Parameters of the radio hannels that affet digital signal transmissions 1.Free spae attenuation - the signal undergoes an average attenuation that depends on the length of the path and signal s frequeny f (1). The exponent γ in relation (1) is alled attenuation index. L pm = [(4πf)/] γ ; (1) - if G t and G r represent the gains of the antennae at the transmitter and reeiver, respetively, and P t is the transmitted power, then the power at the input of the reeiver is given by the Friis s transmission equation [mil]: P r = P t G t G r [/(4 πf)] ; () - by inluding the gains of the two antennae in the free spae attenuation, relation (() expressed in db, beomes: P rm ()[db] = P t [db] - L pm ( )[db]; (3) - the attenuation in (3) is also influened by the geographi parameters of the environment. Aording to the environment where the propagation takes plae, the attenuation index varies. Table 1 show the attenuation indexes for several propagation environments, [rap]: Propagation Environment Attenuation Index, γ Free spae Urban area,7 to 3,5 Shadowed urban area 3 to 5 Diret propagation within buildings 1,6 to 1,8 Table 1 Values of the attenuation index for several propagation environments - the attenuation of radio transmissions also inreases signifiantly in the presene of rain, for frequenies larger than 10 GHz. The method of evaluation of the attenuation due to rain in desribed in [mil].. Log-normal fading - expressions (1) and (3) don t take into aount the fat that the attenuation an be different for two positions situated at the same distane from the transmitter, due to environment fators (buildings, forests et.). Measurements have shown that the attenuation, expressed in db, in different positions at the same distane, has a normal (Gaussian) distribution around the mean value given by (1); the resulted attenuation, in db, is expressed by (4): L p ( ) [db] = L pm ( ) [db] + ΔL pσ [db]; (4) - this variation of the attenuation is alled log-normal fading, beause its value expressed in db (after a logarithm operation), is distributed aording to a Gaussian law. - the probability of the reeived signal s level to be higher than an imposed value an be expressed using the Q(t) funtion and relation (5): u l P () 1 rm P(P () l) Q( ); Q(t) e r du; (5) t 3. Doppler frequeny spread - the radio ommuniations serving mobile stations are affeted by frequeny spreading of the reeived signal, due by the Doppler effet aused by the movement of the mobile station. - onsider a mobile station that moves with speed v and a signal transmitted with the frequeny f, (6.a), that reahes the antenna of the reeiver under an angle θ vs. its normal plane, then the frequeny of the signal entering the reeiver undergoes a frequeny deviation f d, due to the Doppler effet, (7.a). - due to the hange of the inidene angle, and the hange in position of the mobile station, the frequeny deviation f d, modifies its value between 0 and a maximum value f m (7.b), being spread within the range (f p f m, f p + f m ). This shows that the reeived signal undergoes spreading (dispersie) in frequeny, alled Doppler spread. The expression of the reeived signal is given by equation (6.b) st Aos(fpt);a. s r(t) Aos[ (fp f d)t];b. (6) v vfp vfp fd os os ; a. fm for 0 ; b. (7) 1
4. Fading. Classifiation. - multipath propagation affets signifiantly the reeived signal over a radio hannel. The transmitted signal reahes the reeiver s antenna over several propagation paths, determined by the existene of obstales that reflet the transmitted wave. Thus, the reeiver is reahed by the diret wave (if any), that had no obstales in its path, and one or several refleted waves. Beause the propagation paths do not have the same length, the signals that reah the reeiver an have different delays, generating the phenomena of temporal spread. - temporal spread (delay spread) depends on the length of the seondary (refleted) paths and produes inter-symbol interferene, whih beomes signifiant for large delays (of order of one or several symbol periods). Beause temporal spread is a random variable, it is haraterised by its variane σ τ. - in order to haraterize a wide band hannel, the oherene band of the hannel, B, is defined. This is the frequeny range within whih the hannel an be onsidered to be (approximately) uniform, i.e. it has approximately the same attenuation and a linear phase variation. Within this range, two signals having different frequenies have their amplitudes strongly orrelated. The oherene band an be approximated by relation (8) for orrelation fators equalling 0.9 and, respetively, 0.5: B 1/(50σ τ) or B 