What Term Refers to the Extreme Case of Selectivity?

Channel Frequency Selectivity

Robustness to channel frequency selectivity Typically, wideband wireless channels are strongly frequency selective and robustness to frequency selectivity is fundamental to support high throughput advice over wideband channels.

From: 5G Physical Layer , 2018

MIMO-OFDMA Arrangement Level Evaluation

Bruno Clerckx , Claude Oestges , in Mimo Wireless Networks (Second Edition), 2013

Frequency Granularity

Figure 15.2 (a) illustrates the impact of subband size on the performance of SU-MIMO when subband CQI(s) and PMI are reported for every subband (commonly denoted every bit PUSCH 3-two in 3GPP community). It can exist observed that, as the subband size increases, the performance decreases significantly owing to the channel frequency selectivity (of both the channel management and amplitude) within the subband. However, this touch is slightly less pronounced in spatially correlated environment, eastward.g., (0.5,viii), due to the reduced frequency selectivity of the channel direction within the subband. Indeed, as the spatial correlation increases, the spatial directions of the aqueduct become more and more aligned with the eigenvectors of the spatial correlation matrix of the aqueduct, therefore reducing the frequency selectivity of the aqueduct spatial direction. In the extreme case of loftier correlation, the PMI on all subbands is the aforementioned and a unmarried wideband PMI becomes sufficient. Finally, let us note that the frequency selectivity of the channel aamplitude has a higher touch on the operation than that of the channel direction. This tin can exist concluded from the fact that in (0.5,8) scenarios, the throughput performance decreases significantly as the subband size increases despite the fact that the channel direction frequency selectivity is very much reduced due to spatial correlation. A spatially correlated channel nonetheless exhibits significant frequency selectivity of the magnitude of the aqueduct inside a subband and a single CQI per codeword and subband cannot capture accurately such selectivity as the subband size increases.

Effigy 15.2. SU-MIMO performance with realistic feedback mechanisms.

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Wider-Ring "Single-Carrier" Transmission

Erik Dahlman , ... Johan Sköld , in 4G: LTE/LTE-Advanced for Mobile Broadband (Second Edition), 2014

4.1 Equalization against radio-aqueduct frequency selectivity

Historically, the main method to handle signal abuse due to radio-channel frequency selectivity has been to utilise dissimilar forms of equalization [12] at the receiver side. The aim of equalization is to, by different means, recoup for the channel frequency selectivity and thus, at least to some extent, restore the original bespeak shape.

4.1.1 Time-domain linear equalization

The most basic approach to equalization is the time-domain linear equalizer, consisting of a linear filter with an impulse response westward(τ) applied to the received signal (see Figure 4.i).

FIGURE four.1. Full general time-domain linear equalization

Past selecting unlike filter impulse responses, unlike receiver/equalizer strategies tin can be implemented. As an case, in DS-CDMA-based systems a so-called RAKE receiver structure has historically often been used. The RAKE receiver is simply the receiver structure of Figure 4.1 where the filter impulse response has been selected to provide channel-matched filtering:

(4.1) $w\left(\tau \right)=h^{\ast }(-\tau )$

That is, the filter response has been selected as the complex cohabit of the time-reversed channel impulse response. This is also often referred to as a Maximum-Ratio Combining (MRC) filter setting [12].

Selecting the receiver filter co-ordinate to the MRC benchmark—that is, as a channel-matched filter—maximizes the post-filter signal-to-noise ratio (thus the term maximum-ratio combining). However, MRC-based filtering does not provide whatsoever compensation for whatsoever radio-channel frequency selectivity—that is, no equalization. Thus, MRC-based receiver filtering is appropriate when the received signal is mainly impaired past noise or interference from other transmissions but non when a principal part of the overall signal corruption is due to the radio-channel frequency selectivity.

