**2. Sources of difficulties in improvement of short-range CS performance**

Let us clarify the subject of discussion. The term "performance", relatively new in communications, is broader than the term "quality". Furthermore, the evaluation of CS performance has its own groups of the criteria used to compare the systems by their general utility characteristics, further called the "performance" criteria. Another, relatively narrow and stable group of "analytical" criteria is used in the research as a tool permitting to improve one or a part of the performance criteria.

#### **2.1 Basic performance criteria**

The performance criteria determine the required, desired, or real characteristics of the future or existing wireless CS permitting to use these systems for solution of definite tasks in the given conditions, and to evaluate the corresponding benefits, costs, and risks [16]. The main criteria of this group are:


analog signals without coding in real time with the limit bit rate equal to the capacity of the forward channel—the result unfeasible in digital CS with coding. Moreover, the absence of coders radically simplified the construction of FCS transmitters. Initially not noticed, this work initiated a great cycle of research in the optimization of FCS (see e.g. [4–13]) carried out in 1960s in MIT, Bell Lab., Stanford University, NASA, and other research centres. The results of these investigations unambiguously confirmed that modulation and feedback enable a development of simple CS transmitting signals and short codes in real time perfectly and with minimal distortion. Moreover, analytical results of the research determined a way to

However, since the mid-1970s, interest in the research in the FCS theory sharply declined, and, during subsequent decades, only a small number of academic papers were published. The main reason was the lack of practical results, as well as a pessimistic evaluation of the entire direction of research ("The subject itself seems to be a burned out case" [14], p. 324). It is worth adding that, at that time, shortrange transmission was provided by wires, and there was no special need in wireless FCS. At the same time, development of digital technologies, communications and automatics generated a lot of complex theoretical tasks, and the industry required

The situation changed with the appearance of mobile communications and wireless networks (WN) containing a great number of wireless end nodes (EN), each communicating with the base station (BSt) over forward and feedback channels. This renewed interest in FCS was still, however, strictly academic [15] and without any practical results. Having no alternative, currently, all the channels

Apart from the traditional requirements for CS (maximal rate, quality, reliability, range of transmission, etc.), battery-supplied or battery-less low-power transmitters of EN should be minimally complex and minimally energy-consuming, and should satisfy a large number of the other, sufficiently rigorous requirements [16] such as maximal energy efficiency of transmission, optimal utilization of the channel bandwidth, reduction of inter-channel interference, security of transmission, and others. The set of these characteristics is now defined by the general term "performance," and the main task of designers is the improvement of the systems or

The development of the first generations of WN and corresponding FT did not cause any particular difficulties, but each subsequent generation does pose new, increasingly complex problems. One should stress that the performance of the lower, physical (PHY) layer channels EN-BSt dramatically influences the performance of the overall network regardless of the particularities of the higher layers'

The design of the PHY layer channels is carried out almost independently from that of the higher layers and software of WN, and requires thorough knowledge of mathematics, signals processing, communication and information theory, and so on. Nevertheless, even among experienced designers, "the task of changing from a cable to wireless is still seen as a daunting prospect; wireless retains its reputation of being close to black magic. For most designers, it is an area where they have very little ability to change anything, other than the output power" [16]. A similar

This is not an isolated opinion. A large number of recent publications question the capability of the modern theory to provide any noticeable improvement in wireless transmission: "Shannon limit is now routinely being approached within 1 dB on AWGN channels … So is coding theory finally dead? … there is little more to be gained in terms of performance [18]"; "Whether research at the physical layer of

specialists. As a result, most FCS researchers took up these tasks.

of WN employ only the coding principle of transmission.

design of the perfect FCS design.

*Modulation in Electronics and Telecommunications*

channels' performance.

sentiment is expressed in [17].

organization.

**2**


These criteria are used for elaboration of standards and have no analytical tools for a prior evaluation of performance. Instead, each of the listed criteria has a fixed numerical evaluation determining the corresponding requirement to CS. Each standard defines a class of CS with a unique combination of performance criterions granting these systems the ability to solve definite tasks under definite conditions better than other systems. For a new system to have better performance, it should pass a certification which confirms the existence of new qualities. Moreover, to become the standard, it should be manufactured at least by three independent firms [16].

*<sup>Q</sup>*<sup>2</sup> <sup>¼</sup> *<sup>W</sup>sign σ*2 *ξ*

*Perfect Signal Transmission Using Adaptive Modulation and Feedback*

*DOI: http://dx.doi.org/10.5772/intechopen.90516*

is the signal-to-noise ratio (SNR) at the channel output.

*Ebit N<sup>ξ</sup>*

where Φð Þ *x* is the tabulated Gaussian integral:

Φð Þ¼ *x*

<sup>¼</sup> *<sup>F</sup>*<sup>0</sup> *<sup>C</sup>* <sup>2</sup> *C <sup>F</sup>*<sup>0</sup> � 1

time of transmission).

impossible.

**5**

The sources of difficulties are:

and energy-spectral efficiencies.

will not be optimal in another.

<sup>¼</sup> *<sup>W</sup>sign NξF*<sup>0</sup>

The energy efficiency of transmission ("energy per bit") *Ebit* <sup>¼</sup> *<sup>W</sup>signTbit* <sup>¼</sup> *Wsign=C* [J/bit] is determined as the energy necessary for transmission of a single bit of information with a bit rate equal to the capacity of the channel (*Tbit* <sup>¼</sup> <sup>1</sup>*=*2*<sup>C</sup>* is the

The spectral (or bandwidth) efficiency *C=F*<sup>0</sup> describes a number of bits transmitted per second per 1 Hz of the channel bandwidth. The limit values of energy-

� � <sup>¼</sup> *<sup>Q</sup>*<sup>2</sup>

This formula directly follows from (1) and (2) and is convenient for practical applications. Another frequently used but less convenient measure of energy efficiency is defined as *C=W*<sup>0</sup> [bit/s/W], where *W*<sup>0</sup> is the power of transmitter. The example of the expression for the BER computed for two orthogonal signals

> ffiffiffiffiffiffiffiffi *Ebit N<sup>ξ</sup>*

1

0s @

> *z*2 2

spectral efficiencies and SNR are connected by the relationships [23–25]:

transmitting particular bits over channels with AWGN has the form:

BER <sup>¼</sup> Pr*bit* <sup>¼</sup> <sup>1</sup> � <sup>2</sup><sup>Φ</sup>

1 ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> ð *x*

0

i. Impossibility to find, among the infinite set of possible codes, the code minimizing errors of transmission: "The existence of optimal encoding and decoding methods is proved, but there are no methods indicated for the

ii. Impossibility to formulate any expressions for current (not only limit) bit rate

iii. Both the quality of transmission and the results of CSC optimization directly depend on the scenario of the system application (placement of the system,

Implementation of theoretical results is only possible if there is a possibility of at least partial channels identification but a system optimal in one scenario

characteristics of the environment, fading, noise, path loss, etc.).

construction or technical realization of these results [27]."

These relationships are mutually connected by multiple tradeoffs: power-bandwidth tradeoff; tradeoffs between BER and energy efficiency, deployment efficiency-energy efficiency, and many others (see, e.g. [23, 26]). In these conditions, the development of a regular approach for the optimization of CSC is practically

exp

<sup>¼</sup> *SNRCh*, (2)

log <sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> � � *:* (3)

A, (4)

� �*dz:* (5)

Analytical criteria serve a different purpose, and are used to establish conditions that allow the rate, quality, reliability of transmission, and other characteristics of CS to approach their limit or given values in different conditions and under given constraints. These criterions are built on adequate mathematical models of the main components of CS or of the system as a whole, and their values depend on all the basic factors influencing the work of the system. Some of these factors can be regulated by designers who search for their combination that either maximizes the "main" criterion (e.g. the rate of transmission) or minimizes its value (e.g., transmission errors) taking into account existing limitations. The results of research determine the approach to design the best system in a given class under given conditions and limitations.

Analytical evaluations of the quality of transmission are not used in performance criteria but the research results create a rigorous basis for the design of more efficient CSs and for their emergence within new standards. Nevertheless, RF (radio frequency) design "is typically the smallest section of any wireless standard" and "the hardware definition may be less than 5% of the total specification in terms of the number of pages ([16], p. 20).

Currently, the term "improvement of performance" is widely used in communications and pushed out also not strictly defined earlier term "improvement of quality" of the systems, channels, transmission, etc. In this chapter, we use both these terms. It is worth adding that the term "perfect" should be taken literally: the performance of the AFCS discussed below does attain the limits established by information theory. Note that this discussion clarifies not only the terminology, but also the relations between different groups of criteria mutually connected through plural tradeoffs. The analytical results presented below allow for the simultaneous improvement of several performance criteria (to which we will return in the final discussion).

#### **2.2 Basic analytical criteria**

Nowadays, commonly used analytical criteria in CSC performance include the bit rate [bits/s], energy [J/bit], and spectral [bit/s/Hz] efficiencies of transmission, as well as bit error rate (BER). Sets of possible values of energy-spectral efficiencies have upper bounds, and the task of designers is to make the characteristics of the system approach these boundaries under possibly smaller BER. As the basic references, the theory employs limit values of bit rate and efficiencies of the transmission usually computed for linear memory less channels with additive white Gaussian noise (AWGN). So, limit bit rate determines the capacity of the channel (Shannon's formula) as follows:

$$C = F\_0 \log\_2 \left( \mathbf{1} + \frac{W^{\text{sign}}}{N\_\xi F\_0} \right) = F\_0 \log\_2 \left( \mathbf{1} + Q^2 \right), \tag{1}$$

where *Wsign* is the power of signal at the channel output, *Nξ=*2 is the double-side spectral power density of AWGN, 2*F*0is the channel bandwidth, and:

*Perfect Signal Transmission Using Adaptive Modulation and Feedback DOI: http://dx.doi.org/10.5772/intechopen.90516*

$$Q^2 = \frac{W^{\text{sign}}}{\sigma\_\xi^2} = \frac{W^{\text{sign}}}{N\_\xi F\_0} = \text{SNR}^{\text{Ch}},\tag{2}$$

is the signal-to-noise ratio (SNR) at the channel output.

The energy efficiency of transmission ("energy per bit") *Ebit* <sup>¼</sup> *<sup>W</sup>signTbit* <sup>¼</sup> *Wsign=C* [J/bit] is determined as the energy necessary for transmission of a single bit of information with a bit rate equal to the capacity of the channel (*Tbit* <sup>¼</sup> <sup>1</sup>*=*2*<sup>C</sup>* is the time of transmission).

The spectral (or bandwidth) efficiency *C=F*<sup>0</sup> describes a number of bits transmitted per second per 1 Hz of the channel bandwidth. The limit values of energyspectral efficiencies and SNR are connected by the relationships [23–25]:

$$\frac{E^{\rm bit}}{N\_{\xi}} = \frac{F\_0}{C} \left( 2^{\frac{\xi}{F\_0}} - 1 \right) = \frac{Q^2}{\log\_2 \left( 1 + Q^2 \right)}.\tag{3}$$

This formula directly follows from (1) and (2) and is convenient for practical applications. Another frequently used but less convenient measure of energy efficiency is defined as *C=W*<sup>0</sup> [bit/s/W], where *W*<sup>0</sup> is the power of transmitter.

The example of the expression for the BER computed for two orthogonal signals transmitting particular bits over channels with AWGN has the form:

$$\text{BER} = \text{Pr}^{bit} = \mathbf{1} - \mathbf{2}\Phi\left(\sqrt{\frac{E^{bit}}{N\_{\xi}}}\right),\tag{4}$$

where Φð Þ *x* is the tabulated Gaussian integral:

$$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int\_0^x \exp\left(\frac{z^2}{2}\right) dz. \tag{5}$$

These relationships are mutually connected by multiple tradeoffs: power-bandwidth tradeoff; tradeoffs between BER and energy efficiency, deployment efficiency-energy efficiency, and many others (see, e.g. [23, 26]). In these conditions, the development of a regular approach for the optimization of CSC is practically impossible.

