633

# Block Transmission Techniques

Authored by: Paulo Montezuma , Fabio Silva , Rui Dinis

# Frequency-Domain Receiver Design for Doubly Selective Channels

Print publication date:  June  2017
Online publication date:  June  2017

Print ISBN: 9781138700925
eBook ISBN: 9781315169576

10.1201/9781315169576-3

#### Abstract

This chapter starts with a brief introduction to multi-carrier (MC) and single-carrier (SC) modulations. It includes several aspects such as the analytical characterization of each modulation type, and some relevant properties of each modulation. For both modulations special attention is given to the characterization of the transmission and receiving structures, with particular emphasis on the transmitter and receiver performances. MC modulations and their relations with SC modulations are analyzed. Section 1.4 describes the OFDM modulation. Section 1.5 characterizes the basic aspects of the SC-FDE modulation including the linear and iterative FDE receivers. Finally, in Section 1.6, the performance of OFDM and SC-FDE for severely time-dispersive channels is compared.

#### Block Transmission Techniques

This chapter starts with a brief introduction to multi-carrier (MC) and single-carrier (SC) modulations. It includes several aspects such as the analytical characterization of each modulation type, and some relevant properties of each modulation. For both modulations special attention is given to the characterization of the transmission and receiving structures, with particular emphasis on the transmitter and receiver performances. MC modulations and their relations with SC modulations are analyzed. Section 1.4 describes the OFDM modulation. Section 1.5 characterizes the basic aspects of the SC-FDE modulation including the linear and iterative FDE receivers. Finally, in Section 1.6, the performance of OFDM and SC-FDE for severely time-dispersive channels is compared.

#### 3.1  Transmission Structure of a Multicarrier Modulation

An MC system transmits a multicarrier modulated symbol (composed of N symbols on N subcarriers in time N / B). First, a serial to parallel conversion is implemented in order to demultiplex the incoming high-speed serial stream and output several serial streams but of much lower speed. Subsequently, with resort to a constellation mapper, these parallel information bits are then modulated in the specified digital modulation format (phase shift keying (PSK), quadrature amplitude modulation (QAM), etc.). Posteriorly, each of the N modulated symbols is associated onto the respective subcarrier with resort to a bank of N sinusoidal oscillators, disposed in parallel, matched in frequency and phase to the N orthogonal frequencies

. Hence, each subcarrier is centered at frequencies that are orthogonal to each other. Finally, the signals modulated onto the N subcarriers are summed forming the composite MC signal, which is then transmitted through the channel, as shown in Fig. .

Figure 3.1   Transmission structure for multicarrier modulation.

MC modulation transmits a high-speed serial stream at the input, over several streams of lower data rate. As a consequence, the symbol period is extended, resulting in a significant advantage since the transmission becomes more resilient to the multipath environment. This is especially desirable in mobility scenarios, since it allows a reliable signal reception within fast-varying channels.

#### 3.2  Receiver Structure of a Multicarrier Modulation

At the receiver, the received composite signal y(t) is correlated with the set of subcarriers in a sort of a matched filtering operation (the matched filter uses a correlation process to detect the signal). The correlation of y(t) with the

coherent subcarrier 1 is a simple operation which can be expressed as
3.1

where

is the fundamental frequency. From Fourier series definition it can be inferred that all the other frequencies are in fact multiples of the fundamental frequency. Note that when recovering the symbols, the time period of observation of the symbol (i.e., the detection window), corresponds to the time period of integration, which is mandatory to keep the orthogonality, and it consists of the fundamental period . Let us ignore the presence of noise and channel effects. Under these conditions, the received signal y(t) equals the transmitted signal s(t).
3.2

After taking the composite signal and correlating it with the corresponding

coherent subcarrier, the result is integrated from 0 to the fundamental period ,
3.3

which can simply be represented as (taking the summation out of the integral):

3.4

Let us focus on the integration term. The spacing of

between the subcarriers makes them orthogonal over each symbol period. This is a fundamental property expressed as
3.5

Coherent demodulation consists of correlating with

and integrating over the fundamental period . Hence, when we coherently demodulate the subcarrier, all subcarriers are orthogonal except the subcarrier corresponding to the symbol. In other words, if the above result is integrated over the fundamental period , all the terms are zero except when .

