# Pulse code modulation

The pulse-code modulation , shortly PCM is a pulse modulation method using a time and value continuous analog signal into a time- and value-discrete digital signal converts.

## functionality

Figure 1: Sampling, quantization and coding of the original signal curve shown in red for a PCM with 4 bits.

The implementation takes place in the following steps:

1. Sampling of the analog signal by means of pulse amplitude modulation (PAM) with a sampling rate that is constant over time . A time-discrete signal sequence is formed from the time-continuous signal sequence. The Nyquist-Shannon sampling theorem must be fulfilled to maintain the information in the discrete-time sequence . This means that the sampling rate must be more than twice as high as the highest frequency component occurring in the signal curve.
2. Then there is a quantization to discrete values ​​with a finite number of digits. The quantization assigns a certain symbol to a certain range of values.
3. Generation of the digital signal by assigning the individual symbols by means of coding . In many practical applications, the binary code is chosen for PCM .

In electrical circuits , the first step is implemented in the form of a sample-and-hold circuit (SH) and steps two and three in the form of analog-to-digital converters (ADC). In some analog-digital converters, the SH is already integrated as a functional unit.

The number of possible quantization levels obtained when the binary code from the number of bits having a codeword: . The number of quantization levels essentially determines the quantization noise . The larger the quantization levels become, i. H. the smaller n is, the greater the error that occurs. Figure 1 shows a PCM with a dynamic range of only 4 bits, whereby the error is clearly visible. In many applications, a dynamic range of 8 to 24 bits is selected for quantization. ${\ displaystyle n}$${\ displaystyle z}$${\ displaystyle n = 2 ^ {z}}$

## Types of quantization

The quantization can be linear or non-linear.

### Linear quantization

With linear quantization, the value ranges are equally large. This type of PCM is called Linear Pulse Code Modulation (LPCM).

### Nonlinear quantization

With non-linear quantization, larger signal deflections are combined in a larger value range and thus resolved more coarsely. Small signal deflections, however, are quantized with a higher resolution. The advantage is that with fewer bits per sample, less quantization noise can be achieved than with linear quantization. The methods known as A-law and μ-law both use non-linear quantization. They are used in communications technology for the digitization of analog communications signals (voice).

The advantage of digital signal coding, as used by PCM, compared to a time-continuous signal lies in the higher tolerance to interference. The binary coding at the receiver only has to be able to distinguish between a high and a low signal (0 and 1). The different types of modulation (except PCM are pulse amplitude modulation , pulse width modulation , pulse phase modulation , pulse frequency modulation digital modulation) also have a different "resistance" against systematic or random errors. In the case of PCM-modulated signals, in contrast to the other types of modulation, sinusoidal interference ( e.g. mains hum ) can be eliminated by regeneration amplifiers. This is why this method has not only established itself in communications technology, but also in classic analog technology ( high fidelity ).

The disadvantage of PCM coding is the high data transmission rate required ( approx. 1.4 Mbit / s for audio CDs ), which is why adapted and extended PCM methods are used in various applications and the digital information is reduced by means of source coding .

## Extensions

With Differential Pulse Code Modulation ( DPCM ), not the entire binary-coded value is saved, but in the simplest case only the difference to the previous value. This procedure allows smaller word widths and thus higher compression. The so-called delta modulation is a special case of DPCM, where the sampling rate is increased until the quantization is reduced to only 1 bit and the difference of a sample is only 1 bit. Delta modulation is the preliminary stage to delta-sigma modulation , which is used, for example, in higher-quality AD converters for noise shaping and for minimizing quantization noise.

With Adaptive Differential Pulse Code Modulation ( ADPCM ), the scaling of the quantization levels is designed flexibly for data reduction and adapted (adapted) depending on the signal course. The coding algorithm estimates what the next value might look like (this process is also known as prediction ) and adapts the scaling in this way. The difference to the estimated value is transmitted. Depending on the method, forward or backward prediction can be used, which is the basis for Linear Predictive Coding (LPC).

## history

Decisive contributions to the development of pulse-code modulation were the publications by Claude Shannon on the channel capacity of disturbed communication channels and by Karl Küpfmüller on the system theory of electrical communication.

PCM was developed in the 1930s by Bell Labs and by Alec Reeves , who received a patent in 1938 for a PCM system with a sampling rate of 8000 bits per second. It was first used in 1943 in an encrypted telephone system called SIGSALY . In the 1960s, technicians from the Japanese broadcasting company NHK developed recording devices based on PCM with video tape as the carrier medium. The Japanese record label Nippon Columbia was keen to improve the quality of the analogue magnetic tape recordings and rented a recorder from NHK to make test recordings and then developed their own recorder. PCM devices were also developed at the BBC in the early 1970s.

In 1971 the first recording was released under the Denon label that was digitally recorded using the PCM process, followed by works of classical music with European interpreters from 1972 ( Mozart's string quartets KV 421 and 458 with the Smetana Quartet ). 1974 saw the first PCM production in Europe with Bach's Musical Offering ( Paillard Chamber Orchestra ). When the CD was launched in 1982 , Denon already had 400 digital recordings available.

### Telephone networks

The digitized telephone networks are the largest area of ​​application for PCM technology. The electrical speech signal is analogously limited to the frequency range between 300 Hz and 3400 Hz and sampled at a frequency of 8000 Hz. Accordingly, 8000 discrete instantaneous values ​​are measured per second. The range of the signal values ​​to be transmitted is divided into a specific number of quantization intervals. For each instantaneous value it is now determined in which interval it falls. The number of the quantization interval is then transmitted as a binary coded number from the transmitter to the receiver. The larger the number of quantization intervals, the lower the quantization noise. When the long-distance networks were digitized in the 1960s, the number of quantization intervals was chosen so that the quantization distortion is practically inaudible if four conversions from analog to digital and back occur in the course of a telephone connection. That was the case with 128 quantization intervals. Therefore 7 bits are sufficient to indicate the respective interval ( ). 7 bits were to be transmitted for each sample value, which at the sampling frequency of 8000 Hz corresponded to a bit rate of 56 kbit / s. When it became clear that 14 to 15 PCM-to-analog conversions could occur on a worldwide telephone connection, CCITT decided in 1969 to increase the number of quantization intervals. 8-bit PCM has now become an international standard (Recommendation G.711 ). That corresponded to a bit rate of 64 kbit / s. While Europe and most non-European countries also introduced 8-bit PCM in the form of the A-law, North America and Japan stuck to the poorer quality μ-law, which only requires 7-bit PCM. International connections now led to national telephone networks, which digitized differently, which required implementation. The implementation is implemented in that each PCM code word of one law is replaced by the PCM code word of the other law that gives the best match in the reconstruction of the analog signal. With this implementation a problem of the international telephone networks is solved. ${\ displaystyle 2 ^ {7} = 128}$

The intercontinental transmission of long distance calls uses satellite links or submarine cables. In order to use this as economically as possible, PCM is converted to ADPCM with 32 kbit / s, for example. With mobile communications , the frequencies for telephony are scarce and expensive. That is why cellular networks use codecs with an even lower bit rate.