In my recent study of signal processing I came to know that we can compress images/signal using DFT/DCT. If we consider the lossy compression of audio or any image then it follows the same steps:

  1. Apply DCT/DFT on N samples of data.

  2. Collect coefficients which contains most of the signal energy i.e., a signal's DCT representation tends to have more of its energy concentrated in a small number of coefficients when compared to other transforms like the DFT.

But I have doubt that how to get these coefficients. In DFT we have $$ {\tt ORIGINAL:}\ X(e^{jw})=\sum_{n=0}^{N-1} x[n]e^{-j2{\pi}/Nn}\\ {\tt CORRECTED:}\ X[k]=\sum_{n=0}^{N-1} x[n]e^{-j2{\pi}/Nnk} $$

Here, $ \omega $ is a continuous variable. Then how to extract coefficients when $ \omega $ is a continuous variable?

  • $\begingroup$ Welcome to DSP.SE! What you have in your equation is the Discrete-Time Fourier Transform. That is not usually what is used: note that this requires infinite data... which is hard to store. The DFT is something else. $\endgroup$ – Peter K. Dec 2 '15 at 17:37
  • $\begingroup$ sorry for that..I have edited it $\endgroup$ – Virange Dec 2 '15 at 17:41
  • $\begingroup$ Well, let me correct what you've done and you'll see. $\endgroup$ – Peter K. Dec 2 '15 at 17:50
  • 1
    $\begingroup$ Note that you missed out a $k$ (discrete frequency variable) in your edit. $\endgroup$ – Peter K. Dec 2 '15 at 17:51

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