Contents

%Lab 6

%Fourier transform lab

1 Plotting sinusoids

figure %Open figure
fs = 1000; %Sampling frequency: 1000 Hz
dur = 1; %Over 1 second
freq = 2; %Signal frequency
t = 0:1/fs:dur; %Timebase: Sampling over all time in steps of sampling rate
sinW = sin(2*pi*freq*t); %Code for a sine wave of a given freq over time
cosW = cos(2*pi*freq*t); %Cosine sinusoid
%Changing phase: Adding or subtracting a number from 0 to 2 pi
%cosW = cos(2*pi*freq*t); %Cosine sinusoid-pi/2
plot(t,sinW,'color','k','linewidth',4)
hold on
plot(t,cosW,'color','b','linewidth',4)

2 Building up a complex signal from sine waves and showing power

%Then representing it in frequency space
figure
dur = 4; %New duration
t = 0:1/fs:dur; %New time base
freq = 10; %New frequency
signal = sin(2*pi*freq*t); %Another sine wave
subplot(3,1,1)
plot(t,signal,'color','k','linewidth',2)

freq2 = 7; %2nd frequency
signal2 = 0.7.*sin(2*pi*freq2*t); %Another sine wave, at 70% amplitude, ~50% power
subplot(3,1,2)
plot(t,signal2,'color','k','linewidth',2)

jointSignal = signal+signal2; %Linear addition
subplot(3,1,3)
plot(t,jointSignal,'color','k','linewidth',2)
shg

%Make the signal more realistic: Add noise
n = length(t); %How much noise?
noise = randn(1,n); %Just random noise from a normal distribution
signalAndNoise = jointSignal + noise; %Linear addition
figure
subplot(3,1,1)
plot(t,jointSignal,'color','k','linewidth',2)
subplot(3,1,2)
plot(t,signalAndNoise,'color','k','linewidth',2)

%Recovering the original signal frequencies even in relatively
%simple signals - 2 sine waves plus a little bit of noise is
%hopeless from visual inspection.
%But no problem for the Fourier transform:

nyquist = fs/2; %Half the sampling frequency
fSpaceSignal = fft(signalAndNoise)/(length(t)/2); %Scaled fft
fBase = linspace(0,nyquist,floor(length(signalAndNoise)/2)+1);%Matlab indexing off by 1, so we need to correct for this
powerPlot = subplot(3,1,3); %Open the 3rd subplot
%As you recall, the fft gives you the fft of the entire signal, but
%half of that (all beyond Nyquist) will be redundant
%So we plot only half that
halfTheSignal = fSpaceSignal(1:length(fBase));
complexConjugate = conj(halfTheSignal); %We want to plot power!
pow = halfTheSignal.*complexConjugate; %This is power
h = plot(fBase,pow,'color','k','linewidth',2);
xlim([0 20]); ylim([0 1])
powerPlot.XTick = 0:20;
xlabel('Frequency in Hz')
ylabel('Power')

3 Recovering the Phase information of the signal

%In addition to amplitude, the fft returns information about the phase
%of the signals.

%Defining a new simple signal of variable phase but constant frequency
figure
for theta = -2*pi:0.05:2*pi %Doing a couple of loops
dur = 1; %New duration
t = 0:1/fs:dur; %New time base
freq = 20; %New frequency
signal = sin(2*pi*freq*t+theta); %Another sine wave
fftSignal = fft(signal);
fBase = linspace(0,nyquist,floor(length(signal)/2)+1);%Matlab indexing off by 1, so we need to correct for this

subplot(3,1,1)
plot(t,signal)
title('Signal in the time domain')
xlabel('time in seconds')
ylabel('Amplitude')
subplot(3,1,2)
pow = fftSignal(1:length(fBase)).*conj(fftSignal(1:length(fBase)));
plot(fBase,pow)
xlim([0 50])
title('Signal power in the frequency domain')
xlabel('Frequency')
ylabel('Power')
subplot(3,1,3)
[val,ind] = max(pow); %Recover the right index (the one with max power)
phaseAngle = angle(fftSignal(ind)); %The range space of this function is -pi to pi - it wraps around
plot(phaseAngle,1,'o','markerfacecolor','k')
xlim([-pi pi])
ylim([0 1.1])
title(['Phase angle of the signal where power is maximal: ', num2str(phaseAngle,'%1.2f')])
xlabel('Phase angle')
pause(0.1)
end

%To recover the phase information of a given frequency, we can use its
%index, but we need to account for MATLAB's indexing by 1.

4 Aliasing and sampling

%The important thing to keep in mind when thinking about
%sampling and aliasing is that there are 3 relevant frequencies
%The sampling rate of reality, that of the signal and that of your sampling

realSampling = 1000; %Let's say reality updates 1000 times a second
t = 0:1/realSampling:1; %Our new time base - one second
signalFrequency = 10; %10 Hz
samplingRate = [3 4 8 10 11 12 20 41 200];
signal = sin(2*pi*signalFrequency*t);
figure
for ii = 1:length(samplingRate)
subplot(3,3,ii)
plot(t,signal)
hold on
sampledSignal = round(linspace(1,length(t),samplingRate(ii)));
q = plot(t(sampledSignal), signal(sampledSignal),'color','r');
q.Marker = 'o'; %Little circles
q.MarkerFaceColor = 'k'; %Black fill
q.MarkerEdgeColor = 'y'; %Yellow edges
hT = title(['fs = ', num2str(samplingRate(ii))])
hT.FontSize = 26;
%Samples = Measures (of the signal)
end

hT = 
  Text (fs = 3) with properties:

                 String: 'fs = 3'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [0.500001063406717 1.03038674033149 0]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 4) with properties:

                 String: 'fs = 4'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [0.500002316351216 1.03038674033149 0]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 8) with properties:

                 String: 'fs = 8'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [0.500003573285985 1.03038674033149 0]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 10) with properties:

                 String: 'fs = 10'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [0.500001063406717 1.03038674033149 0]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 11) with properties:

                 String: 'fs = 11'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [0.500002316351216 1.03038674033149 0]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 12) with properties:

                 String: 'fs = 12'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [0.500003573285985 1.03038674033149 0]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 20) with properties:

                 String: 'fs = 20'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [1x3 double]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 41) with properties:

                 String: 'fs = 41'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [1x3 double]
                  Units: 'data'

  Use GET to show all properties
hT = 
  Text (fs = 200) with properties:

                 String: 'fs = 200'
               FontSize: 11
             FontWeight: 'bold'
               FontName: 'Helvetica'
                  Color: [0 0 0]
    HorizontalAlignment: 'center'
               Position: [1x3 double]
                  Units: 'data'

  Use GET to show all properties

5 Complex sine waves

figure
trace = exp(i*2*pi*freq2*t) %This will be our trace. This is the term that went into the Fourier transform for one frequency
comet3(t,real(trace),imag(trace)); %Let's do a comet
xlabel('t'); ylabel('real'); zlabel('imaginary')

%Complex exponential = slinky
%Looking at it by turning it on its ear:
%Circle (looking in) or sine wave (from the side)
pause
set(gca,'view',[0 0])
pause
set(gca,'view',[90 0])
axis square

