test_fft.c test fftd.c 
check implies other coefficients should be near zero 
sin 2 th data 
[0] real = 0, imag = 0 
[1] real = 0.707107, imag = 0 
[2] real = 1, imag = 0 
[3] real = 0.707107, imag = 0 
[4] real = 1.22461e-16, imag = 0 
[5] real = -0.707107, imag = 0 
[6] real = -1, imag = 0 
[7] real = -0.707107, imag = 0 
[8] real = -2.44921e-16, imag = 0 
[9] real = 0.707107, imag = 0 
[10] real = 1, imag = 0 
[11] real = 0.707107, imag = 0 
[12] real = 3.67382e-16, imag = 0 
[13] real = -0.707107, imag = 0 
[14] real = -1, imag = 0 
[15] real = -0.707107, imag = 0 
sin 2 th FFT 
cos 0 th = -6.43257e-16, sin 0 th = 0 
cos 1 th = -8.37856e-16, sin 1 th = 2.12744e-16 
cos 2 th = -2.27161e-14, sin 2 th = -8 
cos 3 th = 1.11829e-15, sin 3 th = -1.19295e-15 
cos 4 th = 2.44921e-16, sin 4 th = 0 
cos 5 th = -6.28444e-16, sin 5 th = -7.03108e-16 
cos 6 th = 5.21393e-16, sin 6 th = -5.32907e-15 
cos 7 th = 1.3277e-15, sin 7 th = -2.77099e-16 
cos 8 th = 1.1331e-15, sin 8 th = 0 
cos 9 th = 1.3277e-15, sin 9 th = 2.77099e-16 
cos 10 th = 7.4295e-15, sin 10 th = -8.88178e-16 
cos 11 th = -6.28444e-16, sin 11 th = 7.03108e-16 
cos 12 th = 2.44921e-16, sin 12 th = 0 
cos 13 th = 1.11829e-15, sin 13 th = 1.19295e-15 
cos 14 th = 1.18262e-14, sin 14 th = 8 
cos 15 th = -8.37856e-16, sin 15 th = -2.12744e-16 
antialias and normalize sin 2 th FFT 
cos 0 th = -4.02036e-17, sin 0 th = -0 
cos 1 th = -1.04732e-16, sin 1 th = -2.6593e-17 
cos 2 th = -6.80621e-16, sin 2 th = 1 
cos 3 th = 1.39786e-16, sin 3 th = 1.49119e-16 
cos 4 th = 3.06152e-17, sin 4 th = -0 
cos 5 th = -7.85555e-17, sin 5 th = 8.78885e-17 
cos 6 th = 4.96931e-16, sin 6 th = 2.77556e-16 
cos 7 th = 1.65962e-16, sin 7 th = 3.46373e-17 
check: sin 2 th = 1.0 


cos 2 th data 
[0] real = 1, imag = 0 
[1] real = 0.707107, imag = 0 
[2] real = 6.12303e-17, imag = 0 
[3] real = -0.707107, imag = 0 
[4] real = -1, imag = 0 
[5] real = -0.707107, imag = 0 
[6] real = -1.83691e-16, imag = 0 
[7] real = 0.707107, imag = 0 
[8] real = 1, imag = 0 
[9] real = 0.707107, imag = 0 
[10] real = 3.06152e-16, imag = 0 
[11] real = -0.707107, imag = 0 
[12] real = -1, imag = 0 
[13] real = -0.707107, imag = 0 
[14] real = -4.28612e-16, imag = 0 
[15] real = 0.707107, imag = 0 
cos 2 th FFT 
cos 0 th = 1.99168e-16, sin 0 th = 0 
cos 1 th = -1.25191e-15, sin 1 th = 9.75936e-16 
cos 2 th = 8, sin 2 th = -1.63665e-14 
cos 3 th = 2.11622e-16, sin 3 th = -1.12526e-15 
cos 4 th = 2.44921e-16, sin 4 th = -4.44089e-16 
cos 5 th = 4.8112e-16, sin 5 th = -1.12526e-15 
cos 6 th = 3.9968e-15, sin 6 th = 6.31797e-15 
cos 7 th = 5.59172e-16, sin 7 th = 9.75936e-16 
