In wireless communications, a transmitter put data in radio resources such as time and frequency and a receiver should be able to pull data out. The synchronization performs this role in the receiver of wireless communication systems. Synchronization techniques are an indispensable part of the wireless communication receiver like a conductor of orchestras unifies performers and forms an ensemble. The receiver needs to detect a packet and find an accurate starting position. In addition, it should correct several synchronization errors such as timing offsets (caused by channel delay), carrier frequency offsets (caused by Doppler effect and local oscillator mismatch between transmitter and receiver), phase noises (caused by phase rotation of the channel), and sampling frequency offsets (caused by mismatch between DAC and ADC). In particular, OFDM systems are very sensitive and vulnerable to these synchronization errors. It is a tricky part of the wireless communication system design. In this chapter, we will look into synchronization errors and their effects, review synchronization techniques, and design ML synchronization block in the OFDM system.

10.1 Fundamental Synchronization Techniques for OFDM System

Synchronization techniques can be classified into data-aided methods, decision-directed methods, and non-data-aided methods. The data-aided methods use a reference signal (training symbols, preambles, pilots, etc.) which is known in the receiver side. It provides us with the best synchronization performance among them even if it suffers from the leakage of a bandwidth or data transmission capacity due to overhead signals. Thus, the reference signals for synchronization should be designed to provide high reliability and spend a few radio resources. The decision-directed methods use the detected symbols as the reference signals. They are sensitive to detection errors. The non-data-aided methods (or blind methods) do not depend on a known signal and their complexity is high. It can be used for symbol synchronization of the OFDM system. For example, they make use of autocorrelation between the Cyclic Prefix (CP) and the end of the OFDM symbol. Among the three methods, the data aided methods are widely used in the OFDM systems because of their accuracy.

Many synchronization techniques are based on two important algorithms: autocorrelation and crosscorrelation. As we discussed in Chapters 2 and 4, autocorrelation is based on the similarity between one signal and its time lag version, whereas crosscorrelation is based on the similarity between a transmitted signal and a stored signal in the receiver. In Ref. [1], the non-data-aided method using autocorrelation is introduced. This method exploits the cyclic prefix of the OFDM system. Basically, the cyclic prefix is designed to avoid an inter-symbol interference (ISI). Thus, it is difficult to obtain the ISI free cyclic prefix. In order to use it as a reference signal, more number of OFDM symbols are required and they should be averaged. The autocorrelation, R_r(m), using the CP is defined as follows:

(10.1)

where N, ν, and E are the DFT/IDFT size, the CP length, and the normalized energy, respectively. r(k) is the received OFDM symbol. This method estimates the timing offset using the peak of the autocorrelation function and then calculates the frequency offset using the phase shift between the cyclic prefix and the subcarriers in the end of the OFDM symbol. In the wireless LAN (IEEE802.11a), short and long training symbols are included in the frame. Thus, the data-aided method is possible. For the purpose of coarse time synchronization, the autocorrelation of the periodic short training symbols is performed as follows:

(10.2)

where ν_p is the periodicity factor. In IEEE802.11a, the value of ν_p is 16 (the short training symbol) or 64 (the long training symbol). The data-aided method provides us with more accurate and faster synchronization than the non-data-aided method. The most popular OFDM synchronization is the crosscorrelation using a preamble [2]. It correlates a stored preamble in the receiver with a received symbol. The crosscorrelation R_sr(ν_p) is defined as follows:

(10.3)

where s(k) is the stored preamble. The stored preamble is not affected by noise, fading, and nonlinearity but the received symbol includes them. Thus, it fits well into a low SNR situation. In OFDM systems, coarse timing synchronization and fine timing synchronization are usually based on autocorrelation and crosscorrelation, respectively.

10.2 Synchronization Errors

The OFDM technique is suitable for a high-speed wireless communication system. However, one important condition is that the orthogonality among subcarriers should be maintained. In a practical wireless communication system, it suffers from some imperfections such as nonlinear distortions, thermal noises, and synchronization errors (time and frequency offsets, sampling frequency offsets, and phase offsets). The orthogonality may not be preserved and the system performance may be significantly degraded. Thus, it is very important to compensate for those imperfections. There are many literatures dealing with the effect of time and frequency offsets [3, 4]. In this section, synchronization errors are discussed and the effect of these offsets is investigated.

An OFDMA packet is composed of multiple OFDM symbols. The receiver should find the start position of the OFDM symbols. The transmitted symbols may reach the receiver with different delays. When the channel maximum excess delay is shorter than the guard interval, we can consider several scenarios as shown in Figure 10.7.

