What Does the Shannon Capacity Have to Do With Communications

Fiber Nonlinearity and Capacity: Single-Mode and Multimode Fibers

René-Jean Essiambre , ... Roland Ryf , in Optical Fiber Telecommunications (Sixth Edition), 2013

1.4.4 Analytic formula of fiber capacity

The dominant mechanism responsible for the nonlinear Shannon capacity limit for single polarization signals was established from full simulations to be cross-phase modulation (XPM) [4]. Based on this knowledge, a theory that accounts for XPM in the highly dispersive regime, often referred to as pseudo-linear transmission, has been developed in Ref. [49]. The analytic nonlinear capacity limit C and SE can be expressed as [49]

(1.19) $\mathrm{SE}=\frac{C}{\mathrm{\Delta }f}={\log }_2\left(1+{\left[\frac{n_{\text{sp}}\hslash {\omega }_0\alpha {\mathit{LR}}_s}{P}+4\frac{{\gamma }^2{\mathcal{P}}^{2}L}{|R_s^2{\beta }_2|}\right]}^{-1}\right)\text{,}$

where

(1.20) $\mathcal{P}={\left[{\displaystyle \sum _{n=-N_{\text{ch}}/2(n\ne 0)}^{N_{\text{ch}}/2}}\frac{\kappa }{2\pi }\frac{R_s}{|\mathrm{\Delta }{f}_n|}\right]}^{1/2}P$

and where $\kappa =1$ for a Gaussian constellation and

(1.21) $\kappa =\frac{n^4/5+n^3/2+n^2/3-1/30}{(n^2/3+n/2+1/6{)}^2}-1\text{,}$

for equally spaced n-ring constellations with equal probability of occupation on each ring [4]. The values of κ for n  =   2,   3,   4,   16 are 0.36, 0.5, 0.5733, 0.7433, and 0.8 for an infinite number of rings.

Figure 1.10 shows the nonlinear capacity curves from the full nonlinear simulations for various distances [4, Fig. 35] and the corresponding analytic capacity curves from Eq. (1.19). The parameters used for the analytic capacity Eq. (1.19) not already specified in Table 1.1 or before are: spontaneous emission factor $n_{\text{sp}}=1$ , symbol rate $R_s=100\text{Gbaud}$ , system length L as indicated in Figure 1.10, average signal power P, constellation shaping factor $\kappa =0.74$ for a 16-ring constellation, number of channels $N_{\text{ch}}=5$ , channel spacing $\mathrm{\Delta }f=100\text{GHz}$ and frequency separation between the central and neighboring channels $\mathrm{\Delta }{f}_n=n\mathrm{\Delta }f$ . These parameters give $\mathcal{P}\sim 0.59P$ and the analytical curves of Figure 1.10. The fit between the analytical capacity formula (1.19) and the numerical results is rather good as seen in the figure. The main source of deviation is thought to originate from not taking into account explicitly the reduction in capacity from the ring constellation shape in the linear part of C where a Gaussian constellation is assumed.

Figure 1.10. Maximum SE versus SNR for various distances. Filled circles represent the results of numerical simulations while lines without circles are from the analytic formula of Eq. (1.19).

The maximum value of the nonlinear capacity of Eq. (1.19) or nonlinear capacity limit is given by [49],

(1.22) ${\mathrm{SE}}_{\text{max}}={\log }_2\left\{1+\frac{1}{3L}{\left[\frac{{\left(n_{\text{sp}}\hslash {\omega }_0\alpha \gamma \right)}^2{\mathcal{S}}_B}{|{\beta }_2|}\right]}^{-\frac{1}{3}}\right\}\text{,}$

where ${\mathcal{S}}_B$ is a constant independent of the fiber. At high SNR and on a log scale for the fiber parameters, ${\mathrm{SE}}_{\text{max}}$ behaves as $\propto L^{-1}\text{,}\propto {\alpha }^{-2/3}\text{,}\propto {\gamma }^{-2/3}$ , and $\propto |{\beta }_2{|}^{-1/3}$ , a scaling consistent with the behavior observed in Figures 1.6–1.9. Finally, one should mention that other nonlinear capacity formulas based on FWM have been derived [50,51] that produce similar but non-identical results.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123969606000018

Error-Correction Coding

Michael Parker , in Digital Signal Processing 101 (Second Edition), 2017

12.8 Shannon Capacity and Limit Theorems

No discussion on coding should be concluded without at least a mention of the Shannon capacity theorem and Shannon limit. The Shannon capacity theorem defines the maximum amount of information, or data capacity, which can be sent over any channel or medium (wireless, coax, twister pair, fiber etc.).