trace =
  Column 1
                          1 +                     0i
  Column 2
          0.999032934678125 +    0.0439681183178649i
  Column 3
          0.996133609143173 +    0.0878511965507432i
  Column 4
          0.991307631069507 +     0.131564359092283i
  Column 5
          0.984564334529205 +     0.175023058975276i
  Column 6
          0.975916761938747 +     0.218143241396543i
  Column 7
          0.965381638833274 +     0.260841506289897i
  Column 8
          0.952979341517219 +     0.303035269632774i
  Column 9
          0.938733857653874 +     0.344642923174517i
  Column 10
          0.922672739870115 +     0.385583992277397i
  Column 11
           0.90482705246602 +     0.425779291565073i
  Column 12
          0.885231311332455 +     0.465151078077458i
  Column 13
          0.863923417192835 +     0.503623201635761i
  Column 14
          0.840944582298169 +     0.541121252126876i
  Column 15
          0.816339250717184 +     0.577572703422268i
  Column 16
           0.79015501237569 +     0.612907053652976i
  Column 17
          0.762442511011448 +     0.647055961569444i
  Column 18
           0.73325534622256 +     0.679953378722419i
  Column 19
          0.702649969798849 +     0.711535677209285i
  Column 20
           0.67068557653672 +     0.741741772738739i
  Column 21
           0.63742398974869 +     0.770513242775789i
  Column 22
          0.602929541689025 +     0.797794439538571i
  Column 23
          0.567268949126757 +     0.823532597628427i
  Column 24
          0.530511184306734 +     0.847677936085083i
  Column 25
          0.492727341548292 +     0.870183754669526i
  Column 26
          0.453990499739547 +     0.891006524188368i
  Column 27
          0.414375580993284 +     0.910105970684996i
  Column 28
            0.3739592057378 +     0.927445153334661i
  Column 29
          0.332819544522987 +     0.942990535892864i
  Column 30
          0.291036166828272 +     0.956712051558831i
  Column 31
          0.248689887164855 +     0.968583161128631i
  Column 32
          0.205862608769881 +     0.978580904325472i
  Column 33
          0.162637165194884 +     0.986685944207868i
  Column 34
           0.11909716009487 +     0.992882604569814i
  Column 35
         0.0753268055279327 +     0.997158900260614i
  Column 36
         0.0314107590781282 +     0.999506560365732i
  Column 37
        -0.0125660398833527 +     0.999921044203816i
  Column 38
        -0.0565185344820243 +     0.998401550108975i
  Column 39
         -0.100361714851215 +       0.9949510169813i
  Column 40
         -0.144010782552252 +     0.989576118602651i
  Column 41
         -0.187381314585725 +     0.982287250728689i
  Column 42
          -0.23038942667659 +     0.973098510982127i
  Column 43
         -0.272951935517325 +     0.962027671586086i
  Column 44
         -0.314986519655305 +     0.949096144990295i
  Column 45
         -0.356411878713251 +     0.934328942456612i
  Column 46
          -0.39714789063478 +     0.917754625683981i
  Column 47
         -0.437115766650933 +     0.899405251566371i
  Column 48
         -0.476238203667939 +     0.879316310190556i
  Column 49
         -0.514439533781507 +     0.857526656193652i
  Column 50
          -0.55164587062843 +     0.834078433613171i
  Column 51
         -0.587785252292473 +     0.809016994374947i
  Column 52
         -0.622787780488113 +     0.782390810576588i
  Column 53
         -0.656585755752957 +     0.754251380736104i
  Column 54
         -0.689113808387348 +     0.724653130187047i
  Column 55
         -0.720309024887907 +     0.693653305812805i
  Column 56
         -0.750111069630459 +     0.661311865323652i
  Column 57
         -0.778462301567023 +     0.627691361290701i
  Column 58
         -0.805307885711122 +     0.592856820161059i
  Column 59
         -0.830595899195813 +     0.556875616488188i
  Column 60
         -0.854277431699295 +     0.519817342620709i
  Column 61
         -0.876306680043863 +     0.481753674101716i
  Column 62
         -0.896641036785236 +     0.442758231038902i
  Column 63
         -0.915241172620918 +     0.402906435713663i
  Column 64
         -0.932071112458211 +     0.362275366704546i
  Column 65
         -0.947098304994744 +      0.32094360980721i
  Column 66
         -0.960293685676943 +     0.278991106039229i
  Column 67
         -0.971631732914674 +     0.236498997023725i
  Column 68
         -0.981090517443334 +      0.19354946805086i
  Column 69
         -0.988651744737914 +     0.150225589120757i
  Column 70
         -0.994300790396999 +      0.10661115427526i
  Column 71
         -0.998026728428272 +    0.0627905195293131i
  Column 72
         -0.999822352380809 +     0.018848439715408i
  Column 73
           -0.9996841892833 -    0.0251300954433376i
  Column 74
         -0.997612506361225 -    0.0690600257144054i
  Column 75
         -0.993611310520009 -     0.112856384873481i
  Column 76
         -0.987688340595138 -     0.156434465040231i
  Column 77
         -0.979855052384247 -     0.199709980514407i
  Column 78
         -0.970126596490106 -     0.242599230795407i
  Column 79
         -0.958521789017376 -     0.285019262469976i
  Column 80
         -0.945063075179805 -     0.326888029654943i
  Column 81
         -0.929776485888251 -     0.368124552684678i
  Column 82
         -0.912691587403503 -     0.408649074736349i
  Column 83
         -0.893841424151264 -     0.448383216090032i
  Column 84
          -0.87326245480992 -     0.487250125725332i
  Column 85
         -0.850994481794692 -     0.525174629961296i
  Column 86
         -0.827080574274562 -     0.562083377852131i
  Column 87
         -0.801566984870877 -     0.597904983057519i
  Column 88
         -0.774503060198734 -     0.632570161913125i
  Column 89
         -0.745941145424182 -     0.666011867434251i
  Column 90
         -0.715936483021831 -     0.698165418993473i
  Column 91
         -0.684547105928689 -     0.728968627421411i
  Column 92
         -0.651833725300879 -     0.758361915288722i
  Column 93
         -0.617859613090335 -     0.786288432136619i
  Column 94
         -0.582690479668577 -     0.812694164433094i
  Column 95
         -0.546394346734269 -     0.837528040042142i
  Column 96
         -0.509041415750371 -     0.860742027003944i
  Column 97
         -0.470703932165332 -     0.882291226434953i
  Column 98
         -0.431456045680959 -     0.902133959368203i
  Column 99
         -0.391373666837202 -      0.92023184736587i
  Column 100
         -0.350534320191259 -     0.936549886748192i
  Column 101
         -0.309016994374948 -     0.951056516295154i
  Column 102
         -0.266901989320376 -      0.96372367829001i
  Column 103
         -0.224270760949381 -     0.974526872786577i
  Column 104
         -0.181205763627137 -      0.98344520499533i
  Column 105
         -0.137790290684638 -     0.990461425696651i
  Column 106
        -0.0941083133185149 -      0.99556196460308i
  Column 107
        -0.0502443181797702 -     0.998736956606017i
  Column 108
       -0.00628314396555963 -     0.999980260856137i
  Column 109
         0.0376901826699347 -     0.999289472640589i
  Column 110
         0.0815906115681576 -      0.99666592803403i
  Column 111
          0.125333233564304 -     0.992114701314478i
  Column 112
          0.168833444712734 -     0.985644595148998i
  Column 113
          0.212007109922054 -     0.977268123568193i
  Column 114
          0.254770725683382 -     0.967001487762435i
  Column 115
          0.297041581577035 -     0.954864544746643i
  Column 116
          0.338737920245291 -     0.940880768954226i
  Column 117
          0.379779095521802 -     0.925077206834458i
  Column 118
          0.420085728411807 -     0.907484424541117i
  Column 119
          0.459579860621488 -     0.888136448813544i
  Column 120
          0.498185105339491 -      0.86707070116449i
  Column 121
          0.535826794978996 -     0.844327925502016i
  Column 122
          0.572432125594591 -     0.819952109325452i
  Column 123
          0.607930297694605 -     0.793990398647835i
  Column 124
          0.642252653176584 -      0.76649300680935i
  Column 125
          0.675332808121024 -     0.737513117358174i
  Column 126
          0.707106781186547 -     0.707106781186548i
  Column 127
          0.737513117358174 -     0.675332808121025i
  Column 128
           0.76649300680935 -     0.642252653176585i
  Column 129
          0.793990398647835 -     0.607930297694606i
  Column 130
          0.819952109325452 -     0.572432125594591i
  Column 131
          0.844327925502015 -     0.535826794978996i
  Column 132
           0.86707070116449 -     0.498185105339491i
  Column 133
          0.888136448813545 -     0.459579860621488i
  Column 134
          0.907484424541117 -     0.420085728411806i
  Column 135
          0.925077206834458 -     0.379779095521801i
  Column 136
          0.940880768954225 -     0.338737920245291i
  Column 137
          0.954864544746643 -     0.297041581577035i
  Column 138
          0.967001487762435 -     0.254770725683382i
  Column 139
          0.977268123568194 -     0.212007109922054i
  Column 140
          0.985644595148998 -     0.168833444712733i
  Column 141
          0.992114701314478 -     0.125333233564304i
  Column 142
           0.99666592803403 -    0.0815906115681572i
  Column 143
          0.999289472640589 -    0.0376901826699342i
  Column 144
          0.999980260856137 +    0.0062831439655592i
  Column 145