Another alternative is to select the receiver filter to fully compensate for the radio-channel frequency selectivity. This can be achieved by selecting the receiver-filter impulse response to fulfill the relation:

(four.ii) $h\left(\tau \right)\otimes w\left(\tau \right)=\mathrm{one}$

where "⊗" denotes linear convolution. This selecting of the filter setting, also known equally Nothing-Forcing (ZF) equalization [12], provides full compensation for whatsoever radio-channel frequency selectivity (complete equalization) and thus full suppression of any related signal abuse. Yet, zero-forcing equalization may lead to a big, potentially very large, increase in the noise level after equalization and to an overall degradation in the link performance. This will be the case peculiarly when the channel has big variations in its frequency response.

A third and, in most cases, better alternative is to select a filter setting that provides a trade-off betwixt bespeak corruption due to radio-channel frequency selectivity and dissonance/interference. This can, for case, be done by selecting the filter to minimize the mean-foursquare error between the equalizer output and the transmitted betoken—that is, to minimize:

(4.3) $\epsilon =E\left\{\right|\hat{s}(t)-s(t){|}^2\}.$

This is also referred to as a Minimum Mean-Foursquare Error (MMSE) equalizer setting [12].

In practice, the linear equalizer has nearly oftentimes been implemented as a time-detached FIR filter [22] with L filter taps applied to the sampled received signal, every bit illustrated in Figure 4.2. In full general, the complexity of such a fourth dimension-discrete equalizer grows relatively rapidly with the bandwidth of the signal to be equalized:

Effigy 4.2. Linear equalization implemented every bit a fourth dimension-discrete FIR filter

•

A more wideband indicate is subject to relatively more than radio-channel frequency selectivity or, equivalently, relatively more time dispersion. This implies that the equalizer needs to have a larger bridge (larger length L—that is, more filter taps) to be able to properly compensate for the channel frequency selectivity.

•

A more wideband signal leads to a correspondingly higher sampling rate for the received signal. Thus, the receiver-filter processing needs to be carried out with a correspondingly higher rate.

It can be shown [23] that the time-discrete MMSE equalizer setting $\overline{\mathrm{due\; west}}={[w_0,w_{\mathrm{ane}},\dots ,{\mathrm{west}}_{\mathrm{Fifty}-1}]}^H$ is given past the expression:

(four.4) $\overline{w}=R^{-\mathrm{one}}\overline{p}.$

In this expression, R is the channel-output machine-correlation matrix of size L × L, which depends on the aqueduct impulse response likewise as on the noise level, and $\overline{p}$ is the channel-output/channel-input cross-correlation vector of size L × 1 that depends on the channel impulse response.

In particular, in the case of a large blaster span (big Fifty), the time-domain MMSE equalizer may be of relatively high complexity:

•

The equalization itself (the actual filtering) may be of relatively loftier complexity according to the to a higher place.

•

Calculation of the MMSE equalizer setting, especially the calculation of the inverse of the size L × Fifty correlation matrix R, may be of relatively high complexity.

4.1.2 Frequency-domain equalization

A possible fashion to reduce the complexity of linear equalization is to carry out the equalization in the frequency domain [24], as illustrated in Figure four.3. In such frequency-domain linear equalization, the equalization is carried out block-wise with cake size-Due north. The sampled received bespeak is start transformed into the frequency domain by ways of a size-N DFT. The equalization is then carried out equally frequency-domain filtering, with the frequency-domain filter taps Westward ₀, …, Westward _N−i, for example, being the DFT of the corresponding time-domain filter taps due west ₀, …, w _L−1 of Figure four.2. Finally, the equalized frequency-domain point is transformed back to the time domain by means of a size-N inverse DFT. The block size-N should preferably be selected as Due north = 2 ⁿ for some integer n to allow for computational-efficient radix-2 FFT/IFFT implementation of the DFT/IDFT processing.

Effigy 4.3. Frequency-domain linear equalization

For each processing block of size-N, the frequency-domain equalization basically consists of:

•

A size-N DFT/FFT.

•

Northward complex multiplications (the frequency-domain filter).

•

A size-North inverse DFT/FFT.

Especially in the case of channels with extensive frequency selectivity, implying the need for a large bridge of a time-domain equalizer (large blaster length L), equalization in the frequency domain according to Figure iv.3 can be of significantly less complexity, compared to the time-domain equalization illustrated in Figure 4.2.

