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Introduction

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When you create a radio system, it is necessary to understand the relationship between communication distance and communication quality numerically. Even if there is some error, if you can obtain a numerical value, it will provide the basis for a reliable system. In the world of radio communication, it is normal to allow some safety margin in the numbers. If you don’t allow a margin and use the radio close to its limits, you will experience communication errors.

The C/N ratio is a specific value that indicates communication quality. It is the ratio of the received power C to the noise power N at the receiver. Received power is determined by the transmitted power, propagation loss and receiver performance. In addition, noise is handled as system noise, which is the sum of the signal noise that enters the receiving antenna, the noise in the antenna itself, and noise inside the receiver.
Since the C/N ratio determines the bit error rate of the system, a larger ratio reduces the bit error rate, and as a result, the communication range increases. When communication range is considered, propagation loss is fixed and transmitted power is also regulated at a specified value by the Radio Regulations. Also, if the noise from the antenna to the receiver is thought of as constant, the only means of raising the C/N ratio is to reduce noise inside the receiver. Receiver sensitivity is often considered to be a significant parameter of receiver performance, but for manufacturers, reducing noise power is a key aim.

However, from the point of view of making radio equipment, noise power, transmission data rate and transmitted power are fixed, as is propagation loss, so communication range is unequivocally determined. Therefore communication range is determined by transmission data rate and output power, thus the slower the transmission data rate, the longer the communication range.

In addition, there are various modulation and demodulation methods. However, irrespective of the differences between these methods, the diagnostic criteria of energy per bit to noise power density ratio Eb/N0 exists as a means of understanding the relationship between bit error rate, signal power, and noise power. Eb/N0 plays an important role in system circuit design, and by deciding the required bit error rate and then finding the required received power and required C/N, it is possible to evaluate the circuit.

The approach to circuit design using Eb/N0 is explained in many specialist textbooks and on the Web, but there are no explanations of the meaning of Eb/N0 itself. In this technical article, we will look into the meaning of Eb/N0. But please note that this article cannot explain fully the details of this topic and it is up to the reader to make his own assumptions.

There is also a Java applet to help you understand the explanation visually.

Radio system level diagram

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The C/N ratio is one means of determining the communication quality of radio systems. The “C” in C/N ratio is received power and the “N” is noise power. The relationship in the signal level between the transmitter and receiver, and the noise power at the receiver is as shown in the diagram below.
The received power Pr in this figure corresponds to C. Also Ns correspond to N.

When it is assumed that the radio system is located in free space, the signal level between the transmitter and receiver is as follows when shown as a specific value. The calculation is performed with a decibel value, but you can also use the radio wave propagation characteristic calculation applet on the Circuit Design website.

Numerical example:

Transmitting frequency 400 MHz, transmitted power = 10 dBm (10 mW), communication range 1,000 m, free space propagation loss = 84.5 dB
Transmission cable loss Ltc = 5 dB, transmitting antenna gain Gta = 2.14 dBi
Receiving cable loss Lrc = 5 dB, receiving antenna gain Gra = 2.14 dBi
Received power Pr is

Pr = Pt - Ltc + Gta - Lall + Gra - Lrc = 10 - 5 + 2.14 - 84.5 + 2.14 - 5 = -80.22 dBm

Antenna gain Gta = 2.14 dBi so equivalent isotropically radiated power (EIRP) is as follows.

Peirp = Pt - Ltc + Gta


Propagation loss depends on the communication environment, and it varies significantly according to the following differences.
- Whether communication stations are fixed or mobile.
- Height above ground.
- Urban or countryside
- Indoors or outdoors
- Clear day with/without precipitation.
- Line of sight with/without obstructions
- Other

However, it is not possible to reflect all of this in the calculation, and so a margin is added to a value obtained theoretically, enabling a practical and safe communication system. Various methods have been proposed for finding propagation loss theoretically. However, due to the applications of specified low-power radios, it is not necessary to have a very rigorous value, and in most cases, only a guideline is required. As a method of calculating propagation loss, Circuit Design offers a numerical example calculated in a free space environment as well as an environment with the 2-wave model applied. Propagation loss of a radio system used on the ground in a rural environment is very close to that predicted by the propagation loss curve of the 2-wave model, and it can be used as diagnostic criteria for communication quality. However, when used indoors, there are many variable factors and it cannot be treated so simply.

