When to use FFTshift in MATLAB

May 31, 2017

After much research and experimentation, I have determined the proper way to take forward and inverse Fourier transforms in matlab. The forward transform is:

spectrum = fftshift(fft(ifftshift(signal)));

And the inverse transform is:

signal = fftshift(ifft(ifftshift(spectrum)));

I found this discussion the most helpful.

A thorough explanation

There are two reasonably sane ways to store real data in an array. They only differ in where negative values are placed, so we’ll only use examples where the independent variable (time, frequency, etc.) has positive and negative values, e.g. integers from -5 to 5.

The first way (and the way I prefer) is to place the value for the lowest time value in the first element of the array, and continuing in order. That is,

[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]

I will use the nomenclature of Chris in the discussion linked above, and call this scheme ‘natural’. One could also use ‘monotonic’.

The second way (and the way used by most implementations of the fft algorithm) is to place the value for time (or frequency, or whatever) 0 in the first element of the array. Then continue to the largest time, and then start at the smallest time and proceed to the one just before zero. Using the same example as above,

[0, 1, 2, 3, 4, 5, -5, -4, -3, -2, -1]

Again using Chris’s nomenclature, I will call this the ‘shifted’ form.

These methods both contain all the same data, and it is trivial to convert between them losslessly. Neither is better–they both have their strengths and applications. The only pitfall is using one when the other is expected, as is commonly encountered when taking Fourier transforms.

The first time you plotted an FFT in matlab, you might have noticed that the DC coefficients weren’t where you expected them to be. Instead of being in the center of the plot, they are at the beginning and end. This is because matlab’s fft(), and many other fft implementations, output data in the shifted form, and we expect to view it in the natural form with DC (zero frequency) in the center. Luckily, matlab includes a helpful function called fftshift() to convert shifted data into the natural format. It also includes ifftshift() to undo that shift and convert natural data into the shifted format.

That is only half of the story, though. fft() also expects its input to be in shifted form, with the zero time value in the first element of the array. Many people realize this, and so they use fftshift(fft(fftshift(signal))), but this will cause problems if the number of points is odd. (You can try it for yourself on some familiar function like a rectangle, or a gaussian.)

To see why, remember that fftshift() converts from shifted to natural, and ifftshift() converts from natural to shifted. If we start with natural-format data, and run fftshift() on it, it will be converted from shifted to natural again! We actually want to run ifftshift() to convert it from its natural state into the shifted format that fft() wants.

This subtlety is often missed because fftshift() and ifftshift() appear very similar, and in fact only behave slightly differently (one point) on data with an odd number of points, and they are actually equivalent for even numbers of points.

In the shifted format, the center point (5 above) is the highest time (or frequency) value (see the examples above). fftshift() will move it to the last element in the array, so that it is with all the other points after zero. ifftshift() takes the center point (0 above) and moves it to the first element, because the center element is the 0 time (or frequency) value in the natural format, and must be placed at the beginning for the shifted format. Running one where the other is needed will put the center point at the wrong end of the array, and repeated application will slowly circularly shift the array around. Thus it is important to use the correct function depending on what you are trying to do. This technically means that ifftshift() only needs to be used on data with an odd number of points, but to maintain generality, it is recommended that ifftshift() is always used.

In order to avoid having to remember all this, and to avoid mistakes, you may want to define some functions that will take care of all the shifting for you, such as:

F2 = @(signal) fftshift(fft2(ifftshift(signal)));
IF2 = @(spectrum) fftshift(ifft2(ifftshift(spectrum)));

For a two-dimensional Fourier transform.