How to implement Complex Exponential Function Using Matlab ?

format longE
F = 40E+3;
N = (0:20).';
omega1 = 2*pi*F;                                    % Guessing: 'W1'
omega2 = 4*pi*F;                                    % Guessing: 'w2'
X = exp(1j*omega1*N) + exp(1j*omega2*N)
X = 
000000000000000e+00 + 0.000000000000000e+00i
000000000000000e+00 - 1.165637649496438e-11i
000000000000000e+00 - 2.331275298992875e-11i
000000000000000e+00 - 3.496912948489312e-11i
000000000000000e+00 - 4.662550597985751e-11i
000000000000000e+00 + 2.909640830059826e-10i
000000000000000e+00 - 6.993825896978625e-11i
000000000000000e+00 - 4.308406009455551e-10i
000000000000000e+00 - 9.325101195971502e-11i
000000000000000e+00 + 2.443385770261251e-10i
000000000000000e+00 + 5.819281660119652e-10i
000000000000000e+00 - 4.774661069254125e-10i
000000000000000e+00 - 1.398765179395725e-10i
000000000000000e+00 + 1.977130710462676e-10i
000000000000000e+00 - 8.616812018911102e-10i
000000000000000e+00 - 5.240916129052701e-10i
000000000000000e+00 - 1.865020239194300e-10i
000000000000000e+00 - 1.245896296856808e-09i
000000000000000e+00 + 4.886771540522501e-10i
000000000000000e+00 - 5.707171188851276e-10i
000000000000000e+00 + 1.163856332023930e-09i

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Star Strider 2023-11-3

编辑：Star Strider 2023-11-3

在 MATLAB Online 中打开

O.K., so ‘w1’ and ‘w2’ are actually the same thing.

My code then becomes:

format longE
F = 40E+3;
N = (0:20).';
omega1 = 2*pi*F;                                    % Guessing: 'W1'
omega2 = 2*pi*F;                                    % Guessing: 'w2'
X = exp(1j*omega1*N) + exp(1j*omega2*N)
X = 
000000000000000e+00 + 0.000000000000000e+00i
000000000000000e+00 - 7.770917663309584e-12i
000000000000000e+00 - 1.554183532661917e-11i
000000000000000e+00 - 2.331275298992875e-11i
000000000000000e+00 - 3.108367065323834e-11i
000000000000000e+00 + 1.939760553373217e-10i
000000000000000e+00 - 4.662550597985750e-11i
000000000000000e+00 - 2.872270672970367e-10i
000000000000000e+00 - 6.216734130647667e-11i
000000000000000e+00 + 1.628923846840834e-10i
000000000000000e+00 + 3.879521106746434e-10i
000000000000000e+00 - 3.183107379502750e-10i
000000000000000e+00 - 9.325101195971500e-11i
000000000000000e+00 + 1.318087140308451e-10i
000000000000000e+00 - 5.744541345940735e-10i
000000000000000e+00 - 3.493944086035134e-10i
000000000000000e+00 - 1.243346826129533e-10i
000000000000000e+00 - 8.305975312378718e-10i
000000000000000e+00 + 3.257847693681668e-10i
000000000000000e+00 - 3.804780792567517e-10i
000000000000000e+00 + 7.759042213492868e-10i
format shortE
X
X = 
0000e+00 + 0.0000e+00i
0000e+00 - 7.7709e-12i
0000e+00 - 1.5542e-11i
0000e+00 - 2.3313e-11i
0000e+00 - 3.1084e-11i
0000e+00 + 1.9398e-10i
0000e+00 - 4.6626e-11i
0000e+00 - 2.8723e-10i
0000e+00 - 6.2167e-11i
0000e+00 + 1.6289e-10i
0000e+00 + 3.8795e-10i
0000e+00 - 3.1831e-10i
0000e+00 - 9.3251e-11i
0000e+00 + 1.3181e-10i
0000e+00 - 5.7445e-10i
0000e+00 - 3.4939e-10i
0000e+00 - 1.2433e-10i
0000e+00 - 8.3060e-10i
0000e+00 + 3.2578e-10i
0000e+00 - 3.8048e-10i
0000e+00 + 7.7590e-10i
whos('X')
  Name       Size            Bytes  Class     Attributes

  X         21x1               336  double    complex   

The imaginary parts are very close to zero, so using a more restricted format option, they will appear as zero. Using extended precision, this is readily apparent, and with greater precision or exponential notation (or both) this becomes clear.

EDIT — (3 Nov 2023 at 20:43)

My code does exactly what you asked, both in the original and in the edit. The ‘X’ result is a (21x1) column vector, showing both the real and imaginary parts. The size is (21x1) because ‘N’ includes zero, for a total of 21 elements. If you want it to be exactly (20x1) use the linspace function —

format longE
F = 40E+3;
N = linspace(0, 20, 20).'
N = 20×1
                         0
     1.052631578947368e+00
     2.105263157894737e+00
     3.157894736842105e+00
     4.210526315789473e+00
     5.263157894736842e+00
     6.315789473684211e+00
     7.368421052631579e+00
     8.421052631578947e+00
     9.473684210526315e+00
omega1 = 2*pi*F;
omega2 = 2*pi*F;
X = exp(1j*omega1*N) + exp(1j*omega2*N)
X = 
      2.000000000000000e+00 + 0.000000000000000e+00i
     -1.651586909120879e-01 + 1.993168986016039e+00i
     -1.972722606816205e+00 - 3.291891804969826e-01i
      4.909709741865338e-01 - 1.938800531902734e+00i
      1.891634483443725e+00 + 6.493989382856979e-01i
     -8.033908495827032e-01 + 1.831546653188715e+00i
     -1.758947502506326e+00 - 9.518947859016555e-01i
      1.093896316053294e+00 - 1.674332956650209e+00i
      1.578281018953408e+00 + 1.228425425172970e+00i
     -1.354563142692437e+00 + 1.471447821860900e+00i
     -1.354563142806782e+00 - 1.471447821755638e+00i
      1.578281018857948e+00 - 1.228425425295617e+00i
      1.093896316573241e+00 + 1.674332956310511e+00i
     -1.758947502432355e+00 + 9.518947860383419e-01i
     -8.033908497250313e-01 - 1.831546653126284e+00i
      1.891634483393261e+00 - 6.493989384326954e-01i
      4.909709747886086e-01 + 1.938800531750268e+00i
     -1.972722606790624e+00 + 3.291891806502813e-01i
     -1.651586924591879e-01 - 1.993168985887843e+00i
      2.000000000000000e+00 + 7.759042213492868e-10i
format short
X
X = 
   2.0000 + 0.0000i
  -0.1652 + 1.9932i
  -1.9727 - 0.3292i
   0.4910 - 1.9388i
   1.8916 + 0.6494i
  -0.8034 + 1.8315i
  -1.7589 - 0.9519i
   1.0939 - 1.6743i
   1.5783 + 1.2284i
  -1.3546 + 1.4714i
  -1.3546 - 1.4714i
   1.5783 - 1.2284i
   1.0939 + 1.6743i
  -1.7589 + 0.9519i
  -0.8034 - 1.8315i
   1.8916 - 0.6494i
   0.4910 + 1.9388i
  -1.9727 + 0.3292i
  -0.1652 - 1.9932i
   2.0000 + 0.0000i