Triangulation, Trilateration, or Multilateration? (EE Tip #125)

Local Positioning System (LPS) and GPS (not just the US system) both use several transmitters to enable a receiver to calculate its geographical position. Several techniques are possible, each with its advantages and drawbacks. The important thing in all these techniques is the notion of a direct path (line of sight, or LoS). In effect, if the transmitter signal has not taken the shortest path to the receiver, the distance between them calculated by the receiver will be incorrect, since the receiver does not know the route taken by the radio signal.

Three mathematical techniques are usually used for calculating the position of a receiver from signals received from several transmitters: triangulation, trilateration, and multilateration. The last two are very similar, but should not be confused.


Triangulation (Figure 1) is a very ancient technique, said to date from over 2,500 years ago, when it was used by the Greek philosopher and astronomer Thales of Miletus to measure (with surprising accuracy) the radius of the Earth’s orbit around the Sun.


Figure 1—Triangulation: you are at A, from where you can see B and C. If you know their geographical positions, you can find your own position with the help of a compass.

It allows an observer to calculate their position by measuring two directions towards two reference points. Since the positions of the reference points are known, it is hence possible to construct a triangle where one of the sides and two of the angles are known, with the observer at the third point. This information is enough to defi ne the triangle completely and hence deduce the position of the observer.

Using triangulation with transmitters requires the angle of incidence (angle of arrival, or AoA) of a radio signal to be measured. This can be done using several antennas placed side by side (an array of antennas, for example, Figure 2) and to measure the phase difference between the signals received by the antennas.

Antenna array

Figure 2—An antenna array makes it possible to measure the angle of incidence of a radio signal, and hence its direction.

If the distance between the antennas is small, the incident front of the signal may be considered as straight, and the calculation of the angle will be fairly accurate. It’s also possible to use a directional antenna to determine the position of a transmitter. The antenna orientation producing the strongest signal indicates the direction of the transmitter. All you then have to do is take two measurements from known transmitters in order to be able to apply triangulation.


This technique requires the distance between the receiver and transmitter to be measured. This can be done using a Received Signal Strength Indicator (RSSI), or else from the time of arrival (ToA)—or time of flight (ToF) Figure 3—of the signal, provided that the receiver and transmitter are synchronized — for example, by means of a common timebase, as in GPS.

Arrival time

Figure 3—The length of the arrows corresponds to the arrival time at receiver P of the signals broadcast by three transmitters A, B, and C. It forms a measurement of the distances between the transmitters and the receiver.

Thus, when receiving a signal from a single transmitter, we can situate ourselves on a circle (for simplicity, let’s confi ne ourselves to two dimensions and ideal transmission conditions) with the transmitter at the center. Not very accurate. It gets better with two transmitters — now there are only two positions possible: the two points where the circles around the two transmitters intersect. Adding a third transmitter enables us to eliminate one of these two possibilities (Figure 4).


Figure 4—2-D trilateration. In 3-D, another transmitter has to be added in order to determine a position unambiguously.

When we extend trilateration to three dimensions, the circles become spheres. Now we need to add one more transmitter in order to fi nd the position of the receiver, as the intersection of two spheres is no longer at two points, but is a circle (assuming we ignore the trivial point when they touch). This explains why a GPS needs to “see” at least four satellites to work.


Using a single receiver listening to the signals (pulses, for example) from two synchronized transmitters, it is possible to measure the difference between the arrival times (time difference of arrival, or TDoA) of the two signals at the receiver. Then the principle is similar to trilateration, except that we no longer fi nd ourselves on a circle or a sphere, but on a hyperbola (2D) or a hyperboloid (3D). Here too, we need four transmitters to enable the receiver to calculate its position accurately.

The advantage of multilateration is that the receiver doesn’t need to know at what instant the signals were transmitted—hence the receiver doesn’t need to be synchronized with the transmitters. The signals, and hence the electronics, can be kept simple. The LORAN and DECCA systems, for example, work like this.—Clemens Valens, “Geolocalization without GPS,” Elektor, February 2011.

Wireless Data Links (Part 2): Transmitters and Antennas

If you built your own ham radio “back in the day,” you’ll recall the frustration of putting it together with components that were basic at best.

But as columnist George Novacek points out in the second installment of his series examining wireless data links: “Today you can purchase excellent, reasonably priced low-power gear for data communications off the shelf.”

Transmitter and receiver

Photo 1: SparkFun Electronics’s WRL-10524 transmitter and WRL-10532 receiver are low cost, basic, and work well.

Part 2 of Novacek’s series, appearing in the March issue, looks at transmitters and antennas.