The sources of difficulties are:


These criteria are used for elaboration of standards and have no analytical tools for

Analytical criteria serve a different purpose, and are used to establish conditions that allow the rate, quality, reliability of transmission, and other characteristics of CS to approach their limit or given values in different conditions and under given constraints. These criterions are built on adequate mathematical models of the main components of CS or of the system as a whole, and their values depend on all the basic factors influencing the work of the system. Some of these factors can be regulated by designers who search for their combination that either maximizes the "main" criterion (e.g. the rate of transmission) or minimizes its value (e.g., transmission errors) taking into account existing limitations. The results of research determine the approach to design the best system in a given class under given

Analytical evaluations of the quality of transmission are not used in performance

Currently, the term "improvement of performance" is widely used in communications and pushed out also not strictly defined earlier term "improvement of quality" of the systems, channels, transmission, etc. In this chapter, we use both these terms. It is worth adding that the term "perfect" should be taken literally: the performance of the AFCS discussed below does attain the limits established by information theory. Note that this discussion clarifies not only the terminology, but also the relations between different groups of criteria mutually connected through plural tradeoffs. The analytical results presented below allow for the simultaneous improvement of several performance criteria (to which we will return in the final

Nowadays, commonly used analytical criteria in CSC performance include the bit rate [bits/s], energy [J/bit], and spectral [bit/s/Hz] efficiencies of transmission, as well as bit error rate (BER). Sets of possible values of energy-spectral efficiencies have upper bounds, and the task of designers is to make the characteristics of the system approach these boundaries under possibly smaller BER. As the basic references, the theory employs limit values of bit rate and efficiencies of the transmission usually computed for linear memory less channels with additive white Gaussian noise (AWGN). So, limit bit rate determines the capacity of the channel

> *Wsign NξF*<sup>0</sup>

spectral power density of AWGN, 2*F*0is the channel bandwidth, and:

where *Wsign* is the power of signal at the channel output, *Nξ=*2 is the double-side

<sup>¼</sup> *<sup>F</sup>*<sup>0</sup> log <sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> , (1)

criteria but the research results create a rigorous basis for the design of more efficient CSs and for their emergence within new standards. Nevertheless, RF (radio frequency) design "is typically the smallest section of any wireless standard" and "the hardware definition may be less than 5% of the total specification in terms

a prior evaluation of performance. Instead, each of the listed criteria has a fixed numerical evaluation determining the corresponding requirement to CS. Each standard defines a class of CS with a unique combination of performance criterions granting these systems the ability to solve definite tasks under definite conditions better than other systems. For a new system to have better performance, it should pass a certification which confirms the existence of new qualities. Moreover, to become the

standard, it should be manufactured at least by three independent firms [16].

conditions and limitations.

discussion).

**4**

**2.2 Basic analytical criteria**

(Shannon's formula) as follows:

*C* ¼ *F*<sup>0</sup> log <sup>2</sup> 1 þ

of the number of pages ([16], p. 20).

*Modulation in Electronics and Telecommunications*

iv. The lack of the regular analytical approach to optimization of CSC makes impossible evaluation of the potentially achievable bounds of transmission quality and the search for the most efficient technical solution permitting their achievement.

The analysis that follows is carried out in discrete time. Samples *<sup>x</sup>*ð Þ *<sup>m</sup>* , *<sup>m</sup>* <sup>¼</sup> 1, 2, … of Gaussian input signal *xt* are transmitted iteratively, each in *n* cycles, and independently from previous samples. This permits to reduce the analysis of FCS functioning to transmission of a single sample *x* omitting the upper indices "*m.*"

*Perfect Signal Transmission Using Adaptive Modulation and Feedback*

also assume that the feedback channel Ch2 is realized on high-quality digital components, and that influence of the channel noise on feedback transmission can be neglected. The physical forward channel Ch1 is stationary, memoryless, and its noise *ξk*, *k* ¼ 1, … , *n*, is AWGN with a double-side spectral power density *Nξ=*2. Values *x* of the transmitted sample are held at the input of the sample and hold unit (S&H) during the time *Tn* ¼ *nΔt*<sup>0</sup> sufficient for the transmission of the sample in *n* cycles (*Tn* does not exceed the sampling period *T* ¼ 1*=*2*F*; *Δt*<sup>0</sup> is the duration of the cycle and determines the minimal bandwidth 2*F*<sup>0</sup> ¼ 1*=Δt*<sup>0</sup> of the forward and feedback channels; *F* is the width of the signal baseband). It is assumed that both the FT and BSt have microcontrollers or other signal processing units synchronizing

For every *k*-th cycle of transmission (*k* ¼ 1, … , *n*), microcontroller of the forward transmitter (FT) forms the residual signal *ek* ¼ *x* � *x*^*<sup>k</sup>*�<sup>1</sup> where *x*^*<sup>k</sup>*�<sup>1</sup> is the estimate of the sample computed by the processor of the BSt in previous cycle and delivered to the FT over feedback channel Ch2. Signal *ek* is routed to the input of digitally controlled amplitude (AM) modulator-emitter. The signals emitted by FT

where high-frequency (RF) components are omitted; *A*<sup>0</sup> is the amplitude of the carrier signal; and value of the modulation index *Mk* is set by the corresponding code previously written into the memory of the FT microcontroller, or delivered to the FT from the BSt over the feedback channel. After demodulation, the signal

is routed to the processor of the BSt which computes a new estimate *x*^*<sup>k</sup>* of the

The variable *A* in (7) describes the amplitude *A* ¼ *A*0γ*=d* of the received signal, which depends on the distance *d* between the FT and BSt, and on the channel gain γ dependent on the propagation losses, characteristics of environment, type and gains of antennas, etc. The gains *Lk* in (8) determine the rate of convergence of estimates *x*^*<sup>k</sup>* and their values, like the values *Mk*, are determined additionally free parameters

predicted mean value of the signal *yk* computed in the processor of the BSt in

channel. It also resets the gain *Lk* to the value *Lk*þ<sup>1</sup> and prepares the physical receiver of the station to receive a new signal. Reception of the estimate by the FT initializes the next cycle of the sample transmission: microcontroller computes the residual (estimation error) *ek*þ<sup>1</sup> ¼ *x* � *x*^*k*, resets the gain *Mk* to the value *Mk*þ1, and

The BSt processor stores estimate *x*^*<sup>k</sup>* and sends it to the FT over the feedback

1

*<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> *<sup>x</sup>*^*<sup>k</sup>*�<sup>1</sup> <sup>þ</sup> *Lk* <sup>~</sup>*yk* � <sup>E</sup> <sup>~</sup>*yk*j~*y<sup>k</sup>*�<sup>1</sup>

stored in the memory of the BSt processor. Value E <sup>~</sup>*yk*j~*y<sup>k</sup>*�<sup>1</sup>

<sup>1</sup> <sup>¼</sup> <sup>~</sup>*y*1, … , <sup>~</sup>*yk*�<sup>1</sup>

*yk* ¼ *A*0*Mkek* ¼ *A*0*Mk*ð Þ *x* � *x*^*<sup>k</sup>*�<sup>1</sup> ,ð Þ *x*^<sup>0</sup> ¼ *x*<sup>0</sup> , (6)

~*yk* ¼ *AMkek* þ *ξk*, (7)

,ð Þ *<sup>x</sup>*^<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> *:* (8)

1

denotes a sequence of signals received by

in (8) describes the

and controlling the work of the transmitter and receivers of FCS.

~*yk* have the form of high-frequency pulses of the same duration *Δt*0:

<sup>0</sup> of the samples are assumed to be known. We

The mean value *x*<sup>0</sup> and variance *σ*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90516*

received by the BSt

previous cycle, and ~*y<sup>k</sup>*�<sup>1</sup>

**7**

the base station in previous cycles.

sample according to the Kalman-type equation:

There are many approaches for the improvement of CSC (e.g., [28–30]) but their discussion is beyond the scope of the chapter. We will note the fact stressed in the literature (e.g., in [31, 32]): theoretical bounds determined by the existent methods of analysis are unachievable for real systems. As a result, modern CSCs transmit signals with the necessary performance but nobody can assess the efficiency of their energy, spectral, and other resources utilization. The listed reasons, including scenario-dependent performance, make the development of the perfect CSCs optimally and fully utilizing their energy and spectral resources an unrealistic task.

In the next section, we show that modulation and feedback resolve the listed problems and enable elaboration of the FCS transmitting signals perfectly, as well as permit to improve their performance criteria.

#### **3. Perfect FCS: transmission using feedback and adaptive modulation**

#### **3.1 General principles of FCS transmission**

The novelty of the topic makes us begin by considering sufficiently general but not complex systems to simplify the explanation of the main ideas, mathematical tools, methods, and results. Below we consider point-to-point FCS (block diagram in **Figure 1**) assuming that the input signals are Gaussian and channel noises are AWGN, and high quality feedback channel delivers signals from the BSt to the FT with negligibly small errors. One may add that this block diagram, with different formulations of the tasks, was the subject of both early and later research in this field. The material below does not repeat any of these works but summarizes and clarifies their main ideas, approaches to problem-solving, and results to elucidate the difficulties, which had blocked the development of the theory. We also hope that the reader might appreciate the beauty of these works, which came so close to success, but which are now almost forgotten.

**Figure 1.** *General block diagrams of point-to-point AFCS.*

*Perfect Signal Transmission Using Adaptive Modulation and Feedback DOI: http://dx.doi.org/10.5772/intechopen.90516*

The analysis that follows is carried out in discrete time. Samples *<sup>x</sup>*ð Þ *<sup>m</sup>* , *<sup>m</sup>* <sup>¼</sup> 1, 2, … of Gaussian input signal *xt* are transmitted iteratively, each in *n* cycles, and independently from previous samples. This permits to reduce the analysis of FCS functioning to transmission of a single sample *x* omitting the upper indices "*m.*" The mean value *x*<sup>0</sup> and variance *σ*<sup>2</sup> <sup>0</sup> of the samples are assumed to be known. We also assume that the feedback channel Ch2 is realized on high-quality digital components, and that influence of the channel noise on feedback transmission can be neglected. The physical forward channel Ch1 is stationary, memoryless, and its noise *ξk*, *k* ¼ 1, … , *n*, is AWGN with a double-side spectral power density *Nξ=*2.

Values *x* of the transmitted sample are held at the input of the sample and hold unit (S&H) during the time *Tn* ¼ *nΔt*<sup>0</sup> sufficient for the transmission of the sample in *n* cycles (*Tn* does not exceed the sampling period *T* ¼ 1*=*2*F*; *Δt*<sup>0</sup> is the duration of the cycle and determines the minimal bandwidth 2*F*<sup>0</sup> ¼ 1*=Δt*<sup>0</sup> of the forward and feedback channels; *F* is the width of the signal baseband). It is assumed that both the FT and BSt have microcontrollers or other signal processing units synchronizing and controlling the work of the transmitter and receivers of FCS.

