Parity Schemes Parity Bit Computer Science Essay

Published: Last Edited:

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

A parity bit is an error detection mechanism which can detect an odd number of errors. The stream of data is first divided into blocks of bits, and then the number of 1 bits is counted. Then, a "parity bit" is set, if the number count of one bits is odd then the scheme is called as even parity scheme and if the number count of one bits is even then the scheme is called as odd parity. The principle of the hamming code is, if the tested blocks of bits are overlapping with each other, then in order to detach the error the parity bits are used, and if the error affects even a single bit it is corrected.

The parity scheme also has a limitation. A parity bit is assured only to detect an odd number of bit errors for example one, three, five, and so on. The parity bit seems to be correct if an even number of bits for example two, four, six and so on is flipped and it seems to be correct even if the data is corrupt. On the parity bit mechanism the extensions and variations are horizontal redundancy checks, vertical redundancy checks and "double", "dual" or "diagonal" parity are used in RAID-DP.


A modular arithmetic sum of message code words of a fixed word length is called a checksum of a message for example byte values. In order to detect errors which are consequential in all-zero messages the sum is frequently canceled by means of a one's-complement preceding to transmission as the redundancy information.

Parity bits, check digits, and longitudinal redundancy check are included in Checksum schemes. There are some checksum schemes, such as the Luhn algorithm and the Verhoeff algorithm, which are purposely designed to detect errors frequently introduced by humans in writing down or recollecting identification numbers.

Cyclic redundancy checks

A block of data is considered as the coefficients to a polynomial over a finite field, and then it is divided by a fixed, predestined polynomial. The remainder of this division serves as the redundancy for the message. This is known as cyclic redundancy checks.

The favorable properties of CRC'S are, they are particularly suitable for detecting burst errors. The cyclic redundancy checks are simply implemented in hardware, and they are broadly used in different protocols.


The ISI is the time-domain counter part of the multipath or smearing of one symbol into another subsequent to it. This type of multipath distortions can be elegantly handled by OFDM by adding up a "guard interval" to each symbol. The cyclical or periodic extension of the basic OFDM symbol is guard interval. In other words, the guard interval seems like the rest of the symbol, but it conveys no 'new' information to the receiver.

While no new information is sent to the receiver, the receiver ignores the guard interval and at rest be able to detach and decode the subcarrier. The receiver is capable to eliminate ISI distortion by dumping the unnecessary guard interval, as the guard interval is designed to be longer than any smearing appropriate to the multipath channeled. Therefore, with virtually zero added complexity of the receiver ISI is removed.

As the guard interval reduces the amount of energy which is available at the receiver for channel symbol decoding, it is significant to note down that the dumping of the unnecessary guard interval do have an impact on the performance of noise. As there is no new information present in the guard interval, the data rate is reduced. As a result a high-quality system design makes the guard interval as small as possible through maintaining adequate multipath protection at the same time.

Why guard interval is not used by single carrier systems? By adding the guard interval in between every symbol the Single carrier systems possibly will remove ISI. Conversely, the impact on the data rate for single carrier systems is a lot more rigorous when compared to the impact on the data rate for OFDM. As OFDM uses a package of narrowband subcarriers, and also the frequency width of subcarrier is inversely proportional to the symbol duration it gains a very high data rates with a moderately long symbol period. Accordingly, it has a very little impact on data rate when a short guard interval is added.

A very short duration symbols are used when the Single carrier systems has bandwidths which is equivalent to OFDM. Therefore it has a lot better impact on data rate when a guard interval which is equal to the channel smearing is added.

In conclusion, OFDM is tremendously suitable for wireless communication in classic WLAN deployments in which the multipath is considered as a most important source of distortion. The OFDM performs very well with the arrangement of various narrow subcarriers by means of interleaving and error correction coding, where an extremely simple method for eliminating ISI is given to the receiver by the guard interval. The effective design of reliable and high rate digital wireless communications systems that are required by the conventional single carrier systems are provided by these built in waveforms without any complexities involved.

