1.084961	-1.050122	0.449003
1.085449	-1.126062	0.448883
1.085937	-1.278242	0.448767
1.086426	-1.497075	0.448658
1.086914	-1.770298	0.448559
1.087402	-2.083201	0.448473
1.087891	-2.421277	0.448402
1.088379	-2.772409	0.448347
1.088867	-3.128021	0.448307
1.089355	-3.481529	0.448283
1.089844	-3.827696	0.448278
1.090332	-4.162975	0.448292
1.090820	-4.483087	0.448326
1.091309	-4.781577	0.448384
1.091797	-5.050200	0.448464
1.092285	-5.280251	0.448566
1.092773	-5.462300	0.448690
1.093262	-5.582436	0.448833
1.093750	-5.625906	0.448990
1.094238	-5.585758	0.449159
1.094727	-5.467002	0.449337
1.095215	-5.281183	0.449521
1.095703	-5.038515	0.449706
1.096191	-4.750394	0.449891
1.096680	-4.432288	0.450072
1.097168	-4.103248	0.450250
1.097656	-3.785703	0.450423
1.098145	-3.500421	0.450590
1.098633	-3.261202	0.450751
1.099121	-3.077252	0.450905
1.099609	-2.954529	0.451052
1.100098	-2.896116	0.451190
1.100586	-2.903446	0.451320
1.101074	-2.972491	0.451437
1.101562	-3.091357	0.451542
1.102051	-3.243806	0.451634
1.102539	-3.417844	0.451715
1.103027	-3.609304	0.451788
1.103516	-3.817487	0.451853
1.104004	-4.041250	0.451913
1.104492	-4.272084	0.451967
1.104980	-4.492156	0.452015
1.105469	-4.682515	0.452057
1.105957	-4.828333	0.452092
1.106445	-4.917834	0.452118
1.106934	-4.940403	0.452134
1.107422	-4.887589	0.452137
1.107910	-4.756889	0.452121
1.108398	-4.554541	0.452087
1.108887	-4.291413	0.452031
1.109375	-3.979010	0.451954
1.109863	-3.629967	0.451858
1.110352	-3.256053	0.451745
1.110840	-2.863463	0.451616
1.111328	-2.451393	0.451474
1.111816	-2.017064	0.451323
1.112305	-1.563177	0.451165
1.112793	-1.098093	0.451003
1.113281	-0.629258	0.450839
1.113770	-0.161671	0.450674
1.114258	0.299775	0.450511
1.114746	0.746401	0.450350
1.115234	1.161570	0.450192
1.115723	1.523824	0.450039
1.116211	1.815097	0.449895
1.116699	2.023168	0.449765
1.117187	2.141451	0.449654
1.117676	2.173772	0.449563
1.118164	2.134950	0.449493
1.118652	2.043464	0.449445
1.119141	1.916437	0.449416
1.119629	1.770904	0.449402
1.120117	1.6237
0
Sprinceana
12/18/2016 5:12:03 PM
I've tried and made a research here:

https://engineering.purdue.edu/ME365/Textbook/chapter8.pdf

8
-
26
You  can  use  a  discrete  Fourier transform (DFT) to calculate the Fourier series coefficients 
k
c
provided:
(1)  
you don't alias when sampling, and
(2)  
N samples represent an integer n
umber of periods of the signal, which in turn means that 
your  sample  rate  must  be  directly  linked  to  the  fundamental  frequency  of  the  periodic 
signal.
You will usually use an FFT algorithm to calculate the DFT, and sometimes this algorithm will 
require N 
to be a power of 2, 
q
N2
which further restricts your choice of sample rate.
So basically
fundamental frequency 
=
s
q
p
1f
M
T
2
where M = some integer and 
s
max
max
f
2 f
, where f
is the highest frequency in the si
gnal.
q
s
max
p
2
f
2 f
TM
A  further  restriction  comes  from  the  analog  to  digital  converter  that  you  use  when  sampling 
your signal, which may not always allow you to sample at the rate you wish. Below is a listing 
of  a  MATLAB 
..
m  file  for  calculati
ng  the  Fourier  series  coefficients  of  a  signal  of  period  .1 
second.  Ten  periods  are  used  in  the  transform,  which  is  why  the  tenth  coefficient  is  the  first 
non
-
zero one, as illustrated in Figure 24.
% four.m program to calculate the Fourier Series coeffici
ents 
% of a signal consisting of two sines
%
% First calculate signal 
clg
t=(0:999)/1000;
a=9*cos(2
*pi*10*t) + 5*cos(2*pi*200*t); 
%
% Now FFT result 
-
MATLAB does not require a power of two 
%
y=fft
(a)/1000;
%
% Calculate the first 500 frequencies 
f=(0
:499);
%
%Plot the real and imaginary parts of the first 500 coefficients 

subplot (211);
plot(f,real(y:500)),
’+’)
xlabel (
‘
REAL PART CK
’
) 

8
-
27
subplot(212) 
plot(f,imag(y(1:500)),
’
+
’
) 
xlabel(
‘
FREQUENCY 
–
HERTZ
’
) 
ylabel (
‘
IMAG PART CK
’
)
0
Sprinceana
12/18/2016 5:54:04 PM
http://www.derivative-calculator.net/
http://www.math24.net/differentiation-and-integration-of-fourier-series.htmlhttp://www.math24.net/differentiation-and-integration-of-fourier-series.html
https://www.symbolab.com/solver/derivative-calculator/%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28%5Cfrac%7B3x%2B9%7D%7B2-x%7D%5Cright%29
0
Sprinceana
12/20/2016 7:36:03 PM
Reply: