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the explanation needed is not for the typo,
how can you imagine the period to be complex, the cmplex notation is an easier notation for the periodic function, the phase is a reresentation for the time differene between the periodic signals, what is the relation you imagine between the phase and the period of the signals you represent.
some times we can say that the period of the signal is complex!!
for example, In the case you mentioned we can say the period is complex.
but the problem here is that the signal is complex. It means that you have to had both inphase and quadrature part. so if you have both of them you can say that you have a signal with complex period.
ie. in practice, when you have a I&Q demodulator you have the complex signal.
exp(2*pi*j)=1, then exp(x+2*pi*j)=exp(x) no periodicity at all
if you lost x in your formula, I mean exp(x+2*pi*j*x) then
exp(x+2*pi*j*x)=exp(x)*exp(2*pi*j*x), it is simply a prouduct of two functions, one periodic exp(2*pi*j), the other is real, the product has three cases
1. if x=0, periodic function
2. if x<0, the periodic function is decaying
3. if x>0, the periodic function is increasing
for x increasing the peak of the function is increasing
I cann't believe that the period can be complex, this is up to my understand, any understand may be wrong, please continue discussion, if any other examples please deliver.
I think this is a miss understanding!
from the mathematical point of view we can say a function is periodic if:
f(x+t)=f(x) for all x s.
in which t is the period.
so if look to the problem using this definition then we can find some functions that have complex period.
but if you look to the problem from physical point of view, then we can not imagine complex period.
so I think it is nessary to talk about the definition of periodicity.
It's right unless x is real, when img(exp(-x))=0 because exp(-x) is real.
I think the answer to the main problem is not strictly no, mathematically, it's certainly possible. But the period in general is defined as a real quantity and if you want to visualize a complex period this should not be just like the real period rather a different way to express some special property of a signal.