Multipath propagation

Multipath propagation is the cause of inter-symbol interference i.e. when the wireless signals are transmitted from the transmitter; these signals reach the receiver in many different paths. Due to which the reflection, refraction of the signals and atmospheric effects such as atmospheric deducting and reflections takes place. For example when the signal is transmitted from the transmitter to the receiver the signal may rebound off buildings. As all the paths of the signals which are transmitted from transmitter to the receiver are different this in turn results in delay of few signals at the receiver causes unlike versions of the signal arriving at different times. This delay of the signal spread the part of the given symbol or the complete symbol into the subsequent symbols, in that way it provides with the correct detection of those symbols which cause interference. In addition, various paths of the signal frequently distort the amplitude and phase of the signal thus causing additional interference along with the signal received at the receiver.

Band limited channels

The transmission of a signal from one end to another through a band limited channel also causes inter-symbol interference, where the frequency response is above the cutoff frequency. The frequency components which are above the cutoff frequency are removed by passing a signal through such a channel. Additionally, the channel also attenuates the amplitude of the frequency components which are below the cutoff frequency.

The shape of the signal which is received at the receiver is affected by the filtering of a transmitted signal. The filtering of a rectangular pulse not only affects the nature of the pulse in the first symbol period, but it extends out over the consequent symbol periods as well. The extended pulse of each individual symbol will interfere with subsequent symbols when a message is transmitted through the channel.

Band-limited channels are present in both wired and wireless communications as different to multipath propagation. The drawback is repeatedly forced by the need to operate multiple autonomous signals through the same channel; due to which the part of the total bandwidth is classically allocated to each system. To transmit, the portion of electro-magnetic spectrum may be allocated for the wireless systems, (for example, the FM radio is commonly broadcasted in the frequency range of 87.5 MHz - 108 MHz). The allocation of this frequency range is generally administered by a government agency; in case of the United States this is done by the Federal Communications Commission (FCC). In a wired system, the allocation is decided by the owner of the cable for example optical fiber cable.

For instance, due to the physical properties of the medium the bandwidth is limited, in practical not any of the transmitted signals will propagate above the cutoff frequency in the wired system where the cable is being used.

To avoid the interference which is caused due to the bandwidth limitation, the Communication systems transmits the data over the band-limited channels which typically implement the shaping of the pulse. It is not at all possible to communicate with ISI when the frequency response of the channel is plat and the fixed bandwidth of the shaping filter. Frequently the frequency response of the channel is unknown hence in order to compensate the frequency response the adaptive equalizer is used.

Effects on eye patterns

The study of ISI in a data transmission system or a PCM is, the received wave is applied to the vertical deflection plates of the oscilloscope and a sawtooth wave is applied at the transmitted symbol rate R, 1/T is applied to the horizontal deflection plates. The ensuing display is known as an eye pattern as it resembles the human eye for binary waves. The internal area of the eye pattern is called the eye opening. The information regarding the performance of the relevant system is provided by an eye pattern. The time interval is defined by the width of eye opening above which, with no error from ISI the received wave is sampled. It is obvious that the immediate time at which the eye is open to the widest is considered as the preferred time for sampling.

Since the sampling time is wide-ranging the rate of closure of the eye determines the sensitivity of the system to the timing error.

The margin over the noise at a particular sampling time is defined by the height of the eye after opening.

The graphical representation of the signal characteristics are provided by the eye pattern which can superimpose many samples of signals. In the eye diagram of the binary phase-shift keying system given below 1 represents amplitude of -1 and 0 represents amplitude of +1. The centre of the image represents the current sampling time and the ends of the image represents the earlier and subsequent sampling times. It is clearly seen in the diagram that the different transitions of signal from one sampling time to another sampling time (for example one-to-zero, one-to-one and so on).

The noise margin - in order to cause the error at the receiver the required amount of noise is provided by the distance between the signal and the zero amplitude point at the sampling time; for the proper interpretation of the signal, the signal is sampled anywhere between the points at which zero-to-one and one-to-zero transitions cross with each other. When the signal is less sensitive to the errors at the receiver then these points are said to be better.

