Complex signals is the same as natural signal
You refer to a

sinuasidal signal along with the complex signal derived form it

Using complex numbers to represent a sinusoid is simply a method of conveying magnitude, frequency, and phase. This is true whether we use the exponential notation or the cos(x) + j*sin(x) related by Euler's famous formula. The &quot;j&quot; multiplier is simply a phase shift of +pi/2. Your signal() and time_signal() functions are equivalent and therefore produce the same spectral result.