The DFT window in scenario 2 is located in safe position so that the signal is not contaminated by the previous signal and there is no ISI. The only phase shift occurs. The received signal in frequency domain is expressed as follows:

(10.4)

where Y_k,l, X_k,l, H_k, and W_k,l are the frequency domain received signal, the frequency domain transmitted signal, the channel frequency response, and the Gaussian noise at the kth subcarrier and the lth symbol, respectively. T_d and T_s are the symbol timing offset (the delay to the correct DFT window starting position) and the subcarrier signal interval, respectively. In this case, the subcarrier spacing f_s is 1/NT_s and the signal bandwidth is approximately 1/T_s. On the other hand, the DFT windows in scenario 1 and 3 are located in unsafe position so that ISI occurs and both magnitude and phase are distorted. The received signal in frequency domain is expressed as follows:

(10.5)

where the term means a slight magnitude attenuation because the DFT windows collect smaller samples of the original data. The other sample collected by the DFT window is expressed as the ISI term.

The OFDM system is vulnerable to Carrier Frequency Offset (CFO). The CFO induces the loss of orthogonality among the subcarriers in the OFDM system. The CFO is caused by the Doppler effect and the local oscillator mismatch between a transmitter and a receiver. The wireless communication system designer should consider the CFO and define the OFDM system parameters. The CFO is one of the most common channel impairments in the OFDM system. An accurate estimation of the CFO is a critical part of the synchronization block design in the OFDM system. For example, if we consider that the maximum carrier frequency offset is ±40 ppm, the carrier frequency is 2 GHz, and the subcarrier spacing is 100 kHz, the maximum allowed frequency offset is ±80 kHz. Figure 10.8 illustrates the maximum frequency offset and subcarrier spacing.

The normalized carrier frequency offset ε_CFO is defined as the ratio of CFO Δf to subcarrier spacing f_s. It is composed of three parts as follows:

(10.6)

where ε_I, ε_f, and ε_r are the integer frequency offset, the fractional frequency offset, and the residual frequency offset, respectively. The integer frequency offset literally shifts an index and produces a phase shift. However, it does not affect the orthogonality. The fractional frequency offset produces magnitude attenuation, phase rotation, and inter-carrier interference (ICI). It destroys the orthogonality of the OFDM symbol. The residual frequency offset is less than 5% of the subcarrier spacing [5]. Thus, it does not significantly affect ICI. However, the phase rotation is affected and frequency offset estimation is not perfect. There is always some residual frequency offset. The compensation of the residual frequency offset is carried out separately. The received signal with CFO in frequency domain is expressed as follows:

(10.7)

where N_g is the guard interval. The first term represents the received signal including magnitude attenuation and phase rotation, the second term represents the ICI, and the third term is the Gaussian noise. The ICI affects the orthogonality as well as degrades SNR. The SNR degradation [6] is expressed as follows:

(10.8)

where E_s/N₀ is the symbol to noise ratio. Figure 10.9 illustrates the inter-carrier interference by frequency offset. We can observe the frequency offset Δf at the center subcarrier in the figure. The ICI occurs due to the loss of orthogonality.

The phase noise is caused by many reasons such as carrier phase offset, channel impulse response, residual time, and frequency offset. Although time and frequency synchronizations are finished, the residual time and frequency offsets still exist and they produce the phase noise. The phase noise caused by the frequency offset is constant and time-variant while the phase noise caused by the time offset is not constant and time-invariant. The local oscillator difference between a transmitter and a receiver produces the phase noise as well as the frequency offset. The received signal with the carrier phase offset in frequency domain is expressed as follows:

(10.9)

where φ₀ is the constant phase offset and φ_n,l is the phase offset at the nth sample in the lth symbol. The SNR degradation caused by the phase noise [7] is expressed as follows:

(10.10)

where the phase noise variance is small ( ).

The system clock should be derived from same oscillators but a mismatch between a transmitter oscillator and a receiver oscillator exists. The received signal is sampled at an interval of where δ is Sample Frequency Offset (SFO). The received signal with the SFO in frequency domain is expressed as follows:

(10.11)

As we can observe from (10.11), the effect of the SFO is similar to the effect of the CFO. The SFO affects the symbol timing shift and the SNR loss. The symbol timing shift produces the phase rotation. The SNR degradation [8] caused by the SFO is expressed as follows:

(10.12)

As we can observe from (10.12), the SNR degradation increases in accordance with the subcarrier index k and the SFO δ. It can cause the loss of the orthogonality among the subcarriers.

