Understanding Matlab FFT example

Question

I am new to matlab and FFT and want to understand the Matlab FFT example  For now I have two main questions   1  Why does the x-axis  frequency  end at 500  How do I know that there aren t more frequencies or are they just ignored   2  How do I know the frequencies are between 0 and 500  Shouldn t the FFT tell me  in which limits the frequencies are  Does the FFT only return the amplitude value without the frequency   Thanks for any hint     Example in question   Consider data sampled at 1000 Hz  Form a signal containing a 50 Hz sinusoid of amplitude 0 7 and 120 Hz sinusoid of amplitude 1 and corrupt it with some zero-mean random noise   Fs   1000                       Sampling frequency T   1 Fs                        Sample time L   1000                        Length of signal t    0 L-1  T                   Time vector   Sum of a 50 Hz sinusoid and a 120 Hz sinusoid x   0 7 sin 2 pi 50 t    sin 2 pi 120 t    y   x   2 randn size t          Sinusoids plus noise plot Fs t 1 50  y 1 50   title  Signal Corrupted with Zero-Mean Random Noise   xlabel  time  milliseconds        Converting to the frequency domain  the discrete Fourier transform of the noisy signal y is found by taking the fast Fourier transform  FFT    NFFT   2 nextpow2 L     Next power of 2 from length of y Y   fft y NFFT  L  f   Fs 2 linspace 0 1 NFFT 2 1      Plot single-sided amplitude spectrum  plot f 2 abs Y 1 NFFT 2 1     title  Single-Sided Amplitude Spectrum of y t    xlabel  Frequency  Hz    ylabel   Y f

User · Answer

It sounds like you need to some background reading on what an FFT is (e.g. http://en.wikipedia.org/wiki/FFT). But to answer your questions:

Why does the x-axis (frequency) end at 500?

Because the input vector is length 1000. In general, the FFT of a length-N input waveform will result in a length-N output vector. If the input waveform is real, then the output will be symmetrical, so the first 501 points are sufficient.

Edit: (I didn't notice that the example padded the time-domain vector.)

The frequency goes to 500 Hz because the time-domain waveform is declared to have a sample-rate of 1 kHz. The Nyquist sampling theorem dictates that a signal with sample-rate fs can support a (real) signal with a maximum bandwidth of fs/2.

How do I know the frequencies are between 0 and 500?

See above.

Shouldn't the FFT tell me, in which limits the frequencies are?

No.

Does the FFT only return the amplitude value without the frequency?

The FFT simply assigns an amplitude (and phase) to every frequency bin.

User · Answer

1  Why does the x-axis  frequency  end at 500  How do I know that there aren t more frequencies or are they just ignored    It ends at 500Hz because that is the Nyquist frequency of the signal when sampled at 1000Hz  Look at this line in the Mathworks example   f   Fs 2 linspace 0 1 NFFT 2 1     The frequency axis of the second plot goes from 0 to Fs 2  or half the sampling frequency  The Nyquist frequency is always half the sampling frequency  because above that  aliasing occurs     The signal would  fold  back on itself  and appear to be some frequency at or below 500Hz      2  How do I know the frequencies are between 0 and 500  Shouldn t the FFT tell me  in which limits the frequencies are     Due to  folding  described above  the Nyquist frequency is also commonly known as the  folding frequency    it is physically impossible for frequencies above 500Hz to appear in the FFT  higher frequencies will  fold  back and appear as lower frequencies      Does the FFT only return the amplitude value without the frequency    Yes  the MATLAB FFT function only returns one vector of amplitudes  However  they map to the frequency points you pass to it   Let me know what needs clarification so I can help you further

User · Answer

There are some misconceptions here     Frequencies above 500 can be represented in an FFT result of length 1000   Unfortunately these frequencies are all folded together and mixed into the first 500 FFT result bins   So normally you don t want to feed an FFT a signal containing any frequencies at or above half the sampling rate  as the FFT won t care and will just mix the high frequencies together with the low ones  aliasing  making the result pretty much useless   That s why data should be low-pass filtered before being sampled and fed to an FFT   The FFT returns amplitudes without frequencies because the frequencies depend  not just on the length of the FFT  but also on the sample rate of the data  which isn t part of the FFT itself or it s input   You can feed the same length FFT data at any sample rate  as thus get any range of frequencies out of it   The reason the result plots ends at 500 is that  for any real data input  the frequencies above half the length of the FFT are just mirrored repeats  complex conjugated  of the data in the first half   Since they are duplicates  most people just ignore them   Why plot duplicates   The FFT calculates the other half of the result for people who feed the FFT complex data  with both real and imaginary components   which does create two different halves

User · Answer

The reason why your X-axis plots frequencies only till 500 Hz is your command statement  f   Fs 2 linspace 0 1 NFFT 2 1     Your Fs is 1000  So when you divide it by 2  amp  then multiply by values ranging from 0 to 1  it returns a vector of length NFFT 2 1  This vector consists of equally spaced frequency values  ranging from 0 to Fs 2  i e  500 Hz   Since you plot using  plot f 2 abs Y 1 NFFT 2 1     command  your X-axis limit is 500 Hz

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