$\text{C}=\text{B}{\log }_2\left(1+\text{S}/\text{N}\right)$

where

C is the channel capacity in bits per second (or maximum rate of data)

B is the bandwidth in Hz available for data transmission

S is the received signal power

N is the total channel noise power across bandwidth B

What this says is that higher the signal-to-noise (SNR) ratio and more the channel bandwidth, the higher the possible data rate. This equation sets the theoretical upper limit on data rate, which of course is not fully achieved in practice.

It does not make any limitation on how low the achievable error rate will be. That is dependent on the coding method used.

As a consequence of this, the minimum SNR required for data transmission can be calculated. This is known as the Shannon limit, and it occurs as the available bandwidth goes to infinity.

${\text{E}}_{\text{b}}/{\text{N}}_0=-1.6\text{dB}$

where

E_b is the energy per bit

N₀ is the noise power density in Watts/Hz (N   =   B N₀)

If the E_b/N₀ falls below this level, no modulation method or error-correction method will allow data transmission.

These relationships define maximum theoretical limits, against which the performance of practical modulation and coding techniques can be compared against. As newer coding methods are developed, we are able to get closer and closer to the Shannon limit, usually at the expense of higher complexity and computational rates.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128114537000123

Ultralong-distance undersea transmission systems

Jin-Xing Cai , ... Neal S. Bergano , in Optical Fiber Telecommunications VII, 2020

13.2.1 Advanced modulation formats—increasing channel data rate

Digital coherent detection opens the door for optical communication to use higher order constellations to approach the Shannon capacity. The widely studied modulation formats for undersea systems are quadrature amplitude modulation (QAM) formats and amplitude phase shift keying (APSK) formats with 16, 32, and 64 constellation points. However, increasing the constellation size to create higher order modulation formats results in lower receiver sensitivity such that the achievable distance for the same optical SNR and FEC algorithms rapidly decreases with increasing SE. Until recently, most of the improvements in practical optical coherent systems focused on reducing the implementation penalty of QAM formats using advances in high-speed DSP and hardware including digital-to-analog converters and analog-to-digital converters.

Researchers have studied many modulation and coding techniques to increase receiver sensitivity. Such techniques included trellis-CM, multilevel coding, constellation shaping, capacity achieving binary and nonbinary FEC codes, bit-interleaved coded modulation (BICM) and multidimensional CM. Further reductions in the gap to the Shannon limit now depend on improving the modulation format and coding technique.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128165027000154

Future Research Directions

William Shieh , Ivan Djordjevic , in OFDM for Optical Communications, 2010

12.5 Adaptive Coding in Optical OFDM

Codes on graphs, such as LDPC codes and turbo product codes, have generated much research interest. It has been shown that LDPC codes are able to closely approach Shannon's capacity limits. OFDM is an excellent modulation technique to be used for multiuser access, known as OFDMA. In OFDMA, subsets of subcarriers are assigned to individual users. OFDMA enables time and frequency domain resource partitioning. In the time domain, it can accommodate for the burst traffic (packet data) and enables multiuser diversity. In the frequency domain, it provides further granularity and channel-dependent scheduling. In OFDMA, different numbers of subcarriers can be assigned to different users to support differentiated quality of service (QoS). Each subset of subcarriers can have different kinds of modulation formats and can carry different types of data. The differentiated QoS can be achieved by employing the LDPC codes of different error correction capabilities. Therefore, OFDMA represents an excellent interface between wireless/wireline and optical technologies. Because of their low complexity of decoders and highly regular structure of parity-check matrices, LDPC codes are considered as excellent candidates for rate-adaptive coded OFDM applications. An example of rate-adaptive coding is shown in Figure 12.11. The transmitted data are organized in OFDM packets of duration T _p, each containing n _t OFDM symbols, and every OFDM symbol has n_f subcarriers and occupies the frequency bandwidth of B_f Hz. The cyclic extension guard interval of OFDM is sufficiently long to compensate for chromatic dispersion and maximum differential group delay. The number of OFDM packets and constellation size are determined by channel conditions. When channel conditions are favorable, larger constellations are used and higher data rates are transmitted.

Figure 12.11. OFDM packets for rate-adaptive coded modulation.

