Wave Propagation

Stand next to a still body of water, pick up a pebble, and toss it to the middle of the pond. As the pebble breaks the surface, it displaces the water it comes in immediate contact with. Those water particles displace other water particles as they are pushed out, so you get a rippling effect that grows from the point where the pebble originally made contact out to the edges of the pond. Once the ripples meet the outer edges of the pond, the rings will begin to move back in, and the ripples will continue until the energy created by tossing the pebble into the water runs out.

This principle is the same one on which sound propagates. When sound is created at the source, the vibrations radiate out in waves, like the ripples in a pond, except they travel in every direction. These sound waves bounce off particles of different mediums in order to travel outward. As the sound waves bounce off particles, some of the energy is absorbed so the vibration weakens. The density of a medium plays a part in how fast a sound wave can propagate or move through it. The more tightly that particles of matter are packed together, the faster a sound wave will move through it, but the sound wave will degrade faster as well. So sound travels faster through a solid than a liquid and faster through a liquid than a gas, such as air. As the sound expands across a larger area, the power that was present at the source to create the original compression is dissipated. Thus, the sound becomes less intense as it progresses farther from the source. The sound wave will eventually stop when all the energy present in the wave has been used up moving particles along its path.

Observe the analogy of the pebble in the pond. When the ripples first begin, they will be taller and closer together. As the energy dissipates, the ripples will become shorter and farther apart, until eventually they cease to demonstrate any effect on the water’s surface. This effect can be observed with sound as well. Stand in a canyon and shout out toward the cliffs. The sound you create will travel out and reverberate off the cliffs and return to you. However, what you hear will sound degraded. You may even hear yourself multiple times in an echo, but each time the sound will be weaker than the time before until you can’t hear anything at all. If someone shouted out to you under a body of water, by the time the sound reached your ears, it would have degraded so much from bouncing off all the many particles in the water that you would not be able to articulate what was said.

Technical Properties of Sound

All forms of energy can be measured. Light and sound sources in nature are forms of energy. Acoustical energy consists of fluctuating waves of pressure, generally in the air. One complete cycle of that wave consists of a high- and low-pressure half-cycle. This means that half of a sound wave is made up of the compression of the medium, and the other half is the decompression, or rarefaction, of the medium. Imagine compressing a spring and then letting it go. Now imagine that the spring represents air molecules and your hand is the acoustic energy. Figure 2-1 illustrates the technical properties of a sound wave.

To better understand how sound waves are measured, we must define some terms. Frequency is the rate of the air pressure fluctuation produced by the acoustic energy wave. To be heard by human ears, wave frequencies must be between 20 and 20,000 cycles per second. Although there is more to it than just frequency, in general, the frequency of a sound wave corresponds with the perceived pitch of the sound. The higher the pitch, the higher the frequency. When we are talking about sounds and music, there is not just one single frequency, but many different frequencies overlapping with each other. This means many different frequencies are being represented within a single signal. The bandwidth of a particular signal is the difference between the highest and the lowest frequencies within the signal.

Amplitude is the measure of the magnitude of change in each oscillation of a sound wave. Most often this measurement is peak-to-peak, such as the change between the highest peak amplitude value and the lowest trough amplitude value, which can be negative. Bear in mind the relationship between amplitude and frequency. Two waveforms can have identical frequency with differing amplitude and identical amplitude with differing frequency. For sound waves, the wave oscillations are representative of air molecules moving due to atmospheric pressure changes, so the amplitude of a sound wave is a measure of sound pressure. Sound pressure is measured in pascals or millibars:

1 pascal = 1 newton per square meter

1 millibar = 100 pascals

A newton is the standard international unit for force. It is equal to the amount of net force required to accelerate a mass of one kilogram at a rate of one meter per second squared. Newton’s second law of motion states F = ma, multiplying m (kg) by a (m/s²); the newton is, therefore, N = kg(m/s²).

A frequency spectrum is the complete range of frequencies that can be heard, just as the visible spectrum is the range of light frequencies that can be seen. Devices and equipment can also have frequency spectrums, or ranges.

Basic sine waves cannot have an “average” per se. Rather, it would equal zero because the wave peaks and valleys are symmetrical above and below the reference of zero. What is much more helpful when discussing the average amplitude of a wave over time is called root mean squared (RMS). RMS is often used to calculate a comparable measure of power efficiency, such as in audio amplifiers. RMS measures mean output power to mean input power. RMS is just the squaring of every point of a wave and then finding the average. Squaring a negative value always results in a positive value. RMS gives us a useful average value for discussing audio equipment and comparing amplitude measurements. Basically, RMS is much more similar to the way we hear sound, as opposed to measuring just the peaks of a wave, because we don’t hear just the peaks.