Equation (4) can be rewritten as

3.6

From (6), it is clear that the information symbol

, transmitted by the subcarrier, can be recovered by coherently demodulating the composite signal at the receiver. This is done by locally generating the corresponding l coherent subcarrier, equal in frequency and phase to the subcarrier, and then mix it with the received composite signal. The result is then integrated over the period , and with this process, the respective symbol is recovered. After correlating the received composite signal with each of the N different subcarriers, the N detected information symbols are finally multiplexed into a serial stream through to a parallel to serial operation, as shown in Fig. .

Figure 3.2   Receiving structure for multicarrier modulation.

#### 3.3  Multicarrier Modulations or Single-Carrier Modulations?

In a conventional single carrier modulation, the energy of each symbol is distributed over the total transmission band. The term single carrier implies a unique carrier which occupies the entire communication bandwidth B, and the transmission is performed at a high symbol rate. Considering a bandwidth B, and assuming that one symbol is transmitted every T seconds (in fact, two symbols can be transmitted on different sine and cosine carriers), then the symbol time is given by

. This leads to a symbol rate of . For instance, if a bandwidth of 100 MHz is available, we can transmit symbols at a rate of 100 Mbps, employing a symbol time of . One might think that since an MC modulation scheme transmits N symbols in parallel, it increases the throughput. However, the observation time also increases due to the fact that multicarrier modulation transmits N symbols using N subcarriers within the time period , leading to a symbol rate of . In comparison, a single-carrier scheme transmits one symbol in time period , with a rate of . Obviously, N symbols in N / B time (in the case of an MC) or 1 symbol in 1 / B time (in SC) are both the same with respect to the signal throughput, as illustrated in Fig. .

Figure 3.3   (a) Transmission of N information symbols on N subcarriers in time N / B; (b) transmission of 1 information symbol in 1 / B time.

Previously, we have stated that the overall data rate is the same in multicarrier and single-carrier modulation schemes. We may think that when compared to the SC modulation, the MC modulation is just an extremely complicated system without advantage over an SC system (since both schemes have an overall data rate of B symbols per second). So, if from the symbol rate perspective, the MC system and the SC system are equivalent, what advantages does the much more complex MC system have to offer?

In order to better understand the fundamental advantage of MC modulations, consider a scenario in which the available bandwidth for transmission is B=1024 kHz. An SC system will use the complete bandwidth of 1024 kHz, much greater than the coherence bandwidth of the channel (i.e.,

), which is assumed to be approximately 200 to 300 kHz. In these conditions, since the bandwidth is much greater than the coherence bandwidth, the channel is said to be frequency selective (different frequency components of the signal experience different fading), which implies ISI in the time domain. Therefore, a high bit rate SC digital signal experiences frequency selective fading and ISI occurs, which may result in significant distortion since the symbols interfere with each other, highly distorting the received signal and affecting the reliable detection of the symbols. Now consider an MC system with the same available bandwidth for transmission but with subcarriers. In this case, the bandwidth of each subcarrier is kHz, much less than the coherence bandwidth considered (i.e., kHz). Each subcarrier will then experience frequency flat fading in the frequency domain, and no ISI in the time domain will occur. So, what initially was a wideband radio channel, was divided into several narrowband (ISI-free) subchannels for transmission in parallel.

In comparison with the SC scheme, the overall data rate remains unchanged. However, the much more complex implementation trade-off has a significant advantage: it is possible to implement a ISI free reliable detection scheme at the receiver side. The narrowband subcarriers experience flat fading in the frequency domain, as the bandwidth is less than the coherence bandwidth. Hence, the major motivation behind MC modulations was to convert a frequency selective wideband channel into a non-frequency selective channel. Nevertheless, it is important to note that if the implementation of a coherent modulator is significantly challenging, implementing a bank of N modulators can get extremely complicated in hardware. Since the MC modulation requires a bank of N modulators, proportional to the number of subcarriers. Hence, the modulation, coherent demodulation, and synchronization requirements of the MC modulation scheme led to a very complex system, very susceptible to loss of orthogonality and ICI.

#### 3.4  OFDM Modulations

OFDM was initially proposed by R. Chang in 1966 [7]. His work presented an approach for multiple transmission of signals over a band-limited channel, free of ISI. By dividing the frequency selective channel into several frequency narrowband channels, the smaller individual channels would be subjected to flat fading. Using the Fourier transform, Chang was able to provide a method to guarantee the orthogonality among the parallel channels (or subcarriers), through the summation of sine and cosine. The orthogonality between the subcarriers within an MC modulation is crucial since, as has been seen before, it allows parallel channel data transmission rates equivalent to the bandwidth of the channel, corresponding to half the ideal Nyquist rate. However, due to its complexity, Chang’s system was still hard to implement. The Fourier transforms rely on oscillators whose phase and frequency have to be very precise.

