RADIO SYSTEMS – ETIN15 3 Lecture no: Narrow- and wideband channels Ove Edfors, Department of Electrical and Information technology Ove.Edfors@eit.lth.se 2012-03-19 Ove Edfors - ETIN15 1
Contents • Short review NARROW-BAND CHANNELS WIDE-BAND CHANNELS • Radio signals and complex • What makes a channel notation wide-band? • Large-scale fading • Delay (time) dispersion • Small-scale fading • Narrow- versus wide-band channels • Combining large- and small- scale fading • Noise- and interference- limited links 2012-03-19 Ove Edfors - ETIN15 2
SHORT REVIEW 2012-03-19 Ove Edfors - ETIN15 3
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 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. 2012-03-19 Ove Edfors - ETIN15 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 loss will take place in two scales: P RX ∣ dB G RX ∣ dB Large-scale: shadowing, “slow” 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. 2012-03-19 Ove Edfors - ETIN15 5
RADIO SIGNALS AND COMPLEX NOTATION 2012-03-19 Ove Edfors - ETIN15 6
Simple model of a radio signal • A transmitted radio signal can be written ( ) ( ) = π + φ s t A cos 2 ft Amplitude Frequency Phase • By letting the transmitted information change the amplitude, the frequency, or the phase, we get the tree basic types of digital modulation techniques – ASK (Amplitude Shift Keying) – FSK (Frequency Shift Keying) Constant amplitude – PSK (Phase Shift Keying) 2012-03-19 Ove Edfors - ETIN15 7
The IQ modulator ( ) s t I I-channel ( ) Transmited radio signal (in-phase) π cos 2 f t c ( ) ( ) ( ) = π f s t s t cos 2 f t c I c ( ) ( ) − π s t sin 2 f t Q c o -90 ( ) − π ( ) sin 2 f t s t c Q Q-channel (quadrature) Take a step into the complex domain: s t = s I t j s Q t Complex envelope s t = Re { j 2 f c t } s t e π j 2 f t e Carrier factor c 2012-03-19 Ove Edfors - ETIN15 8
Interpreting the complex notation Complex envelope (phasor) Transmitted radio signal Q ( ) s t s t Re { j 2 f c t } Q s t = s t e ( ) ( ) A t φ t Re { A t e j 2 f c t } j t e = ( ) s t I Re { A t e j 2 f c t t } = I = A t cos 2 f c t t Polar coordinates: By manipulating the amplitude A (t) j t s t = s I t j s Q t = A t e and the phase Φ (t) of the complex envelope (phasor), we can create any type of modulation/radio signal. 2012-03-19 Ove Edfors - ETIN15 9
Example: Amplitude, phase and frequency modulation ( ) ( ) ( ) ( ) = π + φ s t A t cos 2 f t t c ( ) ( ) φ A t t Comment: 00 01 11 00 10 - Amplitude carries information 4ASK - Phase constant (arbitrary) 00 01 11 00 10 - Amplitude constant (arbitrary) 4PSK - Phase carries information 00 01 11 00 10 - Amplitude constant (arbitrary) - Phase slope (frequency) 4FSK carries information 2012-03-19 Ove Edfors - ETIN15 10
A narrowband system described in complex notation (noise free) Channel Receiver Transmitter ( ) ( ) x t y t ( ) ( ) ( ) ( ) ( ) α θ π − π t exp j t exp j 2 f exp j 2 f c c Attenuation Phase ( ) ( ) ( ) ( ) = φ x t A t exp j t In: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = 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 Out: 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. 2012-03-19 Ove Edfors - ETIN15 11
LARGE-SCALE FADING 2012-03-19 Ove Edfors - ETIN15 12
Large-scale fading Basic principle Received power D C d B t n e m e v o A M Position A B C C 2012-03-19 Ove Edfors - ETIN15 13