The quasi-cyclic LDPC codes described in Chapter 6 are excellent candidates for use in rate-adaptive codes. The parity-check matrix of these codes is given as

(12.1) $H=\left[\begin{array}{ccccc}I& I& I& \dots & I\\ I& P^{S[1]}& P^{S[2]}& \dots & P^{S[c-1]}\\ I& P^{2S[1]}& P^{2S[2]}& \dots & P^{2S[c-1]}\\ \dots & \dots & \dots & \dots & \dots \\ I& P^{(r-1)S[1]}& P^{(r-1)S[2]}& \dots & P^{(r-1)S[c-1]}\end{array}\right]$

where P is a p × p permutation matrix, r is the column weight (the number of rows in Eq. (12.1) being used), c is the row weight, and p is a prime. The codeword length is determined by pc, whereas the code rate is lower bounded by

(12.2) $R(r)\ge \frac{\mathit{pc}-\mathit{rp}}{\mathit{pc}}=1-r/c$

For code rate adaptation, we have several options: (1) keep the codeword length fixed (or equivalently keep the row weight c fixed) but change the number of rows r being used, (2) keep r constant but change c, and (3) let both parameters be variables. Note that the parity-check matrix stays quasi-cyclic as the code rate changes. The rate-compatible convolutional codes are already in use in wireless communications. ^{33, 34} However, the code rates of these codes are usually small, puncturing is commonly random, and excessive puncturing yields performance degradation. On the other hand, rate-compatible LDPC codes can be used in a bit-interleaved coded modulation manner, ³⁵ and code rate adaptation can be performed by maximizing the channel capacity. The channel capacity of this scheme can be calculated by employing Ungerboeck's approach ³⁶ :

(12.3) $C_{\text{u}}=I(X;Y)={log}_{2}M-E_{X,Y}\left[{log}_2\frac{\sum _{z\in X}p(y\mid z)}{p(y\mid x)}\right]$

which represents the channel capacity for uniform input distribution (or achievable information rate), where E_{X ,Y} denotes the expectation operator with respect to input and output symbols, and M is the signal constellation size. Transition probabilities p(y|z) are evaluated by employing the training sequences, whereas for the amplified spontaneous emission noise-dominated scenario the transition probability is given by

(12.4) $p(y\mid z)=\frac{1}{2\pi {\sigma }^2}exp\left[-\frac{{\mid y-z\mid }^2}{2\pi {\sigma }^2}\right]$

where σ² is the noise variance. One possible scenario is shown in Figure 12.12 (see Chapter 6 for full description of this scheme). The number of bits per symbol m = log₂ M is determined to maximize the channel capacity expression (Eq. 12.3). The same mapping is employed for the duration of OFDM packets shown in Figure 12.11. Based on optical signal-to-noise ratio estimate, we can determine which code rate is to be used.

Figure 12.12. Code rate adaptive optical OFDM scenario based on multilevel coding and bit-interleaved coded modulation.

Another approach described here is based on adaptive loading, which is already in use in wireless communications. ³⁵ The key idea is to vary the data rate and power assigned to each OFDM subcarrier relative to the subcarrier gain. In adaptive loading, power and rate on each subcarrier are adapted to maximize the total rate of the system using adaptive modulation, such as variable-rate variable-power M-ary QAM or M-ary phase-shift keying. The capacity of the OFDM system with N independent subcarriers of bandwidth B_N and subcarrier gain {g_i , i = 0, …, N − 1} can be evaluated by

(12.5) $C=\underset{P_i:\sum P_i=P}{max}\sum _{i=0}^{N-1}{B}_N{log}_2\left(1+\frac{g_i^2{P}_i}{N_0{B}_N}\right)$

where P_i is the power allocated to ith subcarrier. It can be shown by using the Lagrangian method that the optimum power allocation policy is the water-filling over frequency ³⁷ :

(12.6) $\frac{P_i}{P}=\{\begin{array}{c}1/{\gamma }_{\text{c}}-1/{\gamma }_i,{\gamma }_i\ge {\gamma }_{\text{c}}\\ 0,\mathrm{otherwise}\end{array},{\gamma }_i=g_{i}P/N_0{B}_N$

where γ _i is the signal-to-noise ratio (SNR) of the ith subcarrier, and γ_c is the threshold SNR. By substituting Eq. (12.6) into Eq. (12.5), the following channel capacity expression is obtained:

(12.7) $C={\displaystyle \sum _{i:{\gamma }_i>{\gamma }_{\text{c}}}{B}_N{\log }_2\left(\frac{{\gamma }_i}{{\gamma }_{\text{c}}}\right)}$

The ith subcarrier is used when corresponding SNR is above threshold. The number of bits per ith subcarrier is determined by m_i = ⌊B_N log₂(γ _i /γ_c)⌋, where ⌊ ⌋ denotes the largest integer smaller than the enclosed number.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123748799000125