Received power has an inverse relationship to transmitters. Communication quality is a matter of how well data is decoded without bit errors under weak power, and is determined by the relationship between signal power C and noise power N.

In radio equipment specifications, receiver sensitivity is shown as -110 dBm when BER = 0.1% and so on, indicating that received power Pr must be at least -110 dBm. When the radio is located in a line of sight environment, it is necessary to locate it in a position where a received power of -90 dBm can be obtained, allowing a margin of about 20 dB in relation to this value.

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Caution
In specified low-power radio that has received Technical Standard Conformity Certification, the radio and antenna are integrated, so there is no cable loss. Before shipment, a wattmeter is connected to the antenna terminal to check that output power is at the regulation value. Antenna gain is 2.14 dBi.

 

Receive sensitivity

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The selection criteria for radio equipment include performance, price, configuration etc, but considered from the point of view of performance alone, communication range is obviously an important parameter. The first thing that customers say is that they want long range. At the maximum communication distance, the electric field intensity is fairly low, and the electric field intensity varies significantly due to external factors, resulting in bit errors. In the relationship between receiver sensitivity and communication range, naturally the higher the receiver sensitivity, the longer the communication range.

Incidentally, the oft-mentioned receiver sensitivity is the minimum power value for obtaining the stipulated receiver performance (bit error rate), and it differs according to the conditions of measurement. This is expressed in terms of the dBm when the error rate is a certain percentage. If it is stipulated by the bit error rate, it may also be stipulated by the packet error rate. With digital modulation, the data is sent in packets, so the criterion for receiver sensitivity is the number of packets that actually reach the receiver. Receiver sensitivity is a value measured by the manufacturer using a measuring device. Note that it is not a value that is necessarily valid in the environment of use. Receiver sensitivity is the input power value when random code (PN code) is sent and the stipulated error rate is obtained.
Therefore when comparing performance using the value for receiver sensitivity alone, it is necessary to proceed cautiously. Incidentally, this isn’t regulated by radio regulations.

What are bit errors?

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If for example high level is assigned to “1” in the transmitting end data and low level is assigned to “0” and it reaches the receiver in the form of a radio wave, the data is decoded. However, there is noise in the data which affects the decoding operation. This is shown in the figure below.

The source of this noise is the sum of the signal noise at the input of the receiver and the internal thermal noise inside the receiver. The signal inside of the receiver is amplified and input to the demodulation unit.
The bit decoder (data decoding) samples the centre of the bit data period by voltage detection, comparing the thresholds to determine high and low levels. Here, it is important to determine the value of the moment of the sampling point. (It may also be determined at the zero crossing.) The decoded data contains noise, and this power is constant irrespective of the strength of the signal.

The performance of the carrier wave processor of the receiver differs depending on the device used, and so naturally the amount of noise is different too. The figure below shows how the bits are determined by the bit detector of the receiver. In (1), the noise in the carrier wave processor is low, and the level at the sampling point does not cross the threshold. (2) is the same. However, in (3) the noise power significantly crosses the threshold, and if it occurs at the sampling point, the determination will be wrong. This is the entity known as a bit error.

If you glance at figure (3), your eye/mind can see it as a normal data, but for hardware, discrimination is difficult. Even if noise is removed with a filter, noise in the frequency band causes this kind of thing. In other words, hardware detects everything and does not discriminate between signal and noise.

For this reason, in order to decode without errors, a large carrier-to-noise ratio between signal power C and noise power N is required. Since signal power is decided by the location of installation and receiver performance, it is necessary to reduce the noise power of the receiver in order to eliminate errors.

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Noise has a Gaussian distribution

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A close look at (3) in the figure above shows that the cause of the bit errors is noise which exceeds the threshold at the sampling point, and the result of the determination is that the voltage value is simply reversed. The probability of this noise level in this position has the normal distribution (Gaussian distribution) as shown in the figure below. In system design, probability theory is required for calculating the required received power with the required error rate.

The types of noise that are related to receiver bit error rates include thermal noise, shot noise, flicker noise, natural noise, human noise and so on. Thermal noise is noise that is generated by internal components at warm temperatures. It enters the input unit of the receiver as noise with power of -174 [dBm/Hz] per 1 Hz. Shot noise is internal noise caused by the semiconductors that make up the receiver. Background radiation can cause noise that enters the antenna, while man-made noise is emitted by vehicles, factories and the like. With receiver input noise and internal noise, only the gain of the carrier wave processor is amplified, and input to the demodulator.