cos 8 th = -6.8901e-16, sin 8 th = 0 
cos 9 th = 5.59172e-16, sin 9 th = -9.75936e-16 
cos 10 th = -3.55271e-15, sin 10 th = 1.44071e-14 
cos 11 th = 4.8112e-16, sin 11 th = 1.12526e-15 
cos 12 th = 2.44921e-16, sin 12 th = 4.44089e-16 
cos 13 th = 2.11622e-16, sin 13 th = 1.12526e-15 
cos 14 th = 8, sin 14 th = -4.3586e-15 
cos 15 th = -1.25191e-15, sin 15 th = -9.75936e-16 
antialias and normalize cos 2 th FFT 
cos 0 th = 1.2448e-17, sin 0 th = -0 
cos 1 th = -1.56489e-16, sin 1 th = -1.21992e-16 
cos 2 th = 1, sin 2 th = 7.50492e-16 
cos 3 th = 2.64528e-17, sin 3 th = 1.40658e-16 
cos 4 th = 3.06152e-17, sin 4 th = 5.55112e-17 
cos 5 th = 6.014e-17, sin 5 th = 1.40658e-16 
cos 6 th = 2.77556e-17, sin 6 th = 5.0557e-16 
cos 7 th = 6.98965e-17, sin 7 th = -1.21992e-16 
check: cos 2 th = 1.0 


1.0 + cos 2 th data 
[0] real = 2, imag = 0 
[1] real = 1.70711, imag = 0 
[2] real = 1, imag = 0 
[3] real = 0.292893, imag = 0 
[4] real = 0, imag = 0 
[5] real = 0.292893, imag = 0 
[6] real = 1, imag = 0 
[7] real = 1.70711, imag = 0 
[8] real = 2, imag = 0 
[9] real = 1.70711, imag = 0 
[10] real = 1, imag = 0 
[11] real = 0.292893, imag = 0 
[12] real = 0, imag = 0 
[13] real = 0.292893, imag = 0 
[14] real = 1, imag = 0 
[15] real = 1.70711, imag = 0 
cos 2 th FFT 
cos 0 th = 16, sin 0 th = 0 
cos 1 th = -1.46719e-15, sin 1 th = 8.73365e-16 
cos 2 th = 8, sin 2 th = -1.57389e-14 
cos 3 th = 1.96868e-16, sin 3 th = -1.08278e-15 
cos 4 th = 8.88178e-16, sin 4 th = 0 
cos 5 th = 4.31169e-16, sin 5 th = -1.08278e-15 
cos 6 th = 4.44089e-15, sin 6 th = 6.31841e-15 
cos 7 th = 8.39153e-16, sin 7 th = 8.73365e-16 
cos 8 th = -8.88178e-16, sin 8 th = 0 
cos 9 th = 8.39153e-16, sin 9 th = -8.73365e-16 
cos 10 th = -2.66454e-15, sin 10 th = 1.44067e-14 
cos 11 th = 4.31169e-16, sin 11 th = 1.08278e-15 
cos 12 th = 8.88178e-16, sin 12 th = 0 
cos 13 th = 1.96868e-16, sin 13 th = 1.08278e-15 
cos 14 th = 8, sin 14 th = -4.98614e-15 
cos 15 th = -1.46719e-15, sin 15 th = -8.73365e-16 
antialias and normalize 1.0 + cos 2 th FFT 
cos 0 th = 1, sin 0 th = -0 
cos 1 th = -1.83399e-16, sin 1 th = -1.09171e-16 
cos 2 th = 1, sin 2 th = 6.72049e-16 
cos 3 th = 2.46085e-17, sin 3 th = 1.35347e-16 
cos 4 th = 1.11022e-16, sin 4 th = -0 
cos 5 th = 5.38961e-17, sin 5 th = 1.35347e-16 
cos 6 th = 1.11022e-16, sin 6 th = 5.05516e-16 
cos 7 th = 1.04894e-16, sin 7 th = -1.09171e-16 
check: cos 0 th = 1.0  and  cos 2 th = 1.0 


square wave data, cos series 
[0] real = 1, imag = 0 
[1] real = 1, imag = 0 
[2] real = 1, imag = 0 
[3] real = 1, imag = 0 
[4] real = 0, imag = 0 
[5] real = -1, imag = 0 
[6] real = -1, imag = 0 
[7] real = -1, imag = 0 