However, in that location are ii bug with frequency-domain equalization:

•

The time-domain filtering of Figure 4.2 implements a time-discrete linear convolution. In dissimilarity, frequency-domain filtering according to Figure four.3 corresponds to circular convolution in the time domain. Assuming a time-domain equalizer of length Fifty, this implies that the start L − 1 samples at the output of the frequency-domain equalizer will not be identical to the corresponding output of the time-domain blaster.

•

The frequency-domain filter taps W ₀, …, W _N−ane can be determined past offset determining the pulse response of the corresponding time-domain filter and then transforming this filter into the frequency domain by means of a DFT. Nonetheless, as mentioned in a higher place, determining the MMSE time-domain filter may exist relatively circuitous in the case of a large equalizer length L.

One mode to accost the first issue is to use an overlap in the block-wise processing of the frequency-domain equalizer every bit outlined in Effigy 4.4, where the overlap should be at to the lowest degree Fifty − 1 samples. With such an overlap, the first Fifty − 1 "incorrect" samples at the output of the frequency-domain blaster can exist discarded every bit the corresponding samples are too (correctly) provided equally the last role of the previously received/equalized block. The drawback with this kind of "overlap-and-discard" processing is a computational overhead—that is, somewhat higher receiver complexity.

Effigy four.4. Overlap-and-discard processing

An alternative approach that addresses both of the above bug is to apply cyclic-prefix insertion at the transmitter side (see Figure 4.v). Similar to OFDM, cyclic-prefix insertion in the example of single-carrier transmission implies that a circadian prefix of length N _CP samples is inserted cake-wise at the transmitter side. The transmitter-side block size should be the same as the block size-N used for the receiver-side frequency-domain equalization.

FIGURE 4.v. Circadian-prefix insertion in the instance of unmarried-carrier manual

With the introduction of a circadian prefix, the channel will, from a receiver bespeak of view, appear as a circular convolution over a receiver processing cake of size-N. There is no need for any receiver overlap-and-discard processing. Furthermore, the frequency-domain filter taps can now be calculated straight from an approximate of the sampled channel frequency response without first determining the time-domain equalizer setting. As an instance, in the case of an MMSE equalizer the frequency-domain filter taps can exist calculated according to:

(four.5) $W_m=\frac{H_k^{\ast }}{|H_k{|}^2+{\mathrm{North}}_0}$

where N ₀ is the noise ability and H _k is the sampled channel frequency response. For large equalizer lengths, this calculation is of much lower complexity compared to the time-domain calculation discussed in the previous department.

The drawback of cyclic-prefix insertion in the case of single-carrier manual is the aforementioned equally for OFDM—that is, information technology implies an overhead in terms of both power and bandwidth. 1 method to reduce the relative cyclic-prefix overhead is to increase the cake size-N of the frequency-domain blaster. However, for the block-wise equalization to be authentic, the channel needs to be approximately constant over a time bridge corresponding to the size of the processing block. This constraint provides an upper limit on the block size-N that depends on the charge per unit of the aqueduct variations. Note that this is similar to the constraint on the OFDM subcarrier spacing Δf = 1/T _u depending on the rate of the channel variations, every bit discussed in Affiliate iii.

four.1.three Other equalizer strategies

The previous sections discussed different approaches to linear equalization equally a ways to counteract signal corruption of a wideband signal due to radio-channel frequency selectivity. However, there are also other approaches to equalization:

•

Decision-Feedback Equalization (DFE) [12] implies that previously detected symbols are fed back and used to cancel the contribution of the corresponding transmitted symbols to the overall betoken abuse. Such decision feedback is typically used in combination with time-domain linear filtering, where the linear filter transforms the aqueduct response to a shape that is more than suitable for the determination-feedback phase. Decision feedback tin can also very well be used in combination with frequency-domain linear equalization [24].

