Periodicity properties of complex exponentials in discrete/continuous time

[Jncik]

hello I have a question about the periodicity of complex exponential in discrete time and continuous time

in continuous time we have x(t) = e^(j*ω*t)

while in discrete time we have x[n] = e^j(ω*n)

for the first in order to show whether its periodic we say

x(t+T) = x(t) => ...=> T = 2*k*π/ω and the fundamental period is for k = 0 hence T0 = 2*π/ω i can understand this

for the second my book says that

x[n+N] = x[n] => ... => e^(j*ω*N) = 1 => ω/2*π = m/N

hence ω/2π must be rational

what i don't get is why for the discrete time it must be rational while for the continuous time it can be just 2k*π/ω

I mean, we could find in continuous time the same thing like

Τ*ω = 2*π*m => ω/2π = m/T

hence ω/2π must be rational.. but why don't we say this?

can someone please tell me what's the difference? thanks

 

[tiny-tim]

Hi Jncik!

(try using the X² icon just above the Reply box )

… for the second my book says that

x[n+N] = x[n] => ... => e^(j*ω*N) = 1 => ω/2*π = m/N

hence ω/2π must be rational

what i don't get is why for the discrete time it must be rational while for the continuous time it can be just 2k*π/ω …

because if ω/2π = 1/√2, say, then e^j*ω*N = e^j*π*N/√2 will never be 1

(because j*π*N/√2 will never be a multiple of 2π)

 

[Jncik]

Hi Jncik!

(try using the X² icon just above the Reply box )


because if ω/2π = 1/√2, say, then e^j*ω*N = e^j*π*N/√2 will never be 1

(because j*π*N/√2 will never be a multiple of 2π)

hi thanks for your answer

but why doesn't the same thing apply to the case where it is continuous?

i mean why in continuous time we have no such restrictions?

 

[tiny-tim]

because if N doesn't have to be a whole number, π*N/√2 can be a multiple of 2π …

eg with N = 2√2 ​

 

[Jncik]

i see now, thanks my friend you helped me a lot :)