magnitude spectrum in DFT

Question

I am trying to write a small Discrete Fourier Transformation in Java to find the magnitude spectrum in a clear 400 Hz Sinus Signal (1 second as pcm signed-short)

So first I calculate the DFT for the complex values:

public void berechneDFT(int abtastwerte) {
        int i;

        int N = abtastwerte;
        ReX = new double[N/2+1];
        ImX = new double[N/2+1];

        TextFileOperator tfo = new TextFileOperator(file.substring(0, file.length()-4)+"_DFT.txt");

        try {
            tfo.openOutputStream();
            tfo.writeString("ReX      ImX\n");
        } catch (FileNotFoundException e) {
            e.printStackTrace();
        }

        // real-Anteil berechnen
        for (i=0, ReX[i] = 0, ImX[i] = 0; i <= N/2; i++)
        {
            for(int n=0; n < N; n++)
            {
                ReX[i] += x[n] * Math.cos( (2.0 * Math.PI * n * i) / (double) N);
                ImX[i] += - (x[n] * Math.sin( (2.0 * Math.PI * n * i) / (double) N));
            }

            tfo.writeString(ReX[i] +" "+ImX[i]+"\n");
        }

        x = null;   
        tfo.closeOutputStream();    // flush

        System.out.println("Anteile berechnet.");
    }

And then I try to calculate the magnitude Spectrum:

public void berechneBetragsSpektrum() {

        int N = ReX.length;

        TextFileOperator tfo = new TextFileOperator("betragsspektrum_400hz.txt");
        try {
            tfo.openOutputStream();
        } catch (FileNotFoundException e) {
            e.printStackTrace();
        }
        double powerAtFreq;
        int marker = 0;
        double maxPowerAtFreq = 0;

        for(int i=0; i < N; i++)
        {
            double A1 = ReX[i] * ReX[i];
            double A2 = ImX[i] * ImX[i];

            powerAtFreq = Math.sqrt(A1+A2);

            if(powerAtFreq > maxPowerAtFreq)
            {
                maxPowerAtFreq = powerAtFreq;
                marker = i;
            }

            tfo.writeString(powerAtFreq+"\n");
        }

        tfo.closeOutputStream();
        System.out.println("Stärkste Frequenz: "+(marker)+" Hz");
    }

But for some reason I only get the result of 400 Hz in the 'marker' if I choose to check for all 16000 samples. But shouldn't I see the peak in 400 Hz also if I only choose 800 samples, because with 800 I could see 800/2 = 400 Hz as maximum frequency?

I guess some little thing must be wrong with the code, because if I choose 800 samples I get 20 Hz, for 1600 samples I get 40 Hz which is always 1/40 * sample rate.

What the hell do I miss or did wrong? The results are strange..

Note that if I do the inverse DFT with the complex values I can reconstruct the audio signal again!

Answer 1

The answer for the question is that if you calculate the fourier transforms, magnitude spectrum etc. the indices show relative frequencies which need to be calculated to their correct value.