PERIODOGRAM

This function computes a periodogram to estimate how signal power is distributed across frequencies. It returns a two-column array containing frequency and estimated spectral power.

The periodogram is based on the squared magnitude of the discrete Fourier transform (DFT):

P(f) \propto |X(f)|^2

where X(f) is the DFT of the input signal. Depending on scaling, the output represents either power spectral density or power spectrum.

Excel Usage

=PERIODOGRAM(data, fs, window, nfft, detrend, return_onesided, scaling)

data (list[list], required): 2D range of time-series samples.
fs (float, optional, default: 1): Sampling frequency in hertz (Hz).
window (str, optional, default: “boxcar”): Window name applied before FFT.
nfft (int, optional, default: null): FFT length; null uses input length.
detrend (str, optional, default: “constant”): Detrending method for the signal.
return_onesided (bool, optional, default: true): Return one-sided spectrum for real-valued input.
scaling (str, optional, default: “density”): Spectral scaling mode.

Returns (list[list]): 2D array with columns [frequency, power].

Example 1: Basic periodogram with defaults

Inputs:

data																fs	window	nfft	detrend	return_onesided	scaling
0	1	0	-1	0	1	0	-1	0	1	0	-1	0	1	0	-1	8	boxcar		constant	true	density

Excel formula:

=PERIODOGRAM({0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1}, 8, "boxcar", , "constant", TRUE, "density")

Expected output:

Result
0	0
0.5	0
1	0
1.5	0
2	1
2.5	0
3	0
3.5	0
4	0

Example 2: Periodogram with linear detrending

Inputs:

data																fs	window	nfft	detrend	return_onesided	scaling
0	0.8	0.1	-0.9	-0.2	1.1	0.2	-1	-0.1	0.9	0	-1.1	0.1	1	-0.2	-0.8	8	hann	32	linear	true	density

Excel formula:

=PERIODOGRAM({0,0.8,0.1,-0.9,-0.2,1.1,0.2,-1,-0.1,0.9,0,-1.1,0.1,1,-0.2,-0.8}, 8, "hann", 32, "linear", TRUE, "density")

Expected output:

Result
0	0.000128004
0.25	0.00167162
0.5	0.000885883
0.75	0.000758796
1	0.00260373
1.25	0.00674368
1.5	0.118549
1.75	0.426249
2	0.671639
2.25	0.545337
2.5	0.220244
2.75	0.0313019
3	0.0000948799
3.25	0.00383545
3.5	0.00216765
3.75	0.000520404
4	0.00010619

Example 3: Periodogram returning power spectrum without detrending

Inputs:

data																fs	window	nfft	detrend	return_onesided	scaling
0	1	0	-1	0	1	0	-1	0	1	0	-1	0	1	0	-1	8	boxcar		none	true	spectrum

Excel formula:

=PERIODOGRAM({0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1}, 8, "boxcar", , "none", TRUE, "spectrum")

Expected output:

Result
0	0
0.5	0
1	0
1.5	0
2	0.5
2.5	0
3	0
3.5	0
4	0

Example 4: Two-sided periodogram output

Inputs:

data																fs	window	nfft	detrend	return_onesided	scaling
0	1	0	-1	0	1	0	-1	0	1	0	-1	0	1	0	-1	8	boxcar		constant	false	density

Excel formula:

=PERIODOGRAM({0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1}, 8, "boxcar", , "constant", FALSE, "density")

Expected output:

Result
0	0
0.5	0
1	0
1.5	0
2	0.5
2.5	0
3	0
3.5	0
-4	0
-3.5	0
-3	0
-2.5	0
-2	0.5
-1.5	0
-1	0
-0.5	0

Python Code

import numpy as np
from scipy.signal import periodogram as scipy_periodogram

def periodogram(data, fs=1, window='boxcar', nfft=None, detrend='constant', return_onesided=True, scaling='density'):
    """
    Estimate the power spectral density of a time series using a periodogram.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.periodogram.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        data (list[list]): 2D range of time-series samples.
        fs (float, optional): Sampling frequency in hertz (Hz). Default is 1.
        window (str, optional): Window name applied before FFT. Default is 'boxcar'.
        nfft (int, optional): FFT length; null uses input length. Default is None.
        detrend (str, optional): Detrending method for the signal. Valid options: Constant, Linear, None. Default is 'constant'.
        return_onesided (bool, optional): Return one-sided spectrum for real-valued input. Default is True.
        scaling (str, optional): Spectral scaling mode. Valid options: Density, Spectrum. Default is 'density'.

    Returns:
        list[list]: 2D array with columns [frequency, power].
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        data = to2d(data)

        if fs <= 0:
            return "Error: fs must be greater than 0"
        if detrend not in ("constant", "linear", "none"):
            return "Error: detrend must be 'constant', 'linear', or 'none'"
        if scaling not in ("density", "spectrum"):
            return "Error: scaling must be 'density' or 'spectrum'"
        if not isinstance(data, list) or not all(isinstance(row, list) for row in data):
            return "Error: data must be a 2D list"

        values = []
        for row in data:
            for item in row:
                try:
                    values.append(float(item))
                except (TypeError, ValueError):
                    continue

        if len(values) < 2:
            return "Error: data must contain at least two numeric values"

        nfft_arg = None if nfft is None or nfft <= 0 else nfft
        detrend_arg = False if detrend == "none" else detrend

        freqs, power = scipy_periodogram(
            np.asarray(values, dtype=float),
            fs=fs,
            window=window,
            nfft=nfft_arg,
            detrend=detrend_arg,
            return_onesided=return_onesided,
            scaling=scaling,
        )

        return [[float(freqs[i]), float(power[i])] for i in range(len(freqs))]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

data *

2D range of time-series samples.

Sampling frequency in hertz (Hz).

window

Window name applied before FFT.

nfft

FFT length; null uses input length.

detrend

Detrending method for the signal.

return_onesided

Return one-sided spectrum for real-valued input.

scaling

Spectral scaling mode.