However, it is no easy matter to take these kinds of noise into consideration and analyze the relationship between noise power and bit error rate theoretically. Therefore due to their property of being a continuous murmur, thermal noise and shot noise may be treated as additive white Gaussian noise. (However, noise power is mostly thermal noise.)
Additive white Gaussian noise (AWGN) is white noise with equal power density across all frequency bands, and seen from a time base, the occurrence distribution of the noise amplitude level has a Gaussian distribution with a random, uncorrelated signal.

The power density ND(f) of thermal noise is electric energy per 1 Hz and is calculated with the following formula.

ND(f) = κ*To = 1.38 × 10-23 × 290 = 4.0 × 10-21 [W/Hz]= -174 [dBm/Hz]

In other words, the noise power of the receiver input is at least -174 [dBm/Hz], and this noise is amplified and presented to the demodulator. In order to transmit data, a certain bandwidth which is related to the transmission data rate is required, and the thermal noise falling into the specified bandwidth will be added to the signal.
In addition, the amplifier circuit itself is made up of components that generate noise, and that noise power is also amplified.
Consequently, the noise power output is the sum of this noise as we will explain below.

For example, with communication speed of 2,400 bps and equivalent noise bandwidth of 1,200 Hz (31 dB), the noise power of the carrier wave process input unit is -174 + 31 = -143 [dBm]. Here, only the gain of subsequent circuits is amplified.

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Click the image to go to the applet showing additive white Gaussian noise (AWGN)
(Opens in a new window)

Eb/N0 and bit error rate

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What is Eb/N0?
Eb/N0 is the energy per bit to noise power density ratio and it has the following meaning.

Radio systems differ in modulation method and symbol rate, and these differences are important, for example when evaluating the bit error rate. Even if the modulation method or symbol rates are different, they must be evaluated using the same scale so the Eb/N0 standard is used.
This can be said of various things, but when two things are compared, the same criterion must be used, otherwise it is meaningless. Nevertheless, it’s surprising how common it is to ignore this and make ambiguous assessments.

Eb is signal energy per bit in the baseband signal and is the function of the number of bits that make up the signal energy and symbols in the baseband signal.
N0 is the noise power density in the baseband signal, and this value is obtained from the noise power and bandwidth of the receiver’s demodulator.
Eb/N0 is the ratio of these and it accounts for the differences in modulation methods and enables evaluation with the same standard. If the desired Eb/N0 is determined from the required error rate, the required received power C can be obtained, and from the relationship with propagation loss, the radio system can be better understood.


It is a rather difficult formula, but the bit error rate of the representative modulation methods can be found from Eb/N0 with the following formula. Here the detection method is synchronous detection, and erfc is the complementary error function.

ASK

FSK

BPSK


We have made an applet showing the relationship between the bit error rate of the different modulation methods and Eb/N0.
As the graph shows, the relationship between Eb/N0 and bit error rate changes according to the modulation method and the environment.

For example, with the synchronous detection of MSK modulation, in order to achieve a bit error rate of 0.00001 (1E-5: at the bottom) in an AWGN environment, Eb/N0 of 9.5 dB is required.
From this required Eb/N0, the gain of the receiver and the required received power C and required C/N can be obtained based on the noise factor.

When both radios are located in fixed positions in free space, the Eb/N0 value in an AWGN environment can be used. If the radio equipment is moving or multipath propagation is present, the value for a fading environment is used.

BER

Click the image to go to the applet for obtaining the required BER from the required Eb/N0
(Opens in a new window)

Obtaining the required received power C from Eb/N0

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If the bit error rate required by the radio system is determined, the required C/N can be obtained, but first we will explain the assumptions for finding this value using the figure below.

Received power is a value determined by the propagation environment, but noise power depends largely on the hardware of the receiver. The noise power emitted by specific circuits is often indicated by the temperature of heat sources with equivalent energy. Furthermore, the same approach can be applied in cases where there are several circuits as in the RF block in the figure below, and the entire receiver system can also be addressed with this approach. In the figure below, excluding the antenna, the noise sources are placed in front of the respective circuits (blocks) and the circuits (blocks) themselves are treated as having no noise.
As a receiver system, a system noise temperature control point is provided in front of the RF block for calculation as shown in the diagram below.

For example, the RF noise power Nrf in the RF block in the figure below is calculated as follows from noise temperature Trf, circuit gain Grf, and equivalent noise bandwidth B.