The second image shows the effects of ISI, it is an eye pattern which is operated in a multipath channel of the same system. The loss of definition of the signal transitions shows the effects of receiving belated signals and the signals with distorted versions. The performance of the system is said to be worse, when both noise margin and window in which the signal is sampled is reduced.


The eye diagram of a binary PSK system


The eye diagram of the same system with multipath effects added

Countering ISI

The problem of inter symbol interference is employed by using several techniques that are available in telecommunication and data storage. The impulse response is a design system which is enough that very little energy from one symbol smears into the subsequent symbol.


Successive raised-cosine impulses, representing zero-ISI property

Symbols are separated in time by means of guard periods.

An equalizer is applied at the receiver such that, generally speaking, attempts to unfasten the channel effect by using an inverse filter.

In order to approximate the series of transmitted symbols by using the Viterbi algorithm a sequence detector is applied at the receiver.


The block diagram represents the OFDM system which is modelled using Matlab. A concise explanation of the model is as follows.


Serial to Parallel Conversion

The serial data stream of input is arranged into the word size which is needed for transmission, e.g. 2 bits/word for QPSK, and then it is shifted to a parallel format. By assigning every data word to one carrier the data is then transmitted in parallel.

Modulation of Data

The data is differentially encoded with the preceding signal after it is being transmitted on every carrier, then it is mapped into the format of Phase Shift Keying (PSK). For this reason an extra symbol is added at the begin as the differential encoding requires an initial phase reference. Depending on the modulation method the data on every symbol is mapped to a phase angle. For example, the phase angles used by QPSK are 0, 90, 180, and 270 degrees. The constant amplitude signal is produced by using phase shift keying and was preferred for its ease and to minimize the problems related to amplitude fluctuations suitable to fading. 

Inverse Fourier Transform

After the employment of required spectrum, the equivalent time waveform is found using an inverse fourier transform. To the begin of every symbol the guard period is added. 

Guard Period

By means of two sections the guard period is made. A zero amplitude transmission is the first half of the guard period time and the cyclic extension of the symbol to be transmitted is the other half of the guard period time. This allows the envelope detection to recover the symbol timing with no trouble. As the timing may possibly be precisely determined by the position of the samples it was establish that it was not necessary in whichever of the simulations. 

The symbols are then transformed back to a successive time waveform, after adding the guard. This is called the base band signal meant for the OFDM transmission.


To the transmitted signal a channel model is applied. The signal to noise ratio, multipath is allowed by the model and it also controls the peak power clipping. By adding an identified sum of fair noise to the transmitted signal the signal to noise ratio is to be set. Using an FIR filter the Multipath delay spread is added by simulating the holdup spread. The utmost delay spread is represented by the measurement length of the FIR filter, whereas the reflected signal magnitude is represented by the coefficient amplitude. 


The receiver on the whole does operate reverse to the transmitter. The guard period is detached. To find out the unique transmitted spectrum the FFT of every symbol is considered. Every transmission carrier phase angle is estimated and then transformed back to the word data by demodulating the phase of the received signal. The data words are again transformed back to the original data same as the word size.

OFDM simulation parameters

Most of the simulations which uses configuration are performed on the OFDM signal. If each user is allocated with 8 carriers the 800-carrier system will allow up to 100 users hence it uses the 800-carrier system. The intension behind using the 800-carrier system is that, because of frequency selective fading few carriers may lost, hence when each user is assigned with multiple carries the outstanding carriers will recover the lost data by means of forward error correction. Due to which this method can be used by any system where each user is assigned with a lesser amount of 8 carriers. Therefore the systems with 400 carriers or less than 400 carriers are regarded as excessively small. In order to achieve the sensitivity of OFDM and to avoid the frequency stability errors, more carriers were not being used. As the number of carriers in a system increases it needs more frequency stability.