For every *k*-th cycle of transmission (*k* ¼ 1, … , *n*), microcontroller of the forward transmitter (FT) forms the residual signal *ek* ¼ *x* � *x*^*<sup>k</sup>*�<sup>1</sup> where *x*^*<sup>k</sup>*�<sup>1</sup> is the estimate of the sample computed by the processor of the BSt in previous cycle and delivered to the FT over feedback channel Ch2. Signal *ek* is routed to the input of digitally controlled amplitude (AM) modulator-emitter. The signals emitted by FT ~*yk* have the form of high-frequency pulses of the same duration *Δt*0:

$$y\_k = A\_0 M\_k \varepsilon\_k = A\_0 M\_k (\mathbf{x} - \hat{\mathbf{x}}\_{k-1}), (\hat{\mathbf{x}}\_0 = \mathbf{x}\_0), \tag{6}$$

where high-frequency (RF) components are omitted; *A*<sup>0</sup> is the amplitude of the carrier signal; and value of the modulation index *Mk* is set by the corresponding code previously written into the memory of the FT microcontroller, or delivered to the FT from the BSt over the feedback channel. After demodulation, the signal received by the BSt

$$
\tilde{\jmath}\_k = \mathbf{A} \mathbf{M}\_k \mathbf{e}\_k + \xi\_k,\tag{7}
$$

is routed to the processor of the BSt which computes a new estimate *x*^*<sup>k</sup>* of the sample according to the Kalman-type equation:

$$
\hat{\mathbf{x}}\_{k} = \hat{\mathbf{x}}\_{k-1} + L\_k \left[ \tilde{\mathbf{y}}\_k - \mathbf{E} \left( \tilde{\mathbf{y}}\_k | \tilde{\mathbf{y}}\_1^{k-1} \right) \right], (\hat{\mathbf{x}}\_0 = \mathbf{x}\_0). \tag{8}
$$

The variable *A* in (7) describes the amplitude *A* ¼ *A*0γ*=d* of the received signal, which depends on the distance *d* between the FT and BSt, and on the channel gain γ dependent on the propagation losses, characteristics of environment, type and gains of antennas, etc. The gains *Lk* in (8) determine the rate of convergence of estimates *x*^*<sup>k</sup>* and their values, like the values *Mk*, are determined additionally free parameters stored in the memory of the BSt processor. Value E <sup>~</sup>*yk*j~*y<sup>k</sup>*�<sup>1</sup> 1 in (8) describes the predicted mean value of the signal *yk* computed in the processor of the BSt in previous cycle, and ~*y<sup>k</sup>*�<sup>1</sup> <sup>1</sup> <sup>¼</sup> <sup>~</sup>*y*1, … , <sup>~</sup>*yk*�<sup>1</sup> denotes a sequence of signals received by the base station in previous cycles.

The BSt processor stores estimate *x*^*<sup>k</sup>* and sends it to the FT over the feedback channel. It also resets the gain *Lk* to the value *Lk*þ<sup>1</sup> and prepares the physical receiver of the station to receive a new signal. Reception of the estimate by the FT initializes the next cycle of the sample transmission: microcontroller computes the residual (estimation error) *ek*þ<sup>1</sup> ¼ *x* � *x*^*k*, resets the gain *Mk* to the value *Mk*þ1, and

iv. The lack of the regular analytical approach to optimization of CSC makes impossible evaluation of the potentially achievable bounds of transmission quality and the search for the most efficient technical solution permitting

There are many approaches for the improvement of CSC (e.g., [28–30]) but their discussion is beyond the scope of the chapter. We will note the fact stressed in the literature (e.g., in [31, 32]): theoretical bounds determined by the existent methods of analysis are unachievable for real systems. As a result, modern CSCs transmit signals with the necessary performance but nobody can assess the efficiency of their energy, spectral, and other resources utilization. The listed reasons, including scenario-dependent performance, make the development of the perfect CSCs optimally and fully utilizing their energy and spectral resources an unrealistic

In the next section, we show that modulation and feedback resolve the listed problems and enable elaboration of the FCS transmitting signals perfectly, as well as

**3. Perfect FCS: transmission using feedback and adaptive modulation**

The novelty of the topic makes us begin by considering sufficiently general but not complex systems to simplify the explanation of the main ideas, mathematical tools, methods, and results. Below we consider point-to-point FCS (block diagram in **Figure 1**) assuming that the input signals are Gaussian and channel noises are AWGN, and high quality feedback channel delivers signals from the BSt to the FT with negligibly small errors. One may add that this block diagram, with different formulations of the tasks, was the subject of both early and later research in this field. The material below does not repeat any of these works but summarizes and clarifies their main ideas, approaches to problem-solving, and results to elucidate the difficulties, which had blocked the development of the theory. We also hope that the reader might appreciate the beauty of these works, which came so close to

their achievement.

*Modulation in Electronics and Telecommunications*

permit to improve their performance criteria.

**3.1 General principles of FCS transmission**

success, but which are now almost forgotten.

*General block diagrams of point-to-point AFCS.*

task.

**Figure 1.**

**6**

FCS begins a new cycle of transmission. After *n* cycles, the final estimate *x*^*<sup>n</sup>* is routed to the addressee, while the system recovers its initial state and begins transmission of the next sample.

where *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

tion of the optimization task:

determined by the formula:

where parameter *Q*<sup>2</sup>

channel.

where *P*min

**9**

<sup>0</sup> and *<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> <sup>E</sup> *<sup>x</sup>*j~*y<sup>k</sup>*

It cannot be greater than the power *S*<sup>0</sup> of the transmitter:

*E y*<sup>2</sup> *k* � �≤ *S*<sup>0</sup> or

*Mk* ≤

*<sup>k</sup>* <sup>¼</sup> *AMkPk*�<sup>1</sup> *σ*2 *<sup>ξ</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup>

*Pk* <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

*<sup>k</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>2</sup>

*Mopt <sup>k</sup>* <sup>¼</sup> <sup>1</sup> *A*<sup>0</sup>

ary of MSE values described by the relationship:

*σ*2 *<sup>ξ</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup>

*M*<sup>2</sup> *k=σ*<sup>2</sup>

*M*<sup>2</sup> *kPk*�<sup>1</sup>

*<sup>ξ</sup>Pk*�<sup>1</sup>

MSE is achievable in practice only if these gains *Mk* are set to the values:

ffiffiffiffiffiffiffiffiffi *S*0 *P*min *k*�1

s

*M*<sup>2</sup> *kPk*�<sup>1</sup>

Formula (17) shows that greater values of modulation index *Mk* decrease the MSE *Pk*, that is, improve the quality of transmission. However, the increase of the values *Mk* is limited by condition (15), and the theoretically achievable minimum of

> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffi *P*min *k*�1 <sup>q</sup> ; *<sup>M</sup>opt*

The result of the replacement determines the theoretically achievable lower bound-

*<sup>k</sup>*�<sup>1</sup> is defined by Eq. (17) with the values *Mk* set to the values in (18).

1 *A*<sup>0</sup>

set of permissible values of modulation index *Mk*:

*Lopt*

corresponding minimal values of MSE:

Eq. (8) takes an extremely simple form(12)).

*DOI: http://dx.doi.org/10.5772/intechopen.90516*

1

*Perfect Signal Transmission Using Adaptive Modulation and Feedback*

Free parameters *Mk*, *Lk* in the right-hand side of (13) do not depend explicitly on the previous values of gains *M1,...,Mk*, *L1,...,Lk*, which allows the following formula-

*For each k = 1,...,n, one should find values* of *the gains M1,...,Mk*, *L1,...,Lk*, *which minimize, under additional conditions and constrains, the MSE of transmission* (13)*.* Beginning the first works, the most widely used additional condition was (and still remains) a constraint on the instant or average power of emitted signals *Wsign*

> 1 *n* X*n k*¼1

Without loss of generality, one may assume that the power of the FT transmitter

and the amplitude of emitted signals are connected by the relationship *<sup>S</sup>*<sup>0</sup> <sup>¼</sup> *<sup>A</sup>*<sup>2</sup>

s

this case, substitution of formula (6) into (14) directly gives the expression for the

ffiffiffiffiffiffiffiffiffi *S*0 *Pk*�<sup>1</sup>

According to (13), for every *Mk* satisfying the inequality (15), MSE of transmission depends on the gains *Lk*, which in turn determine the rate of convergence of estimates *x*^*<sup>k</sup>* the input value *x*. The extremum of MSE in the set of *Lk* under definite *Mk*not violating condition (15) can be easily found, and the point of extremum is

> <sup>¼</sup> <sup>1</sup> *AMk*

> > <sup>¼</sup> *Pk*�<sup>1</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup>

The substitution of (16) into (13) gives the following recurrence equation for the

<sup>1</sup> � *Pk Pk*�<sup>1</sup>

*kPk*�<sup>1</sup>

*<sup>ξ</sup>* describes the SNR at the output of the forward

<sup>1</sup> <sup>¼</sup> <sup>1</sup> *σ*0

*E y*<sup>2</sup> *k*

<sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffi *Pk*�<sup>1</sup>

� �≤ *S*0*:* (14)

p *:* (15)

� �*:* (16)

, (17)

, (18)

� � are optimal Bayesian estimates (in this case

<sup>0</sup> .

0. In

#### **3.2 Principles and particularities of FCS and AFCS optimization**

All the results of pioneering and later research in FCS optimization were obtained using *linear* models of FT transmitters, and this section presents the basic idea of these researches, as well techniques and results of FCS optimization for forward transmitters described by the linear model (7).

Models (6)–(8) are not abstract and describe the sequence of transformations of the signal along its transition over the real units and components of FCS, as well the influence of noises and distortions on the final result of transmission. Moreover, each of these models allows calculation of the changes in the statistical characteristics of signals after each subsequent transformation, and considers the most substantial particularities of this process influencing the work of the system.

Apart from the initially known (given) parameters, these models contain free parameters permitting the designers to regulate the work of particular units and improve the performance of the overall system. For the FCS under consideration, these parameters are *M*1, … , *Mn* and *L*1, … , *Ln*. The basic criterion of the transmission quality is the accuracy ("fidelity" in [1]) of recovery of the signal that is MSE of its estimates *Pk* ¼ E ð Þ *x* � *x*^*<sup>k</sup>* <sup>2</sup> h i.

The optimization of FCS begins from the definition of algorithm permitting to compute, using the received data ~*y<sup>k</sup>* <sup>1</sup> , optimal estimates *<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> *<sup>x</sup>*^*<sup>k</sup>* <sup>~</sup>*y<sup>k</sup>* 11 � � minimizing the MSE *Pk* for each *k* ¼ 1, … , *n*. According to Bayesian estimation theory (see [33, 34]), in the Gaussian case, these are conditional averages *<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> *E x*j~*y<sup>k</sup>* 1 � �of random values observed in the presence of AWGN. Moreover, residuals *ek* ¼ *x* � *x*^*<sup>k</sup>* and values ~*yk* of the received signal have zero mean values and are mutually orthogonal [34]:

$$\operatorname{E}\left(e\_{k}|\bar{\mathbf{y}}\_{1}^{k-1}\right) = \operatorname{E}\left(\boldsymbol{\varkappa}|\bar{\mathbf{y}}\_{1}^{k-1}\right) - \hat{\boldsymbol{\varkappa}}\_{k-1} = \mathbf{0};\tag{9}$$

$$\mathbb{E}\left(\tilde{\boldsymbol{\jmath}}\_{k}|\tilde{\boldsymbol{\jmath}}\_{1}^{k-1}\right) = \boldsymbol{\Lambda}\boldsymbol{\mathcal{M}}\_{k}\mathbb{E}\left(\boldsymbol{e}\_{k}|\tilde{\boldsymbol{\jmath}}\_{1}^{k-1}\right) + \mathbb{E}\left(\boldsymbol{\xi}\_{k}|\tilde{\boldsymbol{\jmath}}\_{1}^{k-1}\right) = \mathbf{0};\tag{10}$$

$$\operatorname{E}(e\_k e\_m) = P\_k \delta\_{mk}; \; \operatorname{E}\left(\tilde{\mathcal{Y}}\_k \tilde{\mathcal{Y}}\_m\right) = \left(\sigma\_\xi^2 + A^2 \mathcal{M}\_k^2\right) \delta\_{mk},\tag{11}$$

where δ*mk* = 1 for *m* ¼ *k* and δ*mk* = 0 for *m* 6¼ *k*.