          0.998736956606017 +    0.0502443181797697i
  Column 146
           0.99556196460308 +    0.0941083133185135i
  Column 147
          0.990461425696651 +     0.137790290684637i
  Column 148
           0.98344520499533 +     0.181205763627136i
  Column 149
          0.974526872786577 +      0.22427076094938i
  Column 150
           0.96372367829001 +     0.266901989320375i
  Column 151
          0.951056516295154 +     0.309016994374947i
  Column 152
          0.936549886748193 +     0.350534320191259i
  Column 153
           0.92023184736587 +     0.391373666837202i
  Column 154
          0.902133959368203 +     0.431456045680959i
  Column 155
          0.882291226434954 +     0.470703932165332i
  Column 156
          0.860742027003944 +     0.509041415750371i
  Column 157
          0.837528040042142 +     0.546394346734268i
  Column 158
          0.812694164433094 +     0.582690479668576i
  Column 159
          0.786288432136619 +     0.617859613090334i
  Column 160
          0.758361915288722 +     0.651833725300879i
  Column 161
          0.728968627421412 +     0.684547105928689i
  Column 162
          0.698165418993473 +     0.715936483021831i
  Column 163
          0.666011867434252 +     0.745941145424182i
  Column 164
          0.632570161913125 +     0.774503060198734i
  Column 165
          0.597904983057519 +     0.801566984870876i
  Column 166
           0.56208337785213 +     0.827080574274562i
  Column 167
          0.525174629961295 +     0.850994481794692i
  Column 168
          0.487250125725332 +      0.87326245480992i
  Column 169
          0.448383216090032 +     0.893841424151264i
  Column 170
          0.408649074736349 +     0.912691587403503i
  Column 171
          0.368124552684678 +     0.929776485888251i
  Column 172
          0.326888029654942 +     0.945063075179805i
  Column 173
          0.285019262469976 +     0.958521789017376i
  Column 174
          0.242599230795407 +     0.970126596490106i
  Column 175
          0.199709980514406 +     0.979855052384247i
  Column 176
           0.15643446504023 +     0.987688340595138i
  Column 177
          0.112856384873482 +     0.993611310520008i
  Column 178
         0.0690600257144062 +     0.997612506361225i
  Column 179
          0.025130095443338 +       0.9996841892833i
  Column 180
        -0.0188484397154076 +     0.999822352380809i
  Column 181
        -0.0627905195293127 +     0.998026728428272i
  Column 182
         -0.106611154275259 +     0.994300790396999i
  Column 183
         -0.150225589120756 +     0.988651744737914i
  Column 184
          -0.19354946805086 +     0.981090517443334i
  Column 185
         -0.236498997023724 +     0.971631732914674i
  Column 186
         -0.278991106039229 +     0.960293685676943i
  Column 187
         -0.320943609807208 +     0.947098304994745i
  Column 188
         -0.362275366704545 +     0.932071112458211i
  Column 189
         -0.402906435713663 +     0.915241172620917i
  Column 190
         -0.442758231038901 +     0.896641036785236i
  Column 191
         -0.481753674101716 +     0.876306680043863i
  Column 192
         -0.519817342620709 +     0.854277431699295i
  Column 193
         -0.556875616488188 +     0.830595899195812i
  Column 194
         -0.592856820161059 +     0.805307885711122i
  Column 195
         -0.627691361290701 +     0.778462301567023i
  Column 196
         -0.661311865323651 +      0.75011106963046i
  Column 197
         -0.693653305812805 +     0.720309024887907i
  Column 198
         -0.724653130187047 +     0.689113808387348i
  Column 199
         -0.754251380736104 +     0.656585755752957i
  Column 200
         -0.782390810576589 +     0.622787780488112i
  Column 201
         -0.809016994374947 +     0.587785252292473i
  Column 202
         -0.834078433613171 +      0.55164587062843i
  Column 203
         -0.857526656193652 +     0.514439533781507i
  Column 204
         -0.879316310190556 +     0.476238203667939i
  Column 205
         -0.899405251566372 +     0.437115766650932i
  Column 206
         -0.917754625683981 +      0.39714789063478i
  Column 207
         -0.934328942456612 +      0.35641187871325i
  Column 208
         -0.949096144990295 +     0.314986519655305i
  Column 209
         -0.962027671586086 +     0.272951935517324i
  Column 210
         -0.973098510982127 +     0.230389426676591i
  Column 211
         -0.982287250728688 +     0.187381314585726i
  Column 212
         -0.989576118602651 +     0.144010782552252i
  Column 213
           -0.9949510169813 +     0.100361714851216i
  Column 214
         -0.998401550108975 +    0.0565185344820249i
  Column 215
         -0.999921044203816 +     0.012566039883354i
  Column 216
         -0.999506560365732 -    0.0314107590781278i
  Column 217
         -0.997158900260614 -     0.075326805527933i
  Column 218
         -0.992882604569814 -     0.119097160094869i
  Column 219
         -0.986685944207868 -     0.162637165194884i
  Column 220
         -0.978580904325472 -      0.20586260876988i
  Column 221
         -0.968583161128631 -     0.248689887164855i
  Column 222
         -0.956712051558831 -     0.291036166828271i
  Column 223
         -0.942990535892865 -     0.332819544522987i
  Column 224
         -0.927445153334661 -     0.373959205737801i
  Column 225
         -0.910105970684996 -     0.414375580993284i
  Column 226
         -0.891006524188368 -     0.453990499739547i
  Column 227
         -0.870183754669526 -     0.492727341548291i
  Column 228
         -0.847677936085083 -     0.530511184306734i
  Column 229
         -0.823532597628428 -     0.567268949126756i
  Column 230
         -0.797794439538571 -     0.602929541689025i
  Column 231
          -0.77051324277579 -     0.637423989748689i
  Column 232
         -0.741741772738739 -      0.67068557653672i
  Column 233
         -0.711535677209285 -      0.70264996979885i
  Column 234
         -0.679953378722419 -      0.73325534622256i
  Column 235
         -0.647055961569444 -     0.762442511011448i
  Column 236
         -0.612907053652977 -      0.79015501237569i
  Column 237
         -0.577572703422267 -     0.816339250717184i
  Column 238
         -0.541121252126876 -     0.840944582298169i
  Column 239
          -0.50362320163576 -     0.863923417192835i
  Column 240
         -0.465151078077457 -     0.885231311332456i
  Column 241
         -0.425779291565074 -     0.904827052466019i
  Column 242
         -0.385583992277397 -     0.922672739870115i
  Column 243
         -0.344642923174517 -     0.938733857653874i
  Column 244
         -0.303035269632775 -     0.952979341517219i
  Column 245
         -0.260841506289897 -     0.965381638833274i
  Column 246
         -0.218143241396543 -     0.975916761938747i
  Column 247
         -0.175023058975276 -     0.984564334529205i
  Column 248
         -0.131564359092284 -     0.991307631069506i
  Column 249
        -0.0878511965507435 -     0.996133609143172i
  Column 250
        -0.0439681183178661 -     0.999032934678125i
  Column 251
      -4.28626379701574e-16 -                     1i
  Column 252
         0.0439681183178653 -     0.999032934678125i
  Column 253
         0.0878511965507426 -     0.996133609143173i
  Column 254
          0.131564359092283 -     0.991307631069507i
  Column 255
          0.175023058975275 -     0.984564334529206i
  Column 256
          0.218143241396543 -     0.975916761938747i
  Column 257
          0.260841506289896 -     0.965381638833274i
  Column 258
          0.303035269632774 -     0.952979341517219i
  Column 259
          0.344642923174516 -     0.938733857653874i
  Column 260
          0.385583992277396 -     0.922672739870115i
  Column 261
          0.425779291565073 -     0.904827052466019i
  Column 262
          0.465151078077458 -     0.885231311332455i
  Column 263
          0.503623201635761 -     0.863923417192835i
  Column 264
          0.541121252126875 -     0.840944582298169i
  Column 265
          0.577572703422268 -     0.816339250717184i
  Column 266
          0.612907053652976 -     0.790155012375691i
  Column 267
          0.647055961569444 -     0.762442511011448i
  Column 268
           0.67995337872242 -     0.733255346222559i
  Column 269
          0.711535677209285 -     0.702649969798849i
  Column 270
           0.74174177273874 -     0.670685576536719i
  Column 271
          0.770513242775789 -      0.63742398974869i
  Column 272
          0.797794439538571 -     0.602929541689024i
  Column 273
          0.823532597628427 -     0.567268949126757i
  Column 274
          0.847677936085084 -     0.530511184306734i
  Column 275
          0.870183754669525 -     0.492727341548292i
  Column 276
          0.891006524188368 -     0.453990499739546i
  Column 277
          0.910105970684996 -     0.414375580993283i
  Column 278
          0.927445153334661 -       0.3739592057378i
  Column 279
          0.942990535892865 -     0.332819544522986i
  Column 280
          0.956712051558831 -     0.291036166828272i
  Column 281
          0.968583161128631 -     0.248689887164854i
  Column 282
          0.978580904325472 -     0.205862608769881i
  Column 283
          0.986685944207868 -     0.162637165194883i
  Column 284
          0.992882604569814 -     0.119097160094868i
  Column 285
          0.997158900260614 -    0.0753268055279321i
  Column 286
          0.999506560365732 -    0.0314107590781268i
  Column 287
          0.999921044203816 +    0.0125660398833531i
  Column 288
