RADIO SYSTEMS – ETI 051 Contents 3 • Short review Lecture no: NARROW-BAND CHANNELS WIDE-BAND CHANNELS • Radio signals and complex • Delay (time) dispersion notation • Narrow- versus wide-band • Large-scale fading channels Narrow- and wideband • Small-scale fading • The WSSUS model • Combining large- and small- – Wide-sense stationary (US) channels scale fading – Uncorrelated scatterers (US) – Tapped delay line models • Noise- and interference- limited links • Condensed parameters – Some system functions – Window parameters Ove Edfors, Department of Electrical and Information technology Ove.Edfors@eit.lth.se 2010-03-23 Ove Edfors - ETI 051 1 2010-03-23 Ove Edfors - ETI 051 2 What do we know so far about propagation losses? Two theoretical expressions for ”POWER” [dB] the deterministic propagation loss as functions of distance: L ∣ dB d = { G TX ∣ dB 20log 10 , free space 4 d P TX ∣ dB L ∣ dB 20log 10 h TX h RX , ground plane SHORT REVIEW 2 d P RX ∣ dB G RX ∣ dB There are other models, which we will discuss later. We have discussed shadowing/ diffraction and reflections, but not really made any detailed calculations. 2010-03-23 Ove Edfors - ETI 051 3 2010-03-23 Ove Edfors - ETI 051 4
Statistical descriptions of the mobile radio channel The propagation ”POWER” [dB] loss will change due to movements. G TX ∣ dB P TX ∣ dB L ∣ dB These changes of the propagation RADIO SIGNALS AND loss will take place in two scales: COMPLEX NOTATION P RX ∣ dB Large-scale: shadowing, “slow” G RX ∣ dB changes over many wavelengths . Small-scale: interference, “fast” changes on the scale of a wavelength. Now we are going to approach these variations from a statistical point of view. 2010-03-23 Ove Edfors - ETI 051 5 2010-03-23 Ove Edfors - ETI 051 6 Simple model of a radio signal The IQ modulator ( ) • A transmitted radio signal can be written s t I I-channel ( ) ( ) ( ) Transmited radio signal (in-phase) π cos 2 f t = π + φ s t A cos 2 ft c ( ) ( ) ( ) f = π s t s t cos 2 f t c I c ( ) ( ) − π s t sin 2 f t Amplitude Frequency Phase Q c o • By letting the transmitted information change the -90 ( ) ( ) − π sin 2 f t amplitude, the frequency, or the phase, we get the s t c Q Q-channel tree basic types of digital modulation techniques (quadrature) Take a step into the complex domain: – ASK (Amplitude Shift Keying) s t = s I t j s Q t Complex envelope – FSK (Frequency Shift Keying) s t = Re { j 2 f c t } s t e Constant amplitude – PSK (Phase Shift Keying) j 2 π f t Carrier factor e c 2010-03-23 Ove Edfors - ETI 051 7 2010-03-23 Ove Edfors - ETI 051 8
Interpreting the complex Example: Amplitude, phase and notation frequency modulation ( ) ( ) ( ) ( ) = π + φ s t A t cos 2 f t t Complex envelope (phasor) Transmitted radio signal c ( ) ( ) φ Q A t t Comment: ( ) s t s t 00 01 11 00 10 Re { j 2 f c t } Q s t = s t e ( ) - Amplitude carries information ( ) A t φ t 4ASK Re { A t e j 2 f c t } - Phase constant (arbitrary) j t e = ( ) I s t Re { A t e j 2 f c t t } = I 00 01 11 00 10 = A t cos 2 f c t t - Amplitude constant (arbitrary) 4PSK - Phase carries information Polar coordinates: By manipulating the amplitude A (t) j t 00 01 11 00 10 s t = s I t j s Q t = A t e and the phase Φ (t) of the complex - Amplitude constant (arbitrary) envelope (phasor), we can create any - Phase slope (frequency) type of modulation/radio signal. 4FSK carries information 2010-03-23 Ove Edfors - ETI 051 9 2010-03-23 Ove Edfors - ETI 051 10 A narrowband system described in complex notation (noise free) Transmitter Channel Receiver ( ) ( ) x t y t ( ) ( ) ( ) ( ) ( ) α θ π t exp j t − π exp j 2 f exp j 2 f LARGE-SCALE c c FADING Attenuation Phase ( ) ( ) ( ) ( ) = φ In: x t A t exp j t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Out: = φ π α θ − π y t A t exp j t exp j 2 f t t exp j t exp j 2 f t c c ( ) ( ) ( ) ( ) ( ) ( ) = α φ + θ A t t exp j t t It is the behaviour of the channel attenuation and phase we are going to model. 2010-03-23 Ove Edfors - ETI 051 11 2010-03-23 Ove Edfors - ETI 051 12