Substitution of (10) into (8) results in algorithm computing optimal Bayesian estimates that takes an extremely simple form**:**

$$
\hat{\mathfrak{x}}\_{k} = \hat{\mathfrak{x}}\_{k-1} + L\_{k}\tilde{\mathfrak{y}}\_{k}.\tag{12}
$$

The full transmission-reception algorithm (6)–(8) permits to build a mathematical model of transmission process and to derive the following algorithm for calculation of the mean square error (MSE) of estimates formed by FCS in sequential cycles.

$$\begin{split} P\_k &= \mathbb{E}\left[ \left( \mathbf{x} - \hat{\mathbf{x}}\_k \right)^2 \right] = \mathbb{E}[(\mathbf{1} - AM\_k L\_k) \left( \hat{\mathbf{x}}\_{k-1} - \mathbf{x} \right) + L\_k \xi\_k]^2 = \\ &= \left( \mathbf{1} - AM\_k L\_k \right)^2 P\_{k-1} + L\_k^2 \sigma\_\xi^2, \end{split} \tag{13}$$

*Perfect Signal Transmission Using Adaptive Modulation and Feedback DOI: http://dx.doi.org/10.5772/intechopen.90516*

where *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup> <sup>0</sup> and *<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> <sup>E</sup> *<sup>x</sup>*j~*y<sup>k</sup>* 1 � � are optimal Bayesian estimates (in this case Eq. (8) takes an extremely simple form(12)).

Free parameters *Mk*, *Lk* in the right-hand side of (13) do not depend explicitly on the previous values of gains *M1,...,Mk*, *L1,...,Lk*, which allows the following formulation of the optimization task:

*For each k = 1,...,n, one should find values* of *the gains M1,...,Mk*, *L1,...,Lk*, *which minimize, under additional conditions and constrains, the MSE of transmission* (13)*.*

Beginning the first works, the most widely used additional condition was (and still remains) a constraint on the instant or average power of emitted signals *Wsign* <sup>0</sup> . It cannot be greater than the power *S*<sup>0</sup> of the transmitter:

$$E\left[\mathbf{y}\_k^2\right] \le \mathbf{S}\_0 \quad \text{or} \quad \frac{\mathbf{1}}{n} \sum\_{k=1}^n E\left[\mathbf{y}\_k^2\right] \le \mathbf{S}\_0. \tag{14}$$

Without loss of generality, one may assume that the power of the FT transmitter and the amplitude of emitted signals are connected by the relationship *<sup>S</sup>*<sup>0</sup> <sup>¼</sup> *<sup>A</sup>*<sup>2</sup> 0. In this case, substitution of formula (6) into (14) directly gives the expression for the set of permissible values of modulation index *Mk*:

$$M\_k \le \frac{1}{A\_0} \sqrt{\frac{S\_0}{P\_{k-1}}} = \frac{1}{\sqrt{P\_{k-1}}}.\tag{15}$$

According to (13), for every *Mk* satisfying the inequality (15), MSE of transmission depends on the gains *Lk*, which in turn determine the rate of convergence of estimates *x*^*<sup>k</sup>* the input value *x*. The extremum of MSE in the set of *Lk* under definite *Mk*not violating condition (15) can be easily found, and the point of extremum is determined by the formula:

$$L\_k^{opt} = \frac{AM\_k P\_{k-1}}{\sigma\_\xi^2 + A^2 M\_k^2 P\_{k-1}} = \frac{1}{AM\_k} \left(1 - \frac{P\_k}{P\_{k-1}}\right). \tag{16}$$

The substitution of (16) into (13) gives the following recurrence equation for the corresponding minimal values of MSE:

$$P\_k = \frac{\sigma\_\xi^2 P\_{k-1}}{\sigma\_\xi^2 + A^2 M\_k^2 P\_{k-1}} = \frac{P\_{k-1}}{1 + Q\_k^2 P\_{k-1}},\tag{17}$$

where parameter *Q*<sup>2</sup> *<sup>k</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>2</sup> *M*<sup>2</sup> *k=σ*<sup>2</sup> *<sup>ξ</sup>* describes the SNR at the output of the forward channel.

Formula (17) shows that greater values of modulation index *Mk* decrease the MSE *Pk*, that is, improve the quality of transmission. However, the increase of the values *Mk* is limited by condition (15), and the theoretically achievable minimum of MSE is achievable in practice only if these gains *Mk* are set to the values:

$$\boldsymbol{M}\_{k}^{opt} = \frac{\mathbf{1}}{A\_0} \sqrt{\frac{\mathbf{S}\_0}{P\_{k-1}^{\min}}} = \frac{\mathbf{1}}{\sqrt{P\_{k-1}^{\min}}} ; \quad \boldsymbol{M}\_1^{opt} = \frac{\mathbf{1}}{\sigma\_0} \,, \tag{18}$$

where *P*min *<sup>k</sup>*�<sup>1</sup> is defined by Eq. (17) with the values *Mk* set to the values in (18). The result of the replacement determines the theoretically achievable lower boundary of MSE values described by the relationship:

FCS begins a new cycle of transmission. After *n* cycles, the final estimate *x*^*<sup>n</sup>* is routed to the addressee, while the system recovers its initial state and begins

All the results of pioneering and later research in FCS optimization were obtained using *linear* models of FT transmitters, and this section presents the basic idea of these researches, as well techniques and results of FCS optimization for

Models (6)–(8) are not abstract and describe the sequence of transformations of the signal along its transition over the real units and components of FCS, as well the influence of noises and distortions on the final result of transmission. Moreover, each of these models allows calculation of the changes in the statistical characteristics of signals after each subsequent transformation, and considers the most substantial particularities of this process influencing the work of the system.

Apart from the initially known (given) parameters, these models contain free parameters permitting the designers to regulate the work of particular units and improve the performance of the overall system. For the FCS under consideration, these parameters are *M*1, … , *Mn* and *L*1, … , *Ln*. The basic criterion of the transmission quality is the accuracy ("fidelity" in [1]) of recovery of the signal that is MSE of

The optimization of FCS begins from the definition of algorithm permitting to

the MSE *Pk* for each *k* ¼ 1, … , *n*. According to Bayesian estimation theory (see [33, 34]), in the Gaussian case, these are conditional averages *<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> *E x*j~*y<sup>k</sup>*

dom values observed in the presence of AWGN. Moreover, residuals *ek* ¼ *x* � *x*^*<sup>k</sup>* and values ~*yk* of the received signal have zero mean values and are mutually

1

1

Substitution of (10) into (8) results in algorithm computing optimal Bayesian

The full transmission-reception algorithm (6)–(8) permits to build a mathemat-

<sup>¼</sup> E 1ð Þ � *AMkLk* ð Þþ *<sup>x</sup>*^*<sup>k</sup>*�<sup>1</sup> � *<sup>x</sup> Lkξ<sup>k</sup>* ½ �<sup>2</sup> <sup>¼</sup>

ical model of transmission process and to derive the following algorithm for calculation of the mean square error (MSE) of estimates formed by FCS in

> *kσ*2 *ξ*,

� � <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

� � <sup>þ</sup> <sup>E</sup> *<sup>ξ</sup>k*j~*y<sup>k</sup>*�<sup>1</sup>

� � <sup>¼</sup> <sup>E</sup> *<sup>x</sup>*j~*y<sup>k</sup>*�<sup>1</sup>

� � <sup>¼</sup> *AMk*<sup>E</sup> *ek*j~*y<sup>k</sup>*�<sup>1</sup>

Eð Þ¼ *ekem Pk*δ*mk*; E ~*yk*~*ym*

where δ*mk* = 1 for *m* ¼ *k* and δ*mk* = 0 for *m* 6¼ *k*.

estimates that takes an extremely simple form**:**

<sup>1</sup> , optimal estimates *<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> *<sup>x</sup>*^*<sup>k</sup>* <sup>~</sup>*y<sup>k</sup>*

11 � �

� � <sup>¼</sup> 0; (10)

δ*mk*, (11)

(13)

� � � *<sup>x</sup>*^*<sup>k</sup>*�<sup>1</sup> <sup>¼</sup> 0; (9)

1

*M*<sup>2</sup> *k*

*x*^*<sup>k</sup>* ¼ *x*^*<sup>k</sup>*�<sup>1</sup> þ *Lk*~*yk:* (12)

*<sup>ξ</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup>

� �

minimizing

1 � �of ran-

**3.2 Principles and particularities of FCS and AFCS optimization**

forward transmitters described by the linear model (7).

<sup>2</sup> h i

.

<sup>E</sup> *ek*j~*y<sup>k</sup>*�<sup>1</sup> 1

<sup>E</sup> <sup>~</sup>*yk*j~*y<sup>k</sup>*�<sup>1</sup> 1

transmission of the next sample.

*Modulation in Electronics and Telecommunications*

its estimates *Pk* ¼ E ð Þ *x* � *x*^*<sup>k</sup>*

orthogonal [34]:

sequential cycles.

**8**

*Pk* ¼ E ð Þ *x* � *x*^*<sup>k</sup>*

<sup>2</sup> h i

2

*Pk*�<sup>1</sup> <sup>þ</sup> *<sup>L</sup>*<sup>2</sup>

¼ ð Þ 1 � *AMkLk*

compute, using the received data ~*y<sup>k</sup>*

$$P\_k^{\min} = \sigma\_0^2 \left(\mathbb{1} + Q^2\right)^{-k}; P\_0 = \sigma\_0^2,\tag{19}$$

In the following sections, we show that modulation and feedback may resolve or

Signals *ek* ¼ *x* � *x*^*k*�<sup>1</sup> at the input of AM modulators repeat the error of the sample estimate formed in BSt in previous cycle, and their variance is equal to the MSE *Pk*. This allows the modulation gains to be set, in each cycle, to greater and greater values, which increases SNR at the output of the forward channel and provides superfast

However, externally completely correct additional condition (14) does not count possible saturation of the modulators or emitters, if the signals *ek* ¼ *x* � *x*^*k*�<sup>1</sup> exceed their linear range, and an adequate model of the transmitter is to have the form (see

In real FT, output range ½ � �*A*0, *A*<sup>0</sup> is fixed except in particular cases, and the width of its input range depends on the value of modulation gain *Mk*. Setting the values *Mk* and omitting a consideration of statistics of the signals *ek* ¼ *x* � *x*^*<sup>k</sup>*�<sup>1</sup> excludes a possibility of considering saturation of the FT, which appears, if the signal *yk* ¼ *A*0*Mkek* crosses the boundaries of its output range. It is worth adding that each saturation during the sample transmission distracts its estimate and causes

The probability of the first saturation of FT would appear in the *k*-th cycle and can be easily evaluated: both signals *ek* and *yk* are zero mean Gaussian values, and their variances are known. The not complex calculations yield the following relationship:

> *k* 2*Pk*�<sup>1</sup>

1

� �d*ek* <sup>¼</sup> <sup>1</sup> � <sup>2</sup><sup>Φ</sup>

<sup>1</sup> Þ ¼ Prðj*ek*j≥1*=Mk*<sup>j</sup> <sup>~</sup>*y<sup>k</sup>*�<sup>1</sup>

exp � *<sup>e</sup>*<sup>2</sup>

� �

*Mkek* if *M*^ *<sup>k</sup>*∣*ek*∣ ≤1 sign ð Þ *ek* if *<sup>M</sup>*^ *<sup>k</sup>*∣*ek*∣><sup>1</sup>

*:* (24)

1 *MkPk*�<sup>1</sup>

� �, (25)

at least substantially reduce these problems and make the perfect FCS feasible.

growth of the accuracy of the estimates unachievable without feedback.

(

*yk* ¼ *A*<sup>0</sup>

irreversible loss of information about the sample value.

ð 1*=Mk*

�1*=Mk*

*Static transition characteristic of the transmitter with a finite output range.*

where Φð Þ *x* is a tabulated Gaussian integral (5).

*3.2.1 Influence of FT saturation and its elimination*

*DOI: http://dx.doi.org/10.5772/intechopen.90516*

*Perfect Signal Transmission Using Adaptive Modulation and Feedback*

also **Figure 2**):

Pr*sat*

**Figure 2.**

**11**

*<sup>k</sup>* <sup>¼</sup> Pr <sup>j</sup>*yk*j<sup>≥</sup> *<sup>A</sup>*0<sup>j</sup> <sup>~</sup>*y<sup>k</sup>*�<sup>1</sup>

<sup>¼</sup> <sup>1</sup> � <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*πPk*�<sup>1</sup> p

where SNR *Q*<sup>2</sup> *<sup>k</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>2</sup> *M*<sup>2</sup> *k=σ*<sup>2</sup> *<sup>ξ</sup>* at the forward channel output is constant for each cycle of the sample transmission and takes the values:

$$\text{SNR}\_k = Q\_k^2 = \frac{\mathcal{W}^{\text{ign}}}{\sigma\_\xi^2} = \frac{1}{N\_\xi F\_0} \left(\frac{A\_0 \gamma}{d}\right)^2 = Q^2. \tag{20}$$

*Claim 1:* Relationships (6)–(8) with the parameters *Mk*, *Lk* set, for each *k* ¼ 1, … , *n,* to the values (16) and (18) determine the optimal transmission-reception algorithm, which contains the information permitting us to design optimal FCS transmitting signals with maximal accuracy (minimal MSE), and this boundary is determined by formula (19). Greater accuracy is not feasible.