[8] real = -1, imag = 0 
[9] real = -1, imag = 0 
[10] real = -1, imag = 0 
[11] real = -1, imag = 0 
[12] real = 0, imag = 0 
[13] real = 1, imag = 0 
[14] real = 1, imag = 0 
[15] real = 1, imag = 0 
square wave generate, cos series 
[0] real = 1.03526 
[1] real = 1.02635 
[2] real = 0.997308 
[3] real = 0.928841 
[4] real = 7.51019e-16 
[5] real = -0.928841 
[6] real = -0.997308 
[7] real = -1.02635 
[8] real = -1.03526 
[9] real = -1.02635 
[10] real = -0.997308 
[11] real = -0.928841 
[12] real = -2.43612e-15 
[13] real = 0.928841 
[14] real = 0.997308 
[15] real = 1.02635 
square wave FFT cos series 
cos 0 th = 0, sin 0 th = 0 
cos 1 th = 10.0547, sin 1 th = -3.58659e-14 
cos 2 th = 0, sin 2 th = 0 
cos 3 th = -2.99321, sin 3 th = 1.22318e-14 
cos 4 th = 0, sin 4 th = 0 
cos 5 th = 1.33636, sin 5 th = 2.04036e-15 
cos 6 th = 0, sin 6 th = 0 
cos 7 th = -0.397825, sin 7 th = 4.08733e-15 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = -0.397825, sin 9 th = 1.68148e-14 
cos 10 th = 0, sin 10 th = 0 
cos 11 th = 1.33636, sin 11 th = -2.63662e-15 
cos 12 th = 0, sin 12 th = 0 
cos 13 th = -2.99321, sin 13 th = 1.70108e-14 
cos 14 th = 0, sin 14 th = 0 
cos 15 th = 10.0547, sin 15 th = -1.36825e-14 
antialias and normalize square wave FFT cos
cos 0 th = 0, sin 0 th = -0 
cos 1 th = 1.25683, sin 1 th = 1.38646e-15 
cos 2 th = 0, sin 2 th = -0 
cos 3 th = -0.374151, sin 3 th = 2.98688e-16 
cos 4 th = 0, sin 4 th = -0 
cos 5 th = 0.167045, sin 5 th = -2.92311e-16 
cos 6 th = 0, sin 6 th = -0 
cos 7 th = -0.0497281, sin 7 th = 7.95466e-16 
check cos 1 th = 1.27, cos 3 th = -0.42, cos 5 th = 0.25 
square wave inverse FFT, cos 
rdat 0 = 1 
rdat 1 = 1 
rdat 2 = 1 
rdat 3 = 1 
rdat 4 = 9.71993e-32 
rdat 5 = -1 
rdat 6 = -1 
rdat 7 = -1 
rdat 8 = -1 
rdat 9 = -1 
rdat 10 = -1 
rdat 11 = -1 
rdat 12 = -9.71993e-32 
rdat 13 = 1 
rdat 14 = 1 
rdat 15 = 1 


square wave data, for sin series 
[0] real = 0, imag = 0 
[1] real = 1, imag = 0 
[2] real = 1, imag = 0 
[3] real = 1, imag = 0 
[4] real = 1, imag = 0 
[5] real = 1, imag = 0 
[6] real = 1, imag = 0 
[7] real = 1, imag = 0 
[8] real = 0, imag = 0 
[9] real = -1, imag = 0 
[10] real = -1, imag = 0 
[11] real = -1, imag = 0 
[12] real = -1, imag = 0 
[13] real = -1, imag = 0 
[14] real = -1, imag = 0 
[15] real = -1, imag = 0 
square wave generate, using sin 
[0] real = 0 
[1] real = 0.928841 
[2] real = 0.997308 
[3] real = 1.02635 
[4] real = 1.03526 
[5] real = 1.02635 
[6] real = 0.997308 
[7] real = 0.928841 
[8] real = 1.50204e-15 
[9] real = -0.928841 
[10] real = -0.997308 
[11] real = -1.02635 
[12] real = -1.03526 
[13] real = -1.02635 