Moreover, as was shown before, the complexity of the MC scheme requires a bank of N modulators, proportional to the number of subcarriers. If the implementation of a coherent modulator is significantly challenging, implementing thousands of parallel subcarriers in hardware is extremely difficult, even with state-of-the-art technology. Hence, the modulation, coherent demodulation, and synchronization requirements led to a very complex OFDM analog system, known to be very susceptible to loss of orthogonality and ICI.

In the early 1970s, Weinstein and Ebert [59], proposed a technique that helped to solve the complexity problem of implementing the N modulators and demodulators. With resort to the discrete Fourier transform (DFT), they proposed a method to digitally implement the baseband modulation and demodulation. This approach suppressed the bank modulators and demodulators, highly simplifying the implementation and at the same time ensuring the orthogonality between subcarriers. The DFT converts the information symbols from the time domain to the frequency domain, and the output result is a function of the sampling period

and the number of sample points N. Each of the N frequencies represented in the DFT is a multiple of the fundamental frequency , where the sampling time is given by , with the product corresponding to the total sample time. In its turn, the dual function IDFT converts a signal defined by its frequency components to the corresponding time domain signal, with the duration . According to the well-known result from sampling theorem, a bandlimited signal can be fully reconstructed from the samples at the receiver, as long it is sampled at a rate twice the maximum frequency (Nyquist rate). In order to better understand this, we will take the MC signal, or the MC composite signal y(t) defined by
3.7

and sample it at rate B. The

sample is taken at
3.8

and therefore,

3.9

where the left term of the above equation, x(u) represents the samples of the MC signal, while the right term,

represents the discrete Fourier transform (DFT) of S; this is the DFT of the information symbol. So this powerful result by Weinstein and Ebert [59] shows that there is no need to use N modulators and N demodulators. This is very effective since in order to obtain the samples of the MC transmitted symbol, it is just needed to take the N information symbols, and compute their DFT (assuming the absence of noise).

The processing time can be reduced with resort to the fast Fourier transform (FFT), and the inverse FFT (IFFT). The FFT is a key process to separate the carriers of an OFDM signal. It was developed by Cooley and Tukey [12], and it consists of a very fast algorithm for computing the DFT, capable of reducing the number of arithmetic operations by decreasing the number of complex multiplication operations from

to , for an point IDFT or DFT (with N representing the size of the FFT). This allows a much more practical Fourier analysis since it simply samples the analog composite signal with an analog-to-digital converter (ADC), submitting the resulting samples to the FFT process. The FFT operation at the receiver separates the signal components into the N individual subcarriers and sorts all the signals to recreate the original data stream. On the other hand, the individual digital modulated subcarriers are submitted to the IFFT operation, which forms the composite signal to be transmitted. The IFFT is a conversion process from frequency domain into time domain, so the IFFT can be used at the transmitter to convert frequency domain samples to time domain samples, and hence generate the OFDM symbol.

The FFT is formally described as follows:

3.10

where as its dual, IFFT is given by

3.11

The equations of the FFT and IFFT differ the coefficients they take and the minus sign. Both equations do the same operation, i.e., multiply the incoming signal with a series of sinusoids and separate them into bins. In fact, FFT and IFFT are dual and behave in a similar way. Moreover, the IFFT and FFT blocks are interchangeable.

Fig. illustrates how the use of the IFFT block in the transmitter avoids the need for separate sinusoidal converters (note that IFFT and FFT blocks in the transmitter are interchangeable as long as their duals are used in receiver).

Figure 3.4   Transmission structure for multicarrier modulation with resort to the IFFT block.

#### 3.4.1  Analytical Characterization of the OFDM Modulations

The complex envelope of an OFDM signal, given by (12), is characterized by a sum of blocks (also referred to as bursts), transmitted at a rate

. The duration of each block is , in which denotes the duration of the payload part.
3.12

where

represents the OFDM symbol transmitted on the subcarrier of a given block m, in the frequency domain. Hence, the N data symbols are sent during the block, with the group of complex sinusoids denoting the N subcarriers. Let us consider the OFDM block. During the OFDM block interval, the transmitted signal can be expressed as
3.13

with the pulse shape, r(t), defined as

where

and corresponds to the duration of the “guard interval” used to compensate time-dispersive channels. Therefore r(t) is a rectangular pulse, with a duration that should be greater than T (i.e., ), to be able to deal with the time-dispersive characteristics of the channels. The subcarrier spacing , guarantees the orthogonality between the subcarriers over the OFDM block interval. The different subcarriers are orthogonal during the interval [0, T], which coincides with the effective detection interval, since
3.14

Therefore, for each sampling instant, we may write () as

3.15

In spite of the overlap of the different subcarriers, the mutual influence among them can be avoided. Under these conditions, the bandwidth of each subcarrier becomes small when compared with the coherence bandwidth of the channel (i.e., the individual subcarriers experience flat fading, which allows simple equalization). This means that the symbol period of the subcarriers must be longer than the delay spread of the time-dispersive radio channel.