Moreover, the optimal transmission-reception algorithm permits us to compute the information characteristics of optimal AFCS prior and posterior entropies, as well as the mean amount of information in estimates *x*^*<sup>k</sup>* regarding the values of input samples *x*:

$$H(X) = \frac{1}{2} \log\_2(2\pi e \sigma\_0^2); H(X|\hat{X}\_k) = \frac{1}{2} \log\_2(2\pi e P\_k);\tag{21}$$

$$I(\mathbf{X}, \hat{\mathbf{X}}\_k) = H(\mathbf{X}) - H(\mathbf{X}|\hat{\mathbf{X}}\_k) = \frac{1}{2} \log\_2 \left(\frac{\sigma\_0^2}{P\_k}\right). \tag{22}$$

Taking into the account that the amount of information (22) in estimates is achieved in *k* cycles, that is, during the time *Tk* ¼ *kΔt*0, formulas (22) and (19) permit us to evaluate the current values of bit rate of the signal transmission:

$$\begin{split} R\_{\text{max}}^{\text{FCS}} &= \frac{I\_{\text{max}}\left(\mathbf{X}, \hat{\mathbf{X}}\_k\right)}{k\Delta t\_0} = F\_0 \log\_2\left(\frac{\sigma\_0^2}{P\_k^{\text{min}}}\right) = \\ &= F\_0 \log\_2\left(1 + Q^2\right) = F\_0 \log\_2\left(1 + \frac{W^{\text{sign}}}{N\_\xi F\_0}\right) = C[\text{bit/s}]. \end{split} \tag{23}$$

*Claim 2*. Formula (23) is identical to Shannon's formula (1) and determines the limit bit rate of transmission, that is, the capacity of the system. Moreover, attaining the boundary (23) means that spectral and energy efficiencies of the FCS also attain their limit values and are connected through Shannon's relationship (3). Let us stress that, unlike CSC, the presented relationships determine the approach to the optimal FCS design.

Similar results were obtained in [4, 6, 9] and other works. However, regardless of their correctness, neither the above nor earlier obtained algorithms of transmission in any of their versions could be implemented in practice. Analysis of the reasons showed that the main reason was the omission of saturation effects in the FT.

Another, not less critical reason, and not counted in all formulations of the optimization tasks, has been noted in Section 2, that is, the local dependence of the quality of transmission on the scenario of FCS application. The presented above results confirm this fact directly: values of optimal parameters *Mk*, *Lk* as well as all other characteristics of optimal FCS depend on the distance between the FT and the BSt, so that and changes in the FCS's position or surrounding violate the perfect mode of transmission.

In the following sections, we show that modulation and feedback may resolve or at least substantially reduce these problems and make the perfect FCS feasible.

#### *3.2.1 Influence of FT saturation and its elimination*

*P*min *<sup>k</sup>* <sup>¼</sup> <sup>σ</sup><sup>2</sup>

*M*<sup>2</sup> *k=σ*<sup>2</sup>

cycle of the sample transmission and takes the values:

*SNRk* <sup>¼</sup> *<sup>Q</sup>*<sup>2</sup>

determined by formula (19). Greater accuracy is not feasible.

<sup>2</sup> log <sup>2</sup> <sup>2</sup>*πeσ*<sup>2</sup>

� � <sup>¼</sup> *H X*ð Þ� *H X*j*X*^ *<sup>k</sup>*

¼ *F*<sup>0</sup> log <sup>2</sup>

<sup>¼</sup> *<sup>F</sup>*<sup>0</sup> log <sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>F</sup>*<sup>0</sup> log <sup>2</sup> <sup>1</sup> <sup>þ</sup>

0 � �; *H X*j*X*^ *<sup>k</sup>*

Taking into the account that the amount of information (22) in estimates is achieved in *k* cycles, that is, during the time *Tk* ¼ *kΔt*0, formulas (22) and (19) permit us to evaluate the current values of bit rate of the signal transmission:

> σ2 0 *P*min *k*

*Claim 2*. Formula (23) is identical to Shannon's formula (1) and determines the limit bit rate of transmission, that is, the capacity of the system. Moreover, attaining the boundary (23) means that spectral and energy efficiencies of the FCS also attain their limit values and are connected through Shannon's relationship (3). Let us stress that, unlike CSC, the presented relationships determine the approach to the

Similar results were obtained in [4, 6, 9] and other works. However, regardless of their correctness, neither the above nor earlier obtained algorithms of transmission in any of their versions could be implemented in practice. Analysis of the reasons showed that the main reason was the omission of saturation effects in the FT. Another, not less critical reason, and not counted in all formulations of the optimization tasks, has been noted in Section 2, that is, the local dependence of the quality of transmission on the scenario of FCS application. The presented above results confirm this fact directly: values of optimal parameters *Mk*, *Lk* as well as all other characteristics of optimal FCS depend on the distance between the FT and the BSt, so that and changes in the FCS's position or surrounding violate the perfect

¼

*Wsign NξF*<sup>0</sup> � �

!

*<sup>k</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>2</sup>

*Modulation in Electronics and Telecommunications*

*H X*ð Þ¼ <sup>1</sup>

max <sup>¼</sup> *<sup>I</sup>*max *<sup>X</sup>*,*X*^ *<sup>k</sup>*

*I X*,*X*^ *<sup>k</sup>*

� � *kΔt*<sup>0</sup>

where SNR *Q*<sup>2</sup>

input samples *x*:

*RFCS*

optimal FCS design.

mode of transmission.

**10**

<sup>0</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> � ��*<sup>k</sup>*

<sup>¼</sup> <sup>1</sup> *NξF*<sup>0</sup>

Moreover, the optimal transmission-reception algorithm permits us to compute the information characteristics of optimal AFCS prior and posterior entropies, as well as the mean amount of information in estimates *x*^*<sup>k</sup>* regarding the values of

� � <sup>¼</sup> <sup>1</sup>

� � <sup>¼</sup> <sup>1</sup>

<sup>2</sup> log <sup>2</sup>

*σ*2 0 *Pk* � �

¼ *C*½ � bit*=*s *:*

*Claim 1:* Relationships (6)–(8) with the parameters *Mk*, *Lk* set, for each *k* ¼ 1, … , *n,* to the values (16) and (18) determine the optimal transmission-reception algorithm, which contains the information permitting us to design optimal FCS transmitting signals with maximal accuracy (minimal MSE), and this boundary is

*<sup>k</sup>* <sup>¼</sup> *<sup>W</sup>sign* σ2 *ξ*

; *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

*A*0*γ d* � �<sup>2</sup>

*<sup>ξ</sup>* at the forward channel output is constant for each

<sup>¼</sup> *<sup>Q</sup>*<sup>2</sup>

0, (19)

<sup>2</sup> log <sup>2</sup>ð Þ <sup>2</sup>*πePk* ; (21)

*:* (22)

(23)

*:* (20)

Signals *ek* ¼ *x* � *x*^*k*�<sup>1</sup> at the input of AM modulators repeat the error of the sample estimate formed in BSt in previous cycle, and their variance is equal to the MSE *Pk*. This allows the modulation gains to be set, in each cycle, to greater and greater values, which increases SNR at the output of the forward channel and provides superfast growth of the accuracy of the estimates unachievable without feedback.

However, externally completely correct additional condition (14) does not count possible saturation of the modulators or emitters, if the signals *ek* ¼ *x* � *x*^*k*�<sup>1</sup> exceed their linear range, and an adequate model of the transmitter is to have the form (see also **Figure 2**):

$$y\_k = A\_0 \begin{cases} \mathcal{M}\_k e\_k & \text{if } \hat{\mathcal{M}}\_k |e\_k| \le 1 \\ \text{sign } (e\_k) & \text{if } \hat{\mathcal{M}}\_k |e\_k| > 1 \end{cases} \tag{24}$$

In real FT, output range ½ � �*A*0, *A*<sup>0</sup> is fixed except in particular cases, and the width of its input range depends on the value of modulation gain *Mk*. Setting the values *Mk* and omitting a consideration of statistics of the signals *ek* ¼ *x* � *x*^*<sup>k</sup>*�<sup>1</sup> excludes a possibility of considering saturation of the FT, which appears, if the signal *yk* ¼ *A*0*Mkek* crosses the boundaries of its output range. It is worth adding that each saturation during the sample transmission distracts its estimate and causes irreversible loss of information about the sample value.

The probability of the first saturation of FT would appear in the *k*-th cycle and can be easily evaluated: both signals *ek* and *yk* are zero mean Gaussian values, and their variances are known. The not complex calculations yield the following relationship:

$$\begin{split} \text{Pr}\_{k}^{\text{sat}} &= \text{Pr} \left( |\boldsymbol{\nu}\_{k}| \ge A\_{0} \, |\, \check{\boldsymbol{\jmath}}\_{1}^{k-1} \right) = \text{Pr} (|\boldsymbol{\varepsilon}\_{k}| \ge \mathbf{1}/M\_{k} \, |\, \check{\boldsymbol{\jmath}}\_{1}^{k-1} \,) \\ &= \mathbf{1} - \frac{\mathbf{1}}{\sqrt{2\pi P\_{k-1}}} \int\_{-1/M\_{k}}^{1/M\_{k}} \exp\left( -\frac{\boldsymbol{\sigma}\_{k}^{2}}{2P\_{k-1}} \right) \text{d}\boldsymbol{\varepsilon}\_{k} = \mathbf{1} - 2\Phi \left( \frac{\mathbf{1}}{M\_{k}P\_{k-1}} \right), \end{split} \tag{25}$$

where Φð Þ *x* is a tabulated Gaussian integral (5).

#### **Figure 2.** *Static transition characteristic of the transmitter with a finite output range.*

Substitution of formula (18) into (25) gives the following evaluation of the probability of the first saturation in *k*-th cycle of transmission, beginning with the first cycle:

where parameter *a* satisfies the equation:

*DOI: http://dx.doi.org/10.5772/intechopen.90516*

always in linear mode.

channel:

emitted signals (29).

(adaptive FCS).

*P*min *<sup>k</sup>* ¼

*σ*2

**13**

*<sup>W</sup>sign* <sup>¼</sup> *<sup>A</sup>*<sup>2</sup>

transmission, which takes the form:

*M*<sup>2</sup> *kσ*2 *ν* � �*P*min

> *M*<sup>2</sup> *<sup>k</sup> σ*<sup>2</sup>

*k*�1

*<sup>ν</sup>* <sup>þ</sup> *<sup>P</sup>*min *k*�1

*σ*2 *<sup>ξ</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup>

*σ*2 *<sup>ξ</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup>

where variable *σ*<sup>2</sup>

but also:

*Φ α*ð Þ¼ <sup>1</sup>

*Perfect Signal Transmission Using Adaptive Modulation and Feedback*

ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> ð *α*

0

a. reduces *α* times the amplitude of emitted signals: *A*<sup>0</sup> ! *A*0*=α*,

*<sup>α</sup>*<sup>2</sup> ; *SNRk* <sup>¼</sup> *<sup>Q</sup>*<sup>2</sup>

absence of coders reduces their energy consumption.

exp � *<sup>z</sup>*<sup>2</sup> 2

*Claim 4.* Setting the modulation index to the values (27) prevents the appearance of saturation and the FCS with non-linear FT transmits the signals almost

*Claim 5.* Setting the gains *Mk* to the values (27) not only removes saturation

b. reduces *α*<sup>2</sup> times the power of emitted signals and SNR at the output of the

*<sup>k</sup>* <sup>¼</sup> *<sup>W</sup>sign* σ2 *ξ*

The structure and form of the basic relationships for the MSE, bit rate, and effectiveness remain the same. Changes in Shannon's formula (23) for the capacity and in the other relationships affect only the values of amplitude and power of

*Claim 6*. Reduction of the power of the emitted signal decreases the SNR and capacity of the system but makes the perfect FCS feasible, and these systems transmit signals with limit energy-spectral efficiency optimally, as well as

completely utilizing the frequency and energy resources of the FT. Moreover, the

To distinguish the considered systems and the FCS, and to focus on the new systems, in what follows, we use a new abbreviation to refer to these systems: AFCS

Let us remind that these relationships were derived under assumption that the feedback channels are ideal. This is not an unrealistic suggestion: relatively inexpensive modern CSCs provide virtually error-free short-range transmission. If necessary, the quality of the channel can also be improved by increasing the power of transmitters—the BSt have sufficiently large, if not unlimited energy resource.