[14] real = -0.997308 
[15] real = -0.928841 
square wave FFT sin series 
cos 0 th = 0, sin 0 th = 0 
cos 1 th = -3.98171e-14, sin 1 th = -10.0547 
cos 2 th = 0, sin 2 th = 0 
cos 3 th = -8.5949e-15, sin 3 th = -2.99321 
cos 4 th = 0, sin 4 th = 0 
cos 5 th = -1.07499e-15, sin 5 th = -1.33636 
cos 6 th = 0, sin 6 th = 0 
cos 7 th = -7.55852e-16, sin 7 th = -0.397825 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = 1.39897e-14, sin 9 th = 0.397825 
cos 10 th = 0, sin 10 th = 0 
cos 11 th = 5.77587e-15, sin 11 th = 1.33636 
cos 12 th = 0, sin 12 th = 0 
cos 13 th = 1.3978e-14, sin 13 th = 2.99321 
cos 14 th = 0, sin 14 th = 0 
cos 15 th = 1.64992e-14, sin 15 th = 10.0547 
antialias and normalize square wave FFT sin
cos 0 th = 0, sin 0 th = -0 
cos 1 th = -1.45736e-15, sin 1 th = 1.25683 
cos 2 th = 0, sin 2 th = -0 
cos 3 th = 3.36441e-16, sin 3 th = 0.374151 
cos 4 th = 0, sin 4 th = -0 
cos 5 th = 2.93805e-16, sin 5 th = 0.167045 
cos 6 th = 0, sin 6 th = -0 
cos 7 th = 8.27118e-16, sin 7 th = 0.0497281 
check sin 1 th = 1.27, sin 3 th = 0.42, sin 5 th = 0.25 
square wave inverse FFT, sin series 
rdat 0 = 0 
rdat 1 = 1 
rdat 2 = 1 
rdat 3 = 1 
rdat 4 = 1 
rdat 5 = 1 
rdat 6 = 1 
rdat 7 = 1 
rdat 8 = 0 
rdat 9 = -1 
rdat 10 = -1 
rdat 11 = -1 
rdat 12 = -1 
rdat 13 = -1 
rdat 14 = -1 
rdat 15 = -1 


square wave data, for sin, no zero 
          note: cos values come in 
[0] real = 1, imag = 0 
[1] real = 1, imag = 0 
[2] real = 1, imag = 0 
[3] real = 1, imag = 0 
[4] real = 1, imag = 0 
[5] real = 1, imag = 0 
[6] real = 1, imag = 0 
[7] real = 1, imag = 0 
[8] real = -1, imag = 0 
[9] real = -1, imag = 0 
[10] real = -1, imag = 0 
[11] real = -1, imag = 0 
[12] real = -1, imag = 0 
[13] real = -1, imag = 0 
[14] real = -1, imag = 0 
[15] real = -1, imag = 0 
square wave FFT sin, no zero 
cos 0 th = 0, sin 0 th = 0 
cos 1 th = 2, sin 1 th = -10.0547 
cos 2 th = 0, sin 2 th = 0 
cos 3 th = 2, sin 3 th = -2.99321 
cos 4 th = 0, sin 4 th = 0 
cos 5 th = 2, sin 5 th = -1.33636 
cos 6 th = 0, sin 6 th = 0 
cos 7 th = 2, sin 7 th = -0.397825 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = 2, sin 9 th = 0.397825 
cos 10 th = 0, sin 10 th = 0 
cos 11 th = 2, sin 11 th = 1.33636 
cos 12 th = 0, sin 12 th = 0 
cos 13 th = 2, sin 13 th = 2.99321 
cos 14 th = 0, sin 14 th = 0 
cos 15 th = 2, sin 15 th = 10.0547 
antialias and normalize square wave FFT sin
cos 0 th = 0, sin 0 th = -0 
cos 1 th = 0.25, sin 1 th = 1.25683 
cos 2 th = 0, sin 2 th = -0 
cos 3 th = 0.25, sin 3 th = 0.374151 
cos 4 th = 0, sin 4 th = -0 
cos 5 th = 0.25, sin 5 th = 0.167045 
cos 6 th = 0, sin 6 th = -0 