Large-scale fading Large-scale fading Basic principle More than one shadowing object Signal path in terrain with several diffraction points adding extra attenuation to the pathloss. This is ONE Received power explanation α N D α α C 1 2 d B Movement Total pathloss: L tot = L d × 1 × 2 ×⋯× N A If these are considered Position random and independent, A B C C we should get a normal L tot ∣ dB = L d ∣ dB 1 ∣ dB 2 ∣ dB ⋯ N ∣ dB distribution in the dB domain. Deterministic 2010-03-23 Ove Edfors - ETI 051 13 2010-03-23 Ove Edfors - ETI 051 14 Large-scale fading Large-scale fading Log-normal distribution Fading margin Measurements confirm that in many situations, the large-scale We know that the path loss will vary around the deterministic value fading of the received signal strength has a normal distribution predicted. in the dB domain. ( ) Note dB pdf L | dB scale ”POWER” [dB] We need to design our system with a “margin” allowing us to handle higher path losses than the deterministic prediction. This margin is called a fading margin . P TX ∣ dB L ∣ dB dB Deterministic mean value of path loss, L 0 Increasing the fading margin decreases the probability of outage , |d B which is the probability that our system receive a too low power P RX ∣ dB exp − L ∣ dB − L 0 ∣ dB 2 to operate correctly. 1 pdf L ∣ dB = 2 F ∣ dB 2 2 F ∣ dB Standard deviation F ∣ dB ≈ 4 − 10 dB 2010-03-23 Ove Edfors - ETI 051 15 2010-03-23 Ove Edfors - ETI 051 16
Large-scale fading The Q(.)-function Fading margin (cont.) Upper-tail probabilities Fading margin x Q(x) x Q(x) x Q(x) M Designing the system to handle | dB 4.265 0.00001 3.090 0.00100 1.282 0.10000 an M |d B higher loss than ( ) pdf L 4.107 0.00002 2.878 0.00200 0.842 0.20000 predicted, | dB 4.013 0.00003 2.748 0.00300 0.524 0.30000 lowers the probability of outage. 3.944 0.00004 2.652 0.00400 0.253 0.40000 P out = Pr { L ∣ dB L 0 ∣ dB M ∣ dB } = Q F ∣ dB M ∣ dB 3.891 0.00005 2.576 0.00500 0.000 0.50000 3.846 0.00006 2.512 0.00600 3.808 0.00007 2.457 0.00700 3.775 0.00008 2.409 0.00800 dB L 3.746 0.00009 2.366 0.00900 0| dB 3.719 0.00010 2.326 0.01000 The upper tail probability of a unit variance, 3.540 0.00020 2.054 0.02000 zero-mean, Gaussian (normal) variable: 3.432 0.00030 1.881 0.03000 3.353 0.00040 1.751 0.04000 2 exp − x 2 dx = 1 2 erfc 2 ∞ 2 1 y 3.291 0.00050 1.645 0.05000 Q y = ∫ 3.239 0.00060 1.555 0.06000 y 3.195 0.00070 1.476 0.07000 3.156 0.00080 1.405 0.08000 The complementary error-function 3.121 0.00090 1.341 0.09000 can be found in e.g. MATLAB 2010-03-23 Ove Edfors - ETI 051 17 2010-03-23 Ove Edfors - ETI 051 18 Large-scale fading A numeric example How many dB fading margin, against σ F = 7 dB log-normal fading, do |d B we need to obtain an outage probability of 0.5%? P out = Q F ∣ dB M ∣ dB = = 0.5% 0.005 SMALL-SCALE FADING Consulting the Q(.)-function table (or using a numeric software), we get M ∣ dB = 2.576 ⇒ M ∣ dB = 2.576 ⇒ M ∣ dB = 18 F ∣ dB 7 2010-03-23 Ove Edfors - ETI 051 19 2010-03-23 Ove Edfors - ETI 051 20
Small-scale fading Small-scale fading Two waves Ilustration shown during Lecture 1 Illustration of interference pattern from above Received power [log scale] Wave 1 Movement A B Wave 1 + Wave 2 Position Transmitter Wave 2 A B Reflector λ At least in this case, we can see that the interference pattern changes on the wavelength scale. Many reflectors ... let’s look at a simpler case! 2010-03-23 Ove Edfors - ETI 051 21 2010-03-23 Ove Edfors - ETI 051 22 Small-scale fading Small-scale fading Many incoming waves Rayleigh fading Many incoming waves with Add them up as phasors No dominant component TX RX X independent amplitudes (no line-of-sight) and phases Tap distribution Amplitude distribution 2D Gaussian 3 Rayleigh (zero mean) r φ , 2 r 2 2 r φ , 0.8 2 1 1 r 1 φ r 1 0.6 3 r ( ) ( ) Im a Re a = 0.4 r φ r a , 4 4 r φ , 0.2 3 3 r 4 r φ , 4 0 0 1 2 3 2 exp − r 2 2 pdf r = r ( ) ( ) ( ) ( ) ( ) φ = φ + φ + φ + φ r exp j r exp j r exp j r exp j r exp j 1 1 2 2 3 3 4 4 No line-of-sight 2 component 2010-03-23 Ove Edfors - ETI 051 23 2010-03-23 Ove Edfors - ETI 051 24
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