We should add that the discussed relationships are a particular case of the more general optimal transmission-reception algorithm for the statistically fitted AFCS with noisy feedback [36–39], see also **Table 1** below. It is worth stressing that these algorithms have the same structure and form as those presented in pervious sections. The only but principle difference concerns the expression for the MSE of

� � <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> � ��<sup>1</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> *<sup>σ</sup>*<sup>2</sup>

transmitted from the BSt to FT over the feedback channel with AWGN. Under

*<sup>ν</sup>* ¼ 0, this formula coincides with (17). Analysis of formula (30) gives comprehensive answers to many questions concerning the behavior of feedback systems.

<sup>¼</sup> <sup>1</sup> *NξF*<sup>0</sup>

*γA*<sup>0</sup> *αd* � �<sup>2</sup>

*v*

*<sup>k</sup>*�1, (30)

� � " #*P*min

*σ*2 *<sup>ν</sup>* <sup>þ</sup> *<sup>P</sup>*min *k*�1

*<sup>ν</sup>* is the variance of the errors *vk* in the signals (controls) *x*^*<sup>k</sup>*

<sup>¼</sup> *<sup>Q</sup>*<sup>2</sup>

*:* (29)

� �dz <sup>¼</sup> <sup>1</sup> � *<sup>μ</sup>*

<sup>2</sup> *:* (28)

$$\text{Pr}\_k^{sat} = \mathbf{1} - 2\Phi(\mathbf{1}) \approx \mathbf{1} - \mathbf{0.68} = \mathbf{0.32},$$

and the probability of its appearance during first five cycles of transmission attains the value Pr*sat* 1÷5 <sup>¼</sup> <sup>1</sup> � <sup>1</sup> � *Psat* ð Þ<sup>5</sup> <sup>¼</sup> <sup>1</sup> � <sup>0</sup>*:*68<sup>5</sup> ≈0*:*85 and quickly tends to unity in next cycles.

One should add that MSE of estimates is weakly sensitive to sufficiently rare cases of FT saturation. However, taking into account that each instance of saturation causes a loss of the sample and *I X*, *X*^ *<sup>n</sup>* � � bits of information, the probability of saturation determines the mean percent of erroneous bits in binary sequences delivered to the addressee. These losses can be considered as the BER of transmission (rather bit word error rate—WER but numerically these values are equal). Value of the BER is one of the key characteristics of CSC, and the general tendency in modern communications is to decrease its values to 10�6÷10�<sup>8</sup> and lesser. Therefore, the setting of modulation index requires closest attention. Investigations showed that the severity of this problem can be sufficiently reduced by employing the feedback channel.

*Claim 3*. The linear models of the transmitters that are commonly used in formulations of optimization tasks and constraint on the mean power of emitted signals describe the work of transmitters inadequately. This has been and still remains one of the main reasons why FCSs have not been implemented in practice. The solution of this problem allowed for the application of the "statistical fitting condition" [35].

#### *3.2.2 Statistical fitting condition*

The previous section and formula (25) show that saturation of the FT can be almost eliminated if the gains *Mk* are set to the values that guarantee the probability of saturation not greater than a given small *<sup>μ</sup>*< <1, (e.g., *<sup>μ</sup>* � <sup>10</sup>�4÷10�8) for every sequence ~*y<sup>k</sup>*�<sup>1</sup> <sup>1</sup> :

$$\Pr\_k^{\text{sat}} = \Pr\left( |y\_k| \ge A \diamond |\bar{y}\_1^{k-1}\right) = 1 - \int\_{-1/M\_k}^{1/M\_k} p\left(e\_k |\bar{y}\_1^{k-1}\right) \text{d}e\_k =: 1 - 2\Phi\left(\frac{1}{M\_k P\_{k-1}}\right) \le \mu. \tag{26}$$

Under fulfilled condition (25), the probability of the sample saturation has the value 1 � ð Þ <sup>1</sup> � *<sup>μ</sup> <sup>n</sup>* ≈*nμ*< <1. This means almost always (at the confidence level 1 � *nμ*), the FCS would work in optimal mode without saturations, and the percent of erroneous bits would not be greater than *μ*.

Relationship (26) determines the set of permissible values of the gains *Mk* and its upper boundary. These values make (26) the equation which, after replacement of the values 1*=MkPk*�<sup>1</sup> by the variable *α*, gives the following relationships for the permissible maximal (optimal) values of modulation gains:

$$\mathcal{M}\_k^{opt} = \frac{1}{a\sqrt{P\_{k-1}^{\min}}} ; (k = 1, \dots, n), \tag{27}$$

*Perfect Signal Transmission Using Adaptive Modulation and Feedback DOI: http://dx.doi.org/10.5772/intechopen.90516*

where parameter *a* satisfies the equation:

Substitution of formula (18) into (25) gives the following evaluation of the probability of the first saturation in *k*-th cycle of transmission, beginning with the

*<sup>k</sup>* ¼ 1 � 2Φð Þ1 ≈1 � 0*:*68 ¼ 0*:*32,

≈0*:*85 and quickly tends to unity

� � bits of information, the probability of

and the probability of its appearance during first five cycles of transmission

One should add that MSE of estimates is weakly sensitive to sufficiently rare cases of FT saturation. However, taking into account that each instance of satura-

saturation determines the mean percent of erroneous bits in binary sequences delivered to the addressee. These losses can be considered as the BER of transmission (rather bit word error rate—WER but numerically these values are equal). Value of the BER is one of the key characteristics of CSC, and the general tendency in modern communications is to decrease its values to 10�6÷10�<sup>8</sup> and lesser. Therefore, the setting of modulation index requires closest attention. Investigations showed that the severity of this problem can be sufficiently reduced by employing

*Claim 3*. The linear models of the transmitters that are commonly used in formulations of optimization tasks and constraint on the mean power of emitted signals describe the work of transmitters inadequately. This has been and still remains one of the main reasons why FCSs have not been implemented in practice. The solution of this problem allowed for the application of the "statistical fitting

The previous section and formula (25) show that saturation of the FT can be almost eliminated if the gains *Mk* are set to the values that guarantee the probability of saturation not greater than a given small *<sup>μ</sup>*< <1, (e.g., *<sup>μ</sup>* � <sup>10</sup>�4÷10�8) for every

> *p ek*j~*y<sup>k</sup>*�<sup>1</sup> 1

Under fulfilled condition (25), the probability of the sample saturation has the

1 � *nμ*), the FCS would work in optimal mode without saturations, and the percent

ffiffiffiffiffiffiffiffiffi *P*min *k*�1

Relationship (26) determines the set of permissible values of the gains *Mk* and its upper boundary. These values make (26) the equation which, after replacement of the values 1*=MkPk*�<sup>1</sup> by the variable *α*, gives the following relationships for the

≈*nμ*< <1. This means almost always (at the confidence level

� �d*ek* ¼¼ <sup>1</sup> � <sup>2</sup><sup>Φ</sup>

<sup>q</sup> ;ð Þ *<sup>k</sup>* <sup>¼</sup> 1, *::*, *<sup>n</sup>* , (27)

1 *MkPk*�<sup>1</sup> � �

≤*μ:*

(26)

ð 1*=Mk*

�1*=Mk*

1÷5 <sup>¼</sup> <sup>1</sup> � <sup>1</sup> � *Psat* ð Þ<sup>5</sup> <sup>¼</sup> <sup>1</sup> � <sup>0</sup>*:*68<sup>5</sup>

Pr*sat*

*Modulation in Electronics and Telecommunications*

tion causes a loss of the sample and *I X*, *X*^ *<sup>n</sup>*

first cycle:

attains the value Pr*sat*

the feedback channel.

condition" [35].

sequence ~*y<sup>k</sup>*�<sup>1</sup>

Pr*sat*

**12**

value 1 � ð Þ <sup>1</sup> � *<sup>μ</sup> <sup>n</sup>*

<sup>1</sup> :

*<sup>k</sup>* <sup>¼</sup> Pr <sup>j</sup>*yk*j<sup>≥</sup> *<sup>A</sup>*0<sup>j</sup> <sup>~</sup>*y<sup>k</sup>*�<sup>1</sup>

1 � � <sup>¼</sup> <sup>1</sup> �

of erroneous bits would not be greater than *μ*.

permissible maximal (optimal) values of modulation gains:

*Mopt*

*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> *α*

*3.2.2 Statistical fitting condition*

in next cycles.

$$\Phi(a) = \frac{1}{\sqrt{2\pi}} \int\_0^a \exp\left(-\frac{z^2}{2}\right) \mathrm{d}z = \frac{1-\mu}{2}. \tag{28}$$

*Claim 4.* Setting the modulation index to the values (27) prevents the appearance of saturation and the FCS with non-linear FT transmits the signals almost always in linear mode.

*Claim 5.* Setting the gains *Mk* to the values (27) not only removes saturation but also:


$$\mathcal{W}^{\rm sign} = \frac{A^2}{a^2}; \text{SNR}\_k = \text{Q}\_k^2 = \frac{\mathcal{W}^{\rm sign}}{\sigma\_\xi^2} = \frac{\mathbf{1}}{N\_\xi F\_0} \left(\frac{\chi A\_0}{ad}\right)^2 = \text{Q}^2. \tag{29}$$

The structure and form of the basic relationships for the MSE, bit rate, and effectiveness remain the same. Changes in Shannon's formula (23) for the capacity and in the other relationships affect only the values of amplitude and power of emitted signals (29).

*Claim 6*. Reduction of the power of the emitted signal decreases the SNR and capacity of the system but makes the perfect FCS feasible, and these systems transmit signals with limit energy-spectral efficiency optimally, as well as completely utilizing the frequency and energy resources of the FT. Moreover, the absence of coders reduces their energy consumption.

To distinguish the considered systems and the FCS, and to focus on the new systems, in what follows, we use a new abbreviation to refer to these systems: AFCS (adaptive FCS).

Let us remind that these relationships were derived under assumption that the feedback channels are ideal. This is not an unrealistic suggestion: relatively inexpensive modern CSCs provide virtually error-free short-range transmission. If necessary, the quality of the channel can also be improved by increasing the power of transmitters—the BSt have sufficiently large, if not unlimited energy resource.

We should add that the discussed relationships are a particular case of the more general optimal transmission-reception algorithm for the statistically fitted AFCS with noisy feedback [36–39], see also **Table 1** below. It is worth stressing that these algorithms have the same structure and form as those presented in pervious sections. The only but principle difference concerns the expression for the MSE of transmission, which takes the form:

$$P\_k^{\min} = \frac{\left(\sigma\_\xi^2 + A^2 M\_k^2 \sigma\_\nu^2\right) P\_{k-1}^{\min}}{\sigma\_\xi^2 + A^2 M\_k^2 \left(\sigma\_\nu^2 + P\_{k-1}^{\min}\right)} = \left(\mathbf{1} + Q^2\right)^{-1} \left[\mathbf{1} + Q^2 \frac{\sigma\_\nu^2}{\left(\sigma\_\nu^2 + P\_{k-1}^{\min}\right)}\right] P\_{k-1}^{\min},\tag{30}$$

where variable *σ*<sup>2</sup> *<sup>ν</sup>* is the variance of the errors *vk* in the signals (controls) *x*^*<sup>k</sup>* transmitted from the BSt to FT over the feedback channel with AWGN. Under *σ*2 *<sup>ν</sup>* ¼ 0, this formula coincides with (17). Analysis of formula (30) gives comprehensive answers to many questions concerning the behavior of feedback systems.


