1/f Noise—the flickering candle

October 02, 2013 // By Bruce Trump, Texas Instruments; this, from his EDN column

It’s also called flicker noise, like a flickering candle. Seen on an oscilloscope with a slow sweep it has a wandering baseline (Figure 1) because the high frequency noise rides on larger low frequency content. Pink noise, another metaphoric name, also suggests the stronger low frequency component. Flicker noise seems ever present in physical systems and life science. Weather/climate patterns, for example, have a 1/f component. I won’t attempt to explain why it’s found in semiconductors—deep subject!

The spectrum of flicker noise has a nominal slope of -10 dB/decade, half that of a single R-C pole. Note that it’s the square of the voltage (or power) that declines at a 1/f rate. Noise voltage falls at 1/ (f). The actual slope can vary somewhat but this doesn’t greatly change its behaviour or the conclusions.

A measured spectrum of flicker noise generally looks lumpy, with dips and valleys. You need to average for long periods to get a reasonably smooth plot. The period of 0.1 Hz noise content is 10 seconds, so for a good measurement down to 0.1 Hz you need to average many 10-second periods—five minutes or more. For 0.01 Hz data, take a long lunch. If you repeat the measurement it will likely look different. Noise is noisy and 1/f noise seems noisier than most other noise (did I write that?).

To calculate total noise, VB, over a bandwidth (f1 to f2) we integrate the 1/f function, resulting in the natural logarithm of the frequency ratio, f2/f1.

Points to ponder…

- Each decade of frequency (or other constant ratio of frequencies) contributes equally to total noise. Each successive decade has lower noise density but more bandwidth.

- From the spectral plot, you might infer that 1/f noise grows boundlessly as you measure for increasingly long periods. It does, but very slowly. Noise from 0.1 to 10 Hz doubles (approximately) with a lower bandwidth extended to 3.17e-8 Hz (a one-year period). Add another 6% for ten years.

- It’s challenging, but not impossible, to filter 1/f noise. Flicker noise from 0.1 Hz to 1 kHz (four decades) filtered to 10 Hz (two decades) only reduces the noise by 3 dB. Resistor values must be kept low for low noise which makes capacitor values large for a low frequency cutoff.

Amplifier noise is a combination of 1/f noise and flat (white) noise. The flat noise continues at low frequency but 1/f noise dominates (Figure 2). The 1/f noise continues at high frequency but flat noise dominates. The two blend at the corner frequency, adding randomly to make a 3dB increase.

Amplifier noise is summed over a bandwidth f1 to f2 by integrating the 1/f and flat noise separately over the bandwidth, then combined by the root-sum-of-squares (RSS).

- An N-times increase in flicker noise density increases the corner frequency by N2.

- The total noise from a decade below to a decade above the corner frequency is dominated by the flat-band noise (68%) even though the 1/f noise region “looks bigger.”

You can download an Excel file here that calculates integrated

The 1/f (one-over-f) low frequency noise region of amplifiers seems just a bit mysterious. An EDN reader asked for a discussion of 1/f noise—a challenging topic for a short blog.
Noise, 1/f noise