Parseval’s Theorem

Parseval’s theorem states that the total energy represented in the time domain is the same energy represented in the frequency domain, apart from the normalization fixed by the chosen transform convention.

Parseval Identity for the DFT

For the DFT convention of the section,

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-\mathrm{j}2\pi kn/N}, \qquad x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{\mathrm{j}2\pi kn/N},$$

the corresponding identity is

Note

Law — Parseval’s Theorem for the DFT

$$\sum_{n=0}^{N-1}|x[n]|^2 =\frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^2.$$

Thus the energy of a finite sequence can be calculated from its samples or from its DFT coefficients. In this normalization, frequency bin $k$ accounts for $|X[k]|^2/N$ of the total energy. The theorem is therefore both a check on a DFT calculation and a way to inspect how signal energy is distributed among frequency bins.

The more general inner-product form is

Caution

Key Formula — DFT Parseval relation for two sequences

$$\sum_{n=0}^{N-1}x[n]y^*[n] =\frac{1}{N}\sum_{k=0}^{N-1}X[k]Y^*[k].$$

Setting $y=x$ gives the energy identity. This complex form also shows why the conjugates cannot be omitted for general signals.

Normalization Conventions

The scale factor is not universal; it follows from the selected transform pair. If

$$X_a[k]=a\sum_{n=0}^{N-1}x[n]e^{-\mathrm{j}2\pi kn/N}, \qquad x[n]=\frac{1}{aN}\sum_{k=0}^{N-1}X_a[k]e^{\mathrm{j}2\pi kn/N},$$

where $a\neq0$, then

Caution

Key Formula — Parseval identity with general DFT scaling

$$\sum_{n=0}^{N-1}|x[n]|^2 =\frac{1}{N|a|^2}\sum_{k=0}^{N-1}|X_a[k]|^2.$$

Convention	Forward scale $a$	Inverse scale	Energy relation
Forward unscaled	$1$	$1/N$	$\sum
Unitary or symmetric	$1/\sqrt{N}$	$1/\sqrt{N}$	$\sum
Forward normalized	$1/N$	$1$	$\sum

Common DFT normalizations and their Parseval factors.

Always state or infer the transform normalization before applying Parseval. The unscaled-forward convention is used everywhere else in these notes.

Proof from DFT Orthogonality

For the unscaled forward DFT, expand the squared magnitude and interchange the finite sums:

$$\begin{aligned} \frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^2 &=\frac{1}{N}\sum_{k=0}^{N-1} \sum_{n=0}^{N-1}\sum_{m=0}^{N-1} x[n]x^*[m]e^{-\mathrm{j}2\pi k(n-m)/N}\\ &=\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}x[n]x^*[m] \left[\frac{1}{N}\sum_{k=0}^{N-1} e^{-\mathrm{j}2\pi k(n-m)/N}\right]\\ &=\sum_{n=0}^{N-1}|x[n]|^2. \end{aligned}$$

The bracket is one when $n=m$ and zero otherwise by roots-of-unity orthogonality. In matrix language, the same proof is $\mathbf{F}_N^{\mathsf H}\mathbf{F}_N=N\mathbf{I}$.

Continuous-Time Parseval Relation

With the angular-frequency Fourier pair

$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-\mathrm{j}\omega t}\,\mathrm{d}t, \qquad x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega)e^{\mathrm{j}\omega t}\,\mathrm{d}\omega,$$

Parseval’s theorem becomes

Caution

Key Formula — Continuous-time Parseval identity

$$\int_{-\infty}^{\infty}|x(t)|^2\,\mathrm{d}t =\frac{1}{2\pi}\int_{-\infty}^{\infty} |X(\omega)|^2\,\mathrm{d}\omega.$$

The factor $1/(2\pi)$ belongs to this angular-frequency convention; a different placement of transform normalization changes the displayed scale but not the energy-preservation principle.