•

Maximum-Likelihood (ML) detection, also known equally Maximum-Likelihood Sequence Estimation (MLSE) [25], is strictly speaking not an equalization scheme but rather a receiver approach where the impact of radio-channel time dispersion is explicitly taken into business relationship in the receiver-side detection process. Fundamentally, an ML detector uses the entire received betoken to make up one's mind on the most likely transmitted sequence, taking into business relationship the impact of the fourth dimension dispersion on the received indicate. To implement maximum-likelihood detection the Viterbi algorithm [26] is oftentimes used. ¹ Yet, although maximum-likelihood detection based on the Viterbi algorithm has been extensively used for 2G mobile communication such as GSM, it can be too complex when practical to much wider transmission bandwidths, leading to both much more extensive channel frequency selectivity and much higher sampling rates.

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MIMO in LTE, LTE-Advanced and WiMAX

Bruno Clerckx , Claude Oestges , in Mimo Wireless Networks (Second Edition), 2013

14.iv.3 Open-Loop Transmit Diversity: Space-Fourth dimension/Frequency Coding

Infinite-time/frequency coding has been treated in Chapters vi and 11. For 2 transmit antennas, an O-SFBC relying on the frequency domain version of the Alamouti code (6.141) (where pairs of next subcarriers are coded together instead of ii adjacent fourth dimension instants) is supported by both standards as the basic transmit diversity mode. STBC heavily rests on the quasi-static channel assumption, in lodge to keep the orthogonality of the code and its performance is severely affected past fast-time varying channels. This the reason why SFBC is ordinarily preferred over STBC in high mobility scenarios, assuming that the channel frequency selectivity is not big enough to break the orthogonality of O-SFBC.

For more than than two transmit antennas, transmit diverseness scheme precodes the Alamouti code to convert the physical antennas into two virtual antennas. Hence, for four and eight transmit antennas, infinite-frequency coding is limited to a pair of subcarriers, therefore not guaranteeing total multifariousness. However, the precoder is designed such that all spatial degrees of freedom are exploited every bit much as possible and that information technology provides robustness against transmit spatial correlations. Cheers to a unified blueprint, the receiver can use the same decoding process independently of the number of physical transmit antennas.

Given the deviation in the reference signals designs, 802.16m and LTE have different usages of the OL precoders. 802.16m and LTE accept opted for SFBC combined with precoder cycling (Section 11.5.5) based on precoded pilots and SFBC combined with frequency-switched transmit diversity (FSTD) based on non-precoded CRS, respectively. Through precoder cycling/FSTD and FEC, both approaches benefit from frequency variety on summit of spatial variety by enhancing the frequency selectivity of the channels.

With 802.16m precoder cycling, the precoder is fixed within a RU i and varies on an RU footing owing to the presence of precoded pilots. The precoder ${\mathbf{West}}_k$ on any subcarrier thousand in RU i is fixed to a $n_t\times 2$ matrix ${\mathbf{Thou}}_i$ belonging to a predefined set of matrices (i.eastward., a codebook of matrices). The precoder cycling creates a fixed set of ii virtual antennas beyond all subcarriers within a RU such that the stream of encoded data symbols $c_0\text{,}\dots \text{,}{c}_{T-\mathrm{one}}$ is spread and transmitted over multiple antennas on subcarriers grand and $k+1$ as

(14.one) $\left[\begin{array}{cc}\hfill {\mathbf{c}}_{\mathrm{one\; thousand}}\hfill & \hfill {\mathbf{c}}_{k+1}\hfill \end{array}\right]=\frac{1}{\sqrt{2}}{\mathbf{M}}_i\left[\begin{array}{cc}\hfill c_{\mathrm{grand}}\hfill & \hfill -c_{\mathrm{one\; thousand}+1}^{\ast }\hfill \\ \hfill c_{k+\mathrm{i}}\hfill & \hfill c_k^{\ast }\hfill \end{array}\right]$

where i is the alphabetize of the RU k and $k+1$ belong to and ${\mathbf{c}}_k$ and ${\mathbf{c}}_{k+1}$ are the $n_t\times 1$ codevectors transmitted over subcarrier m and $k+1$ , respectively. The apply of precoded pilots in precoder cycling enables to reduce the pilot overhead just is affected by channel estimation errors due to the RU-level channel estimation constraints.