1/(5σ τ ); (8) - for radio hannels with a mobile station, due to the Doppler spread, a oherene time T, is defined. The T is the time interval within whih the hannel s attenuation and phase harateristis remain (approximately) onstant. It depends on the maximum frequeny deviation (7) and its expression is: T = 0,43/f m ; (9) - the two disturbing effets produe different types of fading. A lassifiation of the types of fading produed by these effets is given in [rap]. - the multipath delay spread produes flat fading or frequeny seletive fading. Denoting the signal bandwidth by B S and the symbol period of the transmission by T S, the onditions of ourrene of the two types of fading are approximately: for flat fading: 1. B S << B or. σ τ << T S ; (10.a) for frequeny seletive fading: 1. B S >> B or. σ τ > >T S (10.b) - for hannel parameters having lose values to transmission s ones, the two types of fading an our simultaneously, in different proportions. - the Doppler spread, together with the multipath propagation auses, in the ase of mobile hannels, fast fading (fading rapid) and slow fading (fading lent). The onditions of ourrene for these fading types are: for fast fading: f d large or. T < T S or 3. the hannel variations are faster than those of the BB signal (11.a) for slow fading: f d small or. T > T S or 3. hannel variations are slower than those of the base-band signal (11.b) 5. Flat fading and frequeny seletive fading - the effets of the multipath propagation depend on the frequeny this is a frequeny seletive phenomenon - example: onsider a QAM transmission within whih s(t) (1) is the transmitted signal, and r(t) (13) is the reeived signal, and for whih the refleted omponent undergoes attenuation b and the delay τ, with respet to the diret wave; indexes i and q denote the in-phase and quadrature signal omponents of the involved signals s(t) vios fpt vqsin fpt; (1); r(t) s(t) bs(t ); (13) - the reeived signal will be: r(t) vi os fpt vq sin fpt bv i(t ) os f p(t ) bv q(t ) sin f p(t ); (14) - using the synhronised arrier for the QAM demodulation for the in-phase branh: p(t) i os fpt; (15) after the multipliation of the arrier with the reeived signal and after LP filtering, operations denoted by *, we get: r(t)*p i(t) v i(t) bv i(t )os fpbv q(t )sin fp (16)
- (16) shows two disturbing effets: the ourrene of interferene between the two modulating signals (third term) and the superposition of the delayed version over the use full signal (seond term). Note that for ertain values of Tp/τ, the seond term is added or subtrated from the value of the useful data signal v i (t), aording to the relation between T p and τ. The extreme situations are given by the equations below: os fp1 fp n; or f p(n1) ; (17) from (17) => r(t)*pi (t) vi (t) bvi (t ) for n f p nt p (18) r(t)*pi (t) vi (t) bvi (t ) for n 1 f p (n1)t p (19) - if the signal frequeny takes the values given by (18), the amplitude of the signal inreases, and if it has the values given by (19), the signal is attenuated; the values of the signal s inreases or dereases depend on the attenuation oeffiient of the signal on the delayed path and on the modulated signal. There should also be noted that, sine the symbol period usually is muh greater than de delay τ, the demodulated baseband signals of the two paths ould be onsidered as synhronous. A n Amplitude f n (i)=(i+1)/τ f n (i)=(i+3)/τ f n (i)=(i+5)/τ f Figure 1 Shemati representation of the frequeny seletiv fading; f n represent the noth frequenies - the frequeny values given by (19) are alled noth frequenies. For fixed radio hannels, these frequenies have fixed values, but for mobile hannels, due to the hange of the position of mobile stations, the values of these frequenies hange. For QAM signals, suh frequenies also our for the term introdued by the quadrature omponent of the modulated signal. Figure 1 presents shematially the frequeny seletive fading phenomenon. - if the frequeny bandwidth of the modulated signal < 1/τ, i.e., the frequeny spaing between a minima and a maxima, then the attenuation is relatively onstant and we have a flat fading. - if the frequeny bandwidth > 1/τ, then the hannel exhibits a frequeny seletive fading phenomenon. - the same phenomenon an our if two refleted waves, delayed in different ways and having a delay differene Δτ, arrive at the reeiver - if the reeiver is reahed by several refleted waves with distint delays, then several groups of noth frequenies our, determined by eah delay time τ i, and the attenuation of eah group is different, aording to the attenuation fator b i of eah refleted wave. 6. Fast