Download Chapter 4: The Science of Digital Media - Digital Audio Representation | THE 100 and more Assignments Theatre in PDF only on Docsity!
Supplement to Chapter 4 of The Science of Digital Media – Digital Audio Representation
Worksheet – Digital Audio > Root Mean Square Amplitude^1
Introduction: The root-mean-square amplitude (RMS amplitude) of a signal relates to the average amplitude of a sound wave over a given period. This period can either be the entire wave or a portion that you have selected. The formula for r , the RMS amplitude of a signal, is as follows:
Let N be the number of samples in an audio signal. Let x i is
the amplitude of the i th^ sample. Then the root-mean-
square amplitude , r , is defined as
=
N
i
r N xi
1
Equation 1
(Different audio processing programs use different terms for what we have called “RMS amplitude.” Adobe Audition uses the term “RMS power”; Sony Sound Forge uses “RMS level.” We prefer the more generic and descriptive term “RMS amplitude.”)
RMS amplitude can be is expressed in terms of sample values or in decibels- full-scale (dBFS). If r is the RMS on the scale of sample values and n is the bit depth of the audio file, then the RMS can be converted to dBFS with the following equation:
20 log n
r
dBFS
Equation 2
Tips for Using MATLAB: We'll implement RMS amplitude using MATLAB’s programming language, which resembles C in many respects.
To create a function in MATLAB, go to File\New\m-file. (You can also use an external text editor like NotePad or WordPad.) In MATLAB, the function file is called an m-file, which contains the code to run a function that can later be called from the MATLAB command window. The following syntax is used to declare a function test that accepts one input argument ( in ) and one output argument ( out ).
(^1) This material is based on work supported by the National Science Foundation under
Grant No. DUE-0340969. This worksheet was written by Todd Martin and Jennifer Burg (burg@wfu.edu).
function out=test(in)
In MATLAB, the input argument is not strongly typed; it can be either a single numerical value or an array, and the function declaration would be the same. In our function, the function receives an array of double s as input, representing the amplitude values of the signal in the time domain.
The body of the function is written in the m-file. Each line (except the declaration of a flow control statement or a function) must be followed by a semicolon, like in C. In our function we will first make sure that there is exactly one input argument when the function is called. To do this, we will use the nargin keyword, as follows:
if(nargin ~= 1) error(‘Incorrect number of input arguments.’); end
This is the general format for all if statements in an m-file. We will next use a for loop to cycle through all of the elements of the input array. A basic for loop in MATLAB is implemented as follows:
N=length(in); runningSum=0; for(i=1:1:N) runningSum = runningSum + (in(i) * in(i)); end
This sum will then need to be divided by the N value, and the square root will need to be taken. The square root can be implemented with the MATLAB sqrt function within the m-file. There is not any end command at the end of the function itself. When the entire function is complete, save the text file as RMS.m and return to the MATLAB command window.
In order to execute the function, you need to include the folder in which you saved RMS.m in the MATLAB path list. MATLAB searches the folders in this list when you call a function in the command window. The path list can be altered by clicking File/Set Path/Add Folder. (Or you can change the current directory to the one that contains the m-file.)
The function can be executed from the command window as follows:
>> r=RMS(signal);
After this command is executed, the variable r contains the RMS amplitude. You don't need to include the conversion to dBFS in the RMS program; we'll look at both values and manually convert between them.
Exercise 1
convert to dBFS, allowing us to compare the result of our MATLAB calculation with what we see in Audition’s statistics window.
There are two ways to obtain the signal for this exercise. If you have Adobe Audition or another equivalent sound processing program, follow the instructions below. If you don't have these programs, skip the following steps and follow the second set of instructions.
Instructions for Adobe Audition Create a 200 Hz wave in Audition by opening the program and clicking File/New. In the New Waveform menu, enter a sampling rate of 1000 Hz and choose Mono and 16-bit resolution.
You should now have a blank waveform window. Click Generate/Tones, and at the Generate Tones menu choose a base frequency of 200 Hz. Make sure that the dB volume meter is set to 0 dB and set the duration to 2 seconds.
You should now have a waveform on your screen. To find the RMS amplitude of the wave according to Audition, click Analyze/Statistics. We're interested in “Total RMS Power” value. Find this value and take note of it.
Total RMS
Power
Now, export the wave as a .wav file. Do this by clicking File/Save As and choosing the file type to be Windows PCM ( *.wav ). Choose the folder in which you would like to save the file and save the file as signal3.wav.
Choose
.wav file
The next step is to import signal3.wav into MATLAB and store it in a variable called signal3. This variable will point to an array of sample values for the wave. To import into MATLAB, first set the current path in MATLAB to the file path of the folder that contains the file. The
>> signal3Adjusted=signal332768;*
Now, use this signal3Adjusted to calculate the RMS. Use the program that you wrote in Exercise 1. Store the result in r.
The result of this computation is in terms of amplitude. In order to convert to dBFS, use Equation 2 with r3 as the r value in the equation. Also, assume a bit depth of 16 bits to mirror the 16-bit depth that Audition uses to calculate the RMS power of this wave. This means that n must be set to 16. In MATLAB, the log10 command can be used for the logarithm. Store the result of this calculation in the variable r3dBFS.
If you have Adobe Audition and created the 200 Hz wave for Exercise 4 in this program, compare the value in r3dBFS and the “Total RMS Power” value shown in the Audition statistics window. (If you have Sound Forge, look in the Statistics window. Other audio processing programs have similar features for RMS.)
