Index Terms—Adaptive modulation, analog transmission, feedback systems, power-bandwidth efficiency. threshold effect

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Abstract –The paper shows a possibility and advantages of con- sideration of the feedback communication systems (FCS) as the estimation systems. There are considered general effects appear- ing in optimal FCS and caused by them changes in limit informa- tion characteristics of the systems. The obtained results enable a transition to practical design of optimal FCS. They remain valid also for the adaptive estimation systems with feedback.

Index Terms—Adaptive modulation, analog transmission, feedback systems, power-bandwidth efficiency. threshold effect

I. I NTRODUCTION

OWADAYS , analogue signal transmission is considered as an initial stage of communications development which has no future in comparison with evident superiority of digital communication systems (CS). However, naither C.E. Shannon, nor founders of information and communication theory, consi- dered digital transmission as more perspective than the analog one. Moreover, in the 1950-1960s, analog communication systems with feedback channels were a subject of very inten- sive and promising researches (e.g. works [1]-[9]). However, results of these researches have no practical application even now. In turn, impressive successes of digital CS theory and applications created, in 1960s, a great field of new general and particular theoretical tasks with granted, not less great interest of the industry to new solutions. This switched the attention of researchers to digital communications, and analog feedback systems (FCS) became the subject of relatively rare theoretical investigations dominated by the great amount of works de- voted to digital CS.

We have studied basic series of works in analog FCS to un- derstand what had hampered implementation of the obtained excellent analytical results and their further development? The answer was somethat unexpected – the main reason was com- monly used linear model of the transmitting part of FCS. The second reason was a solution of optimization task under con- straints on the mean or peak-power of emitted signals. Both these reasons did not allow a consideration of the cases of sa- turation of the transmitters (or overmodulation) which are the sources of abnormal errors.

The particular feature of researches in analog FCS is a pos- sibility to determine analytical form of the mean square error (MSE) of transmission and to derive minimizing MSE optimal transmission-receiving algorithm using methods of applied Bayesian estimation theory [10]. (Let us notice that this is im- possible for digital CS due to impossibility to search minimum of MSE on the set of possible codes). However, MSE does not

Fig. 1. Block diagram of estimation system with adjusted nonlinear observing device [11].

Fig 2. Block-diagram of the analog FCS [13]-[16] .

allow adequate consideration of abnormal errors which cause irreversible loss of information about the samples and increase mean percentage of erroneous bits (bit-error rate - BER) in information streams at the channel and system outputs. Rare but crucially important for practice, abnormal errors add to the value of MSE negligibly small values. They also cannot be included into the varied form of estimating rule. This was the main reason of unsuccessful the probes implementation of the results obtained in the frame of only MSE approach.

In the paper, we give deeper explanation of key ideas of new approach to optimization of FCS which removes the de- scribed above difficulties and enables designing the top- efficient systems working at the upper (Shannon’s) boundaries.

There are discussed not studied before effects appearing in op- timal FSC. The paper develops the results of works [13]-[16].

II. C ONSIDERATION OF ABNORMAL ERRORS IN FCS THEORY

Optimization of FCS is based on the original approach ([11]

,[12] and other works) to optimization of the adaptive feed- back estimation systems (FES) with adaptively adjusted nonli- near observing part (the simplified block diagram of FES is presented in Fig. 1). Particularity of the approach is explicit consideration of always existing possibility of saturation of the analog part of the system.

A. The key ideas of the approach

The main idea is introduction, apart from MSE, of addition- al criterion of FES performance – permissible probability μ

Bayesian Methods in Optimization of Analog Communication Systems with Feedback

Anatoliy Platonov, Senior Member, IEEE

N

x

t

M1 Ch1 DM1

T2

DSPU

Transmission unit (TU) Base station (BS)

S&H

R2 Ch2

Σ ˆk

B B_k,

ˆ

n

x

e

k

s

t,k

s

st k_{t k}_t,_,_,

[t

y

k_k

y

ˆ_k M Qk

Mk

K

t

z^-1

x

R2 x

ˆ_k1 x

ˆ

_n

x

e

k

y

_k ^[^t

y

ykkk

ˆ_k M Qk

z^-1 Estimation

Unit

Calculation of Mk

Calculation

of Pk

x ˆ

n

Pk ^k

P

Pk ˆ_k1

x

2012

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of its saturation for each moment of the system work (statistic- al fitting condition, [11]-[16]). This condition reduces the sets of possible values of the parameters of analogue part to the

“permissible” sets. These sets consist of the parameters gua- ranteeing that, for each time instant, saturation may appear with a probability not greater than prescribed value μ , ( 10

10⁴⁴

). The statistically fitted FES (with parameters from the permissible sets) will be working as the linear systems with a probability not smaller than 1 - μ .