10.3 Synchronization Techniques for OFDM System

The purpose of the OFDM synchronization block is to detect an OFDM symbol, find the time and frequency offset, and maintain the orthogonality of each subcarrier. There are many algorithms to estimate the time and frequency offset. In order to build the synchronization block, the OFDM system designer should consider transmission type, system performance, latency, and complexity and then find the most suitable algorithm. For example, when comparing wireless LAN (IEEE802.11a) with Digital Video Broadcasting-Terrestrial (DVB-T), they have different transmission type. The IEEE802.11a is based on the burst transmission. Its receiver should detect the signal as quickly as it receives the preamble. Thus, the preamble is periodic and suitable for fast synchronization. On the other hand, the DVB-T is based on the continuous transmission and synchronization time is not critical. Averaging some OFDM symbols can be used while adaptive compensation techniques are needed because of the time-varying channel effect.

In the synchronization process, the first step is to detect OFDM symbols in the received signal and find the beginning of the OFDM symbol. As we discussed in Section 10.1, coarse timing synchronization based on autocorrelation acquires rough timing, and then fine timing synchronization based on crosscorrelation improves the timing accuracy. One of the simplest coarse time synchronizations is autocorrelation using the repeated symbols as shown in (10.1). The generalized form with the periodicity factor ν and the separation between two intervals L is expressed as follows:

(10.13)

The time index with the maximum autocorrelation value is as follows:

(10.14)

The correlator output in different symbols has a wide range of values and fluctuates because it is not normalized. Thus, it is difficult to set a threshold level and detect the maximum value. In addition, the correlator output does not drop enough when the correlator window moves away from the preamble. The correlation performance decreases as SNR decreases.

In Ref. [2], T. M. Schmidl and D. C. Cox proposed the power normalized metric which takes the input signal power into account. Thus, the range of the output is decreased so that it is easy to set the threshold level and detect the maximum value. This is the most popular method for the OFDM system synchronization, which is close to the Cramer-Rao bound. They constructed the preamble with two identical segments which has N/2 length and contains a Pseudo-Noise (PN) sequence on the odd frequencies and a null symbol on the even frequencies. Due to the DFT property, we have the following autocorrelation:

(10.15)

and the received energy of the second half symbol is defined as follows:

(10.16)

Thus, the symbol detection metric is defined as follows:

(10.17)

The time index with the maximum autocorrelation value is as follows:

(10.18)

where the time index means the end of the preamble. This method has a low complexity and the autocorrelation can be implemented with iterative formula. However, it is difficult to find the beginning of the OFDM symbol. A plateau effect happens in the autocorrelation output. The cyclic prefix is the copy of the last several subcarriers. In Figure 10.11, the DFT windows in scenarios 1 and 2 produce same correlation result.

Thus, many algorithms tried to overcome this obstacle. One of them is Minn’s algorithm [9]. He has modified the preamble structure consisting of several identical segments. A typical preamble has [A A −A −A] structure where A is the preamble segment with N/4 length. Thus, the autocorrelation and the received energy are defined as follows:

(10.19)

and

(10.20)

This method provides a sharper roll-off than Schmidl and Cox’s power normalized metric. However, it shows us a lower performance under the multipath channel because it uses fewer samples.

In Ref. [10], Maximum Likelihood (ML) signal detection for OFDM synchronization is proposed. The timing offset is estimated as follows:

(10.21)

where ρ is the weighting factor depending on the SNR and is defined as follows:

(10.22)

Likewise, the time index with the maximum value is found as follows:

(10.23)

As we can observe from (10.21) and (10.22), the ML signal detection needs to estimate the SNR value. The SNR can be easily estimated in the continuous transmission such as DVB-T while it is difficult to estimate an accurate SNR in the first preamble. This SNR estimation error increases a false alarm probability. In addition, the ML signal detection has a high complexity due to this SNR calculation.

Although coarse time synchronization is finished, a remaining symbol timing error may still exist. In order to achieve further accuracy, fine time synchronization is necessary. As we briefly discussed in Section 10.1, the crosscorrelation is used for fine time synchronization. It is defined in (10.3). The generalized form with the preamble length K is rewritten as follows:

(10.24)

The received signal r(k) is correlated with the stored preamble s(k) which is not affected by channel impairments and nonlinearity. Likewise, the time index with the maximum value is found as follows:

(10.25)

In a practical wireless communication system, frequency offset estimation is performed before fine time synchronization starts because it cannot manage a signal including frequency offset.

Fractional CFO estimation can be accomplished simultaneously when the symbol boundary is detected. The maximum likelihood CFO estimator [10] is expressed as follows:

(10.26)

where the periodicity factor ν is the block size and the separation L means the distance between two identical blocks. When we consider synchronization using the CP, ν and L are the CP length and the DFT size, respectively. When we consider the preamble [A A] structure where A is the preamble segment with N/2 length, both ν and L are N/2. The acquisition range of the above formula is .