In the majority of simulations the signals which are generated are not scaled to a specific sample rate, as a result it is considered as normalized frequency. To measure the performances the three carrier modulation methods are tested. Due to which the tradeoff between system capacity and system robustness is illustrated. DBPSK is the main durable method which provides the spectral efficiency of 1 bits/Hz. On the other hand by the use of DQPSK which provides the spectral efficiency of 2 bits/Hz and D16PSK which provides the spectral efficiency of 4 bits/Hz the increase of system capacity can be achieved except that the cost of BER goes higher. On every simulation plot it is shown that the modulation method used is BPSK, QPSK, and 16PSK, as the integral part of every OFDM transmission is differential encoding.



In the European market the basis for the Digital Audio Broadcasting (DAB) standard is formed by DAB - OFDM.

The basis for the global ADSL (asymmetric digital subscriber line) standard is formed by ADSL-OFDM.

In the wireless point-to-point configuration and the wireless point-multipoint configuration which uses OFDM technology the Wireless Local Area Networks - development is an ongoing process.

The IEEE 802.11 working group has been published IEEE 802.11a in addition to IEEE 802.11, which summarizes the employ of OFDM in the 5.8-GHz band.


MIMO-OFDM stands for Multiple Input, Multiple Output Orthogonal Frequency Division Multiplexing which is developed by Iospan Wireless. This technology uses numerous antennas to transmit and receive the radio signals. MIMO-OFDM allows the organization of a Broadband Wireless Access (BWA) system to facilitate a Non-Line-of-Sight (NLOS) functionality by the service providers. In particular, the advantage of MIMO-OFDM is, it uses base station antennas which does not contain LOS which in turn results in multipath properties of environment.

According to Iospan, when the radio signals are transmitted from transmitting antenna to receiving antenna, these signals bounce off buildings, trees and other objects. Due to which the numerous "echoes" or "images" of the signal is produced. Therefore, the original signal and the individual echoes arrive at different times at the receiving antenna, which cause the echoes to interfere with each other consequently degrading the quality of the signal.

In order to transmit the data simultaneously in small pieces to the receiver, the MIMO system uses multiple antennas, and the receiver after receiving the small pieces of transmitted data it process the flow of data and place them back together. This process is known as spatial multiplexing. The spatial multiplexing relatively boosts the speed of the data-transmission which is dynamically equivalent to the number of transmitting antennas. In addition to this, the spatial multiplexing technique utilizes the frequency spectrum more efficiently and effectively, as all the data which is transmitted with distinct spatial signatures is in the same frequency band.

The concept of MIMO technology is used by the VOFDM which stands for Vector Orthogonal Frequency Division Multiplexing which is developed by Cisco Systems.

Characteristics and principles of operation


In OFDM, the sub-carrier frequencies are selected in such a way that the sub-carrier frequencies are orthogonal to each other, which means that it eliminates the cross-talk occurred between the sub-channels and the inter-carrier guard bands are also not necessary. The design of transmitter and receiver is significantly simplified or make simpler by this method. This method does not require separate filter for each sub-channel which is dissimilar to conventional FDM.

The sub-carrier spacing required by the orthogonality is \scriptstyle\Delta f \,=\, \frac{k}{T_U}Hertz, where TU is the time in seconds a functional symbol duration (the receiver side window size), and k is a positive integer, characteristically equal to 1. Therefore, the total passband bandwidth for N sub-carriers, is given as B ≈ N·Î”f (Hz).

The high efficiency of spectrum is achieved by means of orthogonality, for the equivalent baseband signal with which the overall symbol rate is equal to the Nyquist rate. Approximately the frequency band which is available can be utilized completely to a great extent. In general OFDM have a 'white' spectrum, which provides kind electromagnetic interference properties with deference to additional co-channel users.

A simple example for orthogonality is as follows: the symbol duration TU = 1 ms would need a sub-carrier spacing of \scriptstyle\Delta f \,=\, \frac{1}{1\,\mathrm{ms}} \,=\, 1\,\mathrm{kHz}(or an integer multiple of that) for orthogonality. The total passband bandwidth of N = 1,000 sub-carriers would result in NΔf = 1 MHz. According to Nyquist rate the necessary bandwidth for this particular symbol time is N/2TU = 0.5 MHz (which means half of the accomplished bandwidth). If a guard interval is applied the requirement of Nyquist bandwidth would be still lesser. The FFT would result in N = 1,000 samples per symbol. when guard interval is not applied, then this would result a sample rate of 1 MHz a base band complex valued signal, which requires a baseband bandwidth of 0.5 MHz in according to Nyquist. conversely, when baseband signal is multiplied with a carrier waveform it produces the passband RF signal (which is a double-sideband quadrature amplitude-modulation) which in turn results in a passband bandwidth of 1 MHz. approximately half of the bandwidth of the same symbol rate can be achieved by the single-side band (SSB) or vestigial sideband (VSB) modulation scheme (that is twice the high efficiency of spectrum for the same length of the symbol).