From (15), the

block should take the form
3.16

where

represents the data symbols of the burst, are the subcarriers, is the center frequency of the subcarrier. It is also assumed that in the interval .

By applying the inverse Fourier transform to both sides of (16), we obtain

3.17

where the center frequency of the

subcarrier is , with a subcarrier spacing of , that assures the orthogonality during the block interval (as stated by ()).

Fig. 5 depicts the PSD of an OFDM signal, as well as the individual subcarrier spectral shapes for

subcarriers and data symbols. As we can see from Fig. 5, when the subcarrier PSD ( ) has a maximum, the adjacent subcarriers have zero-crossings, which achieve null interference between carriers and improve the overall spectral efficiency.

Figure 3.5   The power density spectrum of the complex envelope of the OFDM signal, with the orthogonal overlapping subcarriers spectrum ( N = 16 ) $(N=16)$ .

Since the duration of each symbol is long, a guard interval is inserted between the OFDM symbols to eliminate inter-block interference (IBI). If this guard interval is a cyclic prefix instead of a zero interval, it can be shown that inter-carrier interference (ICI) can also be avoided provided that only the useful part of the block is employed for detection purposes [3]. Therefore, equation (16) is a periodic function in t, with period

, and the complex envelope associated with the guard period can be regarded as a repetition of the multicarrier blocks’s final part, as exemplified in Fig. 6. Thus, it is valid to write
3.18

Consequently, the guard interval is a copy of the final part of the OFDM symbol which is added to the beginning of the transmitted symbol, making the transmitted signal periodic. The cyclic prefix, transmitted during the guard interval, consists of the end of the OFDM symbol copied into the guard interval, and the main reason to do that is on the receiver that integrates over an integer number of sinusoid cycles each multipath when it performs OFDM demodulation with the FFT [12]. The guard interval also reduces the sensitivity to time synchronization problems.

Figure 3.6   MC burst’s final part repetition in the guard interval.

#### 3.4.2  Transmission Structure

Let us now focus on the transmission of the OFDM signal where to simplify it is assumed a noiseless transmission case. Since it is an MC scheme, the incoming high data rate is split into N streams of much lower rate by a serial/parallel converter. The parallel information bits are then modulated with a given digital modulation format, forming the symbols. The data is therefore transmitted by blocks of N complex data symbols with

being chosen from a selected constellation (for example, a PSK constellation, or a QAM). The N individual digital modulated symbols are then submitted to an IFFT operation in order to convert the frequency domain samples to time domain. The output corresponds to the OFDM symbol of (16), and if we sample the OFDM signal with an interval of we get the samples
3.19

where

. Consequently, (19) can be written as
3.20

Hence, referring to the

block, . The IDFT operation can be implemented through an IFFT which is more computationally efficient. At the output of the IFFT, a cyclic prefix of samples, is inserted at the beginning of each block of N IFFT coefficients. It consists of a time-domain cycle extension of the OFDM block, with size larger than the channel impulse response (i.e., the samples assure that the CP length is equal to or greater than the channel length). The cycle prefix is appended between each block, in order to transform the multipath linear convolution into a circular one. Thus, the transmitted block is , and the time duration of an OFDM symbol is times larger than the symbol of an SC modulation. Clearly, the CP is an overhead that costs power and bandwidth since it consists of additional redundant information data. Therefore, the resulting sampled sequence is described by
3.21

After a parallel to serial conversion, this sequence is applied to a digital-to-analog converter (DAC), whose output would be the signal s(t). The signal is upconverted and sent through the channel. The resulting IDFT samples are then submitted to a digital-to-analog conversion operation performed by a DAC. Fig. 7 illustrates a simple OFDM transmission chain block diagram.

Figure 3.7   Basic OFDM transmission chain.