Fractional CFO estimation is performed in time domain while an integer CFO is typically estimated in frequency domain. In Ref. [2], two pilot blocks with differential encoding at identical subcarrier positions are used. In Ref. [11], one pilot block with differential encoding among adjacent subcarriers is exploited. The first scheme has a better performance than the second scheme. However, the system overhead of the first scheme is higher. One simple integer CFO estimator uses frequency domain autocorrelation. It brings good estimation because the integer part of CFO causes a frequency shift of the received signal in frequency domain. Let Y[l, k] be the frequency domain received symbol at the kth subcarrier of the lth OFDM symbol. Two consecutive OFDM symbols are correlated as follows:

(10.27)

where g = 0, ±1, ±2, … means the possible integer subcarrier index and α_k is the kth pilot subcarrier. The integer CFO can be estimated by finding the integer subcarrier index g with the largest value as follows:

(10.28)

This method works well when the pilot assignment is comb-type and the channel is slow fading. Although the fractional CFO and the integer CFO are estimated and compensated, there still is a residual CFO. The residual CFO affects the phase shift but causes a minor SNR degradation. In a time-varying channel, it should be continuously tracked and compensated. Figure 10.15 illustrates synchronization block diagram in the OFDM system.

When we have a preamble and use the data-aided synchronization algorithm, a preamble structure should be designed in consideration of frame synchronization, Automatic Gain Control (AGC), carrier frequency offset estimation, symbol timing synchronization, channel estimation, and fine frequency offset estimation. For example, the preamble of IEEE802.11a contains 10 identical short training symbols, 2 identical long training symbols, and guard interval. Each symbol is used for each synchronization process as shown in Figure 10.16.

The overall synchronization procedure of IEEE802.11a is as follows: Firstly, the receiver observes a wireless channel and detects a frame using a correlation technique within 5.6 µs (=7 × 0.8 µs). In this stage, the frame detection algorithm should be used to avoid false alarms. Secondly, once the frame is detected, the receiver estimates a symbol timing offset and coarse frequency offset within 2.4 µs (=3 × 0.8 µs). In this stage, the receiver compensates the offsets and differentiates preamble symbols from data symbols. Thirdly, channel estimation/equalization and fine frequency offset estimation/compensation are carried out within 8 µs.

10.4 Hardware Implementation of OFDM Synchronization

We briefly investigate the hardware design issues of the OFDM synchronization block in this section. There are many issues to consider when implementing the OFDM synchronization block and choosing its architecture. An OFDM synchronization block requires a real time implementation and a large size memory. Thus, the DSP or FPGA implementation may not be suitable. The ASIC design would be the best solution. Since many wireless communication systems support a high data rate service and they are mobile devices, acquisition speed and power consumption as well as the precision of the estimation are very important parameters. Basically, the synchronization block takes responsibility for estimating and compensating the symbol timing offset and carrier frequency offset. If a burst transmission mode is considered, a frame detection block is included in the symbol timing synchronization.

From (10.21), (10.23), and (10.26), the architecture of the simple ML timing/frequency offset estimator [1] can be designed as shown in Figure 10.17.

In this architecture, the weighting factor ρ depends on the SNR estimation. The synchronization block has a big computational burden in a burst transmission mode. The received signal is fed into autocorrelation and then the argmax block looks for a timing index from the maximum value. Simultaneously, the frequency offset is found at the timing index.

10.5 Problems

1. 10.1. Plot the autocorrelation function of the rectangular signal.
2. 10.2. Describe the properties of autrocorrelation function and crosscorrelation function.
3. 10.3. Plot the crosscorrelation function of the following two signals:
   
   

   where n(t) is the Gaussian noise.

4. 10.4. Plot the crosscorrelation function between two triangular signals of different widths.
5. 10.5. Consider the OFDM system with the following parameters:

   Maximum carrier frequency offset	±30 ppm
   Carrier frequency	5 GHz
   Subcarrier spacing	200 kHz

6. 10.6. Find the maximum allowed frequency offset.
7. 10.7. Describe the SNR degradation caused by CFO, phase noise, and SFO in both a high mobility environment and a low mobility environment.
8. 10.8. There are two types of CFO compensations: The first approach is to use a frequency domain interpolator and the other approach is to use a phase locked loop in time domain. Compare two techniques.
9. 10.9. The preamble structure of wireless LAN (IEEE802.11.a) is different from the preamble structure of UWB (MB-OFDM). Compare their detection probability.
10. 10.10. In LTE standard, there are two types of downlink synchronization signals: Primary Synchronization Signal (PSS) and Secondary Synchronization Signal (SSS). Describe the downlink synchronization process in LTE standard.

References

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