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Following the post and answers below, here is a gist which I think scales the coeffcients correctly and consistently for real- and complex-valued inputs. As defined and explained on Wikipedia :. This means that the number of frequencies in the decomposition changes, and therefore the normalisation should be done relative to NFFT wave components, and not the original N time-points.
Hence, almost all of the examples found online are wrong in the way they normalise the coefficients. Note that this makes no difference then to Parseval's equality, because the added terms in the time-domain are all zero, and therefore their contribution to the now-larger sum is zero too. In the frequency domain though, the added discrete frequencies will have a response to the original signal in general, which gives an intuitive sense of why the obtained coefficients can be quite different indeed between the padded and unpadded transforms.
The code in the OP is therefore incorrect, and instead it appears to be necessary to output both the amplitude and power, as there is no general normalisation coefficient that could accommodate the complex and real cases, with even or odd number of time-points. You can find the Gist here. Check this out:. Results from the Developer Survey are now available. Stack Overflow for Teams — Start collaborating and sharing organizational knowledge.
Create a free Team Why Teams? Learn more. Fourier Transform with Matlab Ask Question. Asked 6 years, 4 months ago. Modified 2 years, 11 months ago. Viewed 3k times. Resampling before continuing. Jonathan H.
Jonathan H Jonathan H 7, 5 5 gold badges 45 45 silver badges 76 76 bronze badges. I suggest comparing the formula given in help fft with the result of fft. Additionally, this does not answer the question of what the discrete amplitudes and power should be for the single-sided case. Add a comment. Sorted by: Reset to default.
Highest score default Trending recent votes count more Date modified newest first Date created oldest first. This is natural in some sense, as moving in time-to-frequency direction can be seen as a projection onto a basis of orthogonal waves with different frequencies, whereas moving in frequency-to-time direction can be seen as a weighted superposition of waves. Matlab uses that normalisation too well, FFTW does. In practice, most applications avoid this redundancy by extracting the DFT coefficients only for positive frequencies.
This leads to complications in the discrete values of the amplitude and power. The amplitudes corresponding to paired DFT coefficients all except the first one and the Nyquist frequency when it exists can simply be doubled and the negative frequency then discarded. Same for the power. Similarly for the power, but note that it is incorrect in that case to compute the discrete power values using the adjusted amplitude values. Therefore it is necessary to compute independently the amplitude and power values from the DFT coefficients another good reason not to scale them.
Code The code in the OP is therefore incorrect, and instead it appears to be necessary to output both the amplitude and power, as there is no general normalisation coefficient that could accommodate the complex and real cases, with even or odd number of time-points.
Please give a citation where this is from. Help Center Help Center. The Fourier transform is a mathematical formula that relates a signal sampled in time or space to the same signal sampled in frequency. In signal processing, the Fourier transform can reveal important characteristics of a signal, namely, its frequency components.
The Fourier transform is defined for a vector x with n uniformly sampled points by. For x and y , the indices j and k range from 0 to n - 1. Consider a sinusoidal signal x that is a function of time t with frequency components of 15 Hz and 20 Hz.
Use a time vector sampled in increments of 1 50 of a second over a period of 10 seconds. Compute the Fourier transform of the signal, and create the vector f that corresponds to the signal's sampling in frequency space. When you plot the magnitude of the signal as a function of frequency, the spikes in magnitude correspond to the signal's frequency components of 15 Hz and 20 Hz. The transform also produces a mirror copy of the spikes, which correspond to the signal's negative frequencies.
To better visualize this periodicity, you can use the fftshift function, which performs a zero-centered, circular shift on the transform. In scientific applications, signals are often corrupted with random noise, disguising their frequency components.
The Fourier transform can process out random noise and reveal the frequencies. For example, create a new signal, xnoise , by injecting Gaussian noise into the original signal, x. Signal power as a function of frequency is a common metric used in signal processing.
Power is the squared magnitude of a signal's Fourier transform, normalized by the number of frequency samples. Compute and plot the power spectrum of the noisy signal centered at the zero frequency. Despite noise, you can still make out the signal's frequencies due to the spikes in power. Using the Fourier transform formula directly to compute each of the n elements of y requires on the order of n 2 floating-point operations.
The fast Fourier transform algorithm requires only on the order of n log n operations to compute. This computational efficiency is a big advantage when processing data that has millions of data points. Many specialized implementations of the fast Fourier transform algorithm are even more efficient when n is a power of 2. Consider audio data collected from underwater microphones off the coast of California. This data can be found in a library maintained by the Cornell University Bioacoustics Research Program.
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