Re: [AUDITORY] Question about level conversions

["Richard F. Lyon" <dicklyon@xxxxxxx>]

Step 2 looks like a wrong sign.  If the tone level is fixed, the tone energy should be an increasing function of duration, but you show a decreasing function of duration.  Did this come from Richards or Ahumada?

The use of units like dB/Hz makes my skin crawl; but I think what you've done with it makes sense for the noise.

In communication theory, we use measures like Eb/N0, that is, energy per bit relative to noise spectral density.  So the E_tone/N0 seems sensibly analogous as a non-dimensional quantity.  It will be easier to see it's non-dimensional if you get out of the dB realm.

By the way, I haven't heard from Al Ahumada for a while, but he was a bit of a bud about 25 years ago, in Silicon Valley.

Dick

On Thu, Mar 9, 2023 at 4:14 PM Alejandro Osses <ale.a.osses@xxxxxxxxx> wrote:

Dear list, hello!

I have a technical question about presentation levels and signal-to-noise ratios (SNRs), as expressed in the current literature and in more classical studies (some of the papers I have recently investigated: Richards (1992, JASA - doi:10.1121/1.402831) and Ahumada et al (1975, JASA - doi:10.1121/1.380453)).

I have some thoughts, which I will share next, please feel free to comment on any of these aspects in terms of maths or concepts, or in terms of conceptual accuracy....

In classical papers (think also of the work by David Green and colleagues), SNRs are often expressed as energy with respect to N0, often reported as E/N0. In this _expression_ E is typically the energy of a target signal (e.g., a tone) which is embedded in a noise, whose spectrum level is N0. I am interested to know the mathematical expressions needed to convert between noise levels to the appropriate target sound level. I will now give an example about how I am currently doing such a conversion.

The example: Let's say I have a Gaussian noise (duration=0.5 s) with frequencies between 0 and 5 kHz, with a total RMS level of 70 dB SPL. On the other hand, my target sounds have a centre frequency of 500 Hz, a duration of 0.1 s, temporally centred in the noise. I want to know the level of the tone, if E_tone/N0 needs to be 11.8 dB (this is actually from Ahumada et al (1975)).

My calculations are as follows:

• Step 1: N0 = 33 dB/Hz (because N0 = lvl_noise - 10*log10(BW) = 70 - 10*log10(5000) = 33 dB/Hz)
• Step 2: The energy of the tone E_tone can be calculated from the normalised level to a duration of 1 second: E_tone = lvl_tone - 10*log10(dur_tone) = lvl_tone + 10 dB (using dur=0.1 s)
• Step 3: At the same time: E_tone needs to have a E_tone/N0 of 11.8 dB or E_tone = N0 + 11.8 dB
• From Step 3 and 4: N0+11.8 = lvl_tone + 10 -----> for the current values: lvl_tone = 33+11.8-10 = 34.8 dB
  

So, my tone will have a level of 34.8 dB SPL for this example.

Am I correct with my calculation method or am I missing any important concept?

Thank you for the feedback,

Alejandro Osses