#### 3.4.3  Reception Structure

At the channel output (after the RF down conversion), the received signal waveform y(t) consists of the convolution of s(t) with the channel impulse response,

, plus the noise signal n(t), i.e.,
3.22

The received signal y(t) is then submitted to an analog-to-digital converter (ADC), and sampled at a rate

. The resulting sequence consists of a set of samples, with the samples being extracted before the demodulation operation. The remaining samples are demodulated through the DFT (performed by an FFT algorithm) to convert each block back to the frequency domain, followed by the baseband demodulation. For a given block, the resulting frequency domain signal , will be
3.23

The OFDM signal detection is based on signal samples spaced by a period of duration T. Due to multipath propagation, the received data bursts overlap leading to a possible loss of orthogonality between the subcarriers, as showed in Fig. 8. However, with resort to a CP of duration

(greater than overall channel impulse response), the overlapping bursts in received samples during the useful interval are avoided, as shown in Fig. .

Figure 3.8   (a) Overlapping bursts due to multipath propagation; (b) IBI cancelation by implementing the cyclic prefix.

Since IBI can be prevented through the CP inclusion, each subcarrier can be regarded individually.

The OFDM receiver structure is implemented employing an N size DFT as shown in Fig. 9.

Figure 3.9   OFDM basic FDE structure block diagram with no space diversity.

Assuming flat fading on each subcarrier and null ISI, the received symbol is characterized in the frequency-domain by

3.24

where

denotes the overall channel frequency response for the subcarrier and represents the additive Gaussian channel noise component.

On the other hand, the frequency-selective channel’s effect, as the fading caused by multipath propagation, can be considered constant (flat) over an OFDM subcarrier if it has a narrow bandwidth (i.e., when the number of subchannels is sufficiently large). Under these conditions, the equalizer only has to multiply each detected subcarrier (each Fourier coefficient) by a constant complex number. This makes equalization far simpler at the OFDM receiver when compared to the conventional single-carrier modulation case. Additionally, from the computation’s point of view, frequency-domain equalization is simpler than the corresponding time-domain equalization, since it only requires an FFT and a simple channel inversion operation. After acquiring the

samples, the data symbols are obtained by processing each one of the N samples (in the frequency domain) with a frequency-domain equalization (FDE), followed by a decision device. Consequently, the FDE is a simple one-tap equalizer [47]. Hence, the channel distortion effects (for an uncoded OFDM transmission) can be compensated by the receiver depicted in Fig. 9, where the equalization process can be accomplished by an FDE optimized under the ZF criterion, with the equalized frequency-domain samples at the subcarrier given by
3.25

Figure 3.10   OFDM receiver structure with a N R x $N_{Rx}$ -branch space diversity.

In (25)

represents the estimated data symbols which are acquired with the set of coefficients , expressed by
3.26

Naturally, the decision on the transmitted symbol in a subcarrier k can be based on

.

Let us consider the case in which we have

-order space diversity. In Fig. 10 a maximal-ratio combining (MRC) [32] diversity scheme is implemented for each subcarrier k. Therefore, the received sample for the receive antenna and the subcarrier is denoted by
3.27

with

denoting the overall channel frequency response between the transmit antenna and the receive antenna for the frequency, denoting the frequency-domain of the transmitted blocks, and denoting the corresponding channel noise. The set of equalized samples , are
3.28

where

is the set of FDE coefficients related to the diversity branch, denoted by
3.29

Finally, by applying (27) and (29) to (28), the corresponding equalized samples can then be given by

3.30

#### 3.5  SC-FDE Modulations

One drawback of the OFDM modulation is the high envelope fluctuations of transmitted signal. Consequently, these signals are more susceptible to nonlinear distortion effects, namely those associated with a nonlinear amplification at the transmitter, resulting in a low power efficiency. This major constraint is even worse in the uplink since more expensive amplifiers and higher power back-off are required at the mobile.

Instead, when an SC modulation is employed with the same constellation symbols, the envelope fluctuations of the transmitted signal will be much lower. Thus, SC modulations are especially adequate for the uplink transmission (i.e., transmission from the mobile terminal to the base station), allowing cheaper user terminals with more efficient high-power amplifiers. Nevertheless, if conventional SC modulations are employed in digital communications systems requiring transmission bit rates of Mbits/s, over severely time-dispersive channels, high signal distortion levels can arise. Therefore, the transmission bandwidth becomes much higher than the channels’ coherence bandwidth. As a consequence, high-complexity receivers will be required to overcome this problem [47].

