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Companding - Compressing + Expanding

2012-02-07T23:15:00.000-08:00

Companding algorithms reduce the dynamic range of an audio signal. In analog systems, this can increase the signal-to-noise ratio (SNR) achieved during transmission, and in the digital domain, it can reduce the quantization error (hence increasing signal to quantization noise ratio). The µ-law algorithm (PCMU) is a companding algorithm, primarily used in the digital telecommunication systems of North America and Japan. And A-law algorithm (PCMA) is the standard companding algorithm, used in European digital communications systems to optimize, i.e., modify, the dynamic range of an analog signal for digitizing.

PCMU for Digitized Signals

PCMU digital implementation takes a maximum sine wave amplitude of ±8159 equal to 3.17 dBm in power representation. The front-end analog circuit and ADCs (Analog-to-Digital Converters) are calibrated to take a (maximum of) 3.17 dBm sine wave and give ±8159 amplitude in digital number representation. The numbers 8159 and above are clipped to 8158 in the process of quantization. In the process of compression, the input range is broken into segments, and each segment will use different intervals. Most segments contain 16 intervals, and the interval size doubles from segment to segment as shown in Figure 1. Large signals use a big quantization step and small signals use a small quantization step. This type of non uniform quantization gives a signal-to-noise ratio (SNR) of 38 to 40 dB for most of the useful signal amplitude range. For full-scale amplitude of 3.17 dBm, signal to quantization is shown as 39.3 dB.

Figure 1

A PCM 14–bit (±8159 amplitude) input is split into eight amplitude segments represented with 3 bits. Each segment is quantized uniformly into 16 levels using 4 bits. The polarity of the input is represented using 1 bit. A sign bit is represented as “1″ for positive numbers and as “0″ for negative numbers. In a two’s complement number format, sign bit-1 is used for negative numbers. After framing 8 bits, the last 7 bits except the sign bit are inverted in PCMU. This bit inversion increases the one’s bit density in transmission systems that help timing and clock recovery circuits in the receiver. Ideal channel noise makes bits, toggle between 01111111 (0x7F) and 11111111 (0xFF), which allows clock recovery to be better even under ideal channel conditions. Bit inversion is achieved by simple exclusive OR (XOR) operation of the encoded output with value 0x7F. This inversion is applied on both positive and negative values of input. At the decoder, the same operation of 7-bit inversion can retrieve the original decoded compressed byte.

PCMU has eight segments (both on the positive side and negative side), numbered from 1 to 8. The intervals are 2,4,8,16,32,64,128 and 256. In PCMU, the maximum interval size is 256, and the corresponding segment number is 8. However, encoded output has only 3 bits for representing the segment number, which means only numbers from 0 to 7, can be used for segment coding. Therefore, in PCMU, maximum and minimum segment numbers are coded as 111(7) and 000(0), respectively.Table 1 lists PCMU encoded and decoded output values for positive numbers and Table 2 lists the output values for some of the negative numbers.

Table 1

Table 2

PCMA for Digitized Signals

PCMA takes a maximum input of ±4096 equal to 3.14dBm in power representation. The front–end analog circuit and analog–to–digital converters (ADCs) are calibrated to take 3.14 dBm and to give ±4096 amplitude in digital number representation. The quantization procedure is the same as with PCMU with few deviations in quantization steps. The step values (intervals) are 128, 64, 32, 16, 8, 4, and 2 (for the last two 16 intervals), which means the last two segments use the same interval 2. Of the 16 total segments for positive and negative numbers, four segments from -63 to 63 use a step size of 2. Hence, PCMA is referred to as 13-segment quantization. PCMA encodes a 13-bit sample number to an 8-bit compressed sample. Unlike in PCMU, where the last 7 bits are inverted, PCMA encoder uses even bits inversion (EBI)., where in, the actual bits in the position 2,4,6 and 8 are inverted. In the A-law tables, (Table 3 and Table 4) bits before inversion and after EBI are also included in separate columns. EBI is performed in software by XOR operation of the encoded output with 0×55. EBI is applied on both positive and negative values of input. At the decoder, the same operation of even bit inversion can retrieve the original decoded compressed byte. EBI in PCMA once again helps clock recovery mechanisms during ideal channel or near zero amplitude signals. Table 3 shows the PCMA Quantization Example for Positive Inputs and Table 4 shows the same output for a few Segments of Negative Inputs.

Table 3

Table 4

Therefore, Companding is a process which helps to transmit and reproduce a high-fidelity voice information in an effective manner (in terms of bandwidth as well as cost).

Analog to Digital Coversion

2012-02-03T04:33:00.002-08:00

The voice frequency (VF) or voice band ranges from approximately 300 Hz to 3.30 kHz. After adding the guard bands (a guard band is an unused part of the radio spectrum between radio bands, for the purpose of preventing interference) of 0-300 Hz and 3.3kHz-4kHz , the net bandwidth that is required for voice transmission will be 4 kHz. Now we need to convert this signal into a digitized one. And the concept behind digitizing a sound signal is Nyquist theorem. Working at Bell Labs, Harry Nyquist discovered that it was not necessary to capture (or send) the entire analog waveform. Only the samples of the wave taken at various points need to be captured to recreate the original one. He also found that in order to reconstruct the original waveform with enough information as the original one, the sampling rate must be at least twice the signal bandwidth.