Nrf = Trf*κ*Grf*B

Standard temperature = 290 K (16.85°C)
κ: Boltzmann constant 1.38 × 10-23 [J/K], To = 290 [K]

There are three types of circuit in the RF block, and the noise temperature Trb of the RF block is

Trb = Trf + Tm / Grf + Tif / (Grf*Gm)

.
The noise power Nrb of the RF block is as follows.

Nrb = κ*Trb*B


The system noise temperature Ts and system noise power Ns are as follows.

Ts = Tin + Trb 、Ns= κ*Ts*B


Next the Eb/N0 in relation to the bit error rate required for the system is determined. The required received power C can be obtained from the following relational expression.

Noise power density No is the system noise power Ns divided by noise bandwidth B.

No = Ns/B

Eb is as follows.

Eb = (Eb/No)*No

Required received power Creq and C/N are as follows.

Creq = Eb*B 、 C/N = Creq/Ns

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For example, the RF block has the configuration in the figure below, and the noise temperature of the block and gain are calculated from the noise temperature and gain of each circuit.

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Moreover, actual FM receivers use nonlinear modulation/demodulation and limiter circuits and the like, making the calculation very complex.

Example of values for obtaining the required received power C from Eb/N0

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As we explained above, Eb/N0 accounts for the different modulation methods, but C/N is related to the symbol rate. Here, to simplify matters, the number of bits making up one symbol is 1 bit (Binary Phase Shift Keying modulation) ie. BPSK.


Conditions:
Environment with signal fading, standard temperature 290 K, bit error rate 2E-3 (0.002), receiver system noise temperature Ts = 1,600 K,
receiver system noise power Ns = -138.8 dBm, equivalent noise bandwidth B = 600 Hz (27.8 dB).

Equipment parameters: Frequency 400MHz, transmitted power 10dBm, transmitting antenna gain Gta 2.14 dBi, receiving antenna gain Gta 2.14dBi, transmission cable loss 0dBm,
BPSK modulation, synchronous detection, communication speed 1200bps,
Environmental conditions: 2-wave model, transmitter height 1.5m, receiver height 1.5m, communication range 3,000m,


◆ Finding the required Eb/N0 in a fading environment from the graph when the bit error rate is 2E-3 (0.002) gives Eb/N0 = 21 dB.
Therefore

Eb = (Eb/No) + No =21 + (-166.6) = -145.6 [dBm-Hz]

Noise power density No is

No = Ns - B = -138.8 - 27.8= -166.6 [dBm/Hz]

Required received power Creq is

Creq = Eb + B = -145.6 + 27.8 = -117.8 [dBm]

C/N is

C/N = Creq - Ns = -117.8 - (-138.8) = 21 [dB]

Theoretically, with BPSK, Eb/N0 and C/N have the same value.

Based on the above, received power of -117 [dBm] is required to achieve a bit error rate of 2E-3 (0.002).


However, with a 400 MHz system, when actual received power Cact is obtained with the conditions above, the result is as follows.

The effective isotropically radiated power PEIRP of the transmitter is

Peirp = 10 + 2.14 = 12.14 [dBm]

When the 2-wave model is applied with a communication distance of 3,000 m, propagation loss Lall is

Lall = 132.04 dB

Receiver gain Gr_all is

Gr_all = 2.14 dB

Actual received power Cact is

Cact = Peirp - Lall + Gr_all = 12.14 - 132.04 + 2.14 = -117.8 [dBm]

The required received power Creq obtained from required Eb/N0 and actual received power Cact match (naturally by reverse calculation). This can be considered a value close enough to the actual performance of the radio equipment.
Applet for calculating the required received power C from the required Eb/N0 (with binary modulation)
Enter parameters for calculation in the yellow fields.

En/N0

Click the image to go to the applet for obtaining the required received power C and required C/N from the required Eb/N0 (with binary)
(Opens in a new window)

Conclusion

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We looked at how to obtain the received power Creq required by a radio system from Eb/N0 by determining the communication quality required for the system. That value matches the performance of the actual radio equipment system. A slower transmission data rate is better from the point of view of noise, as long as it is fast enough for the application purpose. This results in improved receiver sensitivity, longer range, with stable communication.
Communication range tends to be a matter of concern with radio equipment, but even when radio equipment itself has reasonable performance, it’s often the case that its full performance is not realized due to noise generated within or from the surroundings of the radio system. Therefore it’s necessary to go back to basics and review things from a communication system perspective.

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