#### 3.5.1  Transmission Structure

In an SC-FDE modulation, data is transmitted in blocks of N useful modulation symbols

, resulting from a direct mapping of the original data into a selected signal constellation, for example QPSK. Posteriorly, a cyclic prefix with length longer than the channel impulse response is appended, resulting in the transmitted signal . The transmission structure of an SC-FDE scheme is shown in Fig. 11. As we can see the transmitter is quite simple since it does not implement a DFT/IDFT operation. The discrete versions of in-phase and quadrature components are then converted by a DAC onto continuous signals and , which are then combined to generate the transmitted signal
3.31

where r(t) is the support pulse and

denotes the symbol period.

Figure 3.11   Basic SC-FDE transmitter block diagram.

#### 3.5.2  Receiving Structure

The received signal is sampled at the receiver and the CP samples are removed, leading in the time-domain the samples

. As with OFDM modulations, after a size-N DFT results in the corresponding frequency-domain block , with given by
3.32

where

denotes the overall channel frequency response for the frequency of the block, and represents channel noise in the frequency-domain. The receiver structure is depicted in Fig. 12. After the equalizer, the frequency-domain samples referring to the subcarrier, , are given by
3.33

For a zero-forcing (ZF) equalizer the coefficients

are given by (26), i.e.,
3.34

From (34) and (32), we may write (33) as

3.35

Figure 3.12   Basic SC-FDE receiver block diagram.

This means that the channel will be completely inverted. However, in the presence of a typical frequency-selective channel, deep notches in the channel frequency response will cause noise enhancement problems, and as a consequence, there can be a reduction of the signal-to-noise ratio (SNR). This can be avoided by the optimization of the

coefficients under the MMSE criterion. Although the MMSE does not attempt to fully invert the channel effects in the presence of deep fades, the optimization of the coefficients under the MMSE criterion allows to minimize the combined effect of ISI and channel noise, allowing better performances. The mean-square error (MSE), in time-domain, can be described by
3.36

where

3.37

The minimization of

in order to , requires the MSE minimization for each k, which corresponds to impose the following condition,
3.38

which results in the set of optimized FDE coefficients

[27]
3.39

In (39)

denotes the inverse of the SNR, given by
3.40

where

stands for the variance of the real and imaginary parts of the channel noise components , and represents the variance of the real and imaginary parts of the data samples components . The term can be seen as a noise-dependent term that avoids noise enhancement effects for very low values of the channel frequency response. The equalized samples in the frequency-domain , must be converted to the time-domain through an IDFT operation, and the decisions on the transmitted symbols are made upon the resulting equalized samples .

It is possible to extend the SC-FDE receiver for space diversity scenarios. Fig. 13 shows an SC-FDE receiver structure with an

-branch space diversity, where an MRC combiner is applied to each subcarrier k.

Considering the

-order diversity receiver, the equalized samples at the FDE’s output, are given by
3.41

where

is the set of FDE coefficients related to the diversity, which are given by
3.42

with

.

Figure 3.13   Basic SC-FDE receiver block diagram with an N R x $N_{Rx}$ -order space diversity.

#### 3.6  Comparative Analysis between OFDM and SC-FDE

In order to compare OFDM and SC-FDE, refer to the transmission chains of both modulation systems, depicted in Fig. 14. Clearly, the transmission chains for OFDM and SC-FDE are essentially the same, except in the place where the IFFT operation is performed. In the OFDM, the IFFT is placed at the transmitter side to divide the data in different parallel subcarriers. For the SC-FDE, the IFFT is placed in the receiver to convert into the time-domain the symbols at the FDE output. Although there is lower complexity of the transmitter (it does not need the IDFT block), the SC-FDE requires a more complex receiver than OFDM. Consequently, from the point of view of overall processing complexity (evaluated in terms of the number of DFT/IDFT blocks), both schemes are equivalent [54].

Figure 3.14   Basic transmission chain for OFDM and SC-FDE.

Moreover, for the same equalization effort, SC-FDE schemes have better uncoded performance and lower envelope fluctuations than OFDM.

Fig. 15 presents a example of the performance results regarding uncoded OFDM modulations and uncoded SC-FDE modulations with ZF and MMSE equalization, for QPSK signals. The blocks are composed by

data symbols with a cycle prefix of 32 symbols. For simulation purposes, we consider a severely time dispersive channel with 32 equal power taps, with uncorrelated Rayleigh fading on each tap.

Without channel coding, the performance of the OFDM is very close to SC-FDE with ZF equalization. Moreover, SC-FDE has better uncoded performance under the same conditions of average power and complexity demands [26]. It should be noted that these results cannot be interpreted as if OFDM has poor performance, since the OFDM is severely affected by deep-faded subcarriers. Therefore, when combined with error correction codes, OFDM has a higher gain code when compared with SC-FDE [26].