In practice linear statistically fitted FES permit to use Baye- sian methods and to find the parameters of the analog and digi- tal parts of FES minimizing MSE. Solution of this task – op- timal observation-estimation algorithms determine the parame- ters of the analog part and estimation algorithm, as well as optimal rules of the analog part adjusting. This information enables systematic design of optimal FES, whose performance attains theoretically achievable upper boundary (minimal MSE of estimates). Non-linearity of these systems and saturation errors will be excluded at the confidence level ¹ .

One should say that Bayesian methods can’t be applied to optimization of digital CS due to the lack of continuous analyt- ical models of digitizing units and encoders. For this reason, digital theory employs several criterions of the CS perfor- mance. One of them is BER.

As it was shown in [14]-[16], probability of saturation μ plays the same role, as BER in digital CS but was never used in the analysis of analog CS. Deeper analysis of the question showed that just its introduction into the theory (by means of the statistical fitting condition) removes the problems which blocked application of earlier theoretical results [1]-[9], and enables their further development and generalizations.

Similarity of the block diagrams of FES (Fig. 1) and FCS (Fig. 2), as well as coinciding mathematical models of their analogue parts and transmission lines allows direct application of the approach presented in [11],[12] to FCS optimization.

Differences between the technical units, data transfer channels and application of the systems are not essential and require only careful consideration in the parameters of the models.

B. Main models used in FCS and FES analysis.

The block diagram of the point-to-point analog FCS is pre- sented in Fig. 2. The system includes adaptively adjusted peri- pheral transmitting unit (TU) and base station (BS) connected by the forward M1-Ch1-DM1 and feedback T2-Ch2-R2 chan- nels. Both channels are assumed to be stationary, linear and memoryless. The channel noises ξ ,

_t

η are additive white

_n

Gaussian noises (AWGN) with known spectral power densities

,

N N

_ξ _η

. The input signals x are limit-band stationary Gaus-

_t

sian processes with known mean value x and variance

₀

σ .

₀²

The input sample-and-hold unit (S&H) forms sequences of the samples x

^{( )}^m

= ( x mT , ) m 1,2,.. 1,2,.. of the input signal x ,

_t

1/ 2

T F is the sampling period. Each sample is transmitted in n T Δt /

₀

F

₀

/ F cycles ( Δt

⁰

1/ 2 F

⁰

is duration of the single cycle of transmission, 2F determines the channels

₀

bandwidth). As in [14]-[16], we assume that each sample is

Fig. 3. Transition characteristic of adaptive transmitter Σ +M1.

transmitted independently and in the same way. This permits to reduce the analysis of the systems transmitting single sample and omit index m in notations of the input samples x

^{( )}^m

. Si- multaneously, this reduces the analysis to the particular case of the random values estimation considered in [11],[12].

Adaptive transmitter includes the subtracting unit Σ and amplitude modulator M1 with the controlled modulation index

ˆ

k

M . There is assumed that amplitude of the emitted signals cannot be greater than A (saturation level). In this case,

₀

transmitter can be described by the model (see also Fig. 3):

(1) where A f

₀

, ,

₀

φ

_k

are the parameters of the carrier, and

0 0

( -1) k ' d d ' , t t k t k 1,..., n are the current numbers of the transmission cycles. In each cycle, adjusted parameters of the transmitter: modulation index ˆ M and signal ˆ

_k

B at the second

^k

input of the subtractor Σ are set to proper values (see below).

To simplify the form of analytical results the DSB-SC AM is considered, but the obtained results can be easily extended to the other types of AM. This model corresponds to non-linear analog unit of FES block diagram in Fig. 1.

Formed by the subtractor Σ difference signal e

k

ˆ x B

k

modulates the carrier transmitted to BS through the forward channel M1-Ch1-DM1 with AWGN ξ . Signal

_t

s s

_{t k}_{t k}_,_,_,

at the input of demodulator DM1 is described by the relationship:

_{k t}_,

γ

⁰ _{k t}_, _t

s s ξ

r

⁰

,

k t, k

s

_{k t}

γ r

0 k

γ

⁰

s (2) where γ is the channel gain and

₀

r is the distance between TU and BS. Signal y y

_k_k

(“observation”) formed by the demodu- lator DM1 is routed to the input of digital signal processing unit (DSPU) of BS. This part of FCS corresponds to commu- nication line y

_k

y

_k _k

p ξ

k

y

k_k

y

_k

of FES.