Emerging optical communication technologies for 5G

Xiang Liu , Ning Deng , in Optical Fiber Telecommunications VII, 2020

17.4.2 Advanced coherent transmission for high-performance optical core networks

A key element of optical core network is coherent optical transceiver. Recently, probabilistic shaping (PS) has been introduced to further improve coherent optical transceiver performance to better approach the Shannon capacity limit [24,25]. PS with 64QAM has been demonstrated in field trial environment with offline DSP [25]. More recently, a field trial on the use of PS-programmable real-time 200-Gb/s coherent transceivers in a deploying core optical network has been demonstrated, achieving a twofold increase in reach when the PS is activated [26].

Fig. 17.24A shows the schematic of an intelligent intent-driven optical network with a network cloud engine (NCE) consisting of intent, intelligence, automation, and analytics engines, and an intelligent optical layer consisting of reconfigurable optical add-drop multiplexers (ROADMs), advanced optical monitoring, and PS-programmable optical DSP. Fig. 17.24B illustrates the use of format-and-shaping-programmable coherent transceivers to achieve the trade-off between performance and power consumption for a given application or intent.

Figure 17.24. (A) Schematic of an intelligent intent-driven optical network with a network cloud engine (NCE) and an intelligent optical layer; (B) illustration of format-and-shaping-programmable coherent transceivers.

Source: After J. Li, et al., Field trial of probabilistic-shaping-programmable real-time 200-Gb/s coherent transceivers in an intelligent core optical network, in: 2018 Asia Communications and Photonics Conference (ACP), PDP Su2C.1, Hangzhou, China, 2018.

The transmission performances of 200G PDM-PS16QAM, PDM-8QAM, and PDM-16QAM were first compared in a lab environment, as shown in Fig. 17.25A. Evidently, PS16QAM offers the best performance, and doubles the reach of 16QAM at a given optical signal-to-noise ratio (OSNR) penalty of 1.5   dB. Then the transmission performances in the field trial were compared [26]. Fig. 17.25B and C show representative recovered constellations, respectively, for 16QAM after 571-km and PS16QAM after 1142-km transmission.

Figure 17.25. (A) Measured transmission performance comparison among 200G PDM-PS16QAM, PDM-8QAM, and PDM-16QAM in a lab environment; (B) Recovered 200   G PDM-16QAM constellations after 571-km transmission (without loop-back) in the field trial; (C) Recovered 200G PDM-PS16QAM constellations after 1142-km transmission (with loop-back) in the field trial.

Source: After J. Li, et al., Field trial of probabilistic-shaping-programmable real-time 200-Gb/s coherent transceivers in an intelligent core optical network, in: 2018 Asia Communications and Photonics Conference (ACP), PDP Su2C.1, Hangzhou, China, 2018.

Fig. 17.26A and B show the corresponding BER curves. In the BTB case, the required OSNR values at the FEC threshold of 3×10⁻² are 18 and 16.5   dB for PDM-16QAM and PDM-PS16QAM, respectively, which represent 1 and 2.5   dB improvements over the previous PDM-16QAM result. At the FEC threshold, PDM-16QAM requires a received OSNR of 20   dB after 571-km transmission, while PS16QAM requires a received OSNR of 17   dB after 1142-km transmission, confirming that PS16QAM offers doubled reach as compared to 16QAM. The measured average OSNR values after 571-km and 1142-km transmissions are 22.3 and 19.3   dB, respectively. Thus, there is an average OSNR margin of 2.3   dB. Ethernet tester report showed error free over 24   hours with over 2×10¹⁵ bytes received [26].

Figure 17.26. (A) Representative 200G PDM-16QAM performance after the 571-km field test; (B) representative 200G PDM-PS16QAM performance after the 1142-km field test.

Source: After J. Li, et al., Field trial of probabilistic-shaping-programmable real-time 200-Gb/s coherent transceivers in an intelligent core optical network, in: 2018 Asia Communications and Photonics Conference (ACP), PDP Su2C.1, Hangzhou, China, 2018.