Large-scale fading More than one shadowing object Signal path in terrain with several diffraction points adding extra attenuation to the pathloss. This is ONE explanation α N α α 1 2 Total pathloss: L tot = L d × 1 × 2 ×⋯× N If these are considered random and independent, L tot ∣ dB = L d ∣ dB 1 ∣ dB 2 ∣ dB ⋯ N ∣ dB we should get a normal distribution in the dB domain. Deterministic 2012-03-19 Ove Edfors - ETIN15 14
Large-scale fading Log-normal distribution Measurements confirm that in many situations, the large-scale fading of the received signal strength has a normal distribution in the dB domain. ( ) Note dB pdf L | dB scale ”POWER” [dB] P TX ∣ dB L ∣ dB dB Deterministic mean value of path loss, L 0|dB P RX ∣ dB exp − L ∣ dB − L 0 ∣ dB 2 1 pdf L ∣ dB = 2 F ∣ dB 2 2 F ∣ dB Standard deviation F ∣ dB ≈ 4 − 10 dB 2012-03-19 Ove Edfors - ETIN15 15
Large-scale fading Fading margin We know that the path loss will vary around the deterministic value predicted. 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 . Increasing the fading margin decreases the probability of outage , which is the probability that our system receive a too low power to operate correctly. 2012-03-19 Ove Edfors - ETIN15 16
Large-scale fading Fading margin (cont.) Fading margin M Designing the system to handle | dB an M |dB dB higher loss than ( ) pdf L predicted, lowers the probability | dB of outage. P out = Pr { L ∣ dB L 0 ∣ dB M ∣ dB } = Q F ∣ dB M ∣ dB dB L 0| dB The upper tail probability of a unit variance, zero-mean, Gaussian (normal) variable: 2 exp − x 2 dx = 1 2 erfc 2 ∞ 2 1 y Q y = ∫ y The complementary error-function can be found in e.g. MATLAB 2012-03-19 Ove Edfors - ETIN15 17
The Q(.)-function Upper-tail probabilities x Q(x) x Q(x) x Q(x) 4.265 0.00001 3.090 0.00100 1.282 0.10000 4.107 0.00002 2.878 0.00200 0.842 0.20000 4.013 0.00003 2.748 0.00300 0.524 0.30000 3.944 0.00004 2.652 0.00400 0.253 0.40000 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 3.746 0.00009 2.366 0.00900 3.719 0.00010 2.326 0.01000 3.540 0.00020 2.054 0.02000 3.432 0.00030 1.881 0.03000 3.353 0.00040 1.751 0.04000 3.291 0.00050 1.645 0.05000 3.239 0.00060 1.555 0.06000 3.195 0.00070 1.476 0.07000 3.156 0.00080 1.405 0.08000 3.121 0.00090 1.341 0.09000 2012-03-19 Ove Edfors - ETIN15 18
Large-scale fading A numeric example How many dB fading margin, against σ F|dB = 7 dB log-normal fading, do we need to obtain an outage probability of 0.5%? P out = Q F ∣ dB M ∣ dB = = 0.5% 0.005 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 2012-03-19 Ove Edfors - ETIN15 19
SMALL-SCALE FADING 2012-03-19 Ove Edfors - ETIN15 20
Small-scale fading Ilustration shown during Lecture 1 Illustration of interference pattern from above Received power [log scale] Movement A B Position Transmitter A B Reflector Many reflectors ... let’s look at a simpler case! 2012-03-19 Ove Edfors - ETIN15 21
Small-scale fading Two waves Wave 1 Wave 1 + Wave 2 Wave 2 λ At least in this case, we can see that the interference pattern changes on the wavelength scale. 2012-03-19 Ove Edfors - ETIN15 22
Small-scale fading Many incoming waves Many incoming waves with Add them up as phasors independent amplitudes and phases 3 r φ , 2 r 2 2 r φ , 2 1 1 r 1 φ r 1 3 r r φ , 4 4 r φ , 3 3 r 4 r φ , 4 ( ) ( ) ( ) ( ) ( ) 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 2012-03-19 Ove Edfors - ETIN15 23
Small-scale fading Rayleigh fading No dominant component TX RX X (no line-of-sight) Tap distribution Amplitude distribution 2D Gaussian Rayleigh (zero mean) 0.8 0.6 ( ) ( ) = 0.4 Im a Re a r a 0.2 0 0 1 2 3 2 exp − r 2 2 pdf r = r No line-of-sight 2 component 2012-03-19 Ove Edfors - ETIN15 24
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