# perceived loudness of noise vs. sine wave

I am trying to understand what gains I need to use each for a white noise generator and a pink noise generator for a human ear to perceive the loudness to be the same as it is for a 440 Hz sine wave at 1 VRMS.

See the two diagrams below for more details:

• The noise generator is set up as a linear-feedback shift register (LFSR), running at 500 kHz with a depth of 31 bits. For simplicity, assume a flat response in the frequency domain in the human's hearing range. Though most people don't hear up to 20 kHz, let's still assume a range from 20 Hz to 20 kHz.
• To convert the digital noise to audible white noise, a 3-pole low-pass Butterworth filter with a cut-off frequency at 40 kHz and a gain of 1 is used. For simplicity, you may want to assume a perfect low-pass filter with a cut-off at 40 kHz.
• For pink noise, a 3 dB/oct. filter is used with a gain of 1 at 100 Hz.

What would be x and y below?

• for a human ear to perceive is a bit tricky because human hearing is not the same for everyone, but it can be put on a dB scale. The problem is in finding a reference that would fit the general population. Good luck with that... – Sparky256 Dec 28 '18 at 17:42
• I would assume there are some general rules how sound is perceived in terms of loudness/volume. – Hansel Dec 28 '18 at 17:57
• yes there are - see my answer below. – danmcb Dec 28 '18 at 18:03
• Search for "dBA sound level" – Scott Seidman Dec 28 '18 at 23:13

You need to figure this out by taking into account the Fletch Munson equal loudness curves (or the later ISO revision). However the answer also depends also on the actual loudness of your signal after passing through power amp and speaker, which in turn depends on amp gain and speaker efficiency. This is because the difference between 1kHz and other frequencies varies according to SPL.

Of course, there is also some variation from person to person but for the general case, these curves are about the best guide you have.

• OK, so I skimmed through the pages, incl. "equal loudness contours". Looks like I may have no other choice but determine the values experimentally. – Hansel Dec 28 '18 at 18:09
• Note also that program loudness of real program audio is a whole other can of wyrms, EBU BS1770-4 describes one method of producing some numbers that generally works well with typical radio and TV material, but it is not going to work well for either pink or steady state sine in all probability. – Dan Mills Dec 28 '18 at 23:32
• Indeed. Hence the existence of VU and PPM meters amongst other things. In fact it would be helpful if the OP would state their motivation and intent in doing this, as this is an area where context is definitely king. In fact the basic premise is a bit odd - trying to equate a pure sine wave to a noise signal in perceptual loudness is hard, because of the very different characteristics. It's like trying to match the levels of a flute and traffic noise - you would likely ask 10 people and get 7 or 8 different answers. – danmcb Dec 28 '18 at 23:36
• @dbm, I've used a VU meter (LM3916 datasheet,using the precision full-wave peak detector), assuming the perceived loudness is the same for pink and white noise when at 0 dB VU. But the white noise sounds softer. In the end I am looking for the same perceived loudness of white and pink noise as compared to spoken word (maybe going for a 440 Hz sine wave oversimplified my question). – Hansel Dec 29 '18 at 11:49
• @Hansel VU response is specifically NOT peak response. It is an averaged response which is designed to give some guidance about overall "loudness" and was sometimes used by mastering engineers in compiling music compilations (to get the tracks to match each other) or radio DJ's (in the days before compressors ruled the world). PPM however gives "peak" (still with a specific time characteristic) and is for setting recording levels and so on. Some use "VU" to mean both (e.g. para 1, 3916 datasheet) but they are really different things. Google, and you will find. – danmcb Dec 29 '18 at 12:03

Given the gain is 0dB at 100Hz with 3dB/octave pink filter at 1kHz the gain is -10 dB and at 440Hz it is -6.4 dB,

• my experimental results indicated an equal loudness at roughly the same Vpp level before filtering and about 20 dB above the spectral density of the pink noise at equal loudness.

There are many considerations. Here's just a few;

• The -3dB BW of loudness is less than 5kHz

• it is $$\\neq\$$ hearing BW from 20 Hz to 18 kHz over a 40dB equal loudness range
• Loudness of broadband noise energy is measured in $$\dB/\sqrt{Hz}\$$ yet when the peak dB SPL

• how do we measure broad spectral energy loudness to a pure sine tone?

• we are looking here for an SNR of 1 or 0dB meaning signal = noise power
• one could add the total RMS noise energy using average pink noise levels in sub-bands within the 5kHz -3dB BW of given Fletcher Munchen curve at a mid power range.

• You've obviously put a lot of thought into this but I have a hard time understanding your response in detail. When you write "my experimental results indicated an equal loudness ...", I suppose you mean the pink noise has to be amplified by 20 dB to be perceived as having the same loudness as the 440 Hz signal. What does the span of 300 Hz to 5 kHz that you've put in the Fletcher Munchen diagram indicate? What does it mean when you write that the -3 dB bandwidth of loudness is less than 5 kHz? Is that empiric data related to human hearing? – Hansel Dec 29 '18 at 11:40
• I performed the experiment at home in 15 minutes using (free) Audacity to create the noise and graphical pink filter and 440 Hz tone. At first the 440 Hz was so annoying and loud compared to just the pink noise that I started the 440 Hz much lower, but then since pink noise is so familiar and soft, I had to raise the level to slightly more than where I started. THis change in perception of "loudness" is due to the dB/√Hz effect of the tone alone, if you imagine a very small BW 0.1 Hz then after introducing pink noise to compare, and the dB/√BW is now smaller, the tone becomes more silent. – Tony Stewart Sunnyskyguy EE75 Dec 29 '18 at 15:13
• Educate yourself by experiments and the above answer as I did. Audacity has an intuititive menu. Track>add New, Generate>Noise, Analyze> Plot Spectrum, repeat all for Generate >Tone, Play to hear, adjust volume,mute. CHoose graph settings as I did for size to smoothen results. – Tony Stewart Sunnyskyguy EE75 Dec 29 '18 at 15:20
• OK, thanks for the hint. I didn't realize Audacity has the capability to create tracks with a number of different noise colors. I'll give that a try. – Hansel Dec 29 '18 at 17:40
• I forgot to mention ^A to select all of a waveform to Analyze> Then for the advanced reader, try to figure out the software algorithm for noise , You certainly don't need a 500kHz clock with a 44kHz sampled audio. The low frequency can be computed from the PRSG sequence length. – Tony Stewart Sunnyskyguy EE75 Dec 29 '18 at 18:02

Look up "Loudness" in Wikipedia for equal perceived loudness contours

https://en.wikipedia.org/wiki/Loudness

• This would make a good comment to the question, but link-only answers are frowned upon. – Scott Seidman Dec 28 '18 at 18:26