DFT: set freq axis

https://www.physics.ohio-state.edu/~dws/class/8820.uf/project_09/young.pdf
or use Periodogram which plots DFT with an option SampleRate to automatically define the frequency axis (search periodogram in Mathematica).

real data example delta t = 1/10

Clear["Global`*"];Remove["Global`*"]

freq of 30 radians/s divided into 200 time increments.

timeseries:=Table[N[Sin[omega0 delt n]+(RandomReal[]-1/2)],{n,0,M-1}];

delt=1/10;M=128;omega0=6.2;T=M delt;

ListLinePlot[timeseries]

powerspectrum=Abs[Fourier[timeseries]]^2;

The discrete Fourier transform, however, shows a strong peak at 30+1, and a symmetric peak at 201-30, reflecting the frequency component of the original signal near 30/200.

ListLinePlot[powerspectrum,PlotRange->All]

Now put the freq axis on this plot. 
The frequency values are 2 Pi n/T where n is an interger with  0 < n < M-1

omegavals=Table[2 Pi t/T,{t,0,M-1}];

Next plot the power spectrum. For delt = 1 the range of freq is from - Pi to Pi. Howerver, the power spectrum is an even function of omega s wew only plot the range from 0 to Pi.

Dimensions[powerspectrum]
Dimensions[omegavals]

omegavals[[128]]

ListPlot[{Transpose[{omegavals,powerspectrum}],Transpose[{omegavals,powerspectrum}]},PlotRange->{{0,omegavals[[128]]/2},{-1,25}},Joined->{True,False},AxesLabel->{"omega","P(omega)"}]