fading and slow fading. - fast fading and slow fading refer to the speed of variation of the envelope of the reeived signal, i.e. the evolution of the reeived signal s amplitude in time. - for fixed radio hannels, they are aused by the variations of the hannel parameters, but these variations are slow, thus, allowing large symbol period values of the transmission. - for mobile radio hannels, they are aused by the frequeny shift produed by the Doppler effet, the symbol period dereasing with the inrease of the relative speed between transmitter and reeiver. - example: let us analyse the ase of a mobile station that reeives two waves, for whih the arriving angles differ by Θ; the speed of the mobile station is v. - onsidering the transmitted signal to be: s(t) 0os fpt (0) the reeived signal will have the following form: vfp vfp r(t) 0os[t(f p )] 0os[t(fp os )] (1) - the reeived signal deomposed aording to its in-phase and quadrature omponents, i (t) and q (t), is: r(t) i(t)osfpt q(t)sin fpt; a. vfp vfp vfp vfp i(t) 0[os( t) os( tos ); b. q(t) 0[sin( t) sin( tos );. () 3
p - the envelope of the reeived signal is: vf (t) i (t) q(t) 0os[ (1os )] (3) - from (3) the envelope of the reeived signal varies with the frequeny: f f m D (1os ) (4) - the maximum speed of variation is reahed for Θ = 180 and has the value f m, indued by the Doppler effet. - if the frequeny of the envelope s variation has its period larger than the symbol period of the transmission, we have fast fading; if the frequeny of variation is smaller than the symbol frequeny, we have slow fading. 7. Envelope Distribution Laws of the Faded Signal 7.1. N refleted waves the ayleigh distribution - if the reeived signal is omposed of N refleted waves, eah having different but onstant amplitude n, an inidene angle Θ n, and a Doppler shift f dn, then the reeived signal would be expressed by: N vf r(t) n os(f t f dnt); fdn osn (5) n1 - by a similar reasoning as in (0,,4), the expression of the reeived signal deomposed into its I and Q omponents is given by the seond line of (6): r(t) i (t)osf t q (t)sin f t; N N (6) i (t) n osf dnt; q (t) n sin f dnt; n1 n1 - If N has great values, then a. to the Central Limit Theorem, the i (t) and q (t) are random variables with Gaussian distributions and null mean values, and the reeived signal s envelope (t), (7) will be a random variable with a ayleigh distribution [mil]. (t) i (t) q (t) ; (7) P() 0,6065/σ - The probability density of this distribution, i.e. the probability of the envelope to equal, if the average power of the reeived signal equals σ, is given by (8) and is shematially represented in figure. p( ) e ; 0 ; (8) 0 σ σ 3 σ 4σ - the frequeny band of the reeived signal equals [f -f m ; f + f m ] and is determined by the Doppler spread. Figura eprezentarea aproximativă a distribuţiei ayleigh 7.. Diret wave and N refleted waves the ie distribution - the reeived signal is omposed of a diret wave and N refleted wave, eah having different but onstant amplitude n, an inidene angle Θ n, and a Doppler shift f dn, the reeived signal an be expressed as a sum between the diret signal d(t) and a signal r(t) omposed of the refleted waves, whose envelope would take values a. to the ayleigh distribution. - the reeived signal s(t) would be, after its deomposition in the I and Q omponents, expressed by, see also (6): s(t) = d(t) + r(t) = D i (t)osπf t + D q (t)sinπf t + i (t)osπf t + q (t)sinπf t; (9) - Literature shows that the envelope S(t) of the reeived signal: S(t) [Di (t) i (t)] [Dq (t) q (t)] (30) would take values a. to the ie distribution, whih has the probability density funtion: (SA) S A p(s) e I 0( ); for A 0 şis 0; (31) 4
- expression (31) is also expressed in literature in terms of ratio K between the power of the diret signal, of amplitude A, and the sum of the powers of the refleted waves, whih is proportional to σ, see (3.a). I 0 (t) A n t denotes the order-zero modified Bessel funtion, (3.b): K ;a. I0 (t) ;b. (3) n n0 n! - If the power of the diret signal dereases signifiantly (A 0), then the reeived signal is omposed predominantly of refleted signals, and its envelope s values will be distributed aording to the ayleigh distribution. This means that for K 0, the ie distribution degenerates into a ayleigh distribution. 