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Document Type: Prentice Hall
Author: Harold P.E. Stern and Samy A. Mahmoud
Book: Communication Systems: Analysis and Design
ISBN: 0-13-040268-0
NI Supported: No
Publish Date: Apr 5, 2013

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# Theory of Time Division Multiplexing

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### Overview

This tutorial is part of the National Instruments Signal Generator Tutorial series. Each tutorial in this series, will teach you a specific topic of common measurement applications, by explaining the theory and giving practical examples. This tutorial covers the theory of time division multiplexing. For additional signal generator concepts, refer to the Signal Generator Fundamentals main page.

## Time Devision Multiplexing

It's often practical to combine a set of low-bit-rate streams, each with a fixed and pre-defined bit rate, into a single high-speed bit stream that can be transmitted over a single channel. This technique is called time division multiplexing (TDM) and has many applications, including wireline telephone systems and some cellular telephone systems. The main reason to use TDM is to take advantage of existing transmission lines. It would be very expensive if each low-bit-rate stream were assigned a costly physical channel (say, an entire fiber optic line) that extended over a long distance.

Consider, for instance, a channel capable of transmitting 192 kbit/sec from Chicago to New York. Suppose that three sources, all located in Chicago, each have 64 kbit/sec of data that they want to transmit to individual users in New York. As shown in Figure 7-2, the high-bit-rate channel can be divided into a series of time slots, and the time slots can be alternately used by the three sources. The three sources are thus capable of transmitting all of their data across the single, shared channel. Clearly, at the other end of the channel (in this case, in New York), the process must be reversed (i.e., the system must divide the 192 kbit/sec multiplexed data stream back into the original three 64 kbit/sec data streams, which are then provided to three different users). This reverse process is called demultiplexing.

Figure 7-2—Time division multiplexing.

Choosing the proper size for the time slots involves a trade-off between efficiency and delay. If the time slots are too small (say, one bit long) then the multiplexer must be fast enough and powerful enough to be constantly switching between sources (and the demultiplexer must be fast enough and powerful enough to be constantly switching between users). If the time slots are larger than one bit, data from each source must be stored (buffered) while other sources are using the channel. This storage will produce delay. If the time slots are too large, then a significant delay will be introduced between each source and its user. Some applications, such as teleconferencing and videoconferencing, cannot tolerate long delays.

As shown in Example 7-2, the sources that are multiplexed may have different bit rates. When this occurs, each source is assigned a number of time slots in proportion to its transmission rate.

Example 7.1The T1 system for wireline telephone networks

The T1 system is used for wireline long-distance service in North America and is an excellent example of TDM. Speech from a telephone conversation is sampled once every 125 msec and each sample is converted into eight bits of digital data (see Chapter 8 for more details). Using this technique, a transmission speed of 64,000 bits/sec is required to transmit the speech. A T1 line is essentially a channel capable of transmitting at a speed of 1.544 Mbit/sec. This is a much higher transmission speed than a single telephone conversation needs, so TDM is used to allow a single T1 line to carry 24 different speech signals between, say, two different telephone substations (called central offices) within a city. As shown in Figure 7-3, the 1.544 Mbit/sec bit stream is divided into 193-bit frames. The duration of each frame is

corresponding to the period between samples of the speech. Each frame is divided into 24 slots, which are each eight bits wide (corresponding to the number of bits needed to digitize a speech sample). One additional bit at the end of the frame is used for signaling. The eight bits of data corresponding to a sample of the speech are placed into one of the 24 slots in the frame.

For longer distances (say, between two large cities) higher-capacity channels are used and multiple T1 lines are time division multiplexed onto the new channels. A T3 channel for example, has a transmission speed of 44.736 Mbit/sec and uses TDM to carry 28 T1 lines (a total of 672 different speech signals) plus signaling. For more information on this hierarchical multiplexing system, see BeIlamy [7.1].

Figure 7-3—Time division multiplexing on a T1 line.