cos 7 th = 0.25, sin 7 th = 0.0497281 
check sin 1 th = 1.27, sin 3 th = -0.42, sin 5 th = 0.25 
square wave inverse FFT, sin no zero 
rdat 0 = 1 
rdat 1 = 1 
rdat 2 = 1 
rdat 3 = 1 
rdat 4 = 1 
rdat 5 = 1 
rdat 6 = 1 
rdat 7 = 1 
rdat 8 = -1 
rdat 9 = -1 
rdat 10 = -1 
rdat 11 = -1 
rdat 12 = -1 
rdat 13 = -1 
rdat 14 = -1 
rdat 15 = -1 


triangle wave data, cos series 
[0] real = 1, imag = 0 
[1] real = 0.75, imag = 0 
[2] real = 0.5, imag = 0 
[3] real = 0.25, imag = 0 
[4] real = 0, imag = 0 
[5] real = -0.25, imag = 0 
[6] real = -0.5, imag = 0 
[7] real = -0.75, imag = 0 
[8] real = -1, imag = 0 
[9] real = -0.75, imag = 0 
[10] real = -0.5, imag = 0 
[11] real = -0.25, imag = 0 
[12] real = 0, imag = 0 
[13] real = 0.25, imag = 0 
[14] real = 0.5, imag = 0 
[15] real = 0.75, imag = 0 
triangle wave generate, cos series 
[0] real = 0.977507 
[1] real = 0.75159 
[2] real = 0.501724 
[3] real = 0.250988 
[4] real = 8.55373e-18 
[5] real = -0.250988 
[6] real = -0.501724 
[7] real = -0.75159 
[8] real = -0.977507 
[9] real = -0.75159 
[10] real = -0.501724 
[11] real = -0.250988 
[12] real = -2.57621e-16 
[13] real = 0.250988 
[14] real = 0.501724 
[15] real = 0.75159 
triangle wave FFT, cos series 
cos 0 th = 0, sin 0 th = 0 
cos 1 th = 6.56854, sin 1 th = -2.07964e-14 
cos 2 th = 0, sin 2 th = 0 
cos 3 th = 0.809957, sin 3 th = 1.68123e-15 
cos 4 th = 0, sin 4 th = 0 
cos 5 th = 0.361616, sin 5 th = 5.54707e-15 
cos 6 th = 0, sin 6 th = 0 
cos 7 th = 0.259892, sin 7 th = 3.0027e-15 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = 0.259892, sin 9 th = 1.12708e-14 
cos 10 th = 0, sin 10 th = 0 
cos 11 th = 0.361616, sin 11 th = 3.11636e-15 
cos 12 th = 0, sin 12 th = 0 
cos 13 th = 0.809957, sin 13 th = 3.97851e-15 
cos 14 th = 0, sin 14 th = 0 
cos 15 th = 6.56854, sin 15 th = -7.80029e-15 
antialias and normalize triangle wave FFT cos
cos 0 th = 0, sin 0 th = -0 
cos 1 th = 0.821067, sin 1 th = 8.12257e-16 
cos 2 th = 0, sin 2 th = -0 
cos 3 th = 0.101245, sin 3 th = 1.4358e-16 
cos 4 th = 0, sin 4 th = -0 
cos 5 th = 0.045202, sin 5 th = -1.51919e-16 
cos 6 th = 0, sin 6 th = -0 
cos 7 th = 0.0324864, sin 7 th = 5.16758e-16 
check cos 1 th = 0.81, cos 3 th = 0.09, cos 5 th = 0.03 


triangle wave data, sin series 
[0] real = 0, imag = 0 
[1] real = 0.25, imag = 0 
[2] real = 0.5, imag = 0 
[3] real = 0.75, imag = 0 
[4] real = 1, imag = 0 
[5] real = 0.75, imag = 0 
[6] real = 0.5, imag = 0 
[7] real = 0.25, imag = 0 
[8] real = 0, imag = 0 
[9] real = -0.25, imag = 0 
[10] real = -0.5, imag = 0 
[11] real = -0.75, imag = 0 
[12] real = -1, imag = 0 
[13] real = -0.75, imag = 0 
[14] real = -0.5, imag = 0 
