For the majority of humans, there is nothing simpler than listening to sound. You simply, well, listen.
When it becomes necessary to describe the listening experience analytically, however, a host of complex equations and diagrams are required to describe even the simplest of sonic events.
The benefit of mathematical analysis is that it can yield insights that are not apparent through intuition alone. Acoustic signals are easily measured, and the audio components that produce them have characteristics that can be measured.
We do not expect specifications to tell us how a product sounds. This is what listening is for. The main purpose of specifications is to allow us to make sure that we have the right tool for the job, and this information is most often presented in the form of charts and graphs.
But what does this information really mean?
The heart of understanding the specification sheets that describe audio products is the understanding of dependent and independent variables.
The concept is one that most people use every day, though often without realization.
An independent variable is one that describes a series that has a fixed value. For example, the time of day in the city that you live in is an independent variable. Regardless of what happens tomorrow, time will progress like it did today.
What will change are your moment-to-moment activities. These events represent a dependent variable. They depend on time.
If you look at a page in your day planner, you are looking at a plot of activities vs. time.
Time is the independent variable. It is the same on every page of the planner. The scheduled events are the dependent variables, because where you go and what you do depends on what time it is. Most graphs show the relationship between dependent and independent variables.
Now let’s look at a variation on the theme. Let time be the independent variable (it usually is) and let the loudness of the sound system during a show be the dependent variable. The plot might look something like Figure 1.
The horizontal axis represents time (the independent variable) and the vertical axis represents loudness (the dependent variable).
We will call the horizontal axis the x-axis and the vertical axis the y-axis, although any two letters would do. The values on each axis are usually discrete, meaning that they are individual samples, points, or measured values called data points.
The fact that most graphs look like squiggly lines just means that after many data points were taken, they were joined with a line to make it easier to read.
Such two-dimensional plots are found on virtually every good specification sheet in existence. They simply answer the question “What is the value of y when the value of x is this?” Some examples of two-dimensional plots found in audio engineering include:
Each plot shows the value of y for a given value of x. Pretty cool. In math-speak, in each case it can be said that y is a function of x. (We sound smarter when we say it like this.)
From this example, it can be seen that frequency is a very common independent variable in the world of audio and acoustics. The y parameters are said to be frequency-dependent.
In audio and acoustics, almost all parameters that we care to know anything about are frequency-dependent. This means that the answer to virtually any question regarding any of the y parameters is “it depends” — y depends on x.
An example of a frequency-dependent parameter is the setting of a graphic equalizer. In fact, it’s a really good example because it is basically an xy plot of the type that we have been describing.
The x variable is frequency, and the y variable is relative level. The y value depends on the x value. When you look at the front panel of a graphic equalizer, you are looking at an xy graph, which is why it’s called a graphic equalizer.
What Time Is It?
Another common independent variable is time. Many parameters in audio and acoustics are time-dependent. Examples include loudness, temperature and background noise, just to name a few.
Note that Figure 1 just gives us values. It’s still up to us to know what they mean and how to apply them.
Graphs are valuable because they give us some visual feedback regarding trends in the data. For instance, a glance at Figure 3 (later in this article) shows that the loudspeaker’s on-axis directivity is increasing as a function of frequency.
This means that everyone in the room might hear the low-frequency events, like a bass guitar, but only those in front of the loudspeaker will hear the high-frequency events, like the crash of a cymbal.
It’s clear why we would want the directivity of a sound reinforcement loudspeaker to be “frequency-independent.” The directivity of such a device would be a straight horizontal line.
It’s also important to consider the resolution of the graphed data. The closer together we place the points on the x-axis, the less likely it will be that we missed a significant data point when we measured.
For example, we could take the page of a day planner and break the time axis down into hours, minutes, seconds, or even fractions of a second.
Obviously, there is a point of diminishing return on resolution. It must always be appropriate for the data being plotted. If you were plotting the arrival time of the tweeter in the main array to the back of the balcony, then one millisecond resolution would be meaningful.
But that same resolution would be extreme overkill for plotting your daily schedule. What time resolution do I need? Again, it depends!