This real-time field demonstration shows the use of a state-of-the-art probabilistic-shaping-programmable real-time 200-Gb/s coherent transceivers in a deploying intelligent core network to achieve improved performance, energy-efficiency, and protection in case of fiber cuts, showing the benefits of such probabilistic-shaping-programmable transceivers in future high-performance optical core networks.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128165027000191

Transmission system capacity scaling through space-division multiplexing: a techno-economic perspective

Peter J. Winzer , in Optical Fiber Telecommunications VII, 2020

8.3.2 Power-constrained system scaling—parallelism in M and B

In some systems, the overall power allocated to communications is limited. Of the systems mentioned in Section 8.2.2, this applies particularly to two classes of systems that lie on the diametrically opposite end of the transmission distance spectrum: chip-to-chip interfaces (constrained by the maximum possible power a chip can afford for communications) and submarine systems (constrained by the maximum possible electrical supply power for all the optical amplifiers within a submarine cable, which must be powered from high-voltage power feeds at each end of the submersed cable [47]). For such systems, Eq. (8.1) can be used to determine how to best distribute the available power across a system bandwidth B and a number of parallel spatial paths M.

Strictly mathematically, Eq. (8.1) is completely symmetric in polarization, bandwidth, and spatial paths, which suggests the substitution x=2MB,

(8.7) $C=x{\log }_2(1+P/(x{N}_0\left)\right).$

Fig. 8.5A shows the Shannon capacity ( 8.7) as a function of x for the case of a single-span un-amplified link without NLIN (e.g., a free-space optical link) using heterodyne or intradyne coherent detection (shot-noise limited detection with N ₀=hf _c; h is Planck's constant, f _c =193 THz is the system's optical carrier frequency) [3], and three different received signal power levels P are assumed. As the signal energy is being spread across bandwidth and/or spatial paths, the system's capacity monotonically increases with x and approaches its asymptotic value of

Figure 8.5. Scaling of capacity (A) and energy per bit (B) for different received signal power levels on a single-span, unamplified, shot-noise limited coherent detection channel.

(8.8) $\underset{x\to \infty }{\lim }C=(P/N_0){\log }_{2}e=P/(N_0\ln 2).$

Fig. 8.5B shows that the corresponding energy per bit E _b=P/C, cf. Eq. (8.4), normalized by hf _c to photons per bit, monotonically decreases with x and approaches the familiar asymptotically minimum required signal energy per bit of [35]

(8.9) $\underset{x\to \infty }{\lim }{E}_{\mathrm{b}}=\underset{x\to \infty }{\lim }P/C=N_0/{\log }_{2}e=N_0\ln 2.$

Importantly, we note that although the above considerations are performed for the classical Shannon capacity of Eq. (8.1), the same behavior, yet with N ₀=kT (k being Boltzmann's constant and T the applicable noise temperature), is obtained for the maximum achievable capacity and the minimum achievable energy per bit when arbitrary quantum communication techniques are taken into account [3].

While the benefit of bandwidth-spreading modulation is amply known in classical communications engineering [35], the complete analogy to spreading signals within a fixed system bandwidth B across a variable number of parallel spatial paths M may not be as widely appreciated. Which spreading mechanism is best to be used in practice depends on physical and techno-economic considerations, which are unique to each system, as we shall see throughout the remainder of this chapter.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128165027000099

Common Digital Modulation Methods

Tony J. Rouphael , in RF and Digital Signal Processing for Software-Defined Radio, 2009

3.1 Channel Capacity Interpreted

Shannon-Hartley's channel capacity theorem is often applied at the beginning of any waveform and link budget analysis to provide the communication analyst with an upper bound on the data rate given a certain bandwidth and SNR. The achievable data rate, however, greatly depends on many parameters, as will be seen later on in the chapter. It is the influence of these parameters that dictates the deviation of the system data rate from the upper bound.

In this section, we present a simple interpretation of the Shannon-Hartley capacity theorem. A detailed treatment of this subject is beyond the scope of this book; however the interested reader should consult the work done in references [1]–[3]. The intent here is to show the set of assumptions that are normally made when estimating the channel capacity in the presence of AWGN.

Consider a redundancy-free zero-mean baseband signal x(t) with random-like characteristics. In the absence of any coding, the no-redundancy assumption implies that any two samples x(t ₁) and x(t₂ ) will appear to be random if they are sampled at the Nyquist rate or above [4]:

(3.1) $\left|t_1-t_2\right|\ge \frac{1}{2B}$

where B is the bandwidth of the signal. This type of signal is said to be communicating with maximum efficiency over the Gaussian channel. Furthermore, consider the received signal y(t) of the signal x(t) corrupted by noise:

(3.2) $y(t)=x(t)+n(t)$

where n(t) is AWGN with zero mean. The signal and noise are uncorrelated, that is

(3.3) $E\{x(t)n(t)\}=E\{x(t)\}E\{n(t)\}$

where E{.} is the expected value operator as defined in [5]. Consequently, the power of the signal y(t) is the sum of the power of signal x(t) and noise n(t):

