When you get a sequence at the end of the DFT or FFT, you have to be very careful how you read the data.
The horizontal axis in the frequency domain can have 3 interpretations.
First is the digital frequency index k, second the corresponding
digital frequency in radians,
and third the corresponding analogue frequency
in radians/sec. The relationship between the three can be seen as
Since any frequency greater than N/2 (the Nyquist
Frequency or half the sampling frequency)
are redundant for real signals (see the
Twiddle
Factor). This N/2 index corresponds to
and
If you look at the example in the this image
Original Signal
Sampled Signal
From these examples you can see that for an N point sequence, it produces N output samples in the frequency domain. But because of the twiddle factor and the sampling Theorem we only need the first half of the results.
The images below show the actual results of the DFT. As is usually the case a real signal gave rise to a complex frequency response (once again due to the complex part of the twiddle factor. The images also highlight the relationships between the three methods of interpreting the frequency axis. Use whatever one suits your needs best!
One thing that the images do not show is that the frequency response is in fact periodic. Once it reaches N points, the signal (in time AND frequency) repeats itself.
Real part of DFT/FFT results.
Imaginary part of DFT/FFT
results.
On to Decimation in Frequency or back to Butterfly
or back to FFT Contents or back to Main Contents
Or back to the Discrete Fourier Transform contents