^{1}

^{*}

^{1}

We present in this paper a method for enhancing equalization of a dynamic channel. A dynamic channel is characterized and modeled by a high relative velocity between transmitter and receiver and fast changes of environment conditions for wave propagation. Based on Jakes model, an auto-regressive model (AR) [1] for such a dynamic system,
* i.e.*, a time variant channel is developed. More specifically, the enhanced equalization method we are proposing is a combination of a multi-stage time and frequency domain equalizer with a feed-forward loop. The underlined wok presents a unified approach to the equalization method that employs both time and frequency domains data to enhance the equalization scheme. In an OFDM system, the channel coefficients for each tap, in time domain for consecutive blocks, are partially independent thus correlated. Such correlation can improve the channel estimation if it is taken into account. The method in this paper enhances the performance of equalization by dynamically selecting the number of previous OFDM symbols based on the Doppler frequency. In order to decrease the complexity of the system model, we utilize the autocorrelation and Doppler frequency to dynamically select the previous OFDM symbols that will be stored in the memory. In addition to deriving earlier results in a unified manner, the approach presented also leads to enhanced performance results without imposing any restrictions or limitations on the OFDM system such as increasing the number of pilots or cyclic prefix.

There is much on-going research in channel estimation procedures, however, most of it considers a time-invariant channel [^{th} cellular generation standard based on OFDM. The OFDM system is a key broadband wireless technology which supports data rates in excess of 100 Mbps.

Therefore, channel estimation, channel equalization, and the iterative complex algorithms would suffer from high complexity as well as poor performance over fast fading frequency selective channels [

field. One solution to combat the interference is to add the cyclic prefix [

We assume high frequency mobile network technologies, such as but not limited to 4G, and a wireless signal that propagates over rapidly varying channels

occurring at relatively high speeds. Thus, based on the set forth conditions of the channel, we utilize a model known as Jakes model. With Jakes model, the channel is correlated in time due to the Doppler effect. This model enables us to adapt Jakes model for the time varying channel taps, i.e. the underlying channel varies with time. Furthermore, with Jakes model, we are able to estimate the next sample of the channel utilizing the fact that the Auto-regressive model estimates the system based on the previous samples, since the estimation of the next sample of the channel primarily depends on the previous samples. As such, it is obvious that the higher the number of previous samples, the more accurate estimate is obtained. However, the caveat with increasing the number of previous samples is that it causes inefficiency in the system and thus renders this method irrational. Therefore, this paper will introduce the relationship between the number of the previous samples and Doppler shift in which renders the selection of previous number of samples more efficient.

In order to select a precise number of previous OFDM symbols, optimize, and utilize the memory and store said symbols efficiently, this paper takes an advantage of the relationship between the Doppler and the auto-correlation of the subsequent OFDM symbols. This approach will lead to a less complex and more efficient system. The Kalman [

The channel is correlated in time domain due to the Doppler effect. Doppler follows Jakes model where the auto-correlation is not an impulse. Thus, in this case, such a dynamic channel follows Jakes model, and the auto-correlation of Jake model can be easily obtained by the following:

where

Based on the assumption that

wherein

This can be written in a matrix form as follows:

where

Given

Now, we can form the system model matrix based on Equations ((2) and (7)) as follows:

The system model can be written in a vector form as follows:

For a single tap channel, the state space model for the 1^{st} tap is given by:

where,

where h is the channel coefficients at time instant k.

The transition matrix F for the 1^{st} tap is given by:

The previous state is given by:

The covariance matrix for the 1^{st} tap is given by:

The state space model for the 2^{nd} tap is given by:

Finally, for multiple taps channel, the state space model is given by:

The measurement output of the state space model is as follows:

In summary, the state space model and the measurement output for the channel is as follows:

Once we calculate

transform the data to time domain. Next, data is summed and transmitted to a receiver through a channel. At the receiver, data is in time domain thus it is transformed to frequency domain and then decoded in order to retrieve the original data.

Equation below shows a mathematical model of a composite transmitted and received signals in time and frequency domain as follows:

Both equations defined above represent Discrete Fourier Transform (DFT) which transfers the samples from time domain to frequency domain. The Inverse Discrete Fourier Transform which transfers the frequency domain symbols into time domain samples.