Figure 3.15   Performance result for uncoded OFDM and SC-FDE.

Moreover, OFDM symbols are affected by strong envelope fluctuations and excessive peak-to-mean envelope power ratio (PMEPR) which causes difficulties related to power amplification and requires the use of linear amplification at the transmitter. On the other hand, the lower envelope fluctuation of SC signals allows a more efficient amplification. This is a very important aspect for the uplink transmission, where it is desirable to have low-cost and low-consumption power amplifiers. For downlink transmission, since the implementation complexity is gathered at the base stations where the costs and high power consumption are not major constraints, the OFDM schemes are a good option. Considering that both schemes are compatible, it is possible to have a dual-mode system where the user terminal employs an SC-FDE transmitter and an OFDM receiver, while the base station employs an OFDM transmitter and an SC-FDE receiver. Obviously, from Fig. 14, it becomes clear that this approach allows very low complexity mobile terminals the simpler SC transmissions and MC reception schemes.

Previously, it was shown that block transmission techniques, with appropriate cyclic prefixes and employing FDE techniques, are suitable for high data rate transmission over severely time dispersive channels. Typically, the receiver for SC-FDE schemes is a linear FDE, however, it is well known that nonlinear equalizers outperform linear ones [47] [4] [17]. Among nonlinear equalizers the decision feedback equalizer (DFE), is a popular choice since it provides a good tradeoff between complexity and performance. Clearly, the previously described SC-FDE receiver is a linear FDE. Therefore, it would be desirable to design nonlinear FDEs, namely a DFE FDE. An efficient way of doing this is by replacing the linear FDE by an IB-DFE. IB-DFE is a promising iterative FDE technique, for SC-FDE. The IB-DFE receiver can be envisaged as an iterative FDE receiver where the feedforward and the feedback operations are implemented in the frequency domain. Due to the iteration process it tends to offer higher performance than a non-iterative receiver. These receivers can be regarded as low-complexity turbo FDE schemes [56,57], where the channel decoder is not involved in the feedback. True turbo FDE schemes can also be designed based on the IB-DFE concept [5,28]. In this section, we present a detailed study on schemes employing iterative frequency domain equalization.

Although a linear FDE leads to good performance for OFDM schemes, the performance of SC-FDE can be improved if the linear FDE is replaced by an IB-DFE [4]. The receiver structure is depicted in Fig. [2,17].

Figure 3.16   IB-DFE receiver structure (a) without diversity (b) with a N R x $N_{Rx}$ -branch space diversity.

In the case where an

-order space diversity IB-DFE receiver is considered, for the iteration, the frequency-domain block at the output of the equalizer is , with
3.43

where

are the feedforward coefficients associated with the diversity antenna and are the feedback coefficients. is the DFT of the block , with denoting the “hard decision” of from the previous FDE iteration. Considering an IB-DFE with “hard decisions,” it can be shown that the optimum coefficients and that maximize the overall SNR, associated with the samples , are [17]
3.44

and

3.45

respectively, where

denotes the so-called correlation factor, (which is common to all data blocks and diversity branches), and selected to guarantee that
3.46

Although the term “IB-DFE with hard decisions” is often referenced, the term “IB-DFE with blockwise soft decisions” would probably be more adequate, as we will see in the following. It can be seen from (44) and (45), that the correlation factor

is a key parameter for the good performance of IB-DFE receivers, since it gives a blockwise reliability measure of the estimates employed in the feedback loop (associated with the previous iteration). This is done in the feedback loop by taking into account the hard decisions for each block plus the overall block reliability, which reduces error propagation problems. The correlation factor is defined as
3.47

where the block

denotes the data estimates associated with the previous iteration, i.e., the hard decisions associated with the time-domain block at the output of the FDE, = IDFT .

For the first iteration, no information exists about

, which means that , , and coefficients are given by (39) (in this situation the IB-DFE receiver is reduced to a linear FDE). After the first iteration, the feedback coefficients can be applied to reduce a major part of the residual interference (considering that the residual bit error rate (BER) does not assume a high value). After several iterations and for a moderate-to-high SNR, the correlation coefficient will be and the residual ISI will be almost totally canceled. In Fig. 17 is shown the average BER performance evolution for a fading channel. It refers to a transmission system with SC uncoded modulation, employing an IB-DFE receiver with 1, 2, 3, and 4 iterations. For comparative purposes, the corresponding performances of the MFB and additive white Gaussian noise (AWGN) channel are also included.