Example 7.2—TDM with sources having different data rates

Consider the case of three streams with bit rates of 8 kbit/sec,16 kbit/sec, and 24 kbit/sec, respectively. We want to combine these streams into a single high-speed stream using TDM. The high-speed stream in this case must have a transmission rate of 48 kbit/sec, which is the sum of the bit rates of the three sources. To determine the number of time slots to be assigned to each source in the multiplexing process. we must reduce the ratio of the rates, 8:16:24, to the lowest possible form, which in this case is 1:2:3. The sum of the reduced ratio is 6, which will then represent the minimum length of the repetitive cycle of slot assignments in the multiplexing process. The solution is now readily obtained: In each cycle of six time slots we assign one slot to Source A (8 kbit/sec), two slots to Source B (16 kbit/sec), and three slots to Source: C (24 kbit/sec). Figure 7-4 illustrates this assignment, using “a” to indicate data from Source A, “b” to indicate data from Source B, and “c” to indicate data from Source C.

Figure 7-4—Multiplexing input lines with different transmission speeds.

Example 7.3A more complex TDM system

Consider a system with four low-bit-rate sources of 10 kbit/sec, 15 kbit/sec, 20 kbit/sec, and 30 kbit/sec. Determine the slot assignments when the data streams are combined using TDM.

Solution

The rate ratio 10:15:20:30 reduces to 2:3:4:6. The length of the cycle is therefore 2 + 3 + 4 + 6 = 15 slots. Within each cycle of 15 slots, we assign two slots to the 10 kbit/sec source, three slots to the 15 kbit/sec source, four slots to the 20 kbit/sec source, and six slots to the 30 kbit/sec source.

So far we have considered a form of TDM that is based on fixed slot assignments to each of the low-bit-rate data streams. In other words, each stream has predefined slot positions in the combined stream, and the receiver must be aware which slots belong to which input stream. Both transmission ends, the transmitter and the receiver, must be perfectly synchronized to the slot period. For this reason, the technique is usually called synchronous TDM.

There is another important version of TDM, usually referred to as statistical TDM. Statistical TDM is useful for applications in which the low-bit-rate streams have speeds that vary in time. For example, a low-bit-rate stream to a single terminal in a computer network may fluctuate between 2 kbit/sec and 50 kbit/sec during an active connection session (we've all seen variable speeds during Internet connections, for instance). If we assign the stream enough slots for its peak rate (that is, for 50 kbit/sec), then we will be wasting slots when the rate drops well below the peak value. This waste can be especially significant if the system has many variable-speed low-bit-rate streams.

Statistical TDM works by calculating the average transmission rates of the streams to be combined, and then uses a high-speed multiplexing link with a transmission rate that is equal to (or slightly greater than) the statistical average of the combined streams. Since the transmission rates from each source are variable, we no longer assign a fixed number of time slots to each data stream. Rather, we dynamically assign the appropriate number of slots to accommodate the current transmission rates from each stream. Because the combined rate of all the streams will also fluctuate in time between two extreme values, we need to buffer the output of the low-bit-rate streams when the combined rate exceeds the transmission rate of the high-speed link.

With statistical TDM, we are no longer relying on synchronized time slots with fixed assignments for each input stream, as we did with synchronous TDM. So how does the demultiplexer in statistical TDM know which of the received bits belongs to which data stream? Prior to transmission, we divide each stream of bits coming from a source into fixed-size blocks. We then add a small group of bits called a header to each block, with the header containing the addresses of the source and intended user for that block. The block and the header are then transmitted together across the channel. Combined, the block and header are called a packet.

Actually, the header may contain other information besides the source and user addresses, such as extra bits for error control (see Chapter 10) or additional bits for link control (used, for example, to indicate the position of a particular block in a sequence of blocks coming from the same user, or to indicate priority level for a particular message). Extra bits can also be added to the beginning and end of a block for synchronization; a particular pattern of bits, called a start flag, can be used in the header to mark the start of a block, and another particular pattern of bits, called an end flag, can be used to conclude the block. Each block transmitted across the channel thus contains a group of information bits that the user wants, plus additional bits needed by the system to ensure proper transmission. These additional bits, while necessary to system operation, reduce the effective transmission rate on the channel. Figures 7-5 and 7-6 present the statistical TDM technique and the structure of a typical packet.

Figure 7-5—Statistical TDM.

Figure 7-6—Structure of a typical statistical TDM packet.

## Relevant NI products

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For the complete list of tutorials, return to the NI Signal Generator Fundamentals Main page.

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