IFFT operation is defined below:

The time domain

A cyclic prefix refers to the prefixing of a symbol with a repetition of the end. The prefix serves as a guard interval, it eliminates the inter-symbol interference from the previous symbol. Also, the prefix serves as a repetition of the end of the symbol, it allows the linear convolution of a frequency-selective multipath channel to be modelled as circular convolution, which in turn may be transformed to the frequency domain using a discrete Fourier transform. This approach allows for simple frequency-domain processing, such as channel estimation and equalization.

Cyclic Prefixes are used in OFDM in order to combat multipath by making channel estimation easy. As an example, consider an OFDM system which has N subcarriers.

A cyclic prefix (CP) of length is added to the time domain samples to make a more complete OFDM symbol or packet with samples. As mentioned, this cyclic prefix preserves orthogonality between the subcarriers by preventing ISI, i.e. the cyclic prefix provides a time-delay buffer between OFDM symbols seen at the receiver, and must have a duration equal to or greater than the delay spread of the channel. The addition of the CP to the OFDM symbol can be modeled as:

where the length of p is determined by the number of of multipath parameters as follows:

where L is the number of multipath or the length of channel. The reason for Equation (29) is to capture all information in the signal.

For simplicity, let L = 3, p = 2; the received output equation is:

Thus, the received output equation is given as:

where w is the noise and

The received outputs

where

Using Equation (26), the output at the receiver is given by:

The h matrix which is a result of the circular convolution is expressed as:

In this case, we consider the channel under fast fading i.e. the impulse response is changing. Next step is to convert the channel from time domain to the the frequency domain using Fast Fourier Transform (FFT) as shown in Equation (24). The mathematical representation using Equation (35) is given by the following:

We know that that channel in the frequency domain is:

Thus,

which turns out to be a diagonal matrix as shown below:

When compare the matrix in time domain, matrix 36 to the matrix in frequency domain, matrix 40, we can see matrix 40 is reduced in complexity and thus the equalization would much easier since you only have one parameter to deal with for each tap. After we find the channel in frequency domain as shown in matrix 40, we choose an equalization method to equalize the signal. MMSE equalizer in frequency domains proves that it works very well in a fast selective environment. The mathematical representation of the MMSE receiver is shown below:

This section will discuss the proposed method for enhancing the equalization of the channel. More specifically, the enhanced equalization method we are proposing is a combination of a multi-stage time and frequency domain equalizer with a feed-forward loop. Inserting a pilot data at the transmitter and using said pilot data to estimate the channel in time domain. The addition of the feed- forward is utilized to capture a part of the received signal, then extracting the channel information while the signal is being delayed until the information is obtained. The estimated channel is forwarded to the MMSE equalizer. The first stage of the muti-stage equalizer is using pilot data as a measurement data in the frequency domain. The second stage is to convert the measurement data to the time domain. In the third stage, we estimate the channel using the pilot data i.e. the measurement data, i.e. input to the Kalman filter. Taking under consideration that in OFDM system the channel coefficients for each tap, in time domain for- consecutive blocks are not totally independent, but they are correlated; thus such correlation is utilized for time domain channel estimation. Using the KALMAN filter, we can extract this information from previous blocks and use them for the newly arrived one, and perform a better channel estimation. The fourth stage, we use the output of the Kalman filter as a feed-forward to the frequency equalizer for each subcarrier in which the data is converted from time domain to frequency domain. Adding efficiency to the channel by dynamically selecting the number of previous states based on the doppler and the autocorrelation of the channel. Furthermore, in order to evaluate the performance of the proposed method, a comparison is conducted with the following methods:

1) A traditional channel equalization using MMSE and IFFT interpolation.

2) A perfect channel equalization where we assume the channel is perfect.

In this section, simulation is performed to compare the performance between LS and MMSE equalizers at certain conditions explained in the following sections. Also, we validate the performance of the proposed multi-stage equalizer, for high frequency networks and high speed mobility, by comparing its performance with other methods.

the subcarriers. Prior to transmitting the data, the modulated subcarriers are transformed to time domain using IFFT and cascaded together along with guard interval (cyclic prefix) inserted in between each other. The communication time variant channel is modeled with a Gaussian noise added. At the receiver, the symbols are received and guard intervals are removed. After that, the data is converted from time domain to frequency domain. The pilot data positions and their values are fed into the time domain equalizer where the equalizer reduces the inter-symbol interference and retrieves the original data. With the feed- forward, the output of the time domain equalizer which is transformed to a frequency domain as the well as the output data which is still in frequency domain are fed to the frequency domain equalizer to refine the output. After that, the data is converted from serial to parallel and then demodulated.

