How Large Are Signals

🎯 Learning Objectives:

We will 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 term sound and audio indiscriminately. What is the difference here? Let have a look at animation below and the explanation below.
Sound is a pressure disturbance in a physical medium like air that can be perceived by a 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 into audio. The audio signal itself cannot be perceived by hearing. It is just an electronic representation and thus for it to be audible the audio signal would have to be converted into sound using an output transducer such as a loudspeaker. We would not get into the details of physical to analog conversion of sound to audio and how transducers work. We will just assume that we are given an audio signal to work with. Electrical audio signals are analogous to acoustic signals or the pressure waves that it was recorded from. A transducer basically maps the changes in pressure of a wave the changes in the 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 zero volts. When we study the Signal's 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 will take a very simple audio signal, the simplest of them all, the humble sine wave.

Peak Value:

The peak value determines the instantaneous maximum amplitude value within one period of 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 measurements is used in symmetric signals signals that deviate equally far away from zero in both the positive and negative directions.

Peak to Peak Value:

In practical audio engineering particularly for 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 or 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 another in order to reproduce the sound for a sinusoid. The peak to peak value is just 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 percussive instrument's impact can be highly asymmetric.

Average Value:

The average value is based on the average of the numerical values of the amplitude of a signal over a period. How would we calculate the average? Well we sum up a set of values and divide it by the number of values we summed up.
So let us see how we can actually, mathematically express this for signals. We have a signal let us say $f(t)$, which is a function of time. This function could be a simple sine function over time as shown in Figure above and mathematically given as below.

Let us add all of the values of this sine function over the duration of its period. So let us say that this function completes exactly one period in $T$ seconds. We need to add up all of 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. We need to use integral calculus the as a tool to calculate the sum of the values within this period. We integrate $f(t)$ with respect to time t over the range of 0 to T, which is the period. This gives us the sum. So, we just need to divide this by the period itself and that gives us the average.

Obviously this formula does not make much sense if we cannot visualize what is happening.
So let us use a graphing calculator to visualize as given in the following link:

Below is interactive Desmos graph plotter interface. You can follow the steps given below to play with the calculator.

Step 1: Let us start off with the sine function $f(t)=sin(t)$

• Here here you get the familiar sine function that we know and love. Let us consider the x-axis here as the time axis t.

Step 2: The sine function here starts from its period of 0 and ends somewhere at around 6.28, whihc makes sense.

• Insert into the second cell the followings $T=2π$.

Step 3: Now establish an integrate over the range of the period by summing up $f(t)$ with respect to t over the range of $0$ to $T$, which is $2π$.

• To do so, insert in the third cell the following: $y=\int_0^T f(t) dt$
  Note: Here you can already see a problem. The result that you get is zero. That is because the sine wave is exactly symmetrical across the amplitude axis, and you get a peak in the positive amplitude axis and a trough from the negative axis. Since it i s symmetrical the positive and negative values cancel each other out effectively. The sum and in turn the average is zero. The result of zero does not make a lot of sense the average value of a sine function cannot be zero. It is doing some work and besides the positive and negative axes are just frames of reference. So hence to avoid this problem we take the absolute value of the function instead and discard the sign.

Step 4: Let us see what the sine function looks like when we discard the sign and take the absolute value of the function.

• Between second and third cell insert another cell and put the following into it $y=|f(t)|$.
  It is a sort of like a bouncing sine wave constrained to the positive axis.

Step 5: Now modify the cell four by $y=\int_0^T \|f(t)|, dt$, and you see, we get a value like this where the sum is the value of $4$.
Though this may look like a function, with it being represented as a straight line on a graph. It is a single scalar value, since we are reducing the sine function by summing it and removing the reference of time.

Step 6: Now we can divide the function in cell four by the time period as $y=\frac{1}{T}\int_0^T |f(t)| dt$, and that will give us the average of around $0.636$.
The average value calculations though we have described it here is rarely ever used in audio signal measurements. The generally and widely used metric for measuring the average of a signal is called our Mass value or root mean Square value. Since it is quite important to understand the significance of the RMS value why it is used and what it means in the context of audio signals

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