Aliasing and discrete sinusoids

[fisico30]

Hello Forum,

a continuous time, continuous amplitude sinusoid like sin(2pi*f*t) is 2pi periodic:

sin(2pi*f*t)=sin(2pi*f*t+m*2pi)

where m can be any positive or negative integer. Let's sample the sinusoid at a sampling frequency fs (sample interval is ts=1/fs) and get the discrete signal

x[n]=sin(2pi*f*n*ts)=sin(2pi*f*n*ts+2pi*m)=sin{2pi(f+m/(n*ts))*n*ts}

This book says: if we let m be an integer multiple of n, m=k*n, we can replace the ration m/n with k so that

x[n]=sin(2pi*f*n*ts)=sin{2pi(f+kf*s)n*ts}

What happens if m is not an integer multiple of n? Both m and n are integers (I get that), but I don't understand the constraint m=k*n...That does not seem general enough...

thanks
fisico30

 

[jti5017]

well in discrete time, x[n] will not be defined for non integer arguments