History of Telephony: In 1930's long distance carrier systems using FDM were widely in use these were for groups of telephone over long distances. When the telephone user grew with a large increasing rate these trunks which based on cables became a hot issue of concern.
The concept of TDM came in this time of crises but its implementation was retarded due too no suitable component found for its implementation.
The concept of microwave radio was introduced for long-distance transmission as it has a specialty that it had a large band-width but there was a major problem of distortion and noise in it. In 1938 Alec Reeves was the one who came out with the special technique of PCM which performed conversion of analog signals into digital codes. By the help of this technique the problem of noise and distortion was handled very easily.
In 1948 the transistor was invented which helped in the practical implementation of the techniques of TDM and PCM and in parallel it also provided us a tool for combining transmission and switching. In 1965 the telecom industry adopted digital transmission when the first computer controlled switching system was installed. 
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Introduction of Digital Telephony
“The term digital telephony refers to the use of digital technology in the message path of voice communications networks” . In simple words it means digitization of a signal and its transmission over a specific channel. The main problem which occurred when digital telephony was introduced was that would we get the absolute quality of signal which we used to get through analog transmission.
It is basically use of digital electronics in the provision of digital telephone services and systems.
NEED OF DIGITAL TRANSMISSION?
In the beginning of telephony, there were separate pairs of wires for every subscriber between the exchanges.
This technique is very much costly because on increasing individual subscriber lines the cost of network increases accordingly. There was a need of a type of a network that could fully replace the current working network and full fill all the requirements of the end users efficiently.
Digital transmission involves the following steps:
Figure a  General transmission scheme
The transmission will be based on digital data, which is obtained from (generally)
analogue quantities by
1. sampling (Nyquist: sampling with at least twice the maximum frequency), and
2. quantization (introduction of quantization noise through rounding off).
Transmission is the mechanism to transport signals forms one node to another node by the help of a communication medium, e.g. radio link, metallic wires coaxial, fiber optics etc.
Multiple lines are used for long range communications in this modern era. In digital telephony there are techniques by which we can transmit more data between multiple users with very low cost .Those techniques are
- FDM(Frequency-Division Multiplexing)
- TDM(Time-Division Multiplexing)
FDM (Frequency-Division Multiplex)
The method of providing a number of simultaneous channels over a common transmission line by using a different frequency band for a transmission of each channel is called FDM (Frequency-Division Multiplexing)
FDM telephony made it possible to transmit several telephone signals simultaneously on a single wire and thereby transformed the method of telephone transmission. 
In FDM several different type of signal shares a single channel. Each and every transmitted signal is modulated (multiplied) by a different carrier frequency. The technique of multiplying by different carriers is used to avoid overlapping between the modulated signals. These carriers are called as sub carriers. Each signal may use a different kind of modulation(e.g ,DSB-SC,AM,SSB-SC,VSB-SC,or even FM or PM).Each modulated signal spectrum is being separated by guard band which is helpful in avoiding interference .This introduction of a guard band also helps us to separate signal easily at the receivers end.
Till now we have created a modulated signal (base band) which must be further modulated with a high-frequency carrier signal for transmission aver a specific channel.
Now at the receivers end the first thing which is done is demodulation of the transmitted signal to retrieve back the base band signal, which is then passed through a band pass filter to separate each modulated signal.
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By using most appropriate sub carrier each modulated signal is demodulated individually we can get the basic base band signals .
Speech signal which is to be transmitted requires a narrow frequency band. The band which is used in this type of transmission varies from 300 to 3400 Hz. The transmitted speech signal does not use the full bandwidth of the line on which it is being transmitted. Due to the utilization of the full frequency band whole of the bandwidth is further subdivided into consecutive narrower sub bands of 4 kHz each. Now each band is allotted to single different user and without wastage several users can be entertained at the same time.
TDM (Time-Division Multiplex)
TDM is the sharing transmission links among multiple users by assigning time slots to individual users during which they have the entire bandwidth of the system.
This whole concept of time division multiplexing is based on a single thing and that is complete waveform is not required for transmitting signals.
It is enough that we just sample the input (speech signal) at regular intervals and transmits only these samples and at the receivers end the original waveform can be retrieved.
At first we are having a continuous signal we sample it at suitable time intervals keeping on thing in mind that the original signals original shape must not change(get effected to our sampling).If the shape changes then we have to increase the sampling rate its best to keep it above the Nyquist criterion.
Amplitudes can be varied and also the widths or positions of the pulses in proportion to the sample values of the signal; By the help of Pulse-amplitude modulation (PAM), pulse-width modulation (PWM), or pulse-position modulation (PPM).The most important of them all are the PAM and PCM.
PAM (Pulse Amplitude Modulation)
Pulse amplitude modulation is a technique in engineering which is used to describe the analog signal to a pulse type signal conversion where the amplitude of pulse will be denoting the analog information. 
The PAM signal just attained is then converted into a PCM digital signal (base band) which in turn is modulated with a carrier in band pass digital communication systems. Thus as a result the analog to PAM conversion process is the first step in converting an analog waveform (analog signal) to a PCM (digital) signal. 
PCM (Pulse Code Modulation)
PCM is an essentially a special type of analog to digital conversion containing the information in the instantaneous samples of an analog signal and are represented by digital words (bits) in a continuous (serial) bit stream. 
PCM involves three basic steps
The method of sampling is actually conversion of an analog signal to a digital signal simply it can be done by an A/D converter.
Quantization is the process by which an analog signal can be converted into a digital signal by assigning different uniformly distributed levels to the input signal.
In encoding we assign different binary codes to the quantized levels .so that we can form an encrypted signal which is difficult to be detected by other receivers except prescribed ones.
The basic concept of transmission is that we transmit a pulse-modulated signal instead of the original signal. The biggest advantage of doing all of this is that pulse modulation allows simultaneous transmission of several signals on a time shared base.
The PCM process takes time separated samples of an analog signal (sampling) and represents each sample with a digital code related to its amplitude (encoding).
For the last several years, in digital telephony the digital switches, or exchanges, have been replacing the analog systems. Digital exchanges are more reliable than the analog exchanges as we discussed in the above discussion. But we must keep them in close synchronization with each other, for maintaining better performance.
Clarke, A. C., “Extra terrestrial Relays-Can Rocket Stations Give World Wide
Radio Coverage ,” Wireless World ,Vol. LI, No.10, October 1945 , pp. 305-308
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