Power invariant fast Fourier transform (FFT)

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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.

Power invariant Fast Fourier transform (FFT) - time signal (left) and spectrum signal (right) powers are equal
Power invariant Fast Fourier transform (FFT) - time signal (left) and spectrum signal (right) powers are 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.

FFTIFFT
X k = n=0 N1 x n e j2πk n N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBa aaleaacaWGRbaabeaakiabg2da9maaqahabaGaamiEamaaBaaaleaa caWGUbaabeaaaeaacaWGUbGaeyypa0JaaGimaaqaaiaad6eacqGHsi slcaaIXaaaniabggHiLdGccaWGLbWaaWbaaSqabeaacqGHsislcaWG QbGaaGOmaiabec8aWjaadUgadaWcaaqaaiaad6gaaeaacaWGobaaaa aaaaa@4ABD@ x k = 1 N n=0 N1 X n e j2πk n N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGRbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaad6ea aaWaaabCaeaacaWGybWaaSbaaSqaaiaad6gaaeqaaaqaaiaad6gacq GH9aqpcaaIWaaabaGaamOtaiabgkHiTiaaigdaa0GaeyyeIuoakiaa dwgadaahaaWcbeqaaiaadQgacaaIYaGaeqiWdaNaam4Aamaalaaaba GaamOBaaqaaiaad6eaaaaaaaaa@4B6E@
FFT / DFT and IFFT / DFT definitions

The spectrum analysis (FFT) and signal synthesis (IFFT) equations look quite similar apart from the term 1 N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamOtaaaaaaa@3795@ . 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.

FFTIFFT
X k = 1 N n=0 N1 x n e j2πk n N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBa aaleaacaWGRbaabeaakiabg2da9maalaaabaGaaGymaaqaamaakaaa baGaamOtaaWcbeaaaaGcdaaeWbqaaiaadIhadaWgaaWcbaGaamOBaa qabaaabaGaamOBaiabg2da9iaaicdaaeaacaWGobGaeyOeI0IaaGym aaqdcqGHris5aOGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiaaik dacqaHapaCcaWGRbWaaSaaaeaacaWGUbaabaGaamOtaaaaaaaaaa@4C80@ x k = 1 N n=0 N1 X n e j2πk n N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGRbaabeaakiabg2da9maalaaabaGaaGymaaqaamaakaaa baGaamOtaaWcbeaaaaGcdaaeWbqaaiaadIfadaWgaaWcbaGaamOBaa qabaaabaGaamOBaiabg2da9iaaicdaaeaacaWGobGaeyOeI0IaaGym aaqdcqGHris5aOGaamyzamaaCaaaleqabaGaamOAaiaaikdacqaHap aCcaWGRbWaaSaaaeaacaWGUbaabaGaamOtaaaaaaaaaa@4B93@
Power invariant definitions of FFT and IFFT

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 Launch button above!

Calculate the FFT of real and complex time domain signals. Plot time and frequency signals
Calculate the FFT of real and complex time domain signals. Plot time and frequency signals.

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

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

Enter the time domain data – example formats. Or copy-paste values, e.g. from Excel. Enter the time domain data – example formats. Or copy-paste values, e.g. from Excel. Enter the time domain data – example formats. Or copy-paste values, e.g. from Excel. Enter the time domain data – example formats. Or copy-paste values, e.g. from Excel. Enter the time domain data – example formats. Or copy-paste values, e.g. from Excel.
Enter the time domain data - example formats. Or copy-paste values, e.g. from Excel.
Plot time and frequency signals
Plot time and frequency signals

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

Time domainFrequency domain
DC component
DC component
Spectral line at frequency f=0
Spectral line at frequency f=0
Time signal - dirac delta impulse
Time signal - Dirac delta impulse
Constant spectrum
Constant spectrum
Time signal � cosine with frequency fmax/2
Time signal - cosine with frequency f max 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGMbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaakeaacaaIYaaa aaaa@3AB8@
Time signal - cosine with frequency +-fmax/2
Spectral lines at frequencies ± f max 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyySae7aaS aaaeaacaWGMbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaakeaa caaIYaaaaaaa@3CA6@
Time signal � cosine with frequency fmax
Time signal - cosine with frequency f max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@39E2@
Spectral line at frequency fmax
Spectral line at frequency f max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@39E2@
FFT examples (N = 8). The time and frequency domain signals are real and even.
Time domainFrequency domain
Time signal - pulse
Time signal - pulse
Spectrum signal - sinc function
Spectrum signal - Sinc function
FFT examples (N = 32, power invariant). The time and frequency domain signals are real and even.
Time domainFrequency domain
Time signal - complex rotating phasor
Time signal - complex rotating phasor
Single spectral line
Single spectral line
Time signal - ofdm symbol
Time signal - OFDM symbol
Spectrum signal - ofdm symbol
Spectrum signal - OFDM symbol
FFT examples (N = 8). Complex time domain signal.