          0.998401550108975 +    0.0565185344820258i
  Column 289
            0.9949510169813 +     0.100361714851215i
  Column 290
          0.989576118602651 +     0.144010782552252i
  Column 291
          0.982287250728689 +     0.187381314585723i
  Column 292
          0.973098510982127 +      0.23038942667659i
  Column 293
          0.962027671586086 +     0.272951935517324i
  Column 294
          0.949096144990295 +     0.314986519655304i
  Column 295
          0.934328942456613 +     0.356411878713249i
  Column 296
          0.917754625683982 +      0.39714789063478i
  Column 297
          0.899405251566372 +     0.437115766650931i
  Column 298
          0.879316310190557 +     0.476238203667938i
  Column 299
          0.857526656193652 +     0.514439533781506i
  Column 300
          0.834078433613172 +     0.551645870628429i
  Column 301
          0.809016994374948 +     0.587785252292473i
  Column 302
          0.782390810576589 +     0.622787780488111i
  Column 303
          0.754251380736104 +     0.656585755752956i
  Column 304
          0.724653130187048 +     0.689113808387347i
  Column 305
          0.693653305812806 +     0.720309024887906i
  Column 306
          0.661311865323652 +      0.75011106963046i
  Column 307
          0.627691361290701 +     0.778462301567023i
  Column 308
          0.592856820161059 +     0.805307885711122i
  Column 309
          0.556875616488189 +     0.830595899195812i
  Column 310
           0.51981734262071 +     0.854277431699295i
  Column 311
          0.481753674101716 +     0.876306680043863i
  Column 312
          0.442758231038902 +     0.896641036785236i
  Column 313
          0.402906435713664 +     0.915241172620917i
  Column 314
          0.362275366704546 +     0.932071112458211i
  Column 315
          0.320943609807209 +     0.947098304994744i
  Column 316
           0.27899110603923 +     0.960293685676943i
  Column 317
          0.236498997023725 +     0.971631732914674i
  Column 318
          0.193549468050861 +     0.981090517443334i
  Column 319
          0.150225589120757 +     0.988651744737914i
  Column 320
          0.106611154275261 +     0.994300790396999i
  Column 321
         0.0627905195293136 +     0.998026728428272i
  Column 322
         0.0188484397154076 +     0.999822352380809i
  Column 323
        -0.0251300954433371 +       0.9996841892833i
  Column 324
        -0.0690600257144063 +     0.997612506361225i
  Column 325
         -0.112856384873481 +     0.993611310520009i
  Column 326
         -0.156434465040231 +     0.987688340595138i
  Column 327
         -0.199709980514406 +     0.979855052384247i
  Column 328
         -0.242599230795408 +     0.970126596490106i
  Column 329
         -0.285019262469975 +     0.958521789017376i
  Column 330
         -0.326888029654943 +     0.945063075179805i
  Column 331
         -0.368124552684679 +     0.929776485888251i
  Column 332
         -0.408649074736349 +     0.912691587403503i
  Column 333
         -0.448383216090033 +     0.893841424151263i
  Column 334
         -0.487250125725332 +      0.87326245480992i
  Column 335
         -0.525174629961296 +     0.850994481794692i
  Column 336
          -0.56208337785213 +     0.827080574274562i
  Column 337
         -0.597904983057519 +     0.801566984870876i
  Column 338
         -0.632570161913125 +     0.774503060198733i
  Column 339
         -0.666011867434252 +     0.745941145424182i
  Column 340
         -0.698165418993473 +      0.71593648302183i
  Column 341
         -0.728968627421412 +     0.684547105928689i
  Column 342
         -0.758361915288722 +     0.651833725300878i
  Column 343
         -0.786288432136619 +     0.617859613090334i
  Column 344
         -0.812694164433094 +     0.582690479668575i
  Column 345
         -0.837528040042142 +     0.546394346734269i
  Column 346
         -0.860742027003944 +     0.509041415750371i
  Column 347
         -0.882291226434954 +     0.470703932165331i
  Column 348
         -0.902133959368203 +     0.431456045680959i
  Column 349
         -0.920231847365871 +     0.391373666837201i
  Column 350
         -0.936549886748193 +     0.350534320191259i
  Column 351
         -0.951056516295154 +     0.309016994374946i
  Column 352
          -0.96372367829001 +     0.266901989320375i
  Column 353
         -0.974526872786577 +     0.224270760949382i
  Column 354
         -0.983445204995329 +     0.181205763627139i
  Column 355
         -0.990461425696651 +     0.137790290684639i
  Column 356
          -0.99556196460308 +    0.0941083133185162i
  Column 357
         -0.998736956606017 +    0.0502443181797706i
  Column 358
         -0.999980260856137 +   0.00628314396556095i
  Column 359
         -0.999289472640589 -    0.0376901826699334i
  Column 360
          -0.99666592803403 -    0.0815906115681572i
  Column 361
         -0.992114701314478 -     0.125333233564303i
  Column 362
         -0.985644595148998 -     0.168833444712733i
  Column 363
         -0.977268123568194 -     0.212007109922053i
  Column 364
         -0.967001487762435 -     0.254770725683382i
  Column 365
         -0.954864544746643 -     0.297041581577033i
  Column 366
         -0.940880768954226 -     0.338737920245291i
  Column 367
         -0.925077206834458 -     0.379779095521801i
  Column 368
         -0.907484424541117 -     0.420085728411807i
  Column 369
         -0.888136448813545 -     0.459579860621486i
  Column 370
         -0.867070701164491 -      0.49818510533949i
  Column 371
         -0.844327925502015 -     0.535826794978996i
  Column 372
         -0.819952109325452 -     0.572432125594591i
  Column 373
         -0.793990398647837 -     0.607930297694604i
  Column 374
         -0.766493006809351 -     0.642252653176583i
  Column 375
         -0.737513117358174 -     0.675332808121024i
  Column 376
         -0.707106781186547 -     0.707106781186548i
  Column 377
         -0.675332808121024 -     0.737513117358175i
  Column 378
         -0.642252653176586 -     0.766493006809349i
  Column 379
         -0.607930297694606 -     0.793990398647835i
  Column 380
         -0.572432125594591 -     0.819952109325452i
  Column 381
         -0.535826794978996 -     0.844327925502016i
  Column 382
         -0.498185105339493 -     0.867070701164489i
  Column 383
         -0.459579860621489 -     0.888136448813544i
  Column 384
         -0.420085728411807 -     0.907484424541117i
  Column 385
         -0.379779095521801 -     0.925077206834458i
  Column 386
          -0.33873792024529 -     0.940880768954226i
  Column 387
         -0.297041581577036 -     0.954864544746643i
  Column 388
         -0.254770725683383 -     0.967001487762435i
  Column 389
         -0.212007109922054 -     0.977268123568193i
  Column 390
         -0.168833444712733 -     0.985644595148998i
  Column 391
         -0.125333233564306 -     0.992114701314478i
  Column 392
        -0.0815906115681585 -      0.99666592803403i
  Column 393
        -0.0376901826699346 -     0.999289472640589i
  Column 394
        0.00628314396555966 -     0.999980260856137i
  Column 395
         0.0502443181797711 -     0.998736956606017i
  Column 396
         0.0941083133185131 -      0.99556196460308i
  Column 397
          0.137790290684638 -     0.990461425696651i
  Column 398
          0.181205763627138 -      0.98344520499533i
  Column 399
          0.224270760949382 -     0.974526872786577i
  Column 400
          0.266901989320374 -      0.96372367829001i
  Column 401
          0.309016994374947 -     0.951056516295154i
  Column 402
          0.350534320191259 -     0.936549886748192i
  Column 403
          0.391373666837203 -      0.92023184736587i
  Column 404
          0.431456045680961 -     0.902133959368202i
  Column 405
          0.470703932165332 -     0.882291226434954i
  Column 406
          0.509041415750371 -     0.860742027003944i
  Column 407
           0.54639434673427 -     0.837528040042141i
  Column 408
          0.582690479668577 -     0.812694164433093i
  Column 409
          0.617859613090336 -     0.786288432136617i
  Column 410
          0.651833725300878 -     0.758361915288722i
  Column 411
          0.684547105928689 -     0.728968627421411i
  Column 412
          0.715936483021832 -     0.698165418993472i
  Column 413
          0.745941145424184 -      0.66601186743425i
  Column 414
          0.774503060198733 -     0.632570161913125i
  Column 415
          0.801566984870877 -     0.597904983057519i
  Column 416
          0.827080574274562 -      0.56208337785213i
  Column 417
          0.850994481794693 -     0.525174629961294i
  Column 418
           0.87326245480992 -     0.487250125725333i
  Column 419
          0.893841424151264 -     0.448383216090032i
  Column 420
          0.912691587403502 -     0.408649074736352i
  Column 421
          0.929776485888251 -      0.36812455268468i
  Column 422
          0.945063075179804 -     0.326888029654944i
  Column 423
          0.958521789017376 -     0.285019262469977i
  Column 424
          0.970126596490106 -     0.242599230795407i
  Column 425
          0.979855052384246 -     0.199709980514409i
  Column 426
          0.987688340595137 -     0.156434465040232i
  Column 427
          0.993611310520008 -     0.112856384873482i
  Column 428
          0.997612506361225 -    0.0690600257144058i
  Column 429
            0.9996841892833 -    0.0251300954433402i
  Column 430
          0.999822352380809 +    0.0188484397154063i
