How to shift the frequency spectrum?

Question

Lets say we have a spectrum ranging from -X MHz to +X MHz. I would need to correct the frequency error in the spectrum by shifting the zero component to the middle (0 Hz).

If the output (the frequency spectrum) is calculated via FFT, as far as I know I can move the spectrum by adjusting the 'twiddle factors' (or coefficients, for complex data sine and cosine waves).

In the case of a size 1024 FFT (bin indexes from 0 to 1023), 0 Hz component should exist in bin number 511. However, due to possible frequency error the 0 hz component may actually be in bin 510 for example.

I cannot seem to find much information on this. Any help appreciated.

EDIT: Mistake in the question.

Answer 1

If the frequency shift that you want is a multiple of the bin spacing, as in your example, then you can easily effect the shift that you want by just rotating the FFT outputs by the number of bins that you need. In the more common case that the frequency offset is not an integer multiple of the bin spacing, then you can multiply the signal by a complex exponential function before doing the FFT.

So, if you determine that the center frequency component that you speak of is actually located at frequency $f_{offset}$ Hz in your data, and the data is sampled at rate $f_s$ Hz, then to shift the spectrum such that the component of interest is at zero frequency in the FFT output, you would do:

$$ x_{\text{shifted}}[n] = x[n] e^{-\frac{j 2 \pi f_{\text{offset}} n}{f_s}}, \ \ n = 0, 1, \ldots , N-1 $$

$$ Y_{\text{shifted}}[k] = FFT\left[x_{\text{shifted}}[n]\right] $$

Answer 2

well, the simple way is , if u have used the fourier to find the spectra , and u need to know its frequency by how much is it shifted, u can do one thing ..

1) find out the impulse response from that spectra

2) convolve it with a noise

3) see the signal that u obtaine4) take its FFT just to be sure , if it matches with the previous one

4)and see the spectra by averaging in to differnt parts... for this i can give u a algorithm in mathematica software, which is

reflect[a_] := Module[{n = Length[a]},
  RotateRight[a, Floor[n/2]]
  ]

freqAxis[len_] := Module[{},
   If[OddQ[len],
    Range[1, len] - (Ceiling[len/2.]),
    Range[1, len] - (1 + Ceiling[len/2.])
    ]
   ];

colors = {Black, Red, Blue, Brown , ColorData["Legacy", "DarkGreen"], 
   ColorData["Legacy", "Goldenrod"], ColorData["Legacy", "DeepPink"], 
   Cyan, Orange, Purple, ColorData["Legacy", "DeepSkyBlue"], Magenta};

specPlot[pieces_, pieceLen_, color_] := 
 Module[{data, spec, fAxis, pos},
  fAxis = freqAxis[pieceLen];
  data = Partition[Take[mysignal, pieces*pieceLen], pieceLen];
  spec = Total[Abs[Fourier[data]]^2]/pieces;
  spec = reflect[spec];
  Print["valley=", Nearest[spec, 1.0][[1]], " atPos=", 
   pos = Position[spec, Nearest[spec, 1.0][[1]]][[1, 1]], " atFpos=", 
   Position[fAxis, 0][[1, 1]], " atF=", fAxis[[pos]], " firstMax=", 
   Max[Take[spec, Round[pieceLen/2]]], " atF=", 
   fAxis[[Position[spec, Max[Take[spec, Round[pieceLen/2]]]][[1, 
     1]]]], " lastMax=", Max[Take[spec, -Round[pieceLen/2]]], " atF=",
    fAxis[[Position[spec, Max[Take[spec, -Round[pieceLen/2]]]][[1, 
     1]]]]];
  ListLinePlot[Transpose[{fAxis, spec}], PlotStyle -> colors[[color]],
    PlotLabel -> "N = " <> ToString[pieces], PlotRange -> All]
  ]

in this code , i have a argument to take pmsesignal , so u can use ur own signal instead of it..

i am not sure, how well i explained this, but this had worked in my case..

Cheers!