FSTD cycles transmissions over pairs of transmit antennas across subcarriers within a RU such that the transmitted codeword on subcarriers k to $k+3$ for four transmit antennas writes as

(14.2) $\left[\begin{array}{cccc}\hfill {\mathbf{c}}_k\hfill & \hfill {\mathbf{c}}_{k+\mathrm{one}}\hfill & \hfill {\mathbf{c}}_{m+2}\hfill & \hfill {\mathbf{c}}_{\mathrm{thousand}+\mathrm{iii}}\hfill \end{array}\right]=\frac{\mathrm{ane}}{\sqrt{\mathrm{two}}}\left[\begin{array}{cccc}\hfill c_k\hfill & \hfill -c_{k+\mathrm{ane}}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill c_{k+2}\hfill & \hfill -c_{\mathrm{1000}+3}^{\ast }\hfill \\ \hfill c_{k+1}\hfill & \hfill c_k^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill c_{m+3}\hfill & \hfill c_{k+\mathrm{ii}}^{\ast }\hfill \end{array}\right]$

where Alamouti code is performed over $c_k$ and $c_{k+1}$ on antenna 1 and three and over $c_{m+2}$ and $c_{k+\mathrm{iii}}$ on antenna 2 and 4. Alamouti coding is performed respectively over antenna pairs (ane, 3) and (two, 4) (and not over (1, 2) and (3, 4)), in gild to balance the channel estimation errors resulting from the depression CRS density on antennas 3 and 4. The not-precoded CRS allows to switch antennas at the subcarrier level and to perform a wide range of channel interpretation (including interpolation between RBs). Yet, it would be impacted past the relatively high overhead. It is clear that the error matrix of those two codes tin can be rank scarce. Hence, full diversity is not guaranteed, as discussed in Chapter 6.

While the precoder is defined for both four and eight transmit antennas in 802.16m, it is limited to four transmit antennas in LTE and LTE-A. It was indeed shown that an viii-transmit antenna diversity scheme only provides a marginal diversity proceeds over a 4-transmit antenna scheme. Nevertheless, transmit diversity standardized in Rel. eight for two and four transmit antennas can be used even when eight transmit antennas are deployed by using antenna virtualization, where the physical arrays are precoded such that the user effectively perceives but two or iv antennas.

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Electric line communications

L. Lampe , L.T. Berger , in Bookish Press Library in Mobile and Wireless Communications, 2016

16.4.ane.2 Modeling approaches

The effects described above are represented in PLC channel models, which can exist grouped into two general classes. The first form is based on the actual propagation of electromagnetic signals in power line networks taking topology, ability line (eg, cable) properties, and load impedances into business relationship. This course can be referred to equally "physical" models. The second class is based on observations of the propagation effects on the PLC indicate, eg, through CIR and CFR measurements. The resulting models can exist referred to as "empirical" models. These 2 classes tin also be labeled equally "bottom-upward" and "top-downward" [104], which also concisely identify the underlying approaches.

Empirical models

The originally and almost widely used model that incorporates the above-described phenomenological propagation furnishings defines the CFR as [105–107]

(sixteen.1) $\begin{array}{l}H(f)=\sum _{i=1}^N{k}_i{\mathrm{e}}^{-\alpha (f){\ell }_i}{\mathrm{e}}^{-\mathrm{j}2\pi f{\tau }_i},\hfill \end{array}$

where N represents the number of significant signal paths, and chiliad _i , ℓ _i , and τ _i are the gain factor, length, and signal delay for path i, respectively. The parameter α(f) > 0 is the frequency-dependent attenuation coefficient for the considered power line. Nosotros observe that the construction of H(f) in Eq. (16.1) explicitly accounts for multipath propagation and frequency-dependent pathloss. Using curve fitting based on measurements of CFRs, several reference channels with sets of parameters applicable to the BB PLC access domain accept been specified in the Open up PLC European Research Brotherhood (OPERA) project, see [42, Section 2.9]. Parameters for NB PLC channels are provided in [59, App. D].