  Column 431
          0.998026728428272 +    0.0627905195293123i
  Column 432
          0.994300790396999 +      0.10661115427526i
  Column 433
          0.988651744737914 +     0.150225589120758i
  Column 434
          0.981090517443335 +     0.193549468050858i
  Column 435
          0.971631732914674 +     0.236498997023723i
  Column 436
          0.960293685676943 +     0.278991106039229i
  Column 437
          0.947098304994744 +      0.32094360980721i
  Column 438
          0.932071112458212 +     0.362275366704543i
  Column 439
          0.915241172620918 +     0.402906435713661i
  Column 440
          0.896641036785236 +     0.442758231038901i
  Column 441
          0.876306680043864 +     0.481753674101715i
  Column 442
          0.854277431699295 +      0.51981734262071i
  Column 443
          0.830595899195814 +     0.556875616488186i
  Column 444
          0.805307885711123 +     0.592856820161058i
  Column 445
          0.778462301567024 +       0.6276913612907i
  Column 446
          0.750111069630459 +     0.661311865323652i
  Column 447
          0.720309024887906 +     0.693653305812806i
  Column 448
          0.689113808387349 +     0.724653130187046i
  Column 449
          0.656585755752957 +     0.754251380736103i
  Column 450
          0.622787780488112 +     0.782390810576588i
  Column 451
          0.587785252292472 +     0.809016994374948i
  Column 452
          0.551645870628432 +      0.83407843361317i
  Column 453
          0.514439533781507 +     0.857526656193652i
  Column 454
          0.476238203667939 +     0.879316310190556i
  Column 455
          0.437115766650932 +     0.899405251566371i
  Column 456
          0.397147890634779 +     0.917754625683982i
  Column 457
          0.356411878713252 +     0.934328942456612i
  Column 458
          0.314986519655305 +     0.949096144990294i
  Column 459
          0.272951935517325 +     0.962027671586086i
  Column 460
          0.230389426676589 +     0.973098510982127i
  Column 461
          0.187381314585726 +     0.982287250728688i
  Column 462
          0.144010782552253 +     0.989576118602651i
  Column 463
          0.100361714851215 +       0.9949510169813i
  Column 464
         0.0565185344820236 +     0.998401550108975i
  Column 465
         0.0125660398833508 +     0.999921044203816i
  Column 466
        -0.0314107590781273 +     0.999506560365732i
  Column 467
        -0.0753268055279326 +     0.997158900260614i
  Column 468
          -0.11909716009487 +     0.992882604569814i
  Column 469
         -0.162637165194885 +     0.986685944207868i
  Column 470
          -0.20586260876988 +     0.978580904325472i
  Column 471
         -0.248689887164854 +     0.968583161128631i
  Column 472
         -0.291036166828272 +      0.95671205155883i
  Column 473
         -0.332819544522988 +     0.942990535892864i
  Column 474
         -0.373959205737802 +     0.927445153334661i
  Column 475
         -0.414375580993283 +     0.910105970684996i
  Column 476
         -0.453990499739547 +     0.891006524188368i
  Column 477
         -0.492727341548292 +     0.870183754669525i
  Column 478
         -0.530511184306735 +     0.847677936085082i
  Column 479
         -0.567268949126759 +     0.823532597628426i
  Column 480
         -0.602929541689025 +     0.797794439538571i
  Column 481
         -0.637423989748687 +     0.770513242775791i
  Column 482
         -0.670685576536718 +     0.741741772738741i
  Column 483
         -0.702649969798848 +     0.711535677209286i
  Column 484
          -0.73325534622256 +      0.67995337872242i
  Column 485
         -0.762442511011448 +     0.647055961569444i
  Column 486
         -0.790155012375689 +     0.612907053652978i
  Column 487
         -0.816339250717183 +     0.577572703422269i
  Column 488
         -0.840944582298169 +     0.541121252126877i
  Column 489
         -0.863923417192835 +     0.503623201635761i
  Column 490
         -0.885231311332454 +     0.465151078077461i
  Column 491
         -0.904827052466019 +     0.425779291565074i
  Column 492
         -0.922672739870114 +     0.385583992277398i
  Column 493
         -0.938733857653874 +     0.344642923174517i
  Column 494
         -0.952979341517219 +     0.303035269632773i
  Column 495
         -0.965381638833273 +     0.260841506289899i
  Column 496
         -0.975916761938747 +     0.218143241396544i
  Column 497
         -0.984564334529205 +     0.175023058975277i
  Column 498
         -0.991307631069507 +     0.131564359092282i
  Column 499
         -0.996133609143172 +    0.0878511965507456i
  Column 500
         -0.999032934678125 +    0.0439681183178666i
  Column 501
                         -1 +  8.57252759403147e-16i
  Column 502
         -0.999032934678125 -    0.0439681183178649i
  Column 503
         -0.996133609143172 -    0.0878511965507439i
  Column 504
         -0.991307631069507 -     0.131564359092281i
  Column 505
         -0.984564334529206 -     0.175023058975275i
  Column 506
         -0.975916761938747 -     0.218143241396542i
  Column 507
         -0.965381638833274 -     0.260841506289897i
  Column 508
          -0.95297934151722 -     0.303035269632772i
  Column 509
         -0.938733857653875 -     0.344642923174516i
  Column 510
         -0.922672739870115 -     0.385583992277396i
  Column 511
         -0.904827052466019 -     0.425779291565073i
  Column 512
         -0.885231311332455 -     0.465151078077459i
  Column 513
         -0.863923417192836 -     0.503623201635759i
  Column 514
          -0.84094458229817 -     0.541121252126875i
  Column 515
         -0.816339250717184 -     0.577572703422268i
  Column 516
          -0.79015501237569 -     0.612907053652977i
  Column 517
         -0.762442511011449 -     0.647055961569443i
  Column 518
         -0.733255346222561 -     0.679953378722418i
  Column 519
          -0.70264996979885 -     0.711535677209285i
  Column 520
          -0.67068557653672 -      0.74174177273874i
  Column 521
         -0.637423989748689 -      0.77051324277579i
  Column 522
         -0.602929541689029 -     0.797794439538568i
  Column 523
         -0.567268949126757 -     0.823532597628427i
  Column 524
         -0.530511184306737 -     0.847677936085081i
  Column 525
         -0.492727341548291 -     0.870183754669526i
  Column 526
         -0.453990499739552 -     0.891006524188365i
  Column 527
         -0.414375580993285 -     0.910105970684995i
  Column 528
         -0.373959205737804 -      0.92744515333466i
  Column 529
         -0.332819544522986 -     0.942990535892865i
  Column 530
         -0.291036166828277 -     0.956712051558829i
  Column 531
         -0.248689887164856 -     0.968583161128631i
  Column 532
         -0.205862608769885 -     0.978580904325471i
  Column 533
         -0.162637165194883 -     0.986685944207868i
  Column 534
         -0.119097160094876 -     0.992882604569813i
  Column 535
        -0.0753268055279307 -     0.997158900260614i
  Column 536
        -0.0314107590781326 -     0.999506560365731i
  Column 537
         0.0125660398833527 -     0.999921044203816i
  Column 538
         0.0565185344820219 -     0.998401550108975i
  Column 539
          0.100361714851217 -       0.9949510169813i
  Column 540
          0.144010782552248 -     0.989576118602652i
  Column 541
          0.187381314585724 -     0.982287250728689i
  Column 542
          0.230389426676588 -     0.973098510982127i
  Column 543
          0.272951935517327 -     0.962027671586085i
  Column 544
            0.3149865196553 -     0.949096144990296i
  Column 545
           0.35641187871325 -     0.934328942456612i
  Column 546
          0.397147890634778 -     0.917754625683982i
  Column 547
          0.437115766650934 -     0.899405251566371i
  Column 548
          0.476238203667935 -     0.879316310190559i
  Column 549
          0.514439533781506 -     0.857526656193653i
  Column 550
          0.551645870628427 -     0.834078433613173i
  Column 551
          0.587785252292474 -     0.809016994374947i
  Column 552
          0.622787780488108 -     0.782390810576592i
  Column 553
          0.656585755752958 -     0.754251380736102i
  Column 554
          0.689113808387346 -     0.724653130187049i
  Column 555
          0.720309024887907 -     0.693653305812805i
  Column 556
          0.750111069630458 -     0.661311865323653i
  Column 557
          0.778462301567025 -     0.627691361290699i
  Column 558
           0.80530788571112 -     0.592856820161063i
  Column 559
          0.830595899195813 -     0.556875616488188i
  Column 560
          0.854277431699294 -     0.519817342620712i
  Column 561
          0.876306680043864 -     0.481753674101714i
  Column 562
          0.896641036785234 -     0.442758231038905i
  Column 563
          0.915241172620918 -     0.402906435713663i
  Column 564
           0.93207111245821 -     0.362275366704548i
  Column 565
          0.947098304994745 -     0.320943609807208i
  Column 566
          0.960293685676942 -     0.278991106039234i
  Column 567
          0.971631732914675 -     0.236498997023722i
  Column 568
          0.981090517443334 -     0.193549468050863i
  Column 569
          0.988651744737914 -     0.150225589120756i
  Column 570
          0.994300790396999 -     0.106611154275261i
  Column 571
          0.998026728428272 -    0.0627905195293104i
  Column 572
          0.999822352380809 -    0.0188484397154115i
  Column 573
            0.9996841892833 +    0.0251300954433385i
