Development of criteria for determining the probability of error in digital television Текст научной статьи по специальности «Физика»

Radjabov Telman Dadaevich, Graduate student, of the Tashkent University of information technologies Faculty of Telecommunications technology Professor, Doctor of Physics and Mathematics Rakhimov Bakhtiyorjon Nematovich, Graduate student, of the Tashkent University of information technologies Faculty of Telecommunications technology Atadjanov Sherzod Shuxratovitch, Doctor of Technical Sciences, Tashkent University of Information Technologies, head of the department for educational and methodical works with regional branches, applicant University Maxsudov Ravshan Baxtiyor ugli, Graduate student, of the Tashkent University of information technologies Faculty of Telecommunications technology E-mail: umb@tuit.uz, tatuumb@mail.ru

DEVELOPMENT OF CRITERIA FOR DETERMINING THE PROBABILITY OF ERROR IN DIGITAL TELEVISION

Abstract: In this paper, we propose a method for determining and analyzing the probability of a PB error and estimating the noise immunity of signals based on the likelihood theory. The method is relatively simple in estimating the signal-to-noise ratio for signals with quadrature modulation types (for QAM example), it makes it possible to determine the probability of a bit (symbol) error with a sufficiently high accuracy from a small sample and without the need for any reference sequences in the signal, modulation type.

Keywords: modulation, keying, digital television, error probability, orthogonal signal, credibility, equal probability, communication channel, noise.

Digital television typically uses spectrally effective type modulation methods, PSK phase shift keying, FSK frequency shift keying, amplitude shift keying ASK, modulation without breaking phase CPM, differential phase shift keying DPSK, quadrature amplitude modulation QAM, orthogonal modulation with frequency multiplexing of COFDM signals. When all the modulation

methods are used, errors occur in the transmission of the bits of the PB. However, as it is known [1, 196-202], the higher the order of the constellation, the higher requirements are imposed on the transmission channel. Therefore, an adaptive approach to the operation of the communication system as a whole is used in the digital television (also in the GPRS system). The type of modulation is

selected as the result of a trade-off between the desired data rate and the quality of the communication channel. In addition, all digital communication systems have a threshold effect when the system operates on the verge of the corrective power of the channel decoder and a minimal deterioration in the quality of the signal may lead to a disconnection. Therefore, evaluating the proximity of the system to such a breakdown threshold will avoid unexpected effects.

1. Criteria for determining the probability of error PB in digital TV. For channels AWGN, binary symmetric channel (BSC), discrete memory-less channel (DMC) and Q-nary symmetric channel (QSK), the digital signal transmitted during the interval (0, T) is represented as follows:

J s2(t) 0 < t < T for the symbol 1 S (t) [s2(t) 0 < t < T for the symbol 0 The received signal r (t) is distorted due to the noise effect n (t), as well as the non-ideal impulse response of the channel hc (t), and is described by the following formula:

r{t) = (t)* h (t) + n{t) (2)

when hc (t), which does not degrade the quality of the waveform r (t) can be simplified:

r (t) = si (t) + n(t)i = 1,2. 0 < t < T (3) The model for demodulation and detection of a digital signal is shown in (Fig. 1). In this case, demodulation is defined as the restoration of a signal (into an undistorted video pulse), and detection is defined as the decision process regarding the digital value of this signal.

Figure 1. Model of demodulation The demodulation and sampling unit performs signal reconstruction as a preparation for the next necessary step - detection.

At the output of the filter, the noise is AWGN, then the output of step 1 can be described by the following expression.

z (T ) = a (T) + n0 (T )i = 1,2. (4)

where ai (T) is the desired signal component, and a n0 (T) is the noise. The expression (4) can be represented in simplified form as z = ai + n0, where n0

and detection of a digital signal

is the noise component - a random Gaussian variable with zero mean, therefore z (T) - is a random Gaussian variable with an average useful signal a1 or a2, depending on, a binary zero or binary one was transmitted. It is known that the probability density of random Gaussian noise n0 can be expressed as follows [1, 137-138]:

PK) = -

0^/2П

exp

i \2

\°o y

(5)

where &02 - is the noise variance. Using the expressions (4) and (5), we can express the densities of conditional probabilities p(zls^ and p(zls2):

p(z|Si) = -

exp

2

z - a,

'0 j

and

p(z|s2) = -

2

z - a

(6)

(7)

These densities of conditional probabilities are shown in (Fig. 2). The density p(zls^, shown on the right is called the likelihood s1 and shows the probability density of the random variable z (T) under the condition of the s1 symbol transfer, the function p(zIs2) is the likelihood s2 and shows the probability density z(T) the symbol s2. The abscissa, z (T), represents the full range of possible sampling values taken during phase 1 (Fig.1).