(3.4) $\begin{array}{ll}\hfill E\left\{{(x(t)+n(t))}^2\right\}& =E\{x^2(t)\}+E\{n^2(t)\}\hfill \\ \hfill P_y& =P_x+P_n\hfill \end{array}$

Note that any signal voltage content that is within the noise voltage range may not be differentiated from the noise itself. Therefore, we can assume that the smallest signal that can be represented is statistically at noise voltage level, and can be represented by

(3.5) $2^b=\frac{y(t)}{n(t)}$

where b is the number of bits representing a binary number. The relation in (3.5) can be loosely re-expressed as

(3.6) $2^b=\frac{y(t)}{n(t)}=\sqrt{\frac{P_y}{P_n}}=\sqrt{\frac{E\{x^2(t)\}+E\{n^2(t)\}}{E\{n^2(t)\}}}=\sqrt{1+\frac{E\{x^2(t)\}}{E\{n^2(t)\}}}$

Define the SNR as

(3.7) $\mathit{SNR}=\frac{E\{x^2(t)\}}{E\{n^2(t)\}}$

Then the relation in (3.6) becomes

(3.8) $2^b=\sqrt{1+\mathit{SNR}}\Rightarrow b={\log }_2\left(\sqrt{1+\mathit{SNR}}\right)$

In a given time T=1/2B, the maximum number of bits per second that the channel can accommodate is the channel capacity C

(3.9) $C=\frac{b}{T}=2B{\log }_2\left(\sqrt{1+\mathit{SNR}}\right)=B{\log }_2(1+\mathit{SNR})\mathrm{bits}\mathrm{per}\mathrm{second}$

At this point, let's re-examine the assumptions made thus far:

•

The noise is AWGN and the channel itself is memoryless. ¹

•

The signal is random.

•

The signal contains no redundancy—that is, the signal is not encoded and the samples are not related.

However, in reality, the received signal waveform is encoded and shaped and the samples are not completely independent from one another. Therefore, the Shannon capacity equation serves to offer an upper bound on the data rate that can be achieved. Given the channel environment and the application, it is up to the waveform designer to decide on the data rate, encoding scheme, and waveform shaping to be used to fulfill the user's needs. Therefore, when designing a wireless link, it is most important to know how much bandwidth is available, and the required SNR needed to achieve a certain BER.

Example 3-1: Channel Capacity

The voice band of a wired-telephone channel has a bandwidth of 3kHz and an SNR varying between 25dB and 30dB. What is the maximum capacity of this channel at this SNR range?

The capacity can be found using the relation in (3.9). For example, for the case where the SNR=25   dB and 30   dB the capacity is

(3.10) $\begin{array}{c}C=B{log}_2(1+SNR)=3000{log}_2(1+{10}^{25/10})\approx 24.28\text{ kbps}\\ C=B{log}_2(1+SNR)=3000{log}_2(1+{10}^{30/10})\approx 29.9\text{ kbps}\end{array}$

Note that at SNR=25   dB with 3   kHz of channel bandwidth, a data rate of more than 20 Kbps cannot be attained. For the entire SNR range, the capacity is depicted in Figure 3.1.

Figure 3.1. Capacity for a voice band telephone channel with varying SNR

The relation in (3.9) can be further manipulated as

(3.11) $\begin{array}{c}C=SNR\times B{log}_2{(1+SNR)}^{1/SNR}\\ SNR=\frac{C}{B}\frac{1}{{log}_2{(1+SNR)}^{1/SNR}}\end{array}$

The SNR presented in (3.11) can be expressed in terms of energy per bit divided by the noise normalized by the bandwidth as [6]:

(3.12) $\frac{E_b}{N_0}=\frac{B}{C}\mathit{SNR}$

Substituting (3.12) into (3.11) we obtain

(3.13) $\frac{E_b}{N_0}=\frac{1}{{\log }_2(1+\mathit{SNR}{)}^{1/\mathit{SNR}}}$

As the bandwidth grows to infinity, the SNR in Section 3.13 approaches zero and a lower bound on E_b /N ₀ is established under which information, at any given rate, may not be transmitted without error:

(3.14) $\begin{array}{c}{\frac{E_b}{N_0}|}_{\text{lower bound}}=\underset{SNR\to 0}{lim}\frac{1}{{log}_2{(1+SNR)}^{1/SNR}}\\ =\frac{1}{{log}_2\left\{\underset{SNR\to 0}{lim}{(1+SNR)}^{1/SNR}\right\}}=\frac{1}{{log}_{2}e}\approx 0.69\end{array}$

or −1.6   dB. This lower bound on E_b /N ₀ is known as the Shannon limit.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780750682107000035