  Column 574
          0.997612506361225 +    0.0690600257144041i
  Column 575
          0.993611310520008 +     0.112856384873484i
  Column 576
          0.987688340595138 +     0.156434465040227i
  Column 577
          0.979855052384247 +     0.199709980514408i
  Column 578
          0.970126596490106 +     0.242599230795405i
  Column 579
          0.958521789017375 +     0.285019262469978i
  Column 580
          0.945063075179806 +     0.326888029654939i
  Column 581
          0.929776485888251 +     0.368124552684678i
  Column 582
          0.912691587403504 +     0.408649074736347i
  Column 583
          0.893841424151263 +     0.448383216090034i
  Column 584
          0.873262454809922 +     0.487250125725329i
  Column 585
          0.850994481794694 +     0.525174629961293i
  Column 586
          0.827080574274563 +     0.562083377852128i
  Column 587
          0.801566984870878 +     0.597904983057517i
  Column 588
          0.774503060198734 +     0.632570161913124i
  Column 589
          0.745941145424185 +     0.666011867434249i
  Column 590
          0.715936483021833 +     0.698165418993471i
  Column 591
           0.68454710592869 +      0.72896862742141i
  Column 592
           0.65183372530088 +     0.758361915288721i
  Column 593
          0.617859613090338 +     0.786288432136616i
  Column 594
          0.582690479668579 +     0.812694164433092i
  Column 595
          0.546394346734271 +      0.83752804004214i
  Column 596
          0.509041415750373 +     0.860742027003943i
  Column 597
          0.470703932165333 +     0.882291226434953i
  Column 598
          0.431456045680962 +     0.902133959368201i
  Column 599
          0.391373666837205 +     0.920231847365869i
  Column 600
          0.350534320191261 +     0.936549886748192i
  Column 601
          0.309016994374948 +     0.951056516295153i
  Column 602
          0.266901989320376 +      0.96372367829001i
  Column 603
          0.224270760949384 +     0.974526872786576i
  Column 604
          0.181205763627139 +     0.983445204995329i
  Column 605
          0.137790290684639 +     0.990461425696651i
  Column 606
         0.0941083133185148 +      0.99556196460308i
  Column 607
         0.0502443181797728 +     0.998736956606017i
  Column 608
        0.00628314396556138 +     0.999980260856137i
  Column 609
        -0.0376901826699329 +     0.999289472640589i
  Column 610
        -0.0815906115681568 +      0.99666592803403i
  Column 611
         -0.125333233564304 +     0.992114701314478i
  Column 612
         -0.168833444712731 +     0.985644595148998i
  Column 613
         -0.212007109922053 +     0.977268123568194i
  Column 614
         -0.254770725683381 +     0.967001487762435i
  Column 615
         -0.297041581577035 +     0.954864544746643i
  Column 616
         -0.338737920245289 +     0.940880768954226i
  Column 617
         -0.379779095521799 +     0.925077206834459i
  Column 618
         -0.420085728411805 +     0.907484424541118i
  Column 619
         -0.459579860621487 +     0.888136448813545i
  Column 620
         -0.498185105339491 +      0.86707070116449i
  Column 621
         -0.535826794978995 +     0.844327925502016i
  Column 622
          -0.57243212559459 +     0.819952109325453i
  Column 623
         -0.607930297694605 +     0.793990398647836i
  Column 624
         -0.642252653176584 +      0.76649300680935i
  Column 625
         -0.675332808121022 +     0.737513117358176i
  Column 626
         -0.707106781186546 +     0.707106781186549i
  Column 627
         -0.737513117358173 +     0.675332808121025i
  Column 628
          -0.76649300680935 +     0.642252653176585i
  Column 629
         -0.793990398647836 +     0.607930297694605i
  Column 630
         -0.819952109325451 +     0.572432125594593i
  Column 631
         -0.844327925502014 +     0.535826794978998i
  Column 632
          -0.86707070116449 +     0.498185105339491i
  Column 633
         -0.888136448813545 +     0.459579860621488i
  Column 634
         -0.907484424541116 +     0.420085728411809i
  Column 635
         -0.925077206834457 +     0.379779095521803i
  Column 636
         -0.940880768954225 +     0.338737920245292i
  Column 637
         -0.954864544746643 +     0.297041581577035i
  Column 638
         -0.967001487762435 +     0.254770725683381i
  Column 639
         -0.977268123568193 +     0.212007109922057i
  Column 640
         -0.985644595148998 +     0.168833444712735i
  Column 641
         -0.992114701314478 +     0.125333233564305i
  Column 642
          -0.99666592803403 +    0.0815906115681571i
  Column 643
         -0.999289472640589 +    0.0376901826699333i
  Column 644
         -0.999980260856137 -   0.00628314396555746i
  Column 645
         -0.998736956606018 -    0.0502443181797689i
  Column 646
          -0.99556196460308 -    0.0941083133185144i
  Column 647
         -0.990461425696651 -     0.137790290684639i
  Column 648
          -0.98344520499533 -     0.181205763627136i
  Column 649
         -0.974526872786577 -      0.22427076094938i
  Column 650
          -0.96372367829001 -     0.266901989320375i
  Column 651
         -0.951056516295156 -     0.309016994374941i
  Column 652
         -0.936549886748192 -      0.35053432019126i
  Column 653
         -0.920231847365872 -     0.391373666837198i
  Column 654
         -0.902133959368203 -     0.431456045680959i
  Column 655
         -0.882291226434955 -      0.47070393216533i
  Column 656
         -0.860742027003943 -     0.509041415750372i
  Column 657
         -0.837528040042145 -     0.546394346734265i
  Column 658
         -0.812694164433094 -     0.582690479668575i
  Column 659
         -0.786288432136621 -     0.617859613090332i
  Column 660
         -0.758361915288721 -     0.651833725300879i
  Column 661
         -0.728968627421415 -     0.684547105928685i
  Column 662
         -0.698165418993473 -      0.71593648302183i
  Column 663
         -0.666011867434254 -      0.74594114542418i
  Column 664
         -0.632570161913124 -     0.774503060198734i
  Column 665
         -0.597904983057523 -     0.801566984870873i
  Column 666
         -0.562083377852132 -     0.827080574274561i
  Column 667
         -0.525174629961299 -      0.85099448179469i
  Column 668
         -0.487250125725332 -      0.87326245480992i
  Column 669
         -0.448383216090038 -     0.893841424151261i
  Column 670
         -0.408649074736347 -     0.912691587403504i
  Column 671
         -0.368124552684682 -      0.92977648588825i
  Column 672
         -0.326888029654942 -     0.945063075179805i
  Column 673
         -0.285019262469979 -     0.958521789017375i
  Column 674
         -0.242599230795406 -     0.970126596490106i
  Column 675
         -0.199709980514412 -     0.979855052384246i
  Column 676
         -0.156434465040231 -     0.987688340595138i
  Column 677
         -0.112856384873485 -     0.993611310520008i
  Column 678
        -0.0690600257144044 -     0.997612506361225i
  Column 679
        -0.0251300954433424 -       0.9996841892833i
  Column 680
         0.0188484397154076 -     0.999822352380809i
  Column 681
         0.0627905195293101 -     0.998026728428272i
  Column 682
          0.106611154275261 -     0.994300790396999i
  Column 683
          0.150225589120752 -     0.988651744737915i
  Column 684
          0.193549468050863 -     0.981090517443334i
  Column 685
          0.236498997023721 -     0.971631732914675i
  Column 686
           0.27899110603923 -     0.960293685676943i
  Column 687
          0.320943609807208 -     0.947098304994745i
  Column 688
          0.362275366704548 -      0.93207111245821i
  Column 689
          0.402906435713659 -     0.915241172620919i
  Column 690
          0.442758231038902 -     0.896641036785236i
  Column 691
          0.481753674101713 -     0.876306680043865i
  Column 692
          0.519817342620711 -     0.854277431699294i
  Column 693
          0.556875616488185 -     0.830595899195815i
  Column 694
          0.592856820161059 -     0.805307885711122i
  Column 695
          0.627691361290698 -     0.778462301567025i
  Column 696
          0.661311865323653 -     0.750111069630458i
  Column 697
          0.693653305812802 -      0.72030902488791i
  Column 698
          0.724653130187047 -     0.689113808387348i
  Column 699
          0.754251380736102 -     0.656585755752959i
  Column 700
          0.782390810576589 -     0.622787780488111i
  Column 701
          0.809016994374945 -     0.587785252292477i
  Column 702
          0.834078433613173 -     0.551645870628428i
  Column 703
          0.857526656193651 -     0.514439533781509i
  Column 704
          0.879316310190557 -     0.476238203667938i
  Column 705
           0.89940525156637 -     0.437115766650934i
  Column 706
          0.917754625683982 -     0.397147890634778i
  Column 707
          0.934328942456611 -     0.356411878713254i
  Column 708
          0.949096144990295 -     0.314986519655304i
  Column 709
          0.962027671586085 -     0.272951935517327i
  Column 710
          0.973098510982127 -     0.230389426676588i
  Column 711
          0.982287250728688 -     0.187381314585728i
  Column 712
          0.989576118602651 -     0.144010782552251i
  Column 713
            0.9949510169813 -     0.100361714851217i
  Column 714
          0.998401550108975 -    0.0565185344820258i
  Column 715
          0.999921044203816 -    0.0125660398833566i
  Column 716
          0.999506560365732 +    0.0314107590781251i