*Rn* <sup>¼</sup> *I X*,*X*^ *<sup>n</sup>* � � *nΔt*<sup>0</sup>

*Perfect Signal Transmission Using Adaptive Modulation and Feedback*

� � � *I X*,*X*^ *<sup>k</sup>*�<sup>1</sup>

*Δt*<sup>0</sup>

<sup>¼</sup> *<sup>W</sup>sign n NξRn*

channel output, which requires additional measurement.

*<sup>Δ</sup>Rk* <sup>¼</sup> *I X*,*X*^ *<sup>k</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.90516*

spectral efficiencies of transmission, respectively:

*Ebit* <sup>A</sup>*FCS <sup>n</sup> N<sup>ξ</sup>*

case, requires additional measurement of SNR *Q*<sup>2</sup>

received signals, computes optimal estimates *x*^ð Þ<sup>1</sup>

*<sup>P</sup>*^*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> *M* X *M*

*m*¼1

the MSE *Pk* using the known relationship:

as follows:

method is used.

expressed in dB:

**15**

*Rn F*0 ¼ 1 *<sup>n</sup>* log <sup>2</sup> ¼ *F*<sup>0</sup> log <sup>2</sup>

independent of whether the system is optimal or not. This value determines the spectral efficiency of the sample transmission *Rn=F*<sup>0</sup> that is at the AFCS output. The iterative principle of transmission permits us to introduce the measure more informative than (33): the *instant* bit rate, determined by the following relationship:

� �

*σ*2 0 *Pn* ; *ΔRk F*0

which describes the increment of information in sequentially computed estimates *x*^*k*, (*k* ¼ 1, … , *n*). In turn, formulas (33) and (34) define the final and instant

The general expression for the energy efficiency of transmission can be defined

<sup>¼</sup> *nWsign n NξF*0*I X*, *X*^ *<sup>n</sup>*

which shows that, unlike spectral efficiency, this characteristic of the CS performance depends not only on the MSE, but also on the SNR *Q*<sup>2</sup> at the forward

Another particularity of theMSE, which is not currently utilized in communications, is its analytical formulations have empirical analogs, as well as well-studied and widely used methods of their evaluation. In our research, the following method is used. However, as it follows from (36), evaluation of the energy efficiency, in the general

The FT generates and sends to the BSt a testing sequence of M random Gaussian samples *x*ð Þ<sup>1</sup> , … , *x*ð Þ *<sup>M</sup>* � � each generated with the same mean value *x*<sup>0</sup> and variance *σ*<sup>2</sup>

<sup>1</sup> , *<sup>x</sup>*^ð Þ<sup>1</sup>

<sup>2</sup> … , *<sup>x</sup>*^ð Þ *<sup>M</sup>*

using corresponding codes written into memory units of microcontrollers of the FT transmitter and processor of the BSt (or generated by PC). The BSt processes the

ples which, and stored values *x*ð Þ<sup>1</sup> , … , *x*ð Þ *<sup>M</sup>* � �, allow to compute empirical values of

*<sup>x</sup>*ð Þ *<sup>m</sup>* � *<sup>x</sup>*^ð Þ *<sup>m</sup> k* h i<sup>2</sup>

as well as normalized root square (relative error of transmission) <sup>δ</sup> <sup>¼</sup> ffiffiffiffiffi

MSE*k*½ �¼ dB 10 log <sup>10</sup>

next used for the evaluation of the bit rate (35) and energy-spectral efficiency of AFCS. In practice, it is more convenient to compute these values using the MSE

> *P*^*k σ*2 0

!

σ2 0 *Pn* � �

¼ *F*<sup>0</sup> log <sup>2</sup>

¼ log <sup>2</sup>

*Pk*�<sup>1</sup> *Pk*

*Pk*�<sup>1</sup> *Pk*

� � <sup>¼</sup> *nQ*<sup>2</sup>

log <sup>2</sup> *σ*2 0 *Pn*

. In our research, the following

*<sup>n</sup>*�<sup>1</sup>, *<sup>x</sup>*^ð Þ *<sup>M</sup> n* � � of input sam-

,ð Þ *k* ¼ 1, … , *n* (37)

½ � dB *:* (38)

*Pk* <sup>p</sup> *<sup>=</sup>σ*0.

½ � bit*=*s , (33)

½ � bit*=*s , (34)

*:* (35)

, (36)

0

#### **Table 1.**

*Basic relationships for modeling and design of optimal AFCS with non-ideal feedback channel (σ*<sup>2</sup> *<sup>ν</sup>*>0*; index "opt" in the parameters Mopt <sup>k</sup> , <sup>L</sup>opt <sup>k</sup> is omitted).*

The main result is the confirmation that the capacity of FCS and AFCS does not depend on the feedback noise in the initial interval of the sample transmission, and is determined by Shannon's formula (1). However, since the moment *n*<sup>∗</sup> , when the MSE of estimates attains the values of *σ*<sup>2</sup> *<sup>ν</sup>* order, the capacity of AFCS begins to decrease and monotonically tends to zero ([36] and later works).

A summary of the relationships sufficient for the development of a MATLAB model of optimal AFCS and simulation experiments is presented in **Table 1**. Moreover, these seemingly simple relationships were used to design a prototype (demonstrator) of the perfect AFCS, discussed in the next section.

If a relative error of the sample transmission δ ¼ *Δ=σ*<sup>0</sup> is to be attained in minimal time, and feedback is ideal, absolute error *Δ* of the final estimates should be not greater than *σν*, and the corresponding minimal time of transmission is of the order:

$$T\_{\delta} = \frac{n\_{\Delta}}{2F\_0} = -\frac{1}{2F\_0 \log\_2(1+Q^2)} \log\_2\left(\frac{\Delta^2}{\sigma\_0^2}\right) = -\frac{\log\_2 \delta}{C^{\text{Ch1}}}.\tag{31}$$

where *n<sup>Δ</sup>* determines the necessary number of cycles (see e.g. formula (27) in [37] or (22) in [38]). Formula (31) also determines the baseband *F* ≤*F*<sup>δ</sup> ¼ *Cch*<sup>1</sup> *=*2 log <sup>2</sup>δ of the signals that can be transmitted by the AFCS at maximum rate, with the given accuracy and limit energy-spectral efficiency.

#### *3.2.3 Particular role of MSE criterion*

As noted above, the basic criterion for transmission quality is the accuracy (in [1] "fidelity") of the signals' recovery. For the CS transmitting analog signals, this is the MSE of their estimates. The importance of MSE is due to several factors. First of all, for arbitrary linear channels with AWGN, which transmit Gaussian signals, MSE *Pk* determines the amount of information determined by following general relationship ([32, 34], see also (22)):

$$I(X, \hat{X}\_n) = \frac{1}{2} \log\_2 \left(\frac{\sigma\_0^2}{P\_n}\right) \text{ [bit/sample]}.\tag{32}$$

So, if CSs transmit the samples each in *n* cycles, that is, during *Tn* ¼ *nΔt*<sup>0</sup> ¼ *n=*2*F*<sup>0</sup> [s], the final estimates *x*^*<sup>k</sup>* deliver to addressee the amount of information with the bit rate

*Perfect Signal Transmission Using Adaptive Modulation and Feedback DOI: http://dx.doi.org/10.5772/intechopen.90516*

$$R\_n = \frac{I(X, \hat{X}\_n)}{n\Delta t\_0} = F\_0 \log\_2\left(\frac{\sigma\_0^2}{P\_n}\right) \text{ [bit/s]},\tag{33}$$

independent of whether the system is optimal or not. This value determines the spectral efficiency of the sample transmission *Rn=F*<sup>0</sup> that is at the AFCS output.

The iterative principle of transmission permits us to introduce the measure more informative than (33): the *instant* bit rate, determined by the following relationship:

$$
\Delta R\_k = \frac{I(X, \hat{X}\_k) - I(X, \hat{X}\_{k-1})}{\Delta t\_0} = F\_0 \log\_2 \frac{P\_{k-1}}{P\_k} \text{ [bit/s]},\tag{34}
$$

which describes the increment of information in sequentially computed estimates *x*^*k*, (*k* ¼ 1, … , *n*). In turn, formulas (33) and (34) define the final and instant spectral efficiencies of transmission, respectively:

$$\frac{R\_n}{F\_0} = \frac{1}{n} \log\_2 \frac{\sigma\_0^2}{P\_n}; \frac{\Delta R\_k}{F\_0} = \log\_2 \frac{P\_{k-1}}{P\_k}.\tag{35}$$

The general expression for the energy efficiency of transmission can be defined as follows:

$$\frac{E\_n^{\text{bit AFCS}}}{N\_\xi} = \frac{\mathcal{W}\_n^{\text{sign}}}{N\_\xi R\_n} = \frac{n \mathcal{W}\_n^{\text{sign}}}{N\_\xi F\_0 I(X, \hat{X}\_n)} = \frac{nQ^2}{\log\_2 \frac{\sigma\_0^2}{P\_u}},\tag{36}$$

which shows that, unlike spectral efficiency, this characteristic of the CS performance depends not only on the MSE, but also on the SNR *Q*<sup>2</sup> at the forward channel output, which requires additional measurement.

Another particularity of theMSE, which is not currently utilized in communications, is its analytical formulations have empirical analogs, as well as well-studied and widely used methods of their evaluation. In our research, the following method is used. However, as it follows from (36), evaluation of the energy efficiency, in the general case, requires additional measurement of SNR *Q*<sup>2</sup> . In our research, the following method is used.

The FT generates and sends to the BSt a testing sequence of M random Gaussian samples *x*ð Þ<sup>1</sup> , … , *x*ð Þ *<sup>M</sup>* � � each generated with the same mean value *x*<sup>0</sup> and variance *σ*<sup>2</sup> 0 using corresponding codes written into memory units of microcontrollers of the FT transmitter and processor of the BSt (or generated by PC). The BSt processes the received signals, computes optimal estimates *x*^ð Þ<sup>1</sup> <sup>1</sup> , *<sup>x</sup>*^ð Þ<sup>1</sup> <sup>2</sup> … , *<sup>x</sup>*^ð Þ *<sup>M</sup> <sup>n</sup>*�<sup>1</sup>, *<sup>x</sup>*^ð Þ *<sup>M</sup> n* � � of input samples which, and stored values *x*ð Þ<sup>1</sup> , … , *x*ð Þ *<sup>M</sup>* � �, allow to compute empirical values of the MSE *Pk* using the known relationship:

$$\hat{P}\_k = \frac{1}{M} \sum\_{m=1}^{M} \left[ \mathbf{x}^{(m)} - \hat{\mathbf{x}}\_k^{(m)} \right]^2, (k = 1, \dots, n) \tag{37}$$

next used for the evaluation of the bit rate (35) and energy-spectral efficiency of AFCS. In practice, it is more convenient to compute these values using the MSE expressed in dB:

$$\text{MSE}\_k[\text{dB}] = \mathbf{10} \log\_{10} \left( \frac{\hat{P}\_k}{\sigma\_0^2} \right) [\text{dB}]. \tag{38}$$

as well as normalized root square (relative error of transmission) <sup>δ</sup> <sup>¼</sup> ffiffiffiffiffi *Pk* <sup>p</sup> *<sup>=</sup>σ*0.