Submarine line terminal

Arnaud Leroy , Omar Ait Sab , in Undersea Fiber Communication Systems (Second Edition), 2016

Low density parity check codes

LDPC codes, introduced by Gallager in the 1960s [28], are a special class of linear block codes. They were rediscovered thanks to the invention of turbo codes and the upsurge of interest in iterative decoding [12,23] . LDPC codes associated with SISO iterative decoding demonstrated that they can approach the Shannon capacity limit [14,22]. An LDPC code is defined by a sparse parity-check matrix H. Sparse means that the number of zeroes in the parity check matrix H is higher than the number of ones (typically the LDPC codes are designed with the fraction of ones below 1% in the parity check matrix). Each LDPC codeword c satisfies H·c ^T =0. The rows of H are called check nodes and the columns of H are called variable nodes. LDPC code is usually represented graphically by a Tanner graph [29,30]. Figure 14.9 shows a Tanner graph of a Hamming (7,4,3) code and its parity check matrix H.

Figure 14.9. Tanner graph of a Hamming (7,4,3) code.

Iterative SISO decoding is processed between a Variable Nodes decoder and Check Nodes decoder. The belief propagation (BP), also known as sum-product message passing algorithm, is the well-known algorithm to decode LDPC codes [30–32]. Several BP based algorithms have been studied and proposed to improve the performance and decrease the complexity of the iterative decoding of LDPC codes [33–38]. LDPC codes often present an error floor at a lower BER and need an outer hard decision cleanup code to remove it [39,40]. LDPC was introduced in submarine transmission system first with hard decision decoding [21] for 10 and 40   Gbps systems. LDPC codes with soft-decision decoding are currently deployed in 100-Gbps coherent transmission systems. Recently, a class of convolutional LDPC codes, called spatially coupled LDPC codes (SC LDPC), have emerged. SC-LDPC are asymptotically capacity achieving [40–42] and have an attractive encoding/decoding complexity. SC-LDPC codes could be considered as candidates for future submarine optical transmission systems. An example of SC-LDPC code performance is shown in the next section.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128042694000143

Wireless Communication: Concepts, Techniques, Models

Anurag Kumar , ... Joy Kuri , in Wireless Networking, 2008

2.2.2 Channel Capacity with Fading

How does a time varying channel attenuation affect the Shannon capacity formula? If the channel attenuation is h, and the noise is AWGN, then, for transmitted power P_xmt , the channel capacity is given by (2.19):

$W{\log }_2(1+\frac{h{P}_{xmt}}{N_{0}W})$

Suppose that the transmitter is unaware of the extent of the channel fading, and uses a fixed power and a fixed modulation and coding scheme. Suppose also that the fading level varies slowly. Then, for a given level of fading, the receiver must know h in order for the communication to achieve the Shannon capacity. To see this, let us look at Figure 2.4(b). If the channel's power attenuation is h, the received symbols are multiplied by $\sqrt{h}$ . This results in the symbols being "squeezed" together or spread apart. Obviously, the detection thresholds will need to depend on the level of fading.

Suppose that H_k is a stationary and ergodic process. It can then be shown that if the transmitter cannot adapt its coding and modulation, but the receiver can exactly track the fading, then the channel capacity with fading is given by

(2.20) $C_{fading-CSIR}=\int W{\log }_2\left(1+\frac{h{P}_{xmt}}{W{N}_0}\right)gH\left(h\right)dh$

where g_H (·) is the marginal density of the channel attenuation process H_k . For example, g_H (h) is exponential for Rayleigh fading (see Section 2.1.4). The acronym CSIR stands for channel state (or side) information at the receiver. Thus the transmitter can encode at any fixed rate R < C_fading-CSIR and for large enough code blocks the error rate can be made arbitrarily small, provided the receiver can track the channel. It is important to bear in mind that this is an ideal result; to achieve it, the channel fades will have to be averaged over and this will result in large coding delays.

