ad300d52-c397-4e74-9e56-ba1e09f31654 19 6 5 auto

&[DATE] &[TIME] - &[FILENAME]

Smath Limit of "fft" and/or "ifft" Transforms

Fourier complex  from sampled  Y vector [Numeric] Y fft N Y length : n 0 N range D n 1 + el 1 N sqrt / Y k 1 + el e i 2 π * n * N / (* k *^* (k 0 N 1 - sum * : for D 31 line :

Filter 126 - 314 mat transpose :

N 128 :

α 1.5 :

Fourier "ifft" from complex sampled vector [Numeric] v ifft N v length 1 - : n 0 N 1 - range I n 1 + el 1 N sqrt / v k 1 + el e i 2 π * n * N / (* k * -^* (k 0 N 1 - sum * : for I 31 line :

Dirac n 1 N range n Filter rows ≤ Dirac n el Filter n el (α^Re : Dirac n el 0 : if for Dirac 21 line :

V 1.137 1.106 0.228 i * - 1.016 0.439 i * - 0.875 0.617 i * - 0.694 0.748 i * - 0.488 0.823 i * - 0.275 0.839 i * - 0.072 0.796 i * - 0.105 - 0.7 i * - 0.242 - 0.563 i * - 0.328 - 0.397 i * - 0.359 - 0.22 i * - 0.334 - 0.048 i * - 0.257 - 0.102 i * + 0.139 - 0.216 i * + 0.007 0.283 i * + 0.166 0.297 i * + 0.321 0.256 i * + 0.455 0.164 i * + 0.553 0.031 i * + 0.605 0.132 i * - 0.602 0.309 i * - 0.544 0.482 i * - 0.431 0.636 i * - 0.273 0.754 i * - 0.081 0.825 i * - 0.129 - 0.839 i * - 0.341 - 0.793 i * - 0.536 - 0.689 i * - 0.699 - 0.533 i * - 0.816 - 0.337 i * - 0.877 - 0.115 i * - 0.877 - 0.115 i * + 0.816 - 0.337 i * + 0.699 - 0.533 i * + 0.536 - 0.689 i * + 0.341 - 0.793 i * + 0.129 - 0.839 i * + 0.081 0.825 i * + 0.273 0.754 i * + 0.431 0.636 i * + 0.544 0.482 i * + 0.602 0.309 i * + 0.605 0.132 i * + 0.553 0.031 i * - 0.455 0.164 i * - 0.321 0.256 i * - 0.166 0.297 i * - 0.007 0.283 i * - 0.139 - 0.216 i * - 0.257 - 0.102 i * - 0.334 - 0.048 i * + 0.359 - 0.22 i * + 0.328 - 0.397 i * + 0.242 - 0.563 i * + 0.105 - 0.7 i * + 0.072 0.796 i * + 0.275 0.839 i * + 0.488 0.823 i * + 0.694 0.748 i * + 0.875 0.617 i * + 1.016 0.439 i * + 1.106 0.228 i * + 631 mat :

To view Dirac padded vector, tabulate at right margin.

Both transforms "fft, ifft" are equally reasonably greedy... fft(Y),ifft(v)=>Numeric result: Optimiz NONE Auto fft[128] => 8 sec ifft[32] => 17 sec ifft(V>32) fails to compute. ifft(V =32) fails to display. The Smath recursive Σ is in cause. 'Q' display only ifft(V=16). 'Q' ifft(V>16) crashes Smath.

t0 1 time :

Optimiz NONE Auto F Dirac fft :

1 time t0 - 7 sec *

To view 'F', tabulate at right margin. This vector is copy/paste in 'V' ... to isolate.

V V 11611 submatrix :

t0 1 time :

Optimiz NONE Auto Q V ifft :

1 time t0 - 1 sec *

Numeric, Auto Q 1.006717 1.575013 i * - 0.099666 0.472518 i * - 0.107722 0.281551 i * - 0.119273 0.193936 i * - 0.129072 0.13915 i * - 0.136724 0.099351 i * - 0.14467 0.065165 i * - 0.151922 0.034315 i * - 0.160312 0.003235 i * - 0.169551 0.031678 i * + 0.181893 0.073273 i * + 0.200361 0.131045 i * + 0.234208 0.227422 i * + 0.325745 0.447901 i * + 1.235744 1.952916 i * + 151 mat