The main result is the confirmation that the capacity of FCS and AFCS does not depend on the feedback noise in the initial interval of the sample transmission, and is determined by Shannon's formula (1). However, since the moment *n*<sup>∗</sup> , when the

*Basic relationships for modeling and design of optimal AFCS with non-ideal feedback channel (σ*<sup>2</sup>

Initial values *<sup>x</sup>*^<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

Received signal <sup>~</sup>*yk* <sup>¼</sup> *Ayk* <sup>þ</sup> *<sup>ξ</sup><sup>k</sup> <sup>A</sup>* <sup>¼</sup> *<sup>γ</sup>*

Estimate computing *<sup>x</sup>*^*<sup>k</sup>* <sup>¼</sup> *<sup>x</sup>*^*k*�<sup>1</sup> <sup>þ</sup> *Lk*~*yk Lk* <sup>¼</sup> *AMkPk*�<sup>1</sup>

Basic equation for MSE *Pk* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> � ��<sup>1</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> *<sup>σ</sup>*<sup>2</sup>

*<sup>k</sup> is omitted).*

Signal at the modulator input *ek* <sup>¼</sup> *<sup>x</sup>* � *<sup>x</sup>*^*k*�<sup>1</sup> <sup>þ</sup> *vk <sup>α</sup>* : <sup>1</sup>ffiffiffiffi

Emitted signal *yk* <sup>¼</sup> *<sup>A</sup>*0*Mkek <sup>M</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>

A summary of the relationships sufficient for the development of a MATLAB

If a relative error of the sample transmission δ ¼ *Δ=σ*<sup>0</sup> is to be attained in minimal

model of optimal AFCS and simulation experiments is presented in **Table 1**. Moreover, these seemingly simple relationships were used to design a prototype

time, and feedback is ideal, absolute error *Δ* of the final estimates should be not greater than *σν*, and the corresponding minimal time of transmission is of the order:

<sup>2</sup>*F*<sup>0</sup> log <sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> � � log <sup>2</sup>

where *n<sup>Δ</sup>* determines the necessary number of cycles (see e.g. formula (27) in [37] or (22) in [38]). Formula (31) also determines the baseband *F* ≤*F*<sup>δ</sup> ¼

*=*2 log <sup>2</sup>δ of the signals that can be transmitted by the AFCS at maximum rate,

As noted above, the basic criterion for transmission quality is the accuracy (in [1] "fidelity") of the signals' recovery. For the CS transmitting analog signals, this is the MSE of their estimates. The importance of MSE is due to several factors. First of all, for arbitrary linear channels with AWGN, which transmit Gaussian signals, MSE *Pk* determines the amount of information determined by following general

> σ2 0 *Pn* � �

So, if CSs transmit the samples each in *n* cycles, that is, during *Tn* ¼ *nΔt*<sup>0</sup> ¼ *n=*2*F*<sup>0</sup> [s], the final estimates *x*^*<sup>k</sup>* deliver to addressee the amount of information

<sup>2</sup> log <sup>2</sup>

decrease and monotonically tends to zero ([36] and later works).

(demonstrator) of the perfect AFCS, discussed in the next section.

¼ � <sup>1</sup>

with the given accuracy and limit energy-spectral efficiency.

*<sup>ν</sup>* order, the capacity of AFCS begins to

**Algorithm Parameters**

<sup>2</sup>*<sup>π</sup>* <sup>p</sup> <sup>Ð</sup> *α* 0

*σ*2 *ξ*þ*A*2*M*<sup>2</sup> *<sup>k</sup>Pk*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> *AMk* <sup>1</sup> � *Pk Pk*�<sup>1</sup> � �

*v <sup>σ</sup>*<sup>2</sup> ð Þ *<sup>ν</sup>*þ*Pk*�<sup>1</sup> � � 0

*dz* <sup>¼</sup> <sup>1</sup>�*<sup>μ</sup>* 2

*<sup>α</sup>* ffiffiffiffiffiffiffiffiffiffiffiffi *σ*2 *<sup>ν</sup>*þ*Pk*�<sup>1</sup> p

*<sup>ν</sup>*>0*; index*

exp � *<sup>z</sup>*<sup>2</sup> 2 � �

*ασ*<sup>0</sup> ; *Mk=<sup>k</sup>* <sup>≥</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>

*<sup>d</sup> A*<sup>0</sup>

*Pk*�1; *k* ¼ 1, … , *n*

*Δ*2 *σ*2 0 � �

¼ � log <sup>2</sup><sup>δ</sup>

½ � bit*=*sample *:* (32)

*<sup>C</sup>Ch*<sup>1</sup> *:* (31)

MSE of estimates attains the values of *σ*<sup>2</sup>

*<sup>k</sup> , <sup>L</sup>opt*

*Modulation in Electronics and Telecommunications*

*<sup>T</sup><sup>δ</sup>* <sup>¼</sup> *<sup>n</sup><sup>Δ</sup>* 2*F*<sup>0</sup>

*3.2.3 Particular role of MSE criterion*

relationship ([32, 34], see also (22)):

with the bit rate

**14**

*I X*, *X*^ *<sup>n</sup>* � � <sup>¼</sup> <sup>1</sup>

*Cch*<sup>1</sup>

**Table 1.**

*"opt" in the parameters Mopt*

#### **3.3 Adaptive auto-adjusting AFCS to the scenario of application**

The adjusting algorithm uses the "resonance" effect that is increase of MSE, if the values of parameters *Lk*, *Mk* decline from their optimal values (16), (27). The effect is illustrated in **Figure 3** which shows the changes of relative errors of transmission <sup>δ</sup> <sup>¼</sup> ffiffiffiffiffi *Pk* <sup>p</sup> *<sup>=</sup>σ*<sup>0</sup> (normalized root mean square error—RMS) under gains *Mk* set to the values (27) and gain *L*<sup>∗</sup> *<sup>k</sup>* ¼ *Lk*ð Þ 1 þ δ*<sup>L</sup>* , where *Lk* has the value (16) and δ*<sup>L</sup>* is a variable parameter.

**4. Experimental study of AFCS functioning**

*DOI: http://dx.doi.org/10.5772/intechopen.90516*

*Perfect Signal Transmission Using Adaptive Modulation and Feedback*

**Table 1**, for *σ*<sup>2</sup>

given scenario.

**Figure 4.**

*station.*

**17**

mission also attain the limit values.

transmission permits us to write that *R* ¼ *C*):

*Cn F*0 <sup>¼</sup> <sup>3</sup>*:*<sup>32</sup>

*<sup>n</sup>* log <sup>10</sup>

shown in **Figure 4a**, **b**.

The prototype of AFCS was designed on the basis of the optimal transmissionreception algorithm(6)–(8), using and parameters set to the values (16), (18), (or in

At the beginning of every new series of experiments, a self-adjusting algorithm was activated, which set the parameters *Mk* and *Lk* to the values optimal for the

The plots are presented in the decibel scale, and the nearly linear dependence of

Moreover, plots in **Figure 5** allow for a sufficiently accurate evaluation of the characteristics of the system. With this aim, let us rewrite the expression of spectral efficiency (33) in the decibel scale in the form (the confirmed close to perfect

> ¼ � <sup>0</sup>*:*<sup>332</sup> *n*

MSE*n*½ � dB *:* (39)

*σ*2 0 *Pn*

*Layout of PCB modules of prototype of perfect AFCS: (a) forward transmitter integrated with sensor; (b) base*

The main measured characteristic of the prototype was the *dependence of* MSEn½ � dB *on the number of transmission cycles. The experiments were* carried out at different distances between the FT and BSt. Typical dependencies of MSEn½ � dB on *n*

the measured values MSEn½ � dB on the number of cycles means that, on a linear scale, MSE decreases exponentially. According to the results of Section 3.2 (formula (19)), this is possible, if the system transmits signals perfectly, with a bit rate *equal to the capacity* of the system. In this case, spectral and energy efficiencies of trans-

at the distances of 40, 50, and 75 meters are shown in **Figure 5**.

Section 3.1. The layouts of the transmitting (FT) and receiving (BSt) modules are

The transmitter was realized using narrowband adaptive AM modulator followed by the programmable voltage-controlled oscillator VG7050EAN (power 10 dBm, carrier frequency 433.2 MHz). The feedback channel was realized using digital receiver RFM31B-S2 and transmitter RFM23B (power 27 dBm, carrier frequency 868.3 MHz). This ensured virtually ideal feedback transmission of signals in the indoor and outdoor experiments carried out at distances to 100 meters (straight line view, FT with ceramic mini-antennas, BSt with quarter-wave antennas).

*<sup>v</sup>* ¼ 0), and general principles of AFCS transmission described in

To adjust the parameters, the system utilizes two identical testing sequences of Gaussian samples *x*ð Þ<sup>1</sup> , … , *x*ð Þ *<sup>M</sup>* � � whose codes are stored in the memory of the BSt processor and the FT microcontroller (or BSt transmits these sequences to FT over feedback). All the samples in the testing sequence are zero mean Gaussian values with known variance *σ*<sup>2</sup> 0.

In the first cycle, the modulation index *Mopt* <sup>1</sup> is set to the known value 1*=α*σ0, and the FT sends to the BSt the written testing sequence of samples *x*ð Þ<sup>1</sup> , … , *x*ð Þ *<sup>M</sup>* � �, the same one that is stored in the processor of BSt. The BSt processor computes the estimates *x*^ð Þ<sup>1</sup> <sup>1</sup> , … , *<sup>x</sup>*^ð Þ *<sup>M</sup>* 1 � � and values of MSE *<sup>P</sup>*1. It also searches for the minimizing MSE value *<sup>L</sup>*^*opt* <sup>1</sup> and computes the corresponding value *<sup>P</sup>*^min <sup>1</sup> . The computed values *<sup>L</sup>*^*opt* <sup>1</sup> , *<sup>P</sup>*^min <sup>1</sup> , and *<sup>x</sup>*ð Þ<sup>1</sup> , … , *<sup>x</sup>*ð Þ *<sup>M</sup>* � � are stored. Simultaneously, the BSt sends values *<sup>P</sup>*min 1 and estimates *x*^ð Þ<sup>1</sup> <sup>1</sup> , … , *<sup>x</sup>*^ð Þ *<sup>M</sup>* 1 � � to the FT. Reception of these data initiates the second cycle of AFCS adjusting:

In this cycle, the microcontroller of the FT forms the sequence of signals *e* ð Þ *m* <sup>2</sup> ¼ *<sup>x</sup>* � *<sup>x</sup>*^ð Þ *<sup>m</sup>* <sup>1</sup> , (*<sup>m</sup>* <sup>¼</sup> 1, … , M), computes the optimal value of the gain *<sup>M</sup>opt* <sup>2</sup> ¼ 1*=α* ffiffiffiffiffiffiffiffiffi *P*min 1 q , and sets the gain of AM modulator to this value. So adjusted, the FT transmits the sequence of signals *y* ð Þ *m* <sup>2</sup> <sup>¼</sup> *<sup>A</sup>*0*Mopt* <sup>2</sup> *e* ð Þ *m* <sup>2</sup> to the BSt, which processes the received sequence in the same way as in first cycle. The computed values *P*min <sup>2</sup> , *<sup>L</sup>opt* <sup>2</sup> are stored, sequence *x*^ð Þ<sup>1</sup> <sup>1</sup> , … , *<sup>x</sup>*^ð Þ *<sup>M</sup>* 1 � � is replaced by the new sequence *<sup>x</sup>*^ð Þ<sup>1</sup> <sup>2</sup> , … , *<sup>x</sup>*^ð Þ *<sup>M</sup>* 2 � �, and together with *P*min <sup>2</sup> is transmitted to the FT. The receipt of these values initializes the next cycle of adjusting realized according to the same scheme as in the previous cycle. The subsequent cycles repeat these operations. After the M-th cycle, the adjusted AFCS begins nominal functioning.

The duration and frequency of adjustments depend on the dynamics of scenario changes, processors' rate, channel bandwidth, requirements for the accuracy of estimates, environmental characteristics, and other factors.

#### **Figure 3.**

*Changes of mean relative error of estimates depending on the deviation of the gains* Lk *from optimal values* L*opt* k *under fixed optimal gains* M*opt* <sup>1</sup> *.*