Figure 2. Densities of conditional

In Step 2, detection is performed by selecting a hypothesis that is a consequence of a threshold measurement.

Hi

z (T )=

H

(8)

probabilities: p(zls^ and p(zls2) time of the bit Tb. N0 is the spectral power density of noise, because it can be expressed as the noise power N divided by the bandwidth W. Since the bit transmission time Tb and the bit rate Rb are mutually inverse, Tb can be replaced by 1/ Rb:

ST

S/Rb

where H1 and H2 are two possible (binary) hypotheses. The choice of H1 is equivalent to the signal s1 (t) being transmitted, the result of the detection is a digital unit, also H2 is equivalent to the transmission of the signal s2 (t), and hence the result of the detection is a digital zero.

2. Rationing of the functional to evaluate the quality criterion for signal transmission in digital TV

In digital television, the normalized version of the ratio of the average signal power to the average noise power (S/N or SNR) - Eb / N0 is used as the criterion for the quality of transmitted signals.

Eb is the energy of the bit, it can be described as the signal power S multiplied by the transmission

(9)

N0 N/W N/W An important parameter used in digital television is the data transfer rate in bits per second. To represent the bit rate, instead of writing Rb for simplicity, R is written for simplicity. Taking into account (9), it is seen that the ratio Eb / N0 is the S/N ratio normalized to the bandwidth and bit rate:

Eb N

S_ N

W

R

(10)

Therefore, one of the most important quality metrics in digital television is a graph of the probability of occurrence of an erroneous bit Pb from Eb /N0, i.e. the graph of the functional Pb (Eb /N0), the smaller the required ratio is Eb / N0, the more effective is the detection process at a given error probability.

3. Develop criteria for maximum likelihood of receiving signals in digital TV

The decision criterion used in stage 2 (Fig.1) was described by the formula (8) as follows:

H

1

z (T )=

H

The criterion for choosing the threshold y for making a binary solution in expression (8) is based on minimizing the error probability. The calculation of this minimum value of the error Y = Y0 begins with a relationship record of the ratio of the densities of conditional probabilities and the ratio of the priori probabilities of the appearance of the signal. Since the density of the conditional probability p(zIsJ is also called the likelihood function si , formula

H

p(zK) _ > plO p (z|s2 )

(11)

< P(si) H2

there is the likelihood ratio criterion functions. In this inequality P (s2) and P (s2) are priori probabilities of s2 (t) and s 2 (t) signal transmission. It follows from formula (11) that if the ratio of the truth-likeness functions is greater than the ratio of priori probabilities, then hypothesis H1 should be chosen. For P(s2 ) = P(s2) and symmetric right-hypothetical functions p(zIsi),((= 1, 2) ,the substitution of formulas (6) and (7) in (11) yields

H1

/ X > ai + a2 z (T )=< — = Xo

H

(12)

where a1 is the signal component of z(T) in the transmission s1 (t), and a2 is the signal component of z(t) for s2 (t). The threshold Y0, represented by the expression (a1 + a2) / 2 - is the optimal threshold for minimizing the probability of making the wrong decision in this important particular case. This approach is called the minimum error criterion.

For equiprobable signals, the optimal threshold Y0, as shown in (Fig. 2), passes through the intersection of the likelihood functions. From formula (12), it is clear that the decision-making stage consists in an effective choice of the hypothesis corresponding to the signal with the maximum likelihood. If the sample value of the received signal is za

(T)

decision criterion can be considered as a comparison of the likelihood functions p(zaIs^ and p(zaIs2), that is, the more probable value of the transmitted signal corresponds to the largest probability density, while the detector chooses s1 (t) if

p (za |s1 )> p (za |sj (13)

Otherwise, the detector selects s 2 (t). 4. Analytical aspects of error probability estimation and its optimization in digital TV

Theorem. The error probability PB when receiving and transmitting digital signals depends on the energy of the received bit and the power spectral density of the noise, but not on the specific waveform.