The corresponding inner-product form is

$$\int_{-\infty}^{\infty}x(t)y^*(t)\,\mathrm{d}t =\frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega)Y^*(\omega)\,\mathrm{d}\omega.$$

If ordinary frequency in hertz is used instead,

$$X_f(f)=\int_{-\infty}^{\infty}x(t)e^{-\mathrm{j}2\pi ft}\,\mathrm{d}t, \qquad x(t)=\int_{-\infty}^{\infty}X_f(f)e^{\mathrm{j}2\pi ft}\,\mathrm{d}f,$$

then the same identity contains no $2\pi$ factor:

$$\int_{-\infty}^{\infty}|x(t)|^2\,\mathrm{d}t =\int_{-\infty}^{\infty}|X_f(f)|^2\,\mathrm{d}f.$$

The two formulas agree because $\omega=2\pi f$ and $\mathrm{d}\omega=2\pi\,\mathrm{d}f$.

Discrete-Time Fourier Transform Form

For the DTFT pair

$$X(e^{\mathrm{j}\Omega})=\sum_{n=-\infty}^{\infty} x[n]e^{-\mathrm{j}\Omega n}, \qquad x[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi} X(e^{\mathrm{j}\Omega})e^{\mathrm{j}\Omega n}\,\mathrm{d}\Omega,$$

Parseval’s theorem is

Note

Law — Parseval’s theorem for the DTFT

$$\sum_{n=-\infty}^{\infty}|x[n]|^2 =\frac{1}{2\pi}\int_{-\pi}^{\pi} |X(e^{\mathrm{j}\Omega})|^2\,\mathrm{d}\Omega.$$

Any interval of width $2\pi$ may replace $[-\pi,\pi]$ because the DTFT is periodic. The cross form replaces the magnitude squares by $x[n]y^*[n]$ and $X(e^{\mathrm{j}\Omega})Y^*(e^{\mathrm{j}\Omega})$.

Average-Power Forms for Periodic Signals

Parseval also applies to Fourier-series coefficients. For $x(t)=\sum_{r=-\infty}^{\infty}C_r e^{\mathrm{j}r\omega_0t}$,

Caution

Key Formula — Continuous-time Fourier-series Parseval relation

$$\frac{1}{T_0}\int_{t_0}^{t_0+T_0}|x(t)|^2\,\mathrm{d}t =\sum_{r=-\infty}^{\infty}|C_r|^2.$$

For an $N_0$-periodic DT signal, define the synthesis coefficients

$$C[k]=\frac{1}{N_0}\sum_{n=0}^{N_0-1} x[n]e^{-\mathrm{j}2\pi kn/N_0}.$$

Then

Caution

Key Formula — Discrete-time Fourier-series Parseval relation

$$\frac{1}{N_0}\sum_{n=0}^{N_0-1}|x[n]|^2 =\sum_{k=0}^{N_0-1}|C[k]|^2.$$

These are power identities: the left sides average over one period rather than sum energy over all time.

Worked Energy and RMS Checks

Note

Example — Parseval Check for a Four-Point DFT

the section found that

$$x[n]=\{1,2,3,4\} \longleftrightarrow X[k]=\{10,-2+\mathrm{j}2,-2,-2-\mathrm{j}2\}.$$

The time-domain energy is

$$E_x=1^2+2^2+3^2+4^2=30.$$

In frequency,

$$\sum_{k=0}^{3}|X[k]|^2=100+8+4+8=120, \qquad \frac{1}{4}(120)=30.$$

The record RMS is therefore

$$x_{\mathrm{rms}} =\sqrt{\frac{1}{4}\sum_{n=0}^{3}|x[n]|^2} =\sqrt{\frac{1}{4^2}\sum_{k=0}^{3}|X[k]|^2} =\boxed{\sqrt{7.5}}.$$

[!NOTE]

Example — RMS of a Bin-Centred Sinusoid

Let $x[n]=A\cos(2\pi k_0n/N+\phi)$ contain an integer number of cycles, with $k_0$ neither DC nor the Nyquist bin. Its only nonzero DFT coefficients are the conjugate pair at $k_0$ and $N-k_0$, each of magnitude $AN/2$. Parseval gives

$$E_x=\frac{1}{N}\left[2\left(\frac{AN}{2}\right)^2\right] =\frac{NA^2}{2}, \qquad x_{\mathrm{rms}}=\sqrt{\frac{E_x}{N}} =\boxed{\frac{|A|}{\sqrt{2}}}.$$

The familiar sinusoidal RMS value therefore follows directly from its two spectral lines.