A sound wave emanating from a source is like an expanding bubble. The power of the sound would be the energy on the surface of the bubble. As the surface of the bubble expands, that energy is used up in order to move, or vibrate, the air ever farther outward. That means eventually the power that the bubble started with at its source would be expended. The power of sound, or sound’s acoustical power, is a measure of amplitude over time. The sound had a particular measurement at the moment it was emitted from the source, but as it travels, over time the power decreases as the energy present in the sound wave is expended transmitting itself through the medium. The transmission of acoustical energy through a medium is measured in watts. A watt is just the rate of transfer of energy in one second or one joule per second. A joule is equal to the energy expended in applying a force of one newton through a distance of one meter. The following equation is used to calculate sound’s acoustical power:

J=kg*m²/s²=N*4444m=Pa*m³=W*s

where

• N is the newton.

• m is the meter.

• kg is the kilogram.

• s is the second.

• Pa is the pascal.

• W is the watt.

Understanding Attenuation and Noise

When Alexander Graham Bell placed his first audio call to his assistant, Watson, it was only to the next room. Bell’s journal records that the audio was loud but muffled. There are two reasons why the sound was hard to hear: attenuation and noise.

Attenuation, as has previously been discussed, is the degrading of the sound wave as it loses energy over time and space. When the telephone was first made available to the public market, telephone companies had to set up repeaters to account for this loss in signal strength. These repeaters could boost the intensity of the analog wave at different points along the path between two telephones, or nodes. This allowed telephone calls to traverse much longer distances. However, telephone repeaters solved only half the problem.

In audio terms, noise doesn’t necessarily refer to something you hear. Noise can also refer to inaccuracies or imperfections in a signal transmission that was not intended to be present. Noise comes in different forms, like acoustical, digital, or electrical. Noise can be introduced to a signal in many ways, like faulty connections in wiring or external signal interference. In telephony, as audio signals are transmitted across a wire using electrical signals, those same wires can pick up energy from external sources and transmit them along the same current carrying the audio signal. If you are old enough to remember using analog phones, you might also remember hearing a crackling sound in the background during a phone call. That background noise was caused by interference.

When an analog signal must be transmitted a long distance, the signal level is amplified to strengthen it; however, this process also amplifies any noise present in the signal. This amplification could cause the original analog signal to become too distorted to be heard properly at the destination. Therefore, it is very important with analog transmissions to make sure the original audio signal is much stronger than the noise. Electronic filters can also be introduced, which helps remove unwanted frequency from the analog signal. The frequency response may be tailored to eliminate unwanted frequency components from an input signal or to limit an amplifier to signals within a particular band of frequencies. Figure 2-2 illustrates how noise riding on an analog signal can be amplified at repeaters.

Nyquist-Shannon Sampling Theorem

Always keep in mind that digital signals are just long strings of binary numbers. Analog signals must be somehow converted into strings of numbers to be transmitted in the digital realm. When a continuous analog signal is converted to a digital signal, measurements of the analog signal must be taken at precise points. These measurements are referred to as samples. The more frequent the sample intervals, the more accurate the digital representation of the original analog signal. The act of sampling an analog signal for the purpose of reducing it to a smaller set of manageable digital values is referred to as quantizing or quantization. The difference between the resulting digital representation of the original analog signal and the actual value of the original analog signal is referred to as quantization error. Figure 2-3 illustrates how quantizing an analog signal would appear.

In Figure 2-3, each movement upward on the Y-axis is signified by a one-binary digit. Each movement horizontally on the X-axis is signified by a zero-binary digit. Each movement downward on the Y-axis is signified by a negative one-binary digit. In this manner an analog signal can be quantized into a digital signal. However, notice that the binary signal is not exactly the same as the original analog signal. Therefore, when the digital signal is converted back to analog, the sound quality will not be the same. A closer match can be obtained with more samples. The size of each sample is measured in bits per sample and is generally referred to as bit depth. However, it will never be exact because a digital signal cannot ever match the fluid flow of an analog signal. The question becomes, can a digital signal come close enough to an analog signal that it is indistinguishable to the human ear?

When an audio signal has two channels, one for left speakers and one for right speakers, this is referred to as stereo. When an audio signal has only one channel used for both left and right speakers, this is referred to as mono. Using the preceding quantizing information, along with whether an audio signal is stereo or mono, it is relatively simple to calculate the bit rate of any audio source. The bit rate describes the amount of data, or bits, transmitted per second. A standard audio CD is said to have a data rate of 44.1 kHz/16, meaning that the audio data was sampled 44,100 times per second, with a bit depth of 16. CD tracks are usually stereo, using a left and right track, so the amount of audio data per second is double that of mono, where only a single track is used. The bit rate is then 44,100 samples/second × 16 bits/sample × 2 tracks = 1,411,200 bps or 1.4 Mbps.