Proofs of the theorem. In the process of making a binary solution, shown in (Fig. 2), there are two possible errors. An error e will appear when s1 (t) is transmitted, if, due to the channel noise, the level of the transmitted signal z(t) falls below Y0. The probability of this is equal to the following [1, 149-151]:

P (e|s1 ) = P (H2Is1 )=| p (z Is1 )dz (14)

—X

Similarly, the error occurs when s 2 (t) is transmitted, if, due to the channel noise, the level of the transmitted signal z(t) rises above Y0:

P (eIs 2 ) = P (H1Is 2 )= {p (z Is2 )dz

(15)

The probability of an error is equal to the sum of the probabilities of all the possibilities of its appearance. For the binary case, the probability of an erroneous bit can be expressed as follows:

Pb =£P (e s ) = tP (eIst )P (st) (16) i=1 i=1 Combining formulas (14) - (16), we obtain

Pb = P (eIs1 )P (s,) + P (eIs 2 )P ^) (17)

or,

PB = P (H2ls, )P (si) + P (H ils 2 )P (s 2) (18) i.e., when s1 (t) is transmitted, the error occurs when H2 is selected, or when s2 (t) is transmitted, the error occurs when H1 is selected. For equal priori probabilities (i.e. P(s1) = P(s2) = 1/2 ), we have the following:

PB = 1P(H2ls,) + 2P(HilsJ (19)

Using the symmetry of probability densities, we obtain the following:

Pb = P (H2lsi ) = P (HilsJ (20)

The probability of an erroneous bit, PB, is numerically equal to the area under the "tail" of any likelihood function, p (z ls1) or p (zls 2) that "fills" the "wrong" side of the threshold. Thus, to compute PB, the function p(zls^ is integrated from - to to Y0 or p(zls2) from Y0 to to:

Pb = { p (zls 2 )dz (21)

Y0 =(ai +a2 )/2

Here y0 =(a1 + a2)/2 is the optimal threshold from equation (12). replacing the likelihood function p(zls2) by its Gaussian equivalent of formula (7), we have

Pb =

1

exp

2

z - a.

0 J

dz (22)

Y0 =(ai+a2 )/2

We make the substitution u = (z - a2) / A0. Then \du = dz, for z = y0 the boundary of the interval

u = (z - a2) / a0 =(a1 - a2) / 2a0 and

Pb =

I

=(#1 — a2 )/2o

1

exp

2

u 2

du = Q

c \ a, — a

2a,

(23)

0

q (x) is the error integral in the AWGN channel and is used to describe the probability with a Gaussian distribution density. This function is defined as follows:

(24)

1

Q (x ) = -= f exp

V2n X

2

du

Q (x) can not be calculated in an analytical form, it can be calculated by approximation, for x > 3, the approximation function Q (x) is calculated by the following formula:

Q (x ) =

exp

x 2

2

(25)

Suppose that the input of a linear, time-invariant filter, followed by a discredit device (Fig.1), is fed with the known signal s(t) plus the noise of the AWGN n(t). At time t = T, the output signal of the sampling device z (T) consists of the signal component a{ and the noise component n0. The dispersion of the noise at the output (average noise power) is written as ct02 . The ratio of the instantaneous noise power to the average noise power, (S / N )T, at time t = T outside the sampling device in step 1 is equal to the following:

S

N

Jt

(26)

The problem consists of determining the transfer function of the filter H0 (f) with the maximum ratio (S / N)T . The signal at the filter output can be expressed in terms of the filter transfer function H(f) (before optimization) and the Fourier transform of the signal at the input

a, (t)= JH(f )S(f )e2mftdf (27)

—K

where S (f) is the Fourier transform of the signal at the input, s (t). If the two-way spectral noise power density at the input is N0 / 2, then the output noise power can be written as follows:

=N« 02

{ |H(f )\df (28)