This concept became the foundation for PCM (Pulse Code Modulation). PCM is the fundamental thing for any digital transmission or switching technology. As I previously mentioned, the highest frequency content in the voice sample is 4 kHz (4000 samples per second). According to Nyquist criterion, the sampling rate should be at least twice the highest frequency content of the selected signal. So here the sampling rate will be 8000 samples per second. After this, the amplitude of each sample is measured based on a logarithmic scale. This measurement process is called Quantization.

There are two PCM algorithms defined within CCITT G.711, called "A-Law" and "Mu-Law". Mu-Law PCM is used in North America and Japan, and A-Law used in most other countries. In both A-Law and Mu-Law PCM, 8 bit codewords are used to represent each sample's amplitude (voltage value of the voice sample at the sampling instance) To attain this 8 bit quantization, a logarithmic scale is used. The y-axis of this scale is divided into 16 segments called chords ( 8 on the positive side & 8 on the negative side). Within each chord are 16 uniform quantization intervals, or steps. The length of the steps depend on the chord number. For example, chord 1 has 16 steps having a length of Δ each. Chord 2 has 16 steps having a length of 2Δ each. In general, the step size in the n-th chord is 2n−1Δ.(as shown in Figure below)

Now the representation of the 8 bits is as shown below.

The left-most bit is known as the "sign" bit or "polarity" bit, and is a 1 for positive values and a 0 for negative values (both PCM types). This is the Most Significant Bit (MSB) and is transmitted first. The next three bits indicates the chord value, and the final four bits denotes the step value.(as shown in Figure 1). This way of defining an 8 bit pattern for a sample value is called encoding.

Now why do you want to use a logarithmic scale for the quantization process? Dividing the amplitude of the voice signal up into equal positive and negative steps is not an efficient way to encode voice into PCM. This does not take advantage of a natural property of human voice :- voices create low amplitude signals most of the time (people seldom shout on the telephone). That is, most of the energy in human voice is concentrated in the lower end of voice’s dynamic range. To create the highest-quality voice reproduction from PCM, the quantization process must take into account this fact that most voice signals are typically of lower amplitude. To do this the voice coder adjusts the chords and steps so that most of them are in the low-amplitude end of the total encoding range. In this modified scheme, all step sizes are not equal. Step sizes are smaller for lower-amplitude signals. Quantization levels (chords) distributed according to a logarithmic function, instead of linear function gives finer resolution, or smaller quantization steps, at lower signal amplitudes (as shown in Figure 3). Therefore, higher-fidelity reproduction of voice is achieved.

And finally, as each sample is represented using 8 bits, the total bits that needs to be transmitted per second (for the recreation of the original 4kHz band sample) is 8000 samples/sec * 8 bits/sample = 64000 bits transmitted per second or 64Kbps. And therefore the minimum bandwidth required to convey your voice (in digital terms) is 64Kbps.

Lets start with a PSTN call

2012-02-02T23:19:00.000-08:00

The PSTN or the public switched telephone network (PSTN) is the network of the world's circuit-switched telephone networks. You can think of it as a network where each Telephone switch in the network is connected to another switch, either directly or in an indirect way (through other switches). Also the switches are intelligent enough to route your call to a destination switch near to your friend’s or relative’s house, be it a national or an international call. Each of these switches uses the telephone number that you dial, to find the switch where your friend belongs to.

Now when you want to talk to some one over a telephone, you will first dial his/her telephone number. Then the other end’s phone started ringing and if somebody lifts the phone, you will get an answer. Do you know what happened during that tiny gap between the “end of dialing” and “Ringing”? This is the duration where the switches are trying to establish a physical connection between you and the person you dialed through the PSTN network. This physical connection is used to convey your voice data (speech) to the other person and vice versa. Now what does this physical connection looks like? Is it the physical wires that we are talking about? You might have seen only one or two optical cables or electrical cables between the telephone switch boxes, but still it supports thousands of users from your locality. How is this possible? There comes the technology called TDM (Time division multiplexing).

Multiplexing is the method of combining many smaller units of signals carried via separate channels on to a bigger and single channel. Or in other words multiplexing helps to carry multiple users information simultaneously. The unit can be a time slice or a frequency. When it is frequency, that method is called FDM (Frequency division multiplexing) and if it is time, then it is called TDM. Consider your link has a carrying capacity (bandwidth) of 2Mbps or 2 Mega bits per second (i.e. it can carry 2 Megabits every 1 sec), and you want to carry “n” users data simultaneously through this link How do you achieve this? The TDM multiplexer will do the job for you. Each user will be allocated 1/n sec duration of the bandwidth out of this 2Mbps. All users will send there data to the Multiplexer in a parallel fashion. The multiplexer will buffer this information and will arrange them in a serial manner to make the 2Mb of information and sent these data chunks every second. And at the other end a de-multiplexer will separate the information and forward it to multiple destinations. Now you can see, how a single high capacity link can me equipped to carry multiple users data simultaneously.

Same thing is happening in a switch (Digital switch). Your telephones will be wired separately to a switch. The switch will do the multiplexing for you and send your information, as if separate cables are laid for you between the switches.