Between the transmitter and the receiver a very high and accurate synchronization of frequency is required by the OFDM. By means of frequency deviation the sub-carriers are no more orthogonal, which causes the cross-talk between the sub-carriers called inter-carrier interference (ICI). Due to the disparity of transmitter oscillator and receiver oscillator or due to the Doppler shift it causes a typical frequency offset. By means of the receiver the Doppler shift alone can be compensated, but the circumstances are worst when it is in combination with multipath, as it is very difficult to correct the reflections which occur at a variety of frequency offsets. As the speed increases this effect characteristically occurs and goes worst, and hence restricting the high-speed vehicles to use OFDM. There are some techniques which are suggested for the removal of inter-carrier interference ICI suppression, which may increase the complexity of the receiver when used.

Implementation using the FFT algorithm

The use of FFT algorithm on the side of receiver and the inverse FFT on the side of transmitter the orthogonality provides the efficient implementation of modulator and demodulator. Besides few principles and a number of benefits have been identified since 1960s, OFDM is of low-cost digital signal processing components which calculate the FFT effectively hence today it is very popular for wideband communications.

Guard interval for elimination of intersymbol interference

The modulation schemes with low symbol rate is a key principle of OFDM where the symbols are comparatively extensive when compared to the channel time characteristics, which undergo less intersymbol interference caused by multipath propagation, the transmission of various low-rate streams in parallel is more advantageous rather than the transmission of single high rate stream. as the period of each symbol is long, the intersymbol interference is eliminated by inserting the sufficient guard interval in between the symbols of OFDM.

The requirement of pulse shaping filter is eliminated by the guard interval, and it also decreases the problems of sensitivity to time synchronization.

A simple example is as follows: by using conventional single-carrier modulation over a wireless channel when one sends a million symbols per second then every symbol duration will be less than a microsecond or a microsecond. This compels rigorous restriction on synchronization and demand for the elimination of multipath interference. Among one thousand sub-channels when same million symbols are spread per second, the duration of each symbol can be more by a factor of a thousand (which can be one millisecond) which is about the same bandwidth for orthogonality. Let us assume that in between every symbol a guard interval of 1/8 length of the symbol is inserted. When the multi path time spreading is less than the guard interval of 125 micro seconds then the Intersymbol interference can be eliminated. The maximum difference between the lengths of the paths is written as 37.5 kilometers.

During the guard interval the cyclic prefix is transmitted which consists the end of OFDM symbol, after transmitting the OFDM symbol the guard interval is transmitted. The reason for transmitting the end of OFDM symbol including with guard interval is that when OFDM demodulation is performed by means of FFT the integer number of sinusoidal cycles are integrated for every multipath by the receiver.

Simplified equalization

**The effects of frequency-selective channel conditions, for example fading caused by multipath propagation, can be considered as constant (flat) over an OFDM sub-channel if the sub-channel is sufficiently narrow-banded (i.e., if the number of sub-channels is sufficiently large). This makes equalization far simpler at the receiver in OFDM in comparison to conventional single-carrier modulation. The equalizer only has to multiply each detected sub-carrier (each Fourier coefficient) by a constant complex number, or a rarely changed value.