—iX)

Combining formulas (26) and (28), we obtain an expression for (S / N )T:

2

LH (f )S (f )e 2mftdf

S

V N jT

ri , M2 (29)

n0/2LI h (f )l df

The case H (f) = H0 (f) is determined, at which (S / N) reaches a maximum. To do this, we use the Schwarz inequality, one of the forms of which is presented below:

00 00 00 |f1 (x)f2 (x)dx <| |f1 (x)| dx | |f2 (x)| dx (30)

CO

2

Equality is achieved with f1 (x) = kf2 (x), where k is an arbitrary constant, and the symbol "*" denotes a complex conjugate value. If the Schwarz inequality is applied to formula (29) with the substitution of values, we obtain the following inequalities: 2

TO TO TO

{H(f )S(f )eff < { \H(f )2 df J |S(f )2 df (31)

i c \

S_

V N

<

N

00

i S(f )2 df

or

max

v N y

2E

N

(32)

(33)

where the energy E of the input signal s (t) is equal to

k

E = J |S(f )2 df. (34)

—k

To optimize PB in the channel environment and the receiver with the AWGN noise shown in (Fig. 1), we need to select the optimal receiving filter in step 1 and the optimal decision threshold in step 2. For the binary case, the optimal decision threshold is already selected and given by formula (12), and in formula (1.23) it is shown that the error probability at such a threshold is PB = Q [(a1 - a2 )/2a0 ] . For a minimal PB, in general, you need to select a filter (matched) with the maximum argument of the function Q(x). Therefore, you need to determine the maximum (a1 - a2) / 2a0, which is equivalent to the

maximum

( - a2 )2

(35)

where (a1 - a2) is the difference of the desired signal components at the output of the linear filter at time t = T, and the square of this difference signal represents its instantaneous power. In the derivation given in equations (26)-(34), it was shown that 100 matched filters give the maximum possible signal-to-noise ratio equal to 2E / N0. Let us assume that the filter matches the input difference signal h (t)- s2 (t)] . Therefore, for the time t = T, the signal-to-noise ratio at the output:

-Si

N T

(ai - a2 )2 _2Ed

N

and

E =

= {[si (t )- s 2 (t )J dt

(36)

(37)

is the energy of the difference signal at the input of the filter. Equation (36) does not represent the signal-to-noise ratio for any particular transmission, s1 (t) and s 2 (t), this ratio gives the metric of the difference of signals at the filter output.

Combining equations (23) and (36), we obtain the following:

Pb = Q

K

2N

(38)

Equations (38) are important intermediate result involving the energy of the difference signal at the input of the filter. From this equation, we can derive a more general relationship for the energy of the received bit. First, we determine the time correlation coefficient p, which we will use as a measure of the similarity of the two signals s1 (t ) and s 2 (t ). We have a classical correlation expression

1 t

P= — \si (t)s2 (t)dt

Eb o

(39)

and

p = cos0, (40)

where - 1< p < 1. If we consider s1 (t) and s2 (t) as signal vectors s1 and s2, then the more convenient representation p is the formula (40). A vector representation allows you to obtain convenient graphic images. The vectors s1 and s2 are separated by an angle 0; at a small angle, the vectors are sufficiently similar (strongly correlated), and at large angles they differ. The cosine of the angle 0 gives the same normalized correlation metric as the formula (39).

Writing out the expression (37), we obtain the following:

T T T

Ed = Js2 (t)dt + Js22 (t)dt -2js1 (t)s2 (t)dt. (41) 0 0 0

The first two terms in the formula (1.41) represent the energy associated with the bit, Eb:

2

Eb = {5? (t )dt = {5 22 (t )dt (42)

0 0

Substituting equations (39) and (42) into the formula (1.41), we obtain the following:

Ed = Eb + Eb -2pEb = 2Eb (l-p) . (43) Substituting equation (43) into (38), we obtain the following

pb = Q

e (1 -p)

N

(44)

Based on the value of the cross-correlation coefficient, there are three cases of determining the error probability PB in a real system:

1. p = 1. If the signals are represented as vectors, the angle between them will be zero. Since, in a real system, communication signals (alphabet elements) should be as incommensurable as possible so that they can be easily distinguished (detected). This value ofp is practically not used;

2. p = -1. The angle between the signal vectors is 180° and these signals are called antipodal (Fig.3, a).