In this section, we try to explore different channel scenarios, i.e. flat fading channel versus frequency selective channel. Also, we try different types of equalizers, i.e. LS and MMSE. More specifically, we will demonstrate three scenarios with different equalizers techniques, different fading channels, and under certain conditions.

The first scenario is to model a flat fading channel with LS equalizer and analyze the output of said equalizer. The conditions of this scenario are illustrated in

The first scenario demonstrates a flat fading channel as shown in

Sampling time (Ts) | 1 |
---|---|

Signal length | 1e3 |

Channel length (l) | 3 |

Signal to Noise Ratio (SNR) | 20 dB |

Channel taps | [1 1/10 1/100] |

Noise | No |

signal denoted by signal, received signal denoted by rxsignal, and the signal after equalization denoted by det are very close to each other.

Subsequently, the second scenario is to model a frequency selective channel fading, i.e. deep fading, with LS equalizer. The conditions are shown in

As illustrated in

The third scenario is to utilize MMSE Equalizer in place of LS Equalizer with

Sampling time (Ts) | 1 |
---|---|

Signal length | 1e3 |

Channel length (l) | 3 |

Signal to Noise Ratio (SNR) | 20 dB |

Channel taps | [1 0.99 0.95] |

Noise | Yes |

conditions illustrated in

In this section, we validate the performance of the proposed multi-stage equalizer, for high frequency networks and high speed mobility, by comparing its performance with other methods. The channel model used in this paper is Rayleigh channel model with additive white Gaussian noise. The parameters in

Parameters | Values |
---|---|

Bandwidth | 4 MHz |

Total subcarriers | 256 |

Subcarrier spacing | 12 KHz |

Modulation | QAM |

Doppler | [36 200] Hz |

CP length | 5 msec |

SNR | [5:3:40] dB |

Number of pilots | 8 |

thermore, the speed and doppler components are considered to have major contribution to the fading channel.

The simulation is performed at three different speeds, different noise levels with SNR ranging from 5 dB to 40 dB in order to have a wide spectrum of error rate.

While the performance of the proposed method, Multi-stage Equalizer, is being analyzed and explored, said method is performed with respect to two major parameters, Doppler, and number or previous states. In this paper, we emphasize on the relationship between these two parameters in order to improve the

channel equalization.

Channel Equalization has been proved throughout the years to improve the channels performance under certain conditions. Channel equalization has been successfully utilized in many applications such as video, image, voice etc. However, as a result of certain critical conditions such as higher frequency and speed,

the complexity of the system tends to increase and the performance of the system tends to deteriorate. Many Equalization techniques have been proposed to optimize the performance of the wireless channel in such a dynamic environment, however these techniques still use very complex algorithms. In this paper, we have proposed a unique method for enhancing the accurate equalization of time-variant Rayleigh channels under dynamic changes. Significant performance gains were observed for such selective fading channels when compared to the traditional method. Our theoretical and simulation results demonstrate that to achieve such high accurate performance, knowledge of the previous samples of the channel is required however the number of the previous samples depends on the Doppler range. We also compared both LS and MMSE equalizers under different channel conditions. Simulations showed that MMSE equalizer is better than LS under critical channel. In this paper, we present a low-complexity multi-stage equalization algorithm for an OFDM system, utilized for higher frequency network such as but not limited to fourth generation (4G) communication system. The proposed multi-stage algorithm utilizes guard interval in time domain to eliminate inter-symbol-interference. Furthermore, the approach does not require the use of many pilot data for the equalizer improvement, therefore saving the channel bandwidth. To obtain significant performance improvement over the traditional equalizers, a multi-stage algorithm, time domain equalizer Kalman filter, has been adapted in conjunction with frequency domain equalizer, MMSE. Through simulations, the multi-stage algorithm is shown to be superior to the traditional algorithms.

Mawari, R. and Zohdy, M. (2017) Low Complexity Dynamic Channel Equalization in OFDM with High Frequency Mobile Network Technologies. Journal of Signal and Information Processing, 8, 17-41. https://doi.org/10.4236/jsip.2017.82003