  Column 717
          0.997158900260614 +    0.0753268055279304i
  Column 718
          0.992882604569814 +     0.119097160094868i
  Column 719
          0.986685944207868 +     0.162637165194883i
  Column 720
          0.978580904325473 +     0.205862608769878i
  Column 721
          0.968583161128632 +     0.248689887164852i
  Column 722
          0.956712051558831 +      0.29103616682827i
  Column 723
          0.942990535892865 +     0.332819544522986i
  Column 724
          0.927445153334663 +     0.373959205737797i
  Column 725
          0.910105970684997 +     0.414375580993281i
  Column 726
          0.891006524188369 +     0.453990499739545i
  Column 727
          0.870183754669526 +      0.49272734154829i
  Column 728
          0.847677936085083 +     0.530511184306734i
  Column 729
          0.823532597628429 +     0.567268949126754i
  Column 730
          0.797794439538572 +     0.602929541689023i
  Column 731
           0.77051324277579 +     0.637423989748689i
  Column 732
           0.74174177273874 +     0.670685576536719i
  Column 733
          0.711535677209285 +     0.702649969798849i
  Column 734
          0.679953378722419 +     0.733255346222561i
  Column 735
          0.647055961569443 +     0.762442511011449i
  Column 736
           0.61290705365298 +     0.790155012375688i
  Column 737
          0.577572703422271 +     0.816339250717182i
  Column 738
          0.541121252126878 +     0.840944582298167i
  Column 739
          0.503623201635763 +     0.863923417192834i
  Column 740
           0.46515107807746 +     0.885231311332455i
  Column 741
          0.425779291565073 +     0.904827052466019i
  Column 742
          0.385583992277396 +     0.922672739870115i
  Column 743
          0.344642923174516 +     0.938733857653874i
  Column 744
          0.303035269632772 +     0.952979341517219i
  Column 745
          0.260841506289901 +     0.965381638833273i
  Column 746
          0.218143241396546 +     0.975916761938747i
  Column 747
          0.175023058975279 +     0.984564334529205i
  Column 748
          0.131564359092284 +     0.991307631069506i
  Column 749
         0.0878511965507443 +     0.996133609143172i
  Column 750
         0.0439681183178652 +     0.999032934678125i
  Column 751
       -4.9047770029553e-16 +                     1i
  Column 752
        -0.0439681183178662 +     0.999032934678125i
  Column 753
        -0.0878511965507453 +     0.996133609143172i
  Column 754
         -0.131564359092278 +     0.991307631069507i
  Column 755
         -0.175023058975273 +     0.984564334529206i
  Column 756
          -0.21814324139654 +     0.975916761938748i
  Column 757
         -0.260841506289895 +     0.965381638833274i
  Column 758
         -0.303035269632773 +     0.952979341517219i
  Column 759
         -0.344642923174517 +     0.938733857653874i
  Column 760
         -0.385583992277397 +     0.922672739870115i
  Column 761
         -0.425779291565074 +     0.904827052466019i
  Column 762
          -0.46515107807746 +     0.885231311332454i
  Column 763
         -0.503623201635757 +     0.863923417192837i
  Column 764
         -0.541121252126873 +     0.840944582298171i
  Column 765
         -0.577572703422266 +     0.816339250717185i
  Column 766
         -0.612907053652975 +     0.790155012375691i
  Column 767
         -0.647055961569444 +     0.762442511011448i
  Column 768
         -0.679953378722419 +      0.73325534622256i
  Column 769
         -0.711535677209286 +     0.702649969798849i
  Column 770
          -0.74174177273874 +     0.670685576536719i
  Column 771
         -0.770513242775791 +     0.637423989748688i
  Column 772
         -0.797794439538569 +     0.602929541689028i
  Column 773
         -0.823532597628426 +     0.567268949126759i
  Column 774
         -0.847677936085082 +     0.530511184306736i
  Column 775
         -0.870183754669525 +     0.492727341548293i
  Column 776
         -0.891006524188368 +     0.453990499739547i
  Column 777
         -0.910105970684996 +     0.414375580993284i
  Column 778
         -0.927445153334662 +     0.373959205737799i
  Column 779
         -0.942990535892865 +     0.332819544522985i
  Column 780
         -0.956712051558831 +     0.291036166828269i
  Column 781
          -0.96858316112863 +     0.248689887164858i
  Column 782
         -0.978580904325472 +     0.205862608769884i
  Column 783
         -0.986685944207868 +     0.162637165194885i
  Column 784
         -0.992882604569814 +     0.119097160094871i
  Column 785
         -0.997158900260614 +    0.0753268055279329i
  Column 786
         -0.999506560365732 +    0.0314107590781277i
  Column 787
         -0.999921044203816 -     0.012566039883354i
  Column 788
         -0.998401550108975 -    0.0565185344820268i
  Column 789
           -0.9949510169813 -     0.100361714851218i
  Column 790
         -0.989576118602651 -     0.144010782552249i
  Column 791
         -0.982287250728689 -     0.187381314585722i
  Column 792
         -0.973098510982127 -     0.230389426676589i
  Column 793
         -0.962027671586086 -     0.272951935517324i
  Column 794
         -0.949096144990297 -     0.314986519655298i
  Column 795
         -0.934328942456612 -     0.356411878713251i
  Column 796
         -0.917754625683983 -     0.397147890634776i
  Column 797
          -0.89940525156637 -     0.437115766650935i
  Column 798
         -0.879316310190558 -     0.476238203667936i
  Column 799
         -0.857526656193654 -     0.514439533781504i
  Column 800
         -0.834078433613172 -     0.551645870628428i
  Column 801
         -0.809016994374948 -     0.587785252292472i
  Column 802
         -0.782390810576593 -     0.622787780488106i
  Column 803
         -0.754251380736104 -     0.656585755752957i
  Column 804
         -0.724653130187051 -     0.689113808387344i
  Column 805
         -0.693653305812804 -     0.720309024887908i
  Column 806
         -0.661311865323655 -     0.750111069630457i
  Column 807
         -0.627691361290698 -     0.778462301567026i
  Column 808
         -0.592856820161061 -      0.80530788571112i
  Column 809
          -0.55687561648819 -     0.830595899195812i
  Column 810
          -0.51981734262071 -     0.854277431699295i
  Column 811
         -0.481753674101716 -     0.876306680043863i
  Column 812
         -0.442758231038907 -     0.896641036785233i
  Column 813
         -0.402906435713661 -     0.915241172620918i
  Column 814
          -0.36227536670455 -     0.932071112458209i
  Column 815
         -0.320943609807207 -     0.947098304994745i
  Column 816
         -0.278991106039232 -     0.960293685676942i
  Column 817
          -0.23649899702372 -     0.971631732914675i
  Column 818
         -0.193549468050862 -     0.981090517443334i
  Column 819
         -0.150225589120758 -     0.988651744737914i
  Column 820
          -0.10661115427526 -     0.994300790396999i
  Column 821
        -0.0627905195293127 -     0.998026728428272i
  Column 822
        -0.0188484397154137 -     0.999822352380809i
  Column 823
         0.0251300954433398 -       0.9996841892833i
  Column 824
         0.0690600257144019 -     0.997612506361225i
  Column 825
          0.112856384873486 -     0.993611310520008i
  Column 826
          0.156434465040229 -     0.987688340595138i
  Column 827
          0.199709980514406 -     0.979855052384247i
  Column 828
          0.242599230795407 -     0.970126596490106i
  Column 829
          0.285019262469976 -     0.958521789017376i
  Column 830
          0.326888029654937 -     0.945063075179807i
  Column 831
          0.368124552684673 -     0.929776485888253i
  Column 832
          0.408649074736345 -     0.912691587403505i
  Column 833
          0.448383216090029 -     0.893841424151265i
  Column 834
           0.48725012572533 -     0.873262454809922i
  Column 835
          0.525174629961294 -     0.850994481794693i
  Column 836
           0.56208337785213 -     0.827080574274562i
  Column 837
          0.597904983057518 -     0.801566984870877i
  Column 838
          0.632570161913125 -     0.774503060198734i
  Column 839
          0.666011867434247 -     0.745941145424186i
  Column 840
          0.698165418993469 -     0.715936483021835i
  Column 841
          0.728968627421409 -     0.684547105928692i
  Column 842
           0.75836191528872 -     0.651833725300881i
  Column 843
          0.786288432136617 -     0.617859613090336i
  Column 844
          0.812694164433093 -     0.582690479668578i
  Column 845
          0.837528040042141 -      0.54639434673427i
  Column 846
          0.860742027003944 -     0.509041415750371i
  Column 847
          0.882291226434954 -     0.470703932165332i
  Column 848
            0.9021339593682 -     0.431456045680964i
  Column 849
          0.920231847365868 -     0.391373666837207i
  Column 850
          0.936549886748191 -     0.350534320191263i
  Column 851
          0.951056516295153 -     0.309016994374951i
  Column 852
          0.963723678290009 -     0.266901989320378i
  Column 853
          0.974526872786577 -     0.224270760949383i
  Column 854
           0.98344520499533 -     0.181205763627138i
  Column 855
          0.990461425696651 -     0.137790290684638i
  Column 856
           0.99556196460308 -    0.0941083133185135i
  Column 857
          0.998736956606017 -     0.050244318179775i
  Column 858
          0.999980260856137 -   0.00628314396556358i
  Column 859
          0.999289472640589 +    0.0376901826699307i
