1. how do I convert these numbers to dBFS? I know dB are 20*log10(In/out) but I have negative numbers and I don't know what to put inside the logs!

2. how do I compute average RMS loudness?

3. how do I compute peak loudness?

I used matlab to read a wav files, I have a two vectors of numbers between -1 and 1... but I don't really know how to obtain certain loudness measures.

Let's make the problem general enough: suppose that I read a sequence of samples , that is x_1, x_2, ..., x_n

1. how do I convert these numbers to dBFS? I know dB are 20*log10(In/out) but I have negative numbers and I don't know what to put inside the logs!

2. how do I compute average RMS loudness?

3. how do I compute peak loudness?

I would appreciate any help from you guys... I hope somebody will help me!!!
Jp

Wikipedia...  Calculate the RMS value, and then convert that RMS value to dBFS.    Your sum-of-squares value can get really huge!  So in "real life" (with millions of samples in megabyte-size files) , I think most audio editors take reasonable size blocks of samples (whatever "reasonable" is...), calculate a series of RMS values and then take an overall average of those RMS values (like an "average of averages"... Does that make sense?

Wikipedia...  Calculate the RMS value, and then convert that RMS value to dBFS.    Your sum-of-squares value can get really huge!  So in "real life" (with millions of samples in megabyte-size files) , I think most audio editors take reasonable size blocks of samples (whatever "reasonable" is...), calculate a series of RMS values and then take an overall average of those RMS values (like an "average of averages"... Does that make sense?

(i.e. With 16-bits signed integers your "reference" is -32,768 or +32,767.)

... A dB is a ratio of two numbers. In the case of dBFS you are comparing a value to the max. possible, in this case 1. So to get a value in dBFS use 20*log10(x/1). To get the RMS value in dBFS just replace x in the expression with the calculated RMS value.

What course is this for?

Well this is mathematically strange. Does it make sense from the statistical point of view? What does it mean to compute 20*log(RMS/1)? RMS is the sample standard deviation of a signal that has 0 mean (assuming it is scaled so that it belongs to the [-1,1] interval), Now this means that RMS is the square root of the second moment of the signal so
20*log(RMS/1) is the log ratio of the square root of the second moment of a random variable to the peak value... I don't understand the statistical meaning.

I am taking a course in "time series analysis", the focus of the course is on continuous time stochastic processes sampled at a given fixed frequency, so music signals are an example of those processes.  I am taking a course in "time series analysis", the focus of the course is on continuous time stochastic processes sampled at a given fixed frequency, so music signals are an example of those processes.

Quote from: edeimos on 2010-02-09 20:49:16

... A dB is a ratio of two numbers. In the case of dBFS you are comparing a value to the max. possible, in this case 1. So to get a value in dBFS use 20*log10(x/1). To get the RMS value in dBFS just replace x in the expression with the calculated RMS value.

Well this is mathematically strange. Does it make sense from the statistical point of view? What does it mean to compute 20*log(RMS/1)? RMS is the sample standard deviation of a signal that has 0 mean (assuming it is scaled so that it belongs to the [-1,1] interval), Now this means that RMS is the square root of the second moment of the signal so
20*log(RMS/1) is the log ratio of the square root of the second moment of a random variable to the peak value... I don't understand the statistical meaning.

You lost me with the mathematical terminology...    Would it make more sense if you were to calculate the average level?  (i.e. The average of the absolute values?)  From a practical/engineering point of view RMS is just a different way of looking at "average", which relates to power.

...Actually, it sounds vaguely familiar from a business course I took....  Time Series Forecasting, or something?  I don't remember much, and I don't think I ever understood the practical applications...

Now I think I see the problem. You thought that the RMS was taken of a logarithmic signal. In fact, the RMS is calculated from the linear PCM data. Its magnitude is then expressed on a logarithmic scale.

No, from that response, I think you still have it wrong. I don't think anybody has proposed anything here which would logically lead to the use of asymmetric variables.

Moreover - not only has nobody really proposed anything on this thread that would cause the RMS computation to be supplied non-symmetric data, but even if you did, as long as the physical acts of computing square/mean/root make sense in the units of measurement, RMS is still a valid operation. It's not valid on dB figures not because the distribution is asymmetric, but because squaring a decibel is completely meaningless, and generally leads to quantities which have no bearing on anything else.

Would it help if I mentioned that much of the use of RMS figures has little to do with statistics, and much more to do with eg Parseval's Theorem? In electrical engineering, RMS figures are used as a tool to reduce AC problems into DC problems. In signal processing they facilitate signal computation between the time and frequency domains. In the context of loudness, RMS measurements can be almost directly interpreted in terms of signal energy.

From what you say I understand you have problems with mathematics and probability theory... very common among engeneers.

From what you say I understand you have problems with mathematics and probability theory... very common among
engeneers.

Quote from: DVDdoug on 2010-02-10 00:17:09

You lost me with the mathematical terminology...    Would it make more sense if you were to calculate the average level?  (i.e. The average of the absolute values?)  From a practical/engineering point of view RMS is just a different way of looking at "average", which relates to power.

a PCM encoded signal is a sample from a continuous time stochastic process... so the mathematics should be understood. Average is the first order moment of a random variable, RMS is the square root of the second order moment. They describe two different aspects of a symmetric (around zero)  probability distribution, basically they capture central tendency and mean dispersion (read variance if you like).  If you take higher order moments they will capture skewness and kurtosis and so on. But this make sense only for symmetric-around-the-mean random variables.

The sample RMS would make sense (from a probabilistic viewpoint) for a continuous time sinus or square signal, it doesn't make to much sense when you take the signal in dBFS  scale it is not symmetric at all, it has heavy tails and strong asymmetry. So most of the mathematics is just wrong.

I am so amazed how sloppy is the use of mathematics in certain fields.

Time Series Analysis  is the statistic of Stochastic Processes, i.e. dependent data. It's not that funny I know, but all the physics is based on that... quantum physics is probably the most relevant example.

But, still it is strange that the RMS is used on symmetric random variables (such as sinus wave signals) and non-symmetric random variables thinking that they measure the same thing!

Another way of describing RMS is that it's the norm of the vector of all the samples you're working with.