In this video, we’ll take a look at some of the characteristics and properties of an
audio signal.
This means that we are finally leaving the physical domain of sound as perceived as pressure
waves into the analog domain where sound is represented as voltage varying audio signals.
We tend to use the terms “sound” and “audio” indiscriminately.
But what’s the difference here?
Sound is a pressure disturbance in a physical medium like air, that can be perceived by
hearing.
That doesn’t necessarily mean that sound has to be audible.
Infrasonic and ultrasonic sounds are below and above the normal range of human hearing,
but these are sounds nonetheless, imperceptible disturbances in a physical medium.
Audio on the other hand, is an electrical or digital representation of sound.
An input transducer such as a microphone is needed to convert sound to audio.
The audio signal itself cannot be perceived by hearing, it’s just an electronic representation,
and thus is for it to be audible, the audio signal would need to be converted back into
sound using an output transducer such as a loudspeaker.
We won’t get into the details of physical to analog conversion of sound to audio, and
how transducers work.
We’ll just assume that we are given an audio signal to work with.
Electrical audio signals are analogous to the acoustic signal or the pressure wave that
it was recorded from.
A transducer basically maps changes in pressure of a wave to changes in potential difference
or voltage of a signal.
The amplitude values of the signal represent the different voltage levels, and they are
usually bi directional alternating signals with a steady state of 0 volts, and a swing
on either side.
When we study the signals’ amplitude over time, or generally ask the question, how large
an audio signal is, we have several characteristics which define the signal.
To discuss these, we’ll take a very simple audio signal, the simplest of them all, the
humble sine wave.
There is an entire module dedicated to sinusoids and what they are, if you want to learn more.
But we’ll stick to a sine wave because it is the simplest and purest form of an audio
signal.
The peak value determines the instantaneous maximum amplitude value within one
period of the signal.
The peak value can also be the maximum value that is ascertained during any period under
consideration, even though the usual designation here involves the maximum value within one
period.
Usually, the peak value measurement is used with symmetric signals, signals that deviate
equally far from 0 in both a positive and negative direction.
In practical audio engineering, particularly for the recording and transmission of audio
signals, it is the peak-to-peak value that is significant.
This value describes, so to speak, how much “space” (voltage range) the signal “takes
up” in the electrical circuit.
Also, the peak-to-peak value indicates how much a membrane must move in a purely physical
sense from one extreme to the other to record or reproduce the sound.
For a sinusoid, the peak to peak value is twice the peak value.
No difficulty there.
But in the real world, acoustic signals contain a lot of asymmetry.
The voice, for example, or musical instruments like a percussion instrument’s impact can
be highly asymmetric.
The average value is based on the average of the numerical values of the amplitude of
the signal over a period.
How would you calculate the average?
You sum up a set of values, and you divide it by the number of values you summed up.
Quite simple right.
So let’s see how we can mathematically express this for signals.
You have a signal, let’s say f(t), a function of time.
This function could be a simple sine function over time.
Like so.
f(t) = sin(t)
Let’s add all of the values of this sine function over the duration of its period.
So, let’s say that this function completes exactly one period in T seconds, we need to
add up all the values of this sine function within this period.
How do we do that?
This is a continuous function after all, and there are infinitely many points within this
period of time to include.
So we need to use integral calculus as a tool to calculate the sum of all the values within
this period.
So we integrate f(t) with respect to time t, over the range of 0 to capital T which
is the period.
This gives the sum, we just need to divide this by the period itself and that gives us
the average.
Obviously this formula doesn’t make much sense if we can’t visualize what’s happening,
so let’s use a graphing calculator to visualize every step that we took, that’ll lead us
to the average.
I’m going to use Desmos here, and you can check the link down below to play with the
calculator with all the values filled in.
So, let’s start off with a sine function, f(t) = sin(t).
Here you get the familiar sine function that we know and love.
Let’s consider the x axis here as the time axis t.
The sine function here starts its period from 0 and ends somewhere at around 6.28.
This is in fact the value of 2PI, and it makes sense if you’ve seen the module on the origins
of the sine wave, that a sine wave’s period is 2Pi radians.
Now we know what the period of this sine wave is.
It’s 2Pi, so let’s establish that.
Let’s now integrate over the range of the period.
So, I’ll sum up f(t) with respect to t, over the range of 0 to T, which is 2Pi.
Now, you can already see a problem here, the result that you get is zero.
That’s because the sine wave is exactly symmetrical across the amplitude axis, and
you get a peak in the positive amplitude axis and trough in the negative axis.
Since it’s symmetrical, the positive and negative values cancel each other out, and
effectively the sum and in turn the average is zero.
The result of zero doesn’t make a lot of sense.
The average value of a sine function cannot be zero, it’s doing some work and besides
the positive and negative axes are just frames of reference, they don’t mathematically
cancel each other out in the real world.
So hence, to avoid this problem we take the absolute value of the function instead, and
discard the sign.
Let’s see what the sine function looks like when we discard the sign and take the absolute
value of the function.
It’s sort of like a bouncing sine wave constrained to the positive axis.
If we now take a sum of this function, we get a value like this, where the sum is a
value of 4.
Though this may look like a function, with it being represented as a straight line on
the graph, it’s a single scalar value, since we are reducing the sine function into by
summating it, and removing the reference of the time..
Now we can divide this by the time period to get the axis.
And this gives us the average of around 0.636 for a sinusoidal function.
The discussion carries on in the next video, the average value calculations, though I’ve
described it here, is rarely used in audio signal measurements.
The generally and widely used metric for measuring the average of a signal is called RMS value
or Root mean squared value.
Since it's quite important to understand the significance of the RMS value, why it;s used
and what it means in the context of audio signals, I’ve dedicated the whole of the
next video for that.
I’ll see you in that one.