[15] real = -0.25, imag = 0 
triangle wave generate, sin series 
[0] real = 0 
[1] real = 0.250988 
[2] real = 0.501724 
[3] real = 0.75159 
[4] real = 0.977507 
[5] real = 0.75159 
[6] real = 0.501724 
[7] real = 0.250988 
[8] real = 1.71075e-17 
[9] real = -0.250988 
[10] real = -0.501724 
[11] real = -0.75159 
[12] real = -0.977507 
[13] real = -0.75159 
[14] real = -0.501724 
[15] real = -0.250988 
triangle wave FFT 
cos 0 th = 0, sin 0 th = 0 
cos 1 th = -2.41472e-14, sin 1 th = -6.56854 
cos 2 th = 0, sin 2 th = 0 
cos 3 th = 1.55193e-15, sin 3 th = 0.809957 
cos 4 th = 0, sin 4 th = 0 
cos 5 th = 2.31805e-15, sin 5 th = -0.361616 
cos 6 th = 0, sin 6 th = 0 
cos 7 th = 1.08583e-16, sin 7 th = 0.259892 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = 8.00244e-15, sin 9 th = -0.259892 
cos 10 th = 0, sin 10 th = 0 
cos 11 th = 2.69641e-16, sin 11 th = 0.361616 
cos 12 th = 0, sin 12 th = 0 
cos 13 th = 9.02354e-16, sin 13 th = -0.809957 
cos 14 th = 0, sin 14 th = 0 
cos 15 th = 1.09942e-14, sin 15 th = 6.56854 
antialias and normalize triangle wave FFT sin
cos 0 th = 0, sin 0 th = -0 
cos 1 th = -8.22062e-16, sin 1 th = 0.821067 
cos 2 th = 0, sin 2 th = -0 
cos 3 th = 1.53393e-16, sin 3 th = -0.101245 
cos 4 th = 0, sin 4 th = -0 
cos 5 th = 1.61731e-16, sin 5 th = 0.045202 
cos 6 th = 0, sin 6 th = -0 
cos 7 th = 5.06939e-16, sin 7 th = -0.0324864 
check sin 1 th = 0.81, sin 3 th = -0.09, sin 5 th = 0.03 


saw tooth wave data, sin series 
[0] real = 0, imag = 0 
[1] real = 0.125, imag = 0 
[2] real = 0.25, imag = 0 
[3] real = 0.375, imag = 0 
[4] real = 0.5, imag = 0 
[5] real = 0.625, imag = 0 
[6] real = 0.75, imag = 0 
[7] real = 0.875, imag = 0 
[8] real = 0, imag = 0 
[9] real = -0.875, imag = 0 
[10] real = -0.75, imag = 0 
[11] real = -0.625, imag = 0 
[12] real = -0.5, imag = 0 
[13] real = -0.375, imag = 0 
[14] real = -0.25, imag = 0 
[15] real = -0.125, imag = 0 
sawtooth wave generate, sin series 
[0] real = 0 
[1] real = 0.109498 
[2] real = 0.256583 
[3] real = 0.390342 
[4] real = 0.484124 
[5] real = 0.610362 
[6] real = 0.78811 
[7] real = 0.881255 
[8] real = 1.72763e-15 
[9] real = -0.881255 
[10] real = -0.78811 
[11] real = -0.610362 
[12] real = -0.484124 
[13] real = -0.390342 
[14] real = -0.256583 
[15] real = -0.109498 
saw tooth wave FFT, sin series 
cos 0 th = 0, sin 0 th = 0 
cos 1 th = -1.99085e-14, sin 1 th = -5.02734 
cos 2 th = 6.64692e-15, sin 2 th = 2.41421 
cos 3 th = -4.29745e-15, sin 3 th = -1.49661 
cos 4 th = 1.75099e-15, sin 4 th = 1 
cos 5 th = -5.37493e-16, sin 5 th = -0.668179 
cos 6 th = 7.50159e-16, sin 6 th = 0.414214 