In one section, Novacek expands upon the five basic data-transmitter modules—a data encoder, a modulator, a carrier frequency generator, an RF output amplifier, and an antenna:

Low-power data transmitters often integrate the modulator, the carrier frequency generator, and the amplifier into one circuit. A single transistor can do the job. I’ll discuss antennas later. When a transmitter and a receiver are combined into one unit, it’s called a transceiver.

Modulation may not be needed in some simple applications where the mere presence of a carrier is detected to initiate an action. A simple push button will suffice, but this is rarely used as it is subject to false triggering by other transmitters working in the area in the same frequency band.

Digital encoder and decoder ICs are available for simple devices (e.g., garage door openers) or keyless entry where just an on or off output is required from the receiver. These ICs generate a data packet for transmission. If the received packet matches the data stored in the decoder, an action is initiated. Typical examples include Holtek Semiconductor HT12E encoders and HT12D decoders and Freescale Semiconductor MC145026, MC145027, and MC145028 encoder and decoder pairs. For data communications a similar but more advanced scheme is used. I’ll address this when I discuss receivers (coming up in Part 3 of this series).

Novacek’s column goes on to explain modulation types, including OOK and ASK modulation:

OOK modulation is achieved by feeding the Data In line with a 0-to+V-level  datastream. ASK modulation can be achieved by the data varying the transistor biasing to swing the RF output between 100% and typically 30% to 50% amplitude. I prefer to add a separate modulator.

The advantage of ASK as opposed to OOK modulation is that the carrier is always present, thus the receiver is not required to repeatedly synchronize to it. Different manufacturers’ specifications claim substantially higher achievable data rates with ASK rather than OOK.

For instance, Photo 1 shows a SparkFun Electronics WRL-10534 transmitter and a WRL-10532 receiver set for 433.9 MHz (a 315-MHz set is also available), which costs less than $10. It is a bare-bones design, but it works well. When you build supporting circuits around it you can get excellent results. The set is a good starting point for experimentation.

The article also includes tips on a transceiver you can purchase to save time in developing ancillary circuits (XBee), while noting a variety of transceiver, receiver, and transmitter modules are available from manufacturers such as Maxim Integrated, Micrel, and RF Monolithics (RFM).  In addition, the article discusses design and optimization of the three forms of antennas: a straight conductor (monopole), a coil (helical), and a loop.

“These can be external, internal, or even etched onto the PCB (e.g., keyless entry fobs) to minimize the size,” Novacek says.

Do you need advice on what to consider when choosing an antenna for your design?  Find these tips and more in Novacek’s March issue article.

Wireless Data Links (Part 1)

In Circuit Cellar’s February issue, the Consummate Engineer column launches a multi-part series on wireless data links.

“Over the last two decades, wireless data communication devices have been entering the realm of embedded control,” columnist George Novacek says in Part 1 of the series. “The technology to produce reasonably priced, reliable, wireless data links is now available off the shelf and no longer requires specialized knowledge, experience, and exotic, expensive test equipment. Nevertheless, to use wireless devices effectively, an engineer should understand the principles involved.”

Radio communicationsPart 1 focuses on radio communications, in particular low-power, data-carrying wireless links used in control systems.

“Even with this limitation, it is a vast subject, the surface of which can merely be scratched,” Novacek says. “Today, we can purchase ready-made, low-power, reliable radio interface modules with excellent performance for an incredibly low price. These devices were originally developed for noncritical applications (e.g., garage door openers, security systems, keyless entry, etc.). Now they are making inroads into control systems, mostly for remote sensing and computer network data exchange. Wireless devices are already present in safety-related systems (e.g., remote tire pressure monitoring), to say nothing about their bigger and older siblings in remote control of space and military unmanned aerial vehicles (UAVs).”

An engineering audience will find Novacek’s article a helpful overview of fundamental wireless communications principles and topics, including RF circuitry (e.g., inductor/capacitor, or LC, circuits), ceramic surface acoustic wave (SAW) resonators, frequency response, bandwidth, sensitivity, noise issues, and more.

Here is an article excerpt about bandwidth and achieving its ideal, rectangular shape:

“The bandwidth affects receiver selectivity and/or a transmitter output spectral purity. The selectivity is the ability of a radio receiver to reject all but the desired signal. Narrowing the bandwidth makes it possible to place more transmitters within the available frequency band. It also lowers the received noise level and increases the selectivity due to its higher Q. On the other hand, transmission of every signal but a non-modulated, pure sinusoid carrier—which, therefore, contains no information—requires a certain minimum bandwidth. The required bandwidth is determined by the type of modulation and the maximum modulating frequency.

