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The power of the FFT output signal differs from the input signal power due to an asymmetry in the FFT / IFFT definitions. This FFT variant keeps input and output signal powers equal.

The FFT transforms a time signal x to a frequency signal X. And vice versa, the IFFT transforms a frequency signal X to a time signal x.

FFT | IFFT |
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${X}_{k}={\displaystyle \sum _{n=0}^{N-1}{x}_{n}}{e}^{-j2\pi k\frac{n}{N}}$ | ${x}_{k}=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{X}_{n}}{e}^{j2\pi k\frac{n}{N}}$ |

The spectrum analysis (FFT) and signal synthesis (IFFT) equations look quite similar apart from the term $\frac{1}{N}$. This asymmetry leads to a power variation of the time and frequency signal. See Fast Fourier transform (FFT)

Power invariant definitions of the FFT and IFFT are shown below. They might be applied in the context of an OFDM transmission in order to not change the signal power by the IFFT and FFT transformations.

FFT | IFFT |
---|---|

${X}_{k}=\frac{1}{\sqrt{N}}{\displaystyle \sum _{n=0}^{N-1}{x}_{n}}{e}^{-j2\pi k\frac{n}{N}}$ | ${x}_{k}=\frac{1}{\sqrt{N}}{\displaystyle \sum _{n=0}^{N-1}{X}_{n}}{e}^{j2\pi k\frac{n}{N}}$ |

Create your individual signal spectra by using the FFT calculator and check that time and frequency signal powers are equal. Click on the start button above!

Enter the time domain samples. Press *Submit* to calculate the frequency domain result.

- The data size must be a power of 2: 2, 4, 8, 16, 32...
- The format is compatible with Excel: the sequence of complex numbers can be copied from and to spreadsheets.

Note that under this power invariant FFT variation, input and output signal powers are equal.

Some examples illustrate the power invariant Fast Fourier transform. Time signals and corresponding frequency signals are shown.

Time domain | Frequency domain |
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Time domain | Frequency domain |
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Time domain | Frequency domain |
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