cos 7 th = -3.77926e-16, sin 7 th = -0.198912 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = 6.99487e-15, sin 9 th = 0.198912 
cos 10 th = -3.19265e-15, sin 10 th = -0.414214 
cos 11 th = 2.88794e-15, sin 11 th = 0.668179 
cos 12 th = -1.75099e-15, sin 12 th = -1 
cos 13 th = 6.98898e-15, sin 13 th = 1.49661 
cos 14 th = -4.20443e-15, sin 14 th = -2.41421 
cos 15 th = 8.24961e-15, sin 15 th = 5.02734 
antialias and normalize saw tooth wave FFT sin
cos 0 th = 0, sin 0 th = -0 
cos 1 th = -7.28682e-16, sin 1 th = 0.628417 
cos 2 th = 1.52656e-16, sin 2 th = -0.301777 
cos 3 th = 1.68221e-16, sin 3 th = 0.187076 
cos 4 th = 0, sin 4 th = -0.125 
cos 5 th = 1.46903e-16, sin 5 th = 0.0835223 
cos 6 th = -1.52656e-16, sin 6 th = -0.0517767 
cos 7 th = 4.13559e-16, sin 7 th = 0.024864 
check sin 1 th = 0.64, sin 3 th = -0.32, sin 5 th = 0.21 



set up convolution test 
square wave data, a , upper half zero
[0] real = 1, imag = 0 
[1] real = 1, imag = 0 
[2] real = 1, imag = 0 
[3] real = 1, imag = 0 
[4] real = 0, imag = 0 
[5] real = -1, imag = 0 
[6] real = -1, imag = 0 
[7] real = -1, imag = 0 
[8] real = -1, imag = 0 
[9] real = -1, imag = 0 
[10] real = -1, imag = 0 
[11] real = -1, imag = 0 
[12] real = 0, imag = 0 
[13] real = 1, imag = 0 
[14] real = 1, imag = 0 
[15] real = 1, imag = 0 
[16] real = 0, imag = 0 
[17] real = 0, imag = 0 
[18] real = 0, imag = 0 
[19] real = 0, imag = 0 
[20] real = 0, imag = 0 
[21] real = 0, imag = 0 
[22] real = 0, imag = 0 
[23] real = 0, imag = 0 
[24] real = 0, imag = 0 
[25] real = 0, imag = 0 
[26] real = 0, imag = 0 
[27] real = 0, imag = 0 
[28] real = 0, imag = 0 
[29] real = 0, imag = 0 
[30] real = 0, imag = 0 
[31] real = 0, imag = 0 
AT FFT of sauare wave
cos 0 th = 0, sin 0 th = 0 
cos 1 th = 1, sin 1 th = 4.20558 
cos 2 th = 10.0547, sin 2 th = -3.59086e-14 
cos 3 th = 1, sin 3 th = -7.9586 
cos 4 th = 0, sin 4 th = 0 
cos 5 th = 1, sin 5 th = -4.51668 
cos 6 th = -2.99321, sin 6 th = 1.26331e-14 
cos 7 th = 1, sin 7 th = 0.504721 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = 1, sin 9 th = 0.339936 
cos 10 th = 1.33636, sin 10 th = 3.29993e-15 
cos 11 th = 1, sin 11 th = -1.29042 
cos 12 th = 0, sin 12 th = 0 
cos 13 th = 1, sin 13 th = -0.732344 
cos 14 th = -0.397825, sin 14 th = 3.66233e-15 
cos 15 th = 1, sin 15 th = 0.0407965 
cos 16 th = 0, sin 16 th = 0 
cos 17 th = 1, sin 17 th = -0.0407965 
cos 18 th = -0.397825, sin 18 th = 1.62294e-14 
cos 19 th = 1, sin 19 th = 0.732344 
cos 20 th = 0, sin 20 th = 0 
cos 21 th = 1, sin 21 th = 1.29042 
cos 22 th = 1.33636, sin 22 th = -2.7239e-15 
cos 23 th = 1, sin 23 th = -0.339936 