Our example: The OFDM equalization in the above numerical example would require one complex valued multiplication per subcarrier and symbol (i.e., \scriptstyle N \,=\, 1000complex multiplications per OFDM symbol; i.e., one million multiplications per second, at the receiver). The FFT algorithm requires \scriptstyle N \log_2 N \,=\, 10,000complex-valued multiplications per OFDM symbol (i.e., 10 million multiplications per second), at both the receiver and transmitter side. This should be compared with the corresponding one million symbols/second single-carrier modulation case mentioned in the example, where the equalization of 125 microseconds time-spreading using a FIR filter would require, in a naive implementation, 125 multiplications per symbol (i.e., 125 million multiplications per second). FFT techniques can be used to reduce the number of multiplications for an FIR equalizer to a number comparable with OFDM, at the cost of delay between reception and decoding which also becomes comparable with OFDM.

In a sense, improvements in FIR equalization using FFTs or partial FFTs leads mathematically closer to OFDM, but the OFDM technique is easier to understand and implement, and the sub-channels can be independently adapting in other ways than varying equalization coefficients, such as switching between different QAM constellation patterns and error-correction schemes to match individual sub-channel noise and interference characteristics.

Some of the sub-carriers in some of the OFDM symbols may carry pilot signals for measurement of the channel conditions (i.e., the equalizer gain and phase shift for each sub-carrier). Pilot signals and training symbols (Preamble_(communication)) may also be used for time synchronization (to avoid intersymbol interference, ISI), and frequency synchronization (to avoid inter-carrier interference, ICI, caused by Doppler shift).

If differential modulation such as DPSK or DQPSK is applied to each sub-carrier, equalization can be completely omitted, since these non-coherent schemes are insensitive to slowly changing amplitude and phase distortion.

OFDM was initially used for wire, and stationary wireless communications. However with increasing number of applications operating in highly mobile environment, the possibility of using OFDM for such purpose is also investigated. Over the last decade, several researches have been done on how to equalize OFDM transmission over doubly selective channels.

Channel coding and interleaving

OFDM is invariably used in conjunction with channel coding (forward error correction), and almost always uses frequency and/or time interleaving.

Frequency (subcarrier) interleaving increases resistance to frequency-selective channel conditions such as fading. For example, when a part of the channel bandwidth fades, frequency interleaving ensures that the bit errors that would result from those subcarriers in the faded part of the bandwidth are spread out in the bit-stream rather than being concentrated. Similarly, time interleaving ensures that bits that are originally close together in the bit-stream are transmitted far apart in time, thus mitigating against severe fading as would happen when travelling at high speed.

However, time interleaving is of little benefit in slowly fading channels, such as for stationary reception, and frequency interleaving offers little to no benefit for narrowband channels that suffer from flat-fading (where the whole channel bandwidth fades at the same time).

The reason why interleaving is used on OFDM is to attempt to spread the errors out in the bit-stream that is presented to the error correction decoder, because when such decoders are presented with a high concentration of errors the decoder is unable to correct all the bit errors, and a burst of uncorrected errors occurs. A similar design of audio data encoding makes compact disc (CD) playback robust.

A classical type of error correction coding used with OFDM-based systems is convolutional coding, often concatenated with Reed-Solomon coding. Usually, additional interleaving (on top of the time and frequency interleaving mentioned above) in between the two layers of coding is implemented. The choice for Reed-Solomon coding as the outer error correction code is based on the observation that the Viterbi decoder used for inner convolutional decoding produces short errors bursts when there is a high concentration of errors, and Reed-Solomon codes are inherently well-suited to correcting bursts of errors.

Newer systems, however, usually now adopt near-optimal types of error correction codes that use the turbo decoding principle, where the decoder iterates towards the desired solution. Examples of such error correction coding types include turbo codes and LDPC codes, which perform close to the Shannon limit for the Additive White Gaussian Noise (AWGN) channel. Some systems that have implemented these codes have concatenated them with either Reed-Solomon (for example on the MediaFLO system) or BCH codes (on the DVB-S2 system) to improve upon an error floor inherent to these codes at high signal-to-noise ratios.

Adaptive transmission

The resilience to severe channel conditions can be further enhanced if information about the channel is sent over a return-channel. Based on this feedback information, adaptive modulation, channel coding and power allocation may be applied across all sub-carriers, or individually to each sub-carrier. In the latter case, if a particular range of frequencies suffers