Unit DSPU computes the current estimate ˆ x of the sample

_k

according to the Kalman-type equation:

x ˆ

_k

x ˆ

_k¹

L y

_k

[

_k_k

E y ( ( ( (

_k_k

| | | y

¹₁₁^k^k¹¹¹

)] )] ; ( ˆx

₀

x

₀

) (3) where y y

₁₁^k^k¹¹

( ,..., y y

₁₁₁₁

,..., ,..., ,..., y

_k_k₁₁₁

) ) denotes the sequence of observa- tions received at the previous cycles and E y ( y

_k_k_k

| | || y y

₁₁₁^k^k¹¹¹

) ) is the one-cycle prediction of the signal at the demodulator DM1 output. Parameter L determines the rate of algorithm (3)

_k

convergence. Equation (3) represents the mathematical model of digital part of FCS and has the same form as in [11].

In [11],[12] it was shown that, in the Gaussian case, optimal values of the parameter ˆ M do not depend on observation

_k

y y

^k^k

, 0 0

ˆ ˆ

ˆ ( ) if ˆ | | 1

cos(2 )

sgn ( ) if ˆ | | >1

k k k k

t k k

k k k

B B

M x M x

s A f t

x M x S M

d ½

° °

® ¾

° °

¯ ¿

s

k

ˆ

_k 0

/ ˆ

_k

B A M

x x

- -A A

00

x x x

B ˆ

_k

A

⁰

/ M ˆ

_k

ˆ

k

B

A A

00

(3)

and can be computed and set independently. Therefore, apart of the estimate, DSPU computes only control B

_k

B y ( )

₁^k

p

1

)

1

y

k

routed to the TU through the feedback channel R2-Ch2-T2.

Receiver R2 routes the received signals to corresponding input of the subtracting unit Σ which forms the next difference sig-

nal

₁

ˆ

₁

k k

e

x B

. Simultaneously, synchronizing unit of DSPU resets the modulation index to the value ˆ

1

M

k

and the gain L

_k

to new value L

_k₁

, and the next cycle of the sample transmission begins. After n cycles, final estimate of the sam- ple x ˆ

_n

is routed to the addressee, and AFCS begins a transmis- sion of the next sample.

It is important to notice that feedback channel with AWGN does not require detailed modeling. It is sufficient to describe the error of the parameter ˆ B setting as AWGN ν

_k _k

(shown by upper arrows in Figs. 1, 2) with known variance σ and to

²_v

model the feedback chain by the relationship B ˆ

_k₁

B

_k

ν

_k₁

.

C. Optimal transmission-reception algorithm

The presented models allow derivation of analytical form of the MSE of transmission errors P

_k

E x x [( ˆ

_k

) ]

²

for each

1,...,

k n and statistical fitting condition excluding appear- ance of the transmitter saturation at the confidence level ¹ . In turn, identity of the mathematical models of FES and FCS permits to perform full optimization of FSC using approach proposed in [11]. The result of optimization – optimal joint transmission - reception algorithm is as follows [13]-[16]:

a). Receiving algorithm for DSPU of BS, ( k 1,..., n ):

x ˆ

_k

x ˆ

_k¹

L y

_k

y

_k_k_k

; ( ˆx

₀

x

₀

) (4) where gains L are set, in each cycle, to the values

_k

2

1 1

1 2 2

1 1 1

1 1

(1 )

1

k

k k k

P

L P P Q

AM AM Q V

_Q

P

. (5)

b). Transmitter TU adjusting algorithm ( k 1,..., n ):

s

_{t k}_,

A M x

₀

ˆ (

_k

B ^ˆ

_k

) cos(2 S f t

₀

M

_k

) (6) B ^ˆ

_k

B

_k₁

ν

_k

, ˆ

₁

k k

M M

(7)

^B

^k¹

^x ^ˆ

^k¹

;

₁

2 1

1

k

ν k

M

D σ P

; ( M

₀

( D σ

₀

)

¹

). (8) c). Parameters P in (5), (8) equal to minimal MSE (MMSE)

_k

of estimates determined by the relationship:

2 2

2 1 1

(1 )

^v ^k

k k

k

Q P

P Q P

Q

P V V

ª º

« ¬ » ¼ ; ( P

₀

V

₀²

) . (9) Parameter Q in (5), (9) describes the signal-to-noise ratio

²

(SNR) at the channel M1-Ch1-DM1 output:

2 2 2 2

2 1 1

2 2

0

[( ˆ ) ] 1

_Ch

k k

out ξ

A M E x B

W A

Q SNR

α N F

[ [

V V

§ ·

¨ ¸ © ¹ (10)

where W ( / ) A α

²

,is the power of emitted signal and

0 0

/

A A A

_{0 0}_{0 0}₀

γ

₀₀

/ / r r is the amplitude of received signal, respective- ly;

² ^{0 0}

N F

₀

0

/ N F

0₀

ξ ξ

σ . Parameter α (saturation factor) is connected with permissible probability of TU saturation by the equation:

1- 2 ( )

P ) D , where ( ) ) D is Gaussian error function. Ob- servations y

_k

)

y

k_k

in (4) are formed according to the relationship:

y y

_k_k_k

A M x B A A ˆ ˆ

_k_k_k

( ˆ

_k

) ξ

_k

, where A A

₀ ⁰

r

J . (11) Algorithm (4)-(10) contains the basic information permit- ting to design FCS with MMSE of transmission (9).

III. N EW EFFECTS APPEARING IN OPTIMAL FCS

Depending on the task, optimal FCS analysis may have two scenarios: under fixed channel bandwidth F or the system

₀

baseband F . Below, we consider the case of the given F . In

₀

this case, duration of a single cycle Δt

₀

1/ 2 F

₀

, and trans- mission of the sample in n cycles means corresponding nar- rowing of the bandpass by n times.

A. Changes of MMSE versus number of cycles

The results presented below are obtained under condition:

2 0 2 ν

1 2 1

1 1

Ch Ch

inp out

SNR ^V Q SNR

V !! (12)

which describes a typical situation: SNR at the output of the forward channel is small but SNR at the modulator input is large (resources of BS provide the sufficiently small feedback errors). In this case, MMSE (9) can be replaced by the approx- imate relationship [13]-[16]:

2 2 *

0

* *

2 1

ν

(1 ) for 1 1 for

( )

n n

Q n n

n n n n

P σ

V

d d

!

® °

°¯

(13) where threshold point n

^*

is solution of the equation

*

2

P

n

V

_Q

:

2 1

* 0 2

2 2 2 1

2 2

log ( )

1 log

log (1 ) log (1 )

Ch inp Ch

v out

σ SNR

n Q σ SNR

§ ·

¨ ¸

© ¹ . (14)

B. Capacity of forward channel and of FCS as a whole According to (8),(11), E y ( y

_k_k_k

| | || y y

₁₁₁ⁿⁿ¹¹¹

) 0 ) 0 and E y ( y

_k_k_k²²²²

| | || y y

₁₁₁ⁿ¹¹¹

) )

2 2 2

( ) ( / )

_k _ξ

E y

_k

A / D σ

g

y ) (

_k22²

) ( / . Computation of the prior and posterior entropies of received sequences y y

₁₁ⁿⁿ

, and of the mutual amount of information I Y (

₁1₁₁ⁿⁿ

; e

₁1₁₁ⁿⁿ

) ) ^q in y y

₁₁ⁿⁿ

and e gives the relationships:

₁ⁿ

2 2

1 2 2 2

1

( ) ( ) log 2 1

2

n n

k ξ

H Y H Y n A

πeσ α σ

§ ·

¨ ¸

© ¹

¦

1

) ( )

2

1

log 2

₂

2

n n

)

k

1

(( n

)

¦ ((

¦ ^{; (15)}

(

₁

|

₁

) log (2

₂ ²

) 2

n n

H Y n

e S V e

_[

1 1

1ⁿ

|

1₁ⁿ

)

1 1

; (16)

¹ ¹

2

( ; ) [ ( ) ( | )] log (1

2

) 2

n n

I Y n H Y H Y n

e Q

e

1 1 2

1

;

₁₁

) [ ( ) ( | )] log

₂

2

n ₁

) [ ( ) ) ( (

1

n

11

[ ( ) [ ( ) [ ( ) ( ( | )] . (17) If the only information about the input signal is its mean value and variance, then (15) describes maximal, on the set of possible distributions of input sequences e , entropy of the

₁ⁿ

received sequences y y

₁₁ⁿⁿ

. This entropy is additionally max- imized over the mean power of the signals emitted by TU un- der given probability of saturation. The latter means that (17) determines the upper boundary of the amount of information delivered to BS in n cycles of transmission. In this case rela- tionship

1 1

0 2 0 2

0

2 1

0

( )

log ; log 1

(1 )

n n sign

Ch1

I Y W

ch

R F F C

Q N F nΔt

e

[

§ ·

¨ ¸

© ¹

1 1

) )

