Part II. Fading and Diversity Impact of Fading in Detection; Time - PowerPoint PPT Presentation

Part II. Fading and Diversity Impact of Fading in Detection; Time Diversity; Antenna Diversity; Frequency Diversity 1

Simplest Model: Single-Tap Rayleigh Fading Flat fading: single-tap Rayleigh fading H ∼ CN (0 , 1) , Z ∼ CN (0 , N 0 ) V = Hu + Z, Detection: ˆ Detection V = Hu + Z Θ u = a ˆ ˆ θ u = a θ ∈ A � { a 1 , . . . , a M } Detector (Rx) may or may not know the channel coe ffi cients Coherent Detection : Rx knows the realization of H Noncoherent Detection : Rx does not know the realization of H 2

Coherent Detection of BPSK ˆ Detection u = a ˆ ˆ V = Hu + Z Θ = φ ( V, H ) θ p p p u ∈ {± E s } a 0 = + E s , a 1 = − E s H ∼ CN (0 , 1) , Z ∼ CN (0 , N 0 ) Likelihood function: f V,H | Θ ( v, h | θ ) = f V | H, Θ ( v | h, θ ) f H ( h ) ∝ f V | H, Θ ( v | h, θ ) The detection problem is equivalent to binary detection in V = u + ˜ ˜ V � V / h, ˜ ˜ Z � Z / h ∼ CN (0 , N 0 / | h | 2 ) Z, Probability of error conditioned on the realization of H = h : � � �� � 2 √ E s 2 | h | 2 E s P e ( φ �� ; H = h ) = Q = Q N 0 2 √ N 0 / (2 | h | 2 ) 3

Probability of error: �� � 2 | h | 2 E s P e ( φ �� ; H = h ) = Q N 0 P e ( φ �� ) = E H ∼ CN (0 , 1) [ P e ( φ �� ; H )] � �� �� 2 | H | 2 E s = E H ∼ CN (0 , 1) Q N 0 � 1 2 exp( − | H | 2 SNR ) � ≤ E | H | 2 ∼ Exp(0 , 1) Z ∞ 1 1 2 e − t SNR e − t d t = = 2(1 + SNR ) 0 4

Impact of Fading • Let us explore the impact of fading by comparing the performance of coherent BPSK between AWGN and single-tap Rayleigh fading • The average received SNRs are the same: | H | 2 SNR � � E H ∼ CN (0 . 1) = SNR • AWGN: probability of error decays exponentially fast: � √ � ≤ 1 P e ( φ ML ) = Q 2 SNR 2 exp( − SNR ) ������ e − SNR • Rayleigh fading: probability of error decays much slower: � �� �� ≤ 1 1 2 | H | 2 SNR P e ( φ ML ) = E H ∼ CN (0 , 1) Q 2 1+ SNR ������ SNR − 1 5

Availability of channel state information (CSI) at Rx only changes the intercept, but not the slope P e 1 3 dB 10 –2 15 dB 10 –4 10– 6 10 –8 BPSK over AWGN 10 –10 Non-coherent orthogonal 10 –12 Coherent BPSK 10 –14 10 –16 –20 –10 0 10 20 30 40 SNR (dB) 6

Coherent Detection of General QAM Probability of error for -ary QAM M = 2 2 ℓ �� � �� � | h | 2 d 2 3 M − 1 | h | 2 SNR P e ( φ �� ; H = h ) ≤ 4Q = 4Q min 2 N 0 � �� �� 3 M − 1 | H | 2 SNR P e ( φ �� ) ≤ E H ∼ CN (0 , 1) 4Q � � 3 2 exp( − | H | 2 2( M − 1) SNR ) ≤ E | H | 2 ∼ Exp(0 , 1) 2 2( M − 1) SNR ≈ 4( M − 1) = SNR − 1 3 1 + 3 Using general constellation does not change the order of performance (the “slope” on the log P e vs. log SNR plot) Di ff erent constellation only changes the intercept 7

Deep Fade: the Typical Error Event • In Rayleigh fading channel, regardless of constellation size and detection method (coherent/non-coherent), P e ∼ SNR − 1 • This is in sharp contrast to AWGN: P e ∼ exp( − c SNR ) • Why? Let’s take a deeper look at the BPSK case: � � 2 | h | 2 SNR P e ( φ �� ; H = h ) = Q ‣ If channel is good, error probability | h | 2 SNR � 1 = ∼ exp( − c SNR ) ⇒ ‣ If channel is bad, error probability is | h | 2 SNR < 1 = Θ (1) ⇒ | H | 2 > SNR − 1 E | | H | 2 > SNR − 1 � � P e ≡ P {E} = P P | H | 2 < SNR − 1 E | | H | 2 < SNR − 1 � � + P P = 1 − e SNR − 1 ≈ SNR − 1 | H | 2 < SNR − 1 � / P • Deep fade event: {| H | 2 < SNR − 1 } 8

Diversity {| H | 2 < SNR − 1 } Deep fade event: V = Hu + Z • Reception only relies on a single (equivalent) “path” H • If H is in deep fade ⟹ big trouble (low reliability) • Increase the number of “paths” ⟺ Increase diversity ‣ If one path is in deep fade, other paths can compensate! • If there are L indep. paths, the probability of deep fade becomes L Y (1 − P { ���� i �� ���� ���� } ) ≈ 1 − (1 − SNR − 1 ) L ≈ SNR − L 1 − ` =1 • Find independent paths (diversity) over time , space , and frequency to increase diversity! 9