3. p = 0. The angle between the vectors is 90°, such signals are called orthogonal (Fig. 3, b). In order for the two signals to be orthogonal, they should not be correlated during the symbol transmission time, i.e. the following condition must hold:

T

|s1 (t)s2 (t)dt = 0 . (45)

a) b)

Figure 3. Vectors of digital signals: a) antipodal; b) orthogonal

When detecting antipode signals (p = -l) using a matched filter, equation (44) can be written as follows:

Pb = Q

r \2Eh

J N0 J

(46)

Similarly, when detecting orthogonal signals

(P = 0) :

Pb = Q

(47)

Therefore, the error probability PB is dependent on the energy of the input bit and the noise power spectral density, but not on the specific waveform. The theorem is proved.

5. The Energy Advantage Under Canal Coding

For definition of efficiency in the beginning for the circuit 16-level PSK without channel encoding pays off, how much greater (concerning accessible 18.5 dB) value Eb/N0 is required for reception PB = 1010. This additional Eb/N0 is demanded efficiency of channel encoding. Using the following formula it is found E /N0 without use of encoding which will give probability of occurrence of mistake P = 1010.

B

2Q

2Es N

Sin

n M

\

tog2 M

tog2 M

= 10-

(48)

0

B

Calculation of the equation (48) with a trial and error method rather E/N0, will give value for system without channel encod ingE/N0 = 518.25 (27.15 dB) and as each symbol consists from log216=4 the bit's, demanded Eb/N0 (without encoding) = = 518.25/4 = 129.6(21.13 dB). From the equation

, n = 127, k = 106(49)

A- = (log2 m A = dog2 m )f k 1 Et

N o N o ^ n ) N0

it is known, E/N0 = 236.4 (23.74 dB), and therefore, with channel encoding, Eb/N0 (with encoding) = = 236.4/4 = 59.1 (17.72 dB). Efficiency of channel encoding is defined by the following formula:

G foE) = 1^- (dB) -

V 0 / without encoding V 0 / with encoding

(dB) = 21.13 dB-17.72 dB = 3.41 dB For estimation of efficiency of the canal coding compare the attitude Eb/N0 energy, happening to on one bit, to spectral density of the powers of the noise in system with coding and in base system without coding and define the difference in importance Eb/N0 under given probability of the error (fig. 4). This difference measured in decibel and named energy advantage of the code (EAC), can be used for comparison of the different codes.

Figure 4. The

In (table 1) is brought resulting advantage of the coding is brought resulting advantage of the coding for different combination internal and external codes under two importance's of probability of the error - 10-5 and 10-8.

Table 1. - EAC for different codes and combination of the codes [8]

Variant of the coding/ decoding EAC, dB

10-5 10-8

1. Rid-Solomon+Viterbi 6.5...7.5 8.5.9.5

2. Rid-Solomon +bior-thogonal 5...7 7.9

3. Rid-Solomon+short block 4.5.5.5 6.5.7.5

4. Viterbi 4.5.5 5.6.5

5. Block code (hard decision) 3.4 4.5.5.5

6. Convolution code (decoding on threshold) 1.5.3 2.5.4.0

mark of EAC

The conclusion

The final criterion for the quality of the digital television system is often the current bit value (BER) or symbolic error (SER). Their measurement is performed either on the basis of the analysis of the reference sequences that are transmitted together with the signal, or on the basis of data on the occurring errors received from the channel decoder.The technique proposed above can be used not only to determine the signal-to-noise ratio (SNR), but also to calculate the probability of a bit (or symbol) error for all types of digital modulation.

From the method suggested above, it becomes clear that in digital television the probability of error PB does not depend on the type of modulation and on the specific waveform, but only depends on the energy of the input bit and the spectral power density of the noise.

The system of modulation in digital television The method allows to determine the current

and the question of the application of one or the value of the bit error in the receiver much faster and

other modulation methods is universal. The qual- with less error than traditional methods.

ity of the work depends only on technological and

probability-energy parameters of the system.

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