1 1

n n

1

))

1

)

1 1

)

1 111

1 1

(18)

determines the capacity of the forward channel. The single

difference from classical expression is its dependence on satu-

ration factor α in SNR Q . In turn, the mean bit-rate at the

²

optimal FCS output attains maximal

(4)

Fig. 4. Dependence of FCS capacity

R^AS

on number of cycles under different power of received signals (from the top: for W 10 ; 10 ; 10 ; 10 ; 10

² ³ ⁴ ⁵ ⁶

y

3 4 5 6

0 ; 10 ; 10 ; 10³³ ⁴⁴ ⁵⁵ 10 ; 10 ; 1²²

,

2

10

5

V

ξ

, dash lines refer to the capacities

R^Ch¹

, stars - to threshold points).

value and determines the capacity of FCS as a whole:

2 2

0 0 0

2 2

log log

( ; ˆ )

max =

n n

FCS FCS n

n n n

n

σ F σ

F P n P

I X X

R C

T (19)

1 *

0 * 2

2 0 * 2 ν

for 1

log ( 1) for

C

Ch

n n

F n n

n

σ n n σ

d d

!

°

ª º

® « »

° ¬ ¼

¯

[bit/s]

Unlike the constant channel capacity (18), capacity of the system depends on n and diminishes beginning with n n n

^*^*

. This effect is confirmed by the results of simulations (Fig. 4).

C. Spectral-power efficiency of optimal FCS

The spectral (S)-efficiency of FCS R

_n^AS

/ F

₀

[bit/s/Hz] is de- termined by (19) and depends on n in a similar way. The pow- er (P)-efficiency of FCS can be assessed according to the basic definition [17] as normalized to the spectral density of channel noise, energy necessary for delivering of one bit to addressee:

2

0

2 0

log

2

ˆ ˆ

( , ) 2 ( , )

bit FCS

n n

ξ ξ n ξ n

n

E WT nW nQ

N N I X X N F I X X σ

P

[J/bit] . (20)

Substitution of (9) into (20) gives the expression:

1 2

* 2

2 2

* 2

0 *

2 2

ν

for 1 ; log (1 )

for

log ( 1)

bit Ch

bit FCS n

ξ

E Q

n n

N Q

E nQ

N n n

σ n n σ

[

d d

t

°

°° ®

° ª º

° « »

° ¬ ¼

¯

(21)

for theoretically achievable efficiency of transmission. Formu- la (21) and results of simulations (Fig. 5.) show that the “ener- gy of bit” E ^ˆ

_n^{bit AS}

/ N

_ξ

f

_n

( SNR

_out^Ch¹

) grows for n n n

^*^*

.

Claim: Transmission of the input signals with the sampling frequency n times smaller than

^*

F (with the spectrum ex-

⁰

pansion greater than n ) decreases efficiency of transmission.

^*

The obtained results are valid also for estimation systems discussed in the Introduction. For n n n

^*^*

, expressions for the limit characteristics of optimal FCS coincide with correspond- ing results in [1]-[9].

+

Fig. 5. Changes of limit values of P-efficiency

E^ˆ_n^{bit AS}/N_ξ f SNR_n( _out^Ch¹)

in different cycles of transmission (lower trajectory coincides with that for for- ward channel transmission, see also [18]). Equation Chapter (Next) Section 1

R EFERENCES

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[2] C. E. Shannon, "The zero-error capacity of a noisy channel." IRE Trans.

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[3] P. E. Green, “Feedback communication systems,” in Lectures on Com- munication System Theory, McGraw-Hill, NY, 1961 pp. 345-366.

[4] T. J. Goblick, “Theoretical limitations on the transmission of data from analog sources”, IEEE Trans. on Inf. Theory, vol.11, no.4, 1965, pp.

558 – 567.

[5] J. Schalkwijk and T. Kailath, “A coding scheme for additive noise channels with feedback - Part I” IEEE Trans. on In. Theory, vol. 12, April 1966 pp. 172–182,

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[13] Platonov A., “Wysokoefektywne iteracyjne systemy transmisji sygnałów z nadajnikami AM małej mocy (High efficient iterative transmission systems with low-power AM transmiters), Materiały Konferencyne X Poznańskich Warsztatów Telekomunikacyjnych, PWT - 2005, Poznań, 2005, pp. 347-352 (in Polish).

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Index Terms—Adaptive modulation, analog transmission, feedback systems, power-bandwidth efficiency. threshold effect