What is meant by the probability of error?

The concept of probability of error is essential in the context of digital communications and it is also sometimes known as bit error rate or BER. The BER is characterized by a bit error ratio. The bit error ratio is the ratio between the number of bit errors and the total bits transferred. The fundamentals of BER are important parameters in determining the number of bits received in a data stream constituting a communication channel. The bit errors refer to those received bits which are altered due to noise, interference, distortion, and bit synchronization. The error probability of these bits is generally evaluated using the Monte Carlo simulations. Sometimes for simpler models such as Bernoulli's distribution, analytical simulations are also preferred. Some of the other simpler models used in the information theory are the binary symmetric model and additive white gaussian noise model. The parameter, bit error is expressed as a function of $\frac{E_b}{N_o}$ where $E_b$ is the energy per bit and $N_o$ is the noise power spectral density. For instance, in the phase shift modulation process, the bit error rate is expressed as,

$\in =\frac{1}{2}erfc\left(\sqrt{\frac{E_b}{N_o}}\right)$

where, $\in$ represents the bit error rate, and $erfc$ represents the Gauss error function.

The performance of the digital communication system is determined by plotting BER curves. The BER vs received power graphs are often used in optical communication systems, where BER vs signal to ratio (SNR) curves are used for wireless communications. 

Packet error probability

The packet error probability (PER) is the expected value of the packet error ratio. The packet error ratio is the ratio between the number of incorrectly received packets and the total number of received packets. The packet error probability can be expressed as,

$$\begin{gathered} P_E=1-{\left(1-P_e\right)}^N \\ \Rightarrow P_E=1-e^{N\ln \left(1-P_e\right)} \end{gathered}$$

Here, $N$ is the length of a data packet,

$P_E$ is the probability of packet error,

$p_e$ is the bit error probability, which is the expected value of the BER.

The packet error ratio is often characterized by the packet error rate, that is, PER. The PER provides a tool to compute the performance of digital communication systems. This PER is specially used to compute signals when it experiences block fading. Block fading is a process where signal experiences slow coherence time as compared to their transmission rate. 

Parameters that affect the BER

A communication system is normally characterized by transmission and receiver. On the receiver side, the BER is extensively influenced by noise, wireless multipath fading, interferences, distortion, bit synchronization, attenuation, and other various factors. The BER can be improved by using the following approaches:

• By choosing a signal path that provides adequate signal strength
• Selecting a robust modulation scheme
• By selecting a slow modulation scheme
• By using the line coding scheme
• By using channel coding schemes like redundant forward error correction code

On the transmission side, the BER is given by,

$TBER=\frac{Detected bits which are incorrect before error correction}{Total transferred bits}$

Where $TBER$ denotes the transmission bit error rate.

There lies another parameter which is known as the information BER which is generally smaller than transmission BER. The information BER usually represents the decoding error probability. The decoding error probability is given by,

$P_D=\frac{Number of decoded bits that remains incorrect after error rectification}{Total number of decoded bits}$

Where $P_D$ represents the decoding error probability.

BER testing

The bit error rate is analyzed using a bit error rate tester (BERT) device. The BERT device is an electronic instrument that is used to test the quality of signals both for complete systems and single-component systems. The BERT is usually applied to wireless communication, telecommunication systems, and radio communication systems. These devices can also determine the performance of data links, which include fiber optics and wireless links.

There are generally five modules that constitute the BERT equipment, they are:

1. Pattern generator: The pattern generator is an electronic device that is extensively used in digital electronics to generate a digital signal or specific waveform. These wave patterns are externally applied to the device which is under test. 
2. Error detector: Error detectors are specialized equipment with algorithms that are intended for rapid error computations. These error detectors detect the errors generated by the device which is under test.
3. Clock signal generator: Clock signal generators are used to synchronize the working and communication between the pattern generator and the error detector.
4. Digital communication analyzer: These modules display the transmitted and received signals.
5. Electrical to optical converter: These are devices that convert the electrical signals to optical signals or pulses, which are intended to be transmitted by fiber optics. 

Bit error rate stress patterns

The bit error rate testing is applied in communication circuits, and the test results are characterized by a sequence of ones (1) and zeros (0). This sequence is generated by the test pattern generator. There are various types of stress patterns that can be viewed in the BERT machine, they are:

• Pseudorandom binary sequence (PRBS): This is a binary sequence consisting of N bits. These sequences are used to measure signal jitters. Jitter is the deviation of a signal from its periodicity. 
• Quasi random signal source (QRSS): The QRSS generates 20-bit words that repeat every 1048575 words. The QRSS suppresses zero to fourteen times.
• 3 in 24: The 3 in 24 is characterized by the longest string of consecutive zeros.
• 1:7: These stress patterns are also known as 1 in 8. These patterns are characterized by a single 1 in an eight-bit repeating sequence. 
• All ones: This pattern consumes the maximum power in the repeater. This sequence is characterized by all repeating ones.
• All zeros: The pattern is characterized by the maximum number of zeros, or the signal pattern is composed of zeros only. 

Context and Applications

This topic is widely taught in many undergraduate and postgraduate degree courses of:

• Bachelors in Electrical Engineering
• Bachelors in Electrical and Communications Engineering
• Masters in Electrical and Communications Engineering
• Masters in Technology (Communication Systems)

Practice Problems

1. What is the abbreviation of BERT?

1. Bit error rate tester
2. Bit error rate testing
3. Bit error rate test
4. Byte error rate tester

Answer: Option a

Explanation: A bit error rate tester or BERT is an electronic device that is used to test the quality of signals for both single-component systems and complete systems.

2. What is the purpose of a signal clock generator?

1. To generate synchronized signals
2. To generate digital pulses
3. To synchronize the working of pattern generators and error detectors
4. None of these

Answer: Option c

Explanation: Clock signal generators are oscillator-type devices that are used to synchronize circuit operations. The clock signal generator is used to synchronize the operation of error detectors and pattern generators.

3. What does the function $erfc$ represent?

1. Exponential function
2. Bernoulli's function
3. Poisson's function
4. Gauss error function

Answer: Option d

Explanation: The Gauss error function $erfc$ is sometimes represented as $erf\left(z\right)$. The Gauss error function is a complex function of a complex variable. It is a special function that often occurs in probability, statistics, and partial differential equations. 

4. What is the abbreviation for PRBS?

1. Pseudorandom byte sequencing
2. Pseudorandom bit sequencing
3. Pseudorandom bit sequence
4. Pseudorandom byte sequence

Answer: Option c

Explanation: The pseudorandom bit sequence or PRBS is one of the stress patterns generated by the BERT device. These sequences are used to represent the number of signal jitters.

5. What is the correct expression for packet error probability?

1. $P_E=1-e^{N\ln \left(1-P_e\right)}$
2. $P_E=1-\left(1-N\right)$
3. $P_E=1-N\ln \left(1-P_e\right)$
4. $P_E=N\ln \left(1-P_e\right)$

Answer: Option a

Explanation: The packet error probability is the expected value of the packet error ratio. It is given by $P_E=1-e^{N\ln \left(1-P_e\right)}$.