  Column 860
           0.99666592803403 +    0.0815906115681546i
  Column 861
          0.992114701314478 +     0.125333233564302i
  Column 862
          0.985644595148998 +     0.168833444712733i
  Column 863
          0.977268123568194 +     0.212007109922054i
  Column 864
          0.967001487762435 +     0.254770725683382i
  Column 865
          0.954864544746643 +     0.297041581577036i
  Column 866
          0.940880768954227 +     0.338737920245286i
  Column 867
           0.92507720683446 +     0.379779095521797i
  Column 868
          0.907484424541118 +     0.420085728411803i
  Column 869
          0.888136448813546 +     0.459579860621485i
  Column 870
          0.867070701164491 +     0.498185105339489i
  Column 871
          0.844327925502016 +     0.535826794978996i
  Column 872
          0.819952109325452 +     0.572432125594591i
  Column 873
          0.793990398647835 +     0.607930297694606i
  Column 874
          0.766493006809349 +     0.642252653176585i
  Column 875
          0.737513117358177 +     0.675332808121021i
  Column 876
           0.70710678118655 +     0.707106781186545i
  Column 877
          0.675332808121027 +     0.737513117358172i
  Column 878
          0.642252653176586 +     0.766493006809348i
  Column 879
          0.607930297694607 +     0.793990398647834i
  Column 880
          0.572432125594592 +     0.819952109325452i
  Column 881
          0.535826794978997 +     0.844327925502015i
  Column 882
           0.49818510533949 +      0.86707070116449i
  Column 883
          0.459579860621486 +     0.888136448813545i
  Column 884
          0.420085728411811 +     0.907484424541115i
  Column 885
          0.379779095521805 +     0.925077206834457i
  Column 886
          0.338737920245294 +     0.940880768954224i
  Column 887
          0.297041581577037 +     0.954864544746642i
  Column 888
          0.254770725683384 +     0.967001487762435i
  Column 889
          0.212007109922055 +     0.977268123568193i
  Column 890
          0.168833444712734 +     0.985644595148998i
  Column 891
          0.125333233564303 +     0.992114701314478i
  Column 892
         0.0815906115681558 +      0.99666592803403i
  Column 893
         0.0376901826699319 +     0.999289472640589i
  Column 894
       -0.00628314396555525 +     0.999980260856137i
  Column 895
        -0.0502443181797667 +     0.998736956606018i
  Column 896
        -0.0941083133185123 +      0.99556196460308i
  Column 897
         -0.137790290684637 +     0.990461425696651i
  Column 898
         -0.181205763627137 +      0.98344520499533i
  Column 899
         -0.224270760949382 +     0.974526872786577i
  Column 900
         -0.266901989320377 +     0.963723678290009i
  Column 901
         -0.309016994374949 +     0.951056516295153i
  Column 902
         -0.350534320191262 +     0.936549886748191i
  Column 903
         -0.391373666837199 +     0.920231847365872i
  Column 904
         -0.431456045680957 +     0.902133959368204i
  Column 905
         -0.470703932165331 +     0.882291226434954i
  Column 906
          -0.50904141575037 +     0.860742027003944i
  Column 907
         -0.546394346734269 +     0.837528040042142i
  Column 908
         -0.582690479668577 +     0.812694164433094i
  Column 909
         -0.617859613090335 +     0.786288432136618i
  Column 910
          -0.65183372530088 +      0.75836191528872i
  Column 911
         -0.684547105928691 +     0.728968627421409i
  Column 912
         -0.715936483021829 +     0.698165418993475i
  Column 913
         -0.745941145424181 +     0.666011867434253i
  Column 914
         -0.774503060198733 +     0.632570161913126i
  Column 915
         -0.801566984870876 +     0.597904983057519i
  Column 916
         -0.827080574274562 +     0.562083377852131i
  Column 917
         -0.850994481794692 +     0.525174629961295i
  Column 918
         -0.873262454809921 +     0.487250125725331i
  Column 919
         -0.893841424151265 +      0.44838321609003i
  Column 920
         -0.912691587403504 +     0.408649074736346i
  Column 921
          -0.92977648588825 +     0.368124552684681i
  Column 922
         -0.945063075179804 +     0.326888029654945i
  Column 923
         -0.958521789017376 +     0.285019262469977i
  Column 924
         -0.970126596490106 +     0.242599230795408i
  Column 925
         -0.979855052384247 +     0.199709980514407i
  Column 926
         -0.987688340595138 +      0.15643446504023i
  Column 927
         -0.993611310520009 +      0.11285638487348i
  Column 928
         -0.997612506361225 +    0.0690600257144031i
  Column 929
           -0.9996841892833 +    0.0251300954433411i
  Column 930
         -0.999822352380809 -    0.0188484397154054i
  Column 931
         -0.998026728428272 -    0.0627905195293114i
  Column 932
         -0.994300790396999 -     0.106611154275259i
  Column 933
         -0.988651744737915 -      0.15022558912075i
  Column 934
         -0.981090517443334 -     0.193549468050861i
  Column 935
         -0.971631732914675 -     0.236498997023719i
  Column 936
         -0.960293685676943 -     0.278991106039231i
  Column 937
         -0.947098304994746 -     0.320943609807206i
  Column 938
          -0.93207111245821 -     0.362275366704549i
  Column 939
         -0.915241172620919 -      0.40290643571366i
  Column 940
         -0.896641036785237 -       0.4427582310389i
  Column 941
         -0.876306680043864 -     0.481753674101714i
  Column 942
         -0.854277431699299 -     0.519817342620703i
  Column 943
         -0.830595899195816 -     0.556875616488183i
  Column 944
         -0.805307885711125 -     0.592856820161055i
  Column 945
         -0.778462301567026 -     0.627691361290697i
  Column 946
         -0.750111069630462 -     0.661311865323649i
  Column 947
         -0.720309024887909 -     0.693653305812803i
  Column 948
          -0.68911380838735 -     0.724653130187045i
  Column 949
         -0.656585755752958 -     0.754251380736103i
  Column 950
         -0.622787780488113 -     0.782390810576588i
  Column 951
         -0.587785252292479 -     0.809016994374943i
  Column 952
         -0.551645870628435 -     0.834078433613168i
  Column 953
         -0.514439533781511 -      0.85752665619365i
  Column 954
         -0.476238203667943 -     0.879316310190554i
  Column 955
         -0.437115766650936 -     0.899405251566369i
  Column 956
         -0.397147890634783 -      0.91775462568398i
  Column 957
         -0.356411878713253 -     0.934328942456611i
  Column 958
         -0.314986519655306 -     0.949096144990294i
  Column 959
         -0.272951935517326 -     0.962027671586086i
  Column 960
         -0.230389426676597 -     0.973098510982125i
  Column 961
          -0.18738131458573 -     0.982287250728688i
  Column 962
         -0.144010782552257 -      0.98957611860265i
  Column 963
         -0.100361714851219 -       0.9949510169813i
  Column 964
         -0.056518534482028 -     0.998401550108975i
  Column 965
        -0.0125660398833553 -     0.999921044203816i
  Column 966
         0.0314107590781265 -     0.999506560365732i
  Column 967
         0.0753268055279317 -     0.997158900260614i
  Column 968
           0.11909716009487 -     0.992882604569814i
  Column 969
          0.162637165194884 -     0.986685944207868i
  Column 970
          0.205862608769876 -     0.978580904325473i
  Column 971
           0.24868988716485 -     0.968583161128632i
  Column 972
          0.291036166828268 -     0.956712051558832i
  Column 973
          0.332819544522984 -     0.942990535892866i
  Column 974
          0.373959205737798 -     0.927445153334662i
  Column 975
          0.414375580993283 -     0.910105970684996i
  Column 976
          0.453990499739546 -     0.891006524188368i
  Column 977
          0.492727341548292 -     0.870183754669526i
  Column 978
          0.530511184306735 -     0.847677936085083i
  Column 979
          0.567268949126752 -      0.82353259762843i
  Column 980
          0.602929541689021 -     0.797794439538574i
  Column 981
          0.637423989748687 -     0.770513242775792i
  Column 982
          0.670685576536718 -     0.741741772738741i
  Column 983
          0.702649969798848 -     0.711535677209287i
  Column 984
          0.733255346222559 -      0.67995337872242i
  Column 985
          0.762442511011447 -     0.647055961569445i
  Column 986
          0.790155012375691 -     0.612907053652976i
  Column 987
          0.816339250717185 -     0.577572703422267i
  Column 988
          0.840944582298166 -      0.54112125212688i
  Column 989
          0.863923417192833 -     0.503623201635765i
  Column 990
          0.885231311332454 -     0.465151078077462i
  Column 991
          0.904827052466018 -     0.425779291565075i
  Column 992
          0.922672739870114 -     0.385583992277398i
  Column 993
          0.938733857653874 -     0.344642923174518i
  Column 994
          0.952979341517219 -     0.303035269632774i
  Column 995
          0.965381638833274 -     0.260841506289896i
  Column 996
          0.975916761938748 -     0.218143241396541i
  Column 997
          0.984564334529205 -     0.175023058975281i
  Column 998
          0.991307631069506 -     0.131564359092287i
  Column 999
          0.996133609143172 -    0.0878511965507465i
  Column 1000
          0.999032934678125 -    0.0439681183178674i
  Column 1001
                          1 -  1.71450551880629e-15i

Published with MATLAB® R2016a