When the sampling bit depth is increased, quantization noise is reduced so that the signal-to-noise ratio, or SNR, is improved. The relationship between bit depth and SNR is for each 1-bit increase in bit depth, the SNR will increase by 6 dB. Thus, 24-bit digital audio has a theoretical maximum SNR of 144 dB, compared to 96 dB for 16-bit; however, as of 2007 digital audio converter technology is limited to an SNR of about 126 dB (21-bit) because of real-world limitations in integrated circuit design. Still, this approximately matches the performance of the human ear. After about 132 dB (22-bit), you would have exceeded the capabilities of human hearing.

To determine the proper sampling rate that must occur for analog-to-digital conversion to match the audio patterns closely, Harry Nyquist and Claude Shannon came up with a theorem that would accurately calculate what the sampling rate should be. Many audio codecs used today follow the Nyquist-Shannon theorem. Any analog signal consists of components at various frequencies. The simplest case is the sine wave, in which all the signal energy is concentrated at one frequency. In practice, analog signals usually have complex waveforms, with components at many frequencies. The highest frequency component in an analog signal determines the bandwidth of that signal. The higher the frequency, the greater the bandwidth, if all other factors are held constant. The Nyquist-Shannon theorem states that to allow an analog signal to be completely represented in digital form, the sampling rate must be at least twice the maximum cycles per second, based on frequency or bandwidth, of the original signal. This maximum bandwidth is called the Nyquist frequency. If the sampling rate is not at least twice the maximum cycles per second, then when such a digital signal is converted back to analog form by a digital-to-analog converter, false frequency components appear that were not in the original analog signal. This undesirable condition is a form of distortion called aliasing. Sampling an analog input signal at a rate much higher than the minimum frequency required by the Nyquist-Shannon theorem is called over-sampling. Over-sampling improves the quality or the digital representation of the original analog input signal. Under-sampling occurs when the sampling rate is lower than the analog input frequency.

The Nyquist-Shannon theorem led to the Digital Signal 0 (DS0) rate. DS0 was introduced to carry a single digitized voice call. For a typical phone call, the audio sound is digitized at an 8 kHz sample rate using 8-bit pulse-code modulation for each of the 8000 samples per second. This resulted in a data rate of 64 kbps.

Because of its fundamental role in carrying a single phone call, the DS0 rate forms the basis for the digital multiplex transmission hierarchy in telecommunications systems used in North America. To limit the number of wires required between two destinations that need to host multiple calls simultaneously, a system was built in which multiple DS0s are multiplexed together on higher-capacity circuits. In this system, 24 DS0s are multiplexed into a DS1 signal. Twenty-eight DS1s are multiplexed into a DS3. When carried over copper wire, this is the well-known T-carrier system, with T1 and T3 corresponding to DS1 and DS3, respectively.

Outside of North America, other ISDN carriers use a similar Primary Rate Interface (PRI). Japan uses a J1, which essentially uses the same 24 channels as a T1. Most of the world uses an E1, which uses 32 channels at 64 kbps each. ISDN will be covered in more depth in Chapter 5, “Communication Protocols.” However, it is important to note that the same sampling rate used with DS0 is also used in VoIP communications across the Internet. This led to a new issue that in turn opened up a new wave of advancement in the audio communication industry: How could digital audio signals be sent over low-bandwidth networks?

Data Compression Equals Bandwidth Conversion

The answer to the question regarding digital audio signals being sent over low-bandwidth networks came with the development of data compression. Bandwidth is the rate that data bits can successfully travel across a communication path. Transmitting audio can consume a high amount of bandwidth from a finite total available. The higher the sampled frequency of a signal, the larger the amount of data to transmit, resulting in more bandwidth being required to transmit that signal. Data compression can reduce the amount of bandwidth consumed from the total transmission capacity. Compression involves utilizing encoding algorithms to reduce the size of digital data. Compressed data must be decompressed to be used. This extra processing imposes computational or other costs into the transmission hardware.

Lossless and lossy are descriptive terms used to describe whether or not the data in question can be recovered exactly bit-for-bit when the file is uncompressed or whether the data will be reduced by permanently altering or eliminating certain bits, especially redundant bits. Lossless compression searches content for statistically redundant information that can be represented in a compressed state without losing any original information. By contrast, lossy compression searches for nonessential content that can be deleted to conserve storage space. A good example of lossy would be a scenic photo of a house with a blue sky overhead. The actual color of the sky isn’t just one color of blue; there are variances throughout. Lossy compression could replace all the blue variances of the sky with one color of blue, thus reducing the amount of color information needed to replicate the blue sky after decompression. This is, of course, an unsubtle oversimplification of the process, but the concept is the same.

It is worth noting that lossy compression formats suffer from generation loss; repeatedly compressing and decompressing the file will cause it to progressively lose quality. This is in contrast with lossless data compression, where data will not be lost even if compression is repeated numerous times. Lossless compression therefore has a lower limit. A certain amount of data must be maintained for proper replication after decompression. Lossy compression assumes that there is a trade-off between quality and the size of the data after compression. The amount of compression is limited only by the perceptible loss of quality that is deemed acceptable.