How to calculate and apply an inverse FFT: part 1

The inverse Fourier transform (inverse FFT or iFFT) reverses the operation of the Fourier transform and derives a time-domain representation from a frequency-domain dataset.

In early 2024, EE World published a series on the Fourier transform, which can convert a time-domain signal to the frequency domain (Figure 1, red arrow). The process is reversible (Figure 1, blue arrow), and we’ll close out the year looking the inverse Fourier transform.

Your time-domain waveform looks continuous. Will we be covering a discrete version of the inverse Fourier transform?
Yes, we will focus exclusively on the inverse discrete Fourier transform (IDFT) and the inverse fast Fourier transform (IFFT), which speeds up IDFT execution. Here’s the IDFT formula:
where X(k) represents our frequency-domain dataset, N is the number of data points in X(k), and x(n) represents the time-domain dataset we want to construct.

Could you provide a simple example of how the inverse FFT works?
Let X(k) equal 0, 1, 2, 3, so N equals 4, and we can simplify equation 1:
We can start by solving for x(n=0). With n equaling zero, all our exponents for all possible values of k (0, 1, 2, 3) are zero, and equation 2 becomes trivial:
The other three equations are not so trivial.

How do we handle the nonzero exponents?
The trick is to remember Euler’s formula:
Figure 2 shows all four equations with annotations in red, and our IDFT dataset is x(n) = 1.5, -0.5-j0.5, -0.5, -0.5+0.5j.

Euler's equation example — Figure 2. Euler’s formula helps evaluate terms with nonzero exponents.

How do we automate this process?
As in our earlier series on the Fourier transform, we’ll use Microsoft Excel. Excel uses a version of the fast Fourier transform (FFT) that requires that the number of points N in a dataset must be a power of 2. We have four in the sample dataset we presented above, so we can start with that.

How do we get started?
Figure 3 shows how to proceed. Enter column headings as shown on the top left and then enter the input dataset in cells A2 through A5. Then select the Data tab and choose Data Analysis from the Data ribbon. A list of available analysis tools will appear. Choose Fourier Analysis, and the box in the top right of Figure 3 will appear. Check Inverse on the bottom left of the box and enter the data input range (A2 through A5). Then, click the Output Range button, and enter the cell range where Excel should write the results (B2 to B5 in this example). Click OK and Excel will generate the results as shown in the bottom right. As you can see, Excel’s results agree with our manual calculations. (If you encounter problems executing these steps, including not being able to find Data Analysis on the Data ribbon, see part 1 and part 2 of our earlier Fourier analysis series for more detail.)