What is Pink Noise and how is it different from White Noise?

Utpal Kumar 3 minute read DATASCIENCE June 28, 2023

Introduction

Pink noise — also called 1/f noise or flicker noise — is a random signal with equal energy per octave. The name is an analogy to pink light, whose power spectrum falls off with frequency. It turns up all over nature (and is a favorite of computer-music composers), which makes it worth understanding alongside its flatter cousin, white noise.

The one distinction

It’s all in the power spectrum $S(f)$:

\[S_{\text{white}}(f) \propto \text{const}, \qquad S_{\text{pink}}(f) \propto \frac{1}{f}\]

White noise spreads power equally across all frequencies (flat spectrum). Pink noise puts more power at low frequencies, decreasing by half each octave — equal power per octave rather than per hertz.

On log-log axes, white noise is flat; pink noise is a straight downward line (slope −1).

How is pink noise different from white noise?

Both are random signals, but they differ across three dimensions:

Aspect	White noise	Pink noise
Spectral distribution	Flat — equal power at all frequencies	Power halves per octave (1/f) — more low-frequency energy
Correlation	Independent samples, no correlation	Long-range correlations; each sample depends on the past
Perceived sound	Hiss/static; masks background, tests audio gear	Balanced, “soothing”; used in audio, acoustics, sleep/relaxation

Check your understanding

Compared with white noise, where does pink noise concentrate its power?

Synthesis of white noise in Python

To generate white noise in Python, we can use the NumPy library, which provides a function called numpy.random.randn() to generate white noise.

## Generating White Noise
import numpy as np
import matplotlib.pyplot as plt

def generate_white_noise(samples):
    white_noise = np.random.randn(samples)
    return white_noise

# Generate 1000 samples of pink noise
samples = 1000
white_noise = generate_white_noise(samples)

# Compute the frequency spectrum using FFT
spectrum = np.fft.fft(white_noise)
freq = np.fft.fftfreq(samples)

# Consider only positive frequencies
positive_freq = freq[:samples // 2]
positive_spectrum = spectrum[:samples // 2]

# Plotting the pink noise
fig, ax = plt.subplots(2, 1, figsize=(10,6))
ax[0].plot(white_noise)
ax[0].set_title("White Noise")
ax[0].set_xlabel("Time")
ax[0].set_ylabel("Amplitude")

ax[1].plot(positive_freq, np.abs(positive_spectrum))
ax[1].set_xlabel("Frequency")
ax[1].set_ylabel("Magnitude")

plt.savefig('white_noise_synthesis.png', dpi=300, bbox_inches='tight')
plt.close()

Synthesis of pink noise in Python

To generate pink noise in Python, we can use the NumPy library, which provides a function called numpy.random.randn() to generate white noise. Then, we can then apply a filter to the white noise signal to convert it into pink noise.

## Generating Pink Noise
import numpy as np
import matplotlib.pyplot as plt
np.random.seed(10250)

def generate_pink_noise(samples):
    b = [0.049922035, -0.095993537, 0.050612699, -0.004408786]
    a = [1, -2.494956002, 2.017265875, -0.522189400]

    pink_noise = np.random.randn(samples)
    pink_noise = np.convolve(pink_noise, b)
    pink_noise = np.convolve(pink_noise, a, mode='valid')
    return pink_noise / 0.091420942

# Generate 1000 samples of pink noise
samples = 1000
pink_noise = generate_pink_noise(samples)

# Compute the frequency spectrum using FFT
spectrum = np.fft.fft(pink_noise)
freq = np.fft.fftfreq(samples)

# Consider only positive frequencies
positive_freq = freq[:samples // 2]
positive_spectrum = spectrum[:samples // 2]

# Plotting the pink noise
fig, ax = plt.subplots(2, 1, figsize=(10,6))
ax[0].plot(pink_noise)
ax[0].set_title("Pink Noise")
ax[0].set_xlabel("Time")
ax[0].set_ylabel("Amplitude")

ax[1].plot(positive_freq, np.abs(positive_spectrum))
ax[1].set_xlabel("Frequency")
ax[1].set_ylabel("Magnitude")

plt.savefig('pink_noise_synthesis.png', dpi=300, bbox_inches='tight')
plt.close()

The trick is the filter: white noise (flat spectrum) is passed through the IIR filter defined by coefficients b and a, whose frequency response approximates a 1/f roll-off — turning the flat spectrum into the pink one.

Recap

Without scrolling up — what separates the two?

White noise: $S(f) \propto \text{const}$ — flat spectrum, independent samples, hiss-like.
Pink noise: $S(f) \propto 1/f$ — more low-frequency power, long-range correlations, “balanced” sound.
In code: generate white noise with np.random.randn, then filter it (the b/a coefficients) to impose the 1/f roll-off that makes it pink.

References

"1/f noise" in Music: Music from 1/f Noise — Voss & Clarke, 1978, The Journal of the Acoustical Society of America, 63(1), 258–263.

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