“For example, AM radios carry maximum 5-kHz audio and, consequently, need 10-kHz bandwidth to accommodate the carrier with its two 5-kHz sidebands. Therefore, AM broadcast stations have to be spaced a minimum of 20 kHz apart. However, narrowing the bandwidth will lead to the loss of parts of the transmitted information. In a data-carrying systems, it will cause a gradual increase of the bit error rate (BER) until the data becomes useless. At that point, the bandwidth must be increased or the baud rate must be decreased to maintain reliable communications.

“An ideal bandwidth would have a shape of a rectangle, as shown in Figure 1 by the blue trace. Achieving this to a high degree with LC circuits can get quite complicated, but ceramic resonators used in modern receivers can deliver excellent, near ideal results.”

Figure 1: This is the frequency response and bandwidth of a parallel resonant LC circuit. A series circuit graph would be inverted.

Figure 1: This is the frequency response and bandwidth of a parallel resonant LC circuit. A series circuit graph would be inverted.

To learn more about control-system wireless links, check out the February issue now available for membership download or single-issue purchase. Part 2 in Novacek’s series discusses transmitters and antennas and will appear in our March issue.

RS-422/-485 Serial Interface for PCI Express

SealevelThe 7802e is a PCI Express serial interface adapter that provides eight serial ports individually configurable for RS-422 or RS-485 communications. The adapter is well suited for applications including test and measurement, security systems, and broadcast.

The board’s high-performance 16C950 UART includes 128-byte FIFOs for error-free operation in high-speed serial applications. The 16C950 UART also supports 9-bit framing and is software compatible with legacy UART applications.

A PCI Express link supplies the 7802e’s 62.5-MHz clock. This ultra-high speed clock is divided by a flexible 8-bit clock prescalar. In RS-485 mode, the transmitter is automatically enabled in hardware, eliminating the need for application software control. This enables the 7802e to be used with standard serial applications without the risk of bus contention and data corruption.

All Sealevel PCI Express serial adapters include SeaCOM software for Windows and Linux OSes. The adapters also include WinSSD, a full-featured application for testing and diagnostics including bit error rate testing (BERT), throughput monitoring, loopback tests, and test pattern message transmissions.

The 7802e costs $469.

Sealevel Systems, Inc.

Simple Guitar Transmitter (EE Tip #102)

You need a guitar amplifier to play an electric guitar. The guitar must be connected with a cable to the amplifier, which you might consider an inconvenience. Most guitar amplifiers operate off the AC power line. An electric guitar fitted with a small transmitter offers several advantages. You can make the guitar audible via an FM tuner/amplifier, for example. Both the connecting cable and amplifier are then unnecessary. With a portable FM broadcast radio or, if desired, a boombox, you can play in the street or in subway.

Source: Elektor 3/2009

Source: Elektor 3/2009

stations (like Billy Bragg). In that case, everything is battery-powered and independent of a fixed power point. (You might need a permit, though.)

Designing a transmitter to do this is not necessary. A variety of low-cost transmitters are available. The range of these devices is often not more than around 30′, but that’s likely plenty for most applications. Consider a König FMtrans20 transmitter. After fitting the batteries and turning it on, you can detect a carrier signal on the radio. Four channels are available, so it should always be possible to find an unused part of the FM band. A short cable with a 3.5-mm stereo audio jack protrudes from the enclosure. This is the audio input. The required signal level for sufficient modulation is about 500 mVPP.

If a guitar is connected directly, the radio’s volume level will have to be high to get sufficient sound. In fact, it will have to be so high that the noise from the modulator will be quite annoying. Thus, a preamplifier for the guitar signal is essential.

To build this preamplifier into the transmitter, you first have to open the enclosure. The two audio channels are combined. This is therefore a single channel (mono) transmitter. Because the audio preamplifier can be turned on and off at the same time as the transmitter, you also can use the transmitter’s on-board power supply for power. In our case, that was about 2.2 V. This voltage is available at the positive terminal of an electrolytic capacitor. Note that 2.2 V is not enough to power an op-amp. But with a single transistor the gain is already big enough and the guitar signal is sufficiently modulated. The final implementation of the modification involves soldering the preamplifier circuit along an edge of the PCB so that everything still fits inside the enclosure. The stereo cable is replaced with a 11.8″ microphone cable, fitted with a guitar plug (mono jack). The screen braid of the cable acts as an antenna as well as a ground connection for the guitar signal. The coil couples the low-frequency signal to ground, while it isolates the high-frequency antenna signal. While playing, the cable with the transmitter just dangles below the guitar, without being a nuisance. If you prefer, you can also secure the transmitter to the guitar with a bit of double-sided tape.

—Gert Baars, “Simple Guitar Transmitter,” Elektor,  080533-1, 3/2009.