Figure 3.17   Uncoded BER perfomance for an IB-DFE receiver with four iterations.

From the results, we can see that the

required for BER= is around 15.5 dB for the iteration (that corresponds to the linear SC-FDE), decreasing to 11 dB after only three iterations, being clear that the use of the iterative receiver allows a significant performance improvement. Also, the asymptotic BER performance becomes close to the MFB after a few iterations.

It should be noted that (43) can be written as

3.48

where

and (as stated before, can be considered as the blockwise reliability of the estimates ).

*****

#### 3.7.2  IB-DFE with Soft Decisions

To improve the IB-DFE performance it is possible to use “soft decisions,”

, instead of “hard decisions,” . Under these conditions, the “blockwise average” is replaced by “symbol averages” [28]. This can be done by using = DFT instead of = DFT , where denotes the average symbol values conditioned to the FDE output from the previous iteration, . To simplify the notation, is replaced by in the following equations.

For QPSK constellations, the conditional expectations associated with the data symbols for the

iteration are given by
3.49

with the log-likelihood ratio (LLR) of the “in-phase bit" and the “quadrature bit," associated with

and , respectively, given by and , respectively, with
3.50

where the signs of

and define the hard decisions and , respectively. In (49), and denote the reliabilities related to the “in-phase bit" and the “quadrature bit" of the symbol, and are given by
3.51

and

3.52

respectively. Therefore, the correlation coefficient employed in the feedforward coefficients will be given by

3.53

Obviously, for the first iteration

and, consequently, . Therefore, the receiver with “blockwise reliabilities" (hard decisions) and the receiver with “symbol reliabilities" (soft decisions) employ the same feedforward coefficients; however, in the first the feedback loop uses the “hard-decisions" on each data block, weighted by a common reliability factor, whereas in the second the reliability factor changes from bit to bit. From the performance results shown in Fig. 18, we observe clear BER improvements when we employ “soft decisions” instead of “hard decisions” in IB-DFE receivers.

Figure 3.18   Improvements in uncoded BER performance accomplished by employing “soft decisions" in an IB-DFE receiver with four iterations.

The IB-DFE receiver can be implemented in two different ways, depending on whether the channel decoding output is outside or inside the feedback loop. In the first case the channel decoding is not performed in the feedback loop, and this receiver can be regarded as a low complexity turbo equalizer implemented in the frequency domain. Since this is not a true “turbo" scheme, we will call it “conventional IB-DFE." In the second case the IB-DFE can be regarded as a turbo equalizer implemented in the frequency domain and therefore we will denote it as “turbo IB-DFE." For uncoded scenarios it only makes sense to employ conventional IB-DFE schemes. However, it is important to point out that in coded scenarios we could still employ a “conventional IB-DFE" and perform the channel decoding procedure after all the iterations of the IB-DFE. However, since the gains associated with the iterations are very low at low-to-moderate SNR values, it is preferable to involve the channel decoder in the feedback loop, i.e., to use the “turbo IB-DFE."

Figure 3.19   SISO channel decoder for soft decisions.

The most common way to perform detection in digital transmission systems with channel coding is to consider separately the channel equalization and channel decoding operations. However, using a different approach in which both operations are executed in conjunction, it is possible to achieve better performance results. This can be done employing turbo-equalization systems where channel equalization and channel decoding processes are repeated in an iterative way, with “soft decisions” being traversed through them. Turbo equalizers were firstly proposed for time-domain receivers. However, turbo equalizers can be implemented in the frequency-domain that, as conventional turbo equalizers, use “soft decisions” from the channel decoder output in the feedback loop.

The main difference between “conventional IB-DFE" and “turbo IB-DFE" is in the decision device: in the first case the decision device is a symbol-by-symbol soft-decision (for the QPSK constellation this corresponds to the hyperbolic tangent, as in (49)); for the turbo IB-DFE a SISO channel decoder (soft-in, soft-out) is employed in the feedback loop. The SISO block can be implemented as defined in [58], and provides the LLRs of both the “information bits" and the “coded bits." The input of the SISO block are the LLRs of the “coded bits" at the FDE output, given by

and . It should be noted that the data bits must be encoded, interleaved, and mapped into symbols before transmission. The receiver scheme is illustrated in Fig. 19. At the receiver side the equalized samples are demapped by a soft demapper followed by a deinterleaver providing the LLRs of the “coded bits” to the SISO channel decoder. The SISO operation is proceeded by a interleaver and after that a soft mapper provides the desired “soft decisions.”

Coherent refers to equal in frequency and phase to the

carrier.