In Problem 2.6 we see that $C_{fading-CSIR}\le W{\log }_2\left(1+\frac{\text{E}\left(H\right)P_{xmt}}{W{N}_0}\right)$ , that is, the capacity with fading is less than that with no fading with the same average SNR. With fading, there will be times when the SNR is higher than the average and times when the SNR will be lower than the average. Yet this result shows that the resulting channel capacity is less than that without fading, as long as the same average SNR is maintained.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978012374254450003X

MIMO Propagation and Channel Modeling

Ernst Bonek , in MIMO, 2011

2.1 Introduction

The channel is that part of a communication system that cannot be engineered. Propagation is at the heart of any wireless system; it sets the ultimate limits for other fields of communications engineering. However, we engineers should not see the propagation channel only as a limitation. We can devise methods to exploit the opportunities that this channel offers. In particular, we can put to work the spatial domain for our goal to improve wireless information transfer. If "Smart Antenna" was the catchword of the past in the area where signal processing meets electromagnetics, "MIMO" has taken its place today. Or is it just hype? While it is true that MIMO systems can, in theory, provide the seemingly paradoxical arbitrary multiplication of Shannon's capacity, and while nobody doubts that MIMO will be the enabling technology for high-speed wireless data, many questions remain.

The flood of MIMO papers in the past ten years, presumably intended to shed light on the topic, has led to quite a considerable confusion about what can actually be achieved. This paper will try to clarify the role of the propagation channel.

MIMO may offer three different benefits, namely beamforming gain, spatial diversity, and spatial multiplexing (Figure 2.1).

Figure 2.1. Benefits that MIMO offers in principle.

By beamforming, the transmit and receive antenna patterns can be focused into a specific angular direction by the appropriate choice of complex baseband antenna weights. The more correlated the antenna signals, the better for beamforming. Under Line of Sight (LOS) channel conditions, the Receiver Rx and Transmitter Tx gains may add up, leading to an upper limit of $m\cdot n$ for the beamforming gain of a MIMO system ( $n$ and $m$ being the number of antenna elements of Rx and of Tx, respectively).

Multiple replicas of the radio signal from different directions in space give rise to spatial diversity, which can be used to increases the transmission reliability of the fading radio link. For a spatially white MIMO channel, that is, completely uncorrelated antenna signals, the diversity order is limited to $m\cdot n$ . Spatial correlation will reduce the diversity order and is therefore an important channel characteristic.

Most MIMO systems are designed for moving terminals, so we will have to discuss the topic "correlation" both in space and in time. MIMO channels can support parallel data streams by transmitting and receiving on orthogonal spatial channels (spatial multiplexing). The number of usefully multiplexed streams depends on the rank of the instantaneous channel matrix $H$ , which, in turn, depends on the spatial properties of the radio environment. The spatial multiplexing gain may reach $\min \left(m,n\right)$ in a sufficiently rich scattering environment. So we have to consider what sufficiently rich scattering means and what its relative importance compared to SNR is.

Knowledge about the state of the instantaneous MIMO channel is essential for the system engineers. Whether this Channel State Information (CSI) is available at the receiver, the transmitter or both, further whether this information relates to the actual, instantaneous channel or is only information about a statistical average of the channel, will result in entirely different receive and transmit strategies.

Beamforming, diversity, and multiplexing are rivaling techniques. To highlight the role of the propagation channel, the threefold trade-off between beamforming, diversity, and multiplexing can be broken down into several dichotomous trade-offs [Wei03], see Figure 2.2.

Figure 2.2. How directivity or diversity at the receive and transmit ends of a MIMO link determine to which extent the channel supports beamforming, diversity, or multiplexing.

First, the optimal trade-off between beamforming on one hand and diversity/multiplexing on the other hand is mainly dictated by the channel properties, similar to the trade-off between beamforming and diversity in the MISO case. A directive channel favors beamforming, a diverse (nondirective) channel allows for diversity and/or multiplexing. Second, there is the trade-off between diversity and multiplexing. Partly, this trade-off is also dictated by the channel. If one side of the MIMO link is purely directive no multiplexing is possible, but diversity can be exploited at the uncorrelated link end. However, if both link-ends of the MIMO channel are (partly) decorrelated both features, diversity and multiplexing, are possible. In this case the optimal trade-off between diversity and multiplexing [ZT03] is purely determined by system requirements, i.e., desired data rate and reliability of transmission. High data rates can be achieved by employing multiplexing to full extent, high reliability is attained by diversity.

The partial overlap of the ellipses in Figure 2.2 indicates that there is a gradual transition between the pure realizations of a certain MIMO benefit. In summary, it is the propagation environment that determines what can be gained by MIMO techniques; however, this does not mean that it will be gained in actual MIMO operation. The idea that "proper" signal processing can convert any channel into $\min \left(m,n\right)$ parallel channels, still voiced by parts of the MIMO community, is far too optimistic and must be dismissed.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123821942000022

What Does the Shannon Capacity Have to Do With Communications

Source: https://www.sciencedirect.com/topics/engineering/shannon-capacity

Share this post