Reception in noise presence

Reception numerica in noise presence

1) Characteristics of the numerical reception in noise presence:

We suppose that a real numerical channel is affection from AWGN, it can be represented on a base of ¥ payers whom the base to n payers contains on which the useful process is represented. The noise n_r(t) represented on the first N payers contains remaining information on the process whilenthe (t) represented on the ¥ - N payers is independent is from the useful process that from n_r(t), it comes called noise insignificant and neglected in the remaining trattazione. The noise is represented therefore in the space of marks them from the carrier where the n_j they have density of probability of 1° the order, analogous it marks received them r_r(t) is described from the observable carrier with that they are v.a. continuous in how much sum of a v.a. discreet and a v.a. it continues.

 

2) Demodulation and decision in noise presence:

It supposes itself that the numerical modulation has been lacking in codifies and using a transformation series-parallel, has associated to every element of the constellation one binary word with bit. The outline of the numerical demodulatore is:

tothe escape of the bench of correlatori the members r _j of observable carrier r ₀ have themselves that for via of the noise does not coincide with no carrier of the constellation, have themselves in fact that the probability to receive oncer _j that has been emitted s_d is and being the n_j statistically independent is had .

A decisore is therefore necessary which applying a decision criterion it associates r₀ to one of the carriers of the constellation.

 

3) Criteria of decision:

CRITERION MAP (Maximum A posteriori probability)

For every carrier s_d that it can be emitted from the modulator it calculates the conditioned probability dopodichè chooses the carrier s_j if every for the ¹ j, of it turns out that the probability of wrong decision is minimal and therefore the criterion is optimal. The in kind is not notes while instead they are the then applying the theorem to it of Bayes has therefore observing that the single numerator is function of obtains a formulation alternative of second criterion MAP which is chosen s_j if every for the ¹ j Remembering then that is reached 3ª the formulation of second criterion MAP which chooses s_j if every for the ¹j that in short it implies the subdivision of the space of marks them in regions of decision weighed with the probability of emission of every carrier of the constellation. From this last formulation of Criterion MAP evince that it is verified if is had the maximum of the vectorial function with therefore has the following alternative outline of numerical demodulatore with decisore .

CRITERION MILILITER (Maximum Likelihood = Maximum verosimiglianza)

It is a criterion subottimo according to which every for the ¹ is chosen s _j if j which banally the second formulation corresponds alternative which chooses s_j if every for the ¹j that is chooses the carrier s_j of the constellation that more close turns out to be to r₀ . In this case being absent the probabilities of emission of the single carriers, the decision regions have all the same area.

 

4) Probability of correct and wrong decision:

Remembering that the decision criteria do not make other that to associate to every carrier of the constellation a decision region, it follows some that the probability of corrected decision conditioned to the issued carrier s_d is the integral of on the associatedD region _d to s_d , has that is while the probability of corrected decision is and the probability of error P_and = 1-P_C Finally the probability of P error_and is the probability that is a wrong bit and is function of codifies of used modulation, as an example in the case of antipodale modulation .

Linear esteem of marks them continuous

5) Characteristics of the continuous esteem:

The received process is the sum of a stationary or ciclostazionario useful process with null medium continuous realization s(t) to valor and spectral density famous P(f) which a P(f) power corresponds, and of a noise Gaussian additive with null statistical medium realization n(t) to valor and spectral density N(f), beginning from r(t) the appraiser supplies a continuous esteem linear of the single comprised useful part s(t) in r(t), the esteem error store clerk is . For a data moment t_{the 0} performances of the appraiser are function of s(t) = s and come characterized through the conditioned polarization , the conditioned variance of the error and the medium quadratic error conditioned whose expected values are respective the polarization, the variance of the error and the quadratic error medium . An adimensional relative pointer called quality of the continuous esteem has itself moreover that in short is a relationship beacon-noise.

 

6) Characteristics of appraisers LTI:

The appraiser must necessarily introduce distortion in how much must favor the transit of the spectral members of r(t) for which he marks them useful prevails on the noise and hindering instead the transit of the members for which the noise prevails on marks them. Admitting the single linear distortion of amplitude the following outline is had equivalent for the appraiser whom it evidences as the continuous esteem is is composed from 4 addends

where n(t) and n_and(t) they are disturb while s_and(t) is a disturbance if .

One finds that the characteristics of the appraiser sono , , , being P_and the power of the disturbance due to the linear distortion of amplitude, the power of the single noise in escape and the power of the process of esteem LTI. It is observed that the optimization of the quality of the esteem is invariant regarding the presence of a constant factor in the employed function of transfer, this result comes employed for ricondursi always to the case particular of noise Gaussian additive white man by means of one sbiancatore to put in front to the appraiser and one sbiancatore complementary to postplace.

15) Typology of appraisers LTI:

Optimal linear appraiser of Wiener-Hopf

Diminishing the quality of the esteem is maximized, such optimization is carried out from the filter of Wiener-Hopf with , antitransforming finds a real impulsive answer and equal therefore antimotive therefore the quadripolo is not physically realizable, then uses a filter whose impulsive answer approximates that one of Wiener-Hopf per t > 0 but it is null for t < 0.

Linear appraiser with optimal cut of band

We consider for semplicità a noise white man with spectral density N₀ , the linear appraiser with optimal cut of band has function of rectangular transfer that is worth g_t0 for the frequencies comprised between f₁ and f₂ for which the useful process prevails on the noise and is worth elsewhere instead 0. The quality that turns out some is and is maximized for the physically realizable filter introduces an attenuation of 6dB to the cut frequencies and determines a relationship greater beacon-noise regarding that it would be had with the rectangular filter.

Not polarized linear appraiser

If we have a process received with useful part closely limited in the B band = f_M â " f_m, we can reduce the noise generating but distortion using a filter with band B₁₂ < B.

If the scope is instead to reduce the distortion, the appraiser must be polarized that is does not have to be a filter with frequencies of cut that coincide with the ends of the useful band, in such case in fact the quality of the esteem comes to coincide with the relationship beacon-noise to the Q escape.

Such appraiser is famous also in absence of acquaintance of the spectral density of useful power and its quality is gained immediately, is therefore preferred to the other appraisers also because to growing of the quality of the esteem, it turns out to you differ little.