8. Effets of the final radio frequeny amplifiers - in order to ensure power effiieny, final radio amplifiers are used lose to the saturation region of their transfer harateristi P o = f(p i ). - under these irumstanes, the more the output power gets loser to the maximum allowed value, the more non-linear the power transfer harateristi beomes, and the amplifier enters the saturation region of its transfer harateristi. - for these ases, the transfer harateristi an no longer be approximated with a linear harateristi, i.e., U o = a U i, but an be approximated with a polynomial transfer funtion of type (35) that inludes the relative delays between the omponents of the output signal: - to point out the effets of this non-linearity upon the output signal, we approximate the transfer (amplifiation) harateristi by a polynomial funtion (33), whih would inlude the relative delays of the output signal s omponents : U 0 (t) = a U i (t) + b U i (t-τ ) + U 3 i (t-τ 3 ) + ; a,b, onstants; (33) - the modulated signal that has to be amplified is: s i (t) = (t)os(ω p t + φ(t)); (34) - if the modulated signal (34) has a onstant envelope (t) = 0, its phase is denoted by Φ(t), and the phase shifts inserted by the delays t 1 and t are denoted by φ 1 (t) and φ (t), then the output signal of suh an amplifier (hard-limiting) will be: b 0 b s 0 o(t) a0os (t) os[( (t) 1)] (35) 40os[3( (t) )] 3 0os( (t) ); (t) wpt (t) - after some trigonometri manipulations (35) an be expressed as: b s 0 o(t) [a 3os (t)] os (t) [3sin (t)] sin (t) b 0 os[( (t) 1 )] 4 0 os[3( (t) )] (36) 3 sin b 0 0 a 9 9a os os[ (t) artg ] (a 3) os b 0 os[( (t) 1 )] 4 0 os[3 ( (t) )]; - relation (36) shows that the output signal has a baseband spetral omponent, b 0 /, a spetral omponent on the seond harmoni of the arrier signal, the seond term, and a spetral omponent on the third harmoni of the arrier signal, the last term. - the presene of these terms shows that the non-linearity of the amplifier inserts spetral omponents outside the useful (allowed) frequeny band, that are not present in the input signal; this phenomenon is alled spetral regrowth. The levels of these omponents derease for more linear amplifiers, onstant b and very small, and for small variations of the modulated signal s envelope around its average value, i.e. small PAP values. - oming bak to the first term of (36), this is plaed in the allowed frequeny band, but both its amplitude and phase are modified (distorted). - if the envelope (t) is not onstant, the signal at the output of a non-linear amplifier is expressed by (37); in (37) only the spetral omponents in the allowed frequeny band are inluded s e (t) = u((t)) os[ω p t + v((t)) + φ(t)]; (37) 5
- the variation of (t), u((t)), is denoted as the AM/AM harateristi, while the variation of the phase is denoted as the AM/PM harateristi of the amplifier, both indiating the ways in whih the variation of the modulated signal that has to be amplified, affet the amplitude and phase of the output (amplified) signal. - the degree of distortion inserted by suh an amplifier dereases with the derease of the PAP value of the modulated signal. Summarizing, the nonlinear harateristi of the final F amplifier lead to the following onsequenes: 1. distortion of the signal s envelope - by the harateristi AM/AM. distortion of the signal s phase by the harateristi AM/PM 3. spetral regrowth of some omponents outside the allowed frequeny band, whih were removed by the transmitter s filters before the final amplifiation; - the first two onsequenes affet the quality of the signal that reahes the reeiver, leading to an inreased biterror rate after the demodulation. - the third onsequene generates some undesired signals that interfere with the transmissions in the neighbouring frequeny bands. - to derease the amplitudes of the omponents outside the allowed frequeny band, whih our due to the spetral regrowth, as well as for dereasing the distortions of the signal in the allowed frequeny band, two aspets should be taken into aount: the derease of the non-linearity degree of the final power amplifier; the employment of modulations that ensure an amplitude of the modulated signal as onstant as possible; - the derease of the degree of non-linearity of the final amplifier an be ahieved by the following approahes: lowering the operation point in the linear region of the amplifier s harateristi P o = f(p i ). This proess requires the derease of the output power, alled output bak-off ( repliere a ieşirii ) and denoted with B o [db], that is obtained by dereasing the power of the input signal, alled input bak-off ( repliere a intrării ) and denoted with B i [db]. employment of some methods to ompensate the non-linearity, e.g., the method alled linear amplifiation with nonlinear omponents (LINC). employment of amplifiers with AM/AM harateristis as linear as possible, and with AM/PM harateristis as onstant as possible. 6