References [108, 109] present empirical values for distance-dependent attenuation as well as coupling losses for NB PLC in the CENELEC-A band over LV distribution lines, which tin exist considered equally a single-tap version of Eq. (16.1). Furthermore, a uncomplicated two-tap channel model has been proposed in [102, 110]. More specifically, the CIR is modeled as

(16.ii) $\begin{array}{l}\hfill h(t)=h\delta (t)+h\delta (t-\tau ),\end{array}$

where δ is the Dirac-delta function and τ is the relative delay of the 2d path, and both taps have the same gain h. It has been recognized in [102, 110] that the boilerplate channel gain and the root-mean-foursquare filibuster spread (RMS-DS) are negatively correlated, lognormally distributed random variables. Thus, both h and τ are obtained from a randomly chosen boilerplate aqueduct gain to generate a sample CIR. It is argued that this uncomplicated model captures the essence of channel frequency-selectivity for NB PLC and can also be used for comparative analysis of BB PLC systems [111].

For in-business firm BB PLC and based on measurements of a full of 144 CFRs in the 30–100   MHz band in different types of homes, Tlich et al. [112] have proposed a random generator for CFRs. Nine classes of channels with their associated boilerplate attenuation models are defined, and the multipath nature is included through distributions for width, superlative, and number of peaks and notches in the CFR.

The measurement results reported in [112] were as well used in [113, 114] to present a different random CFR generator that closely matches the expression in Eq. (xvi.ane). In particular, the parameters North, g _i , and ℓ _i are considered random variables, and their distribution parameters for the different classes identified in [112] have been obtained through a bend-plumbing fixtures procedure [115].

Another empirical CFR generator has been presented in [116]. While verification that the generator creates representative CFRs in a statistical sense is based on results from a measurement campaign, the actual method of generation follows a physical model, as described in the adjacent paragraph. Furthermore, it considers the presence of fourth dimension-varying load impedances in LPTV channels. The CFR generators from [113, 116] are available every bit MATLAB code from the authors' websites.

Concrete models

This modeling approach describes the electric properties of a manual line through the specification of line parameters, line length, the position of branches etc., and makes use of transmission line theory (TLT) to obtain CFRs or CIRs for specific PLC links. The underlying assumption is that the electro-magnetic (EM) field associated with the PLC signal has a transverse EM (TEM) or quasi-TEM structure [100]. Most physical models correspond power line elements and connected loads through ABCD or S-parameter matrices or equivalently through their propagation constants and reflection coefficients, respectively [117]. These are and so interconnected to produce the CFR, eg, [118–124]. Alternatively, one can consider a finite number Due north of paths and compute their gain factors g _i in Eq. (sixteen.1) following the transmission and reflection coefficients along those paths, as it is illustrated for ii paths in Fig. 16.8; eg, [125, 126].

Physical models also allow for incorporation and analysis of the effects of different wiring practices [120], time-varying load impedances [118], and multiconductor lines [121]. The latter makes use of multiconductor manual line (MTL) theory [100], and its application can be extended to obtain channel realizations for the utilize of multiple conductors for MIMO transmission [127], which we will discuss farther in Department 16.5. Through the application of statistical models for concrete parameters of the considered grid topology, such as number of branches, co-operative lengths, load impedances, etc., CFR/CIR random generators tin can be developed [119, 128]. Furthermore, the use of physical models is essential to capture interdependencies of channel responses for different links in the aforementioned network when considering cooperative or relay transmission [129–131].

While physical models are able to accurately represent the propagation of PLC signals over power lines, it may be considered a drawback that a significant amount of input data about the underlying grid structure and line elements, besides as some computational resource, are required to obtain CFR or CIR results. Hence, channel emulator software modules presented in [132, 133] (based on [124]) and [134], which include the computation of transmission line parameters, can exist useful. Nosotros note that concrete models are as well discussed in the relevant IEEE standardization documents [82, App. F, 59, App. D].

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