cos 24 th = 0, sin 24 th = 0 
cos 25 th = 1, sin 25 th = -0.504721 
cos 26 th = -2.99321, sin 26 th = 1.63792e-14 
cos 27 th = 1, sin 27 th = 4.51668 
cos 28 th = 0, sin 28 th = 0 
cos 29 th = 1, sin 29 th = 7.9586 
cos 30 th = 10.0547, sin 30 th = -1.35715e-14 
cos 31 th = 1, sin 31 th = -4.20558 
triangle wave data, b , upper half zero 
[0] real = 1, imag = 0 
[1] real = 0.75, imag = 0 
[2] real = 0.5, imag = 0 
[3] real = 0.25, imag = 0 
[4] real = 0, imag = 0 
[5] real = -0.25, imag = 0 
[6] real = -0.5, imag = 0 
[7] real = -0.75, imag = 0 
[8] real = -1, imag = 0 
[9] real = -0.75, imag = 0 
[10] real = -0.5, imag = 0 
[11] real = -0.25, imag = 0 
[12] real = 0, imag = 0 
[13] real = 0.25, imag = 0 
[14] real = 0.5, imag = 0 
[15] real = 0.75, imag = 0 
[16] real = 0, imag = 0 
[17] real = 0, imag = 0 
[18] real = 0, imag = 0 
[19] real = 0, imag = 0 
[20] real = 0, imag = 0 
[21] real = 0, imag = 0 
[22] real = 0, imag = 0 
[23] real = 0, imag = 0 
[24] real = 0, imag = 0 
[25] real = 0, imag = 0 
[26] real = 0, imag = 0 
[27] real = 0, imag = 0 
[28] real = 0, imag = 0 
[29] real = 0, imag = 0 
[30] real = 0, imag = 0 
[31] real = 0, imag = 0 
BT FFT of sauare wave
cos 0 th = 0, sin 0 th = 0 
cos 1 th = 1, sin 1 th = 2.85769 
cos 2 th = 6.56854, sin 2 th = -2.05955e-14 
cos 3 th = 1, sin 3 th = -4.77997 
cos 4 th = 0, sin 4 th = 0 
cos 5 th = 1, sin 5 th = -1.30835 
cos 6 th = 0.809957, sin 6 th = 1.88313e-15 
cos 7 th = 1, sin 7 th = -1.5291 
cos 8 th = 0, sin 8 th = 0 
cos 9 th = 1, sin 9 th = -0.61149 
cos 10 th = 0.361616, sin 10 th = 6.12629e-15 
cos 11 th = 1, sin 11 th = -0.695224 
cos 12 th = 0, sin 12 th = 0 
cos 13 th = 1, sin 13 th = -0.166844 
cos 14 th = 0.259892, sin 14 th = 2.76244e-15 
cos 15 th = 1, sin 15 th = -0.224704 
cos 16 th = 0, sin 16 th = 0 
cos 17 th = 1, sin 17 th = 0.224704 
cos 18 th = 0.259892, sin 18 th = 1.07559e-14 
cos 19 th = 1, sin 19 th = 0.166844 
cos 20 th = 0, sin 20 th = 0 
cos 21 th = 1, sin 21 th = 0.695224 
cos 22 th = 0.361616, sin 22 th = 3.07146e-15 
cos 23 th = 1, sin 23 th = 0.61149 
cos 24 th = 0, sin 24 th = 0 
cos 25 th = 1, sin 25 th = 1.5291 
cos 26 th = 0.809957, sin 26 th = 3.71328e-15 
cos 27 th = 1, sin 27 th = 1.30835 
cos 28 th = 0, sin 28 th = 0 
cos 29 th = 1, sin 29 th = 4.77997 
cos 30 th = 6.56854, sin 30 th = -7.71703e-15 
cos 31 th = 1, sin 31 th = -2.85769 
convolution of a with b 
real[0] = 1 
real[1] = 1.75 
real[2] = 2.25 
real[3] = 2.5 
real[4] = 1.5 
real[5] = -0.5 
real[6] = -2.25 
real[7] = -3.75 
real[8] = -5 
real[9] = -5.5 
real[10] = -5.25 
real[11] = -4.25 
real[12] = -1.5 
real[13] = 2.25 
real[14] = 5.25 
real[15] = 7.5