Energy and Power Spectral Density

Parseval’s theorem also identifies where energy lies in frequency. For a continuous-time energy signal, define the energy spectral density (ESD) by

Caution

Key Formula — Energy spectral density

$$S_{xx}^{(E)}(\omega)=|X(\omega)|^2, \qquad E_x=\frac{1}{2\pi}\int_{-\infty}^{\infty} S_{xx}^{(E)}(\omega)\,\mathrm{d}\omega.$$

An energy signal has finite total energy and zero average power, so ESD gives energy per unit angular frequency.

A power signal generally has infinite total energy, so $|X(\omega)|^2$ is not used as an ordinary finite-energy density. Its power spectral density (PSD) is instead obtained from the Fourier transform of its time-averaged autocorrelation:

Caution

Key Formula — Power spectral density

$$S_{xx}^{(P)}(\omega)=\mathcal{F}\{R_{xx}(\tau)\}, \qquad P_x=\frac{1}{2\pi}\int_{-\infty}^{\infty} S_{xx}^{(P)}(\omega)\,\mathrm{d}\omega.$$

This autocorrelation–spectrum statement is the Wiener–Khinchin theorem, developed in the section. ESD and PSD are therefore related ideas but apply to different signal classes: integrating the ESD returns total energy, whereas integrating the PSD returns average power, with the scale factor required by the Fourier convention.

For a finite $N$-sample DFT record under the convention above, useful discrete allocations are

Caution

Key Formula — Energy and mean square from DFT bins

$$E_k=\frac{|X[k]|^2}{N}, \qquad E_x=\sum_{k=0}^{N-1}E_k, \qquad P_N=\frac{E_x}{N} =\frac{1}{N^2}\sum_{k=0}^{N-1}|X[k]|^2.$$

If the samples represent a rectangularly windowed record at sampling frequency $F_s$, a common two-sided periodogram is

$$\widehat{S}_{xx}[k]=\frac{|X[k]|^2}{NF_s}.$$

With $\Delta f=F_s/N$ it satisfies

$$\sum_{k=0}^{N-1}\widehat{S}_{xx}[k]\,\Delta f=P_N.$$

For a nonrectangular window, replace $N$ in the density denominator by $\sum_n|w[n]|^2$. A one-sided PSD for real data doubles conjugate-pair bins but does not double DC or the even-$N$ Nyquist bin. These scaling details are essential when a plotted FFT magnitude is to be interpreted as a physical density rather than as unnormalized coefficients.

Parseval also gives the output energy of an LTI system without reconstructing the waveform. If $Y(\omega)=H(\omega)X(\omega)$, then

$$E_y=\frac{1}{2\pi}\int_{-\infty}^{\infty} |H(\omega)|^2|X(\omega)|^2\,\mathrm{d}\omega.$$

This form is widely used in noise, filter, and bandwidth calculations.

Tip

Exam Focus — Parseval Quick Review

• For this book’s DFT convention, $\sum_n|x[n]|^2=N^{-1}\sum_k|X[k]|^2$; a unitary DFT instead has equal unscaled sums on both sides.

• CT angular frequency and the DTFT introduce $1/(2\pi)$ in the frequency integral; a CT transform written in hertz does not.

• Fourier-series Parseval relations equate one-period average power to the sum of squared series-coefficient magnitudes.

• ESD $=|X|^2$ for an energy signal; PSD is the Fourier transform of autocorrelation for a power signal or WSS process.

• Parseval is a rapid DFT check and converts energy, mean square, RMS, and filtered-output calculations between domains.