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Fading Channels: Capacity, BER and Diversity

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Fading Channels: Capacity, BER and Diversity Fading Channels: Capacity, BER and Diversity Master Universitario en Ingenier´ıade Telecomunicaci´on I. Santamar´ıa Universidad de Cantabria Introduction Capacity BER Diversity Conclusions Contents Introduction Capacity BER Diversity Conclusions Fading Channels: Capacity, BER and Diversity 0/48 Introduction Capacity BER Diversity Conclusions Introduction I We have seen that the randomness of signal attenuation (fading) is the main challenge of wireless communication systems I In this lecture, we will discuss how fading affects 1. The capacity of the channel 2. The Bit Error Rate (BER) I We will also study how this channel randomness can be used or exploited to improve performance diversity ! Fading Channels: Capacity, BER and Diversity 1/48 Introduction Capacity BER Diversity Conclusions General communication system model sˆ[n] bˆ[n] b[n] s[n] Channel Channel Source Modulator Channel ⊕ Demod. Encoder Decoder Noise I The channel encoder (FEC, convolutional, turbo, LDPC ...) adds redundancy to protect the source against errors introduced by the channel I The capacity depends on the fading model of the channel (constant channel, ergodic/block fading), as well as on the channel state information (CSI) available at the Tx/Rx Let us start reviewing the Additive White Gaussian Noise (AWGN) channel: no fading Fading Channels: Capacity, BER and Diversity 2/48 Introduction Capacity BER Diversity Conclusions AWGN Channel I Let us consider a discrete-time AWGN channel y[n] = hs[n] + r[n] where r[n] is the additive white Gaussian noise, s[n] is the transmitted signal and h = h ejθ is the complex channel j j I The channel is assumed constant during the reception of the whole transmitted sequence I The channel is known to the Rx (coherent detector) I The noise is white and Gaussian with power spectral density N0=2 (with units W/Hz or dBm/Hz, for instance) I We will mainly consider passband modulations (BPSK, QPSK, M-PSK, M-QAM), for which if the bandwith of the lowpass signal is W the bandwidth of the passband signal is 2W Fading Channels: Capacity, BER and Diversity 3/48 Introduction Capacity BER Diversity Conclusions Signal-to-Noise-Ratio I Transmitted signal power: P 2 I Received signal power: P h j j I Total noise power: N = 2WN0=2 = WN0 I The received SNR is P h 2 SNR = γ = j j WN0 I In terms of the energy per symbol P = Es =Ts I We assume Nyquist pulses with β = 1 (roll-off factor) so W = 1=Ts I Under these conditions 2 2 Es Ts h Es h SNR = j j = j j Ts N0 N0 Fading Channels: Capacity, BER and Diversity 4/48 Introduction Capacity BER Diversity Conclusions I For M-ary modulations, the energy per bit is 2 Es Eb h Eb = SNR = j j log(M) ) N0 log(M) I To model the noise we generate zero-mean circular complex 2 Gaussian random variables with σ = N0 2 r[n] CN(0; σ ) or r[n] CN(0; N0) ∼ ∼ I The real and imaginary parts have variance σ2 N σ2 = σ2 = = 0 I Q 2 2 Fading Channels: Capacity, BER and Diversity 5/48 Introduction Capacity BER Diversity Conclusions Average SNR and Instantaneous SNR Sometimes, we will find useful to distinguish between the average and the instantaneous signal-to-noise ratio P Average SNR =γ ¯ = ) WN0 P Instantaneous SNR =γ ¯ h 2 = h 2 ) j j WN0 j j I AWGN channel: N0 and h are assumed to be known (coherent detection), therefore we typically take (w.l.o.g) h = 1 SNR =γ ¯ = P , and the SNR at the receiver is WN0 constant) I Fading channels: The instantaneous signal-to-noise-ratio SNR = γ =γ ¯ h 2 is a random variable j j Fading Channels: Capacity, BER and Diversity 6/48 Introduction Capacity BER Diversity Conclusions Capacity AWGN Channel I Let us consider a discrete-time AWGN channel y[n] = hs[n] + r[n] where r[n] is the additive white Gaussian noise, s[n] is the transmitted symbol and h = h ejθ is the complex channel j j I The channel is assumed constant during the reception of the whole transmitted sequence I The channel is known to the Rx (coherent detector) The capacity (in bits/seg or bps) is given by the well-known Shannon's formula C = W log (1 + SNR) = W log (1 + γ) 2 where W is the channel bandwidth, and SNR = Pjhj ; with P WN0 2 being the transmit power, h the power channel gain and N0=2 the power spectral densityj (PSD)j of the noise Fading Channels: Capacity, BER and Diversity 7/48 Introduction Capacity BER Diversity Conclusions I Sometimes, we will find useful to express C in bps/Hz (or bits/channel use) C = log (1 + SNR) A few things to recall 1. Shannon's coding theorem proves that a code exists that achieves data rates arbitrarily close to capacity with vanishingly small probability of bit error I The codewords might be very long (delay) I Practical (delay-constrained) codes only approach capacity 2. Shannon's coding theorem assumes Gaussian codewords, but digital communication systems use discrete modulations (PSK,16-QAM) Fading Channels: Capacity, BER and Diversity 8/48 Introduction Capacity BER Diversity Conclusions 10 Shaping gain (1.53 dB) 8 Gaussian inputs log(1+SNR) 64-QAM 6 16-QAM 4 Capacity (bps/Hz) 4-QAM (QPSK) 2 Pe (symbol) = 10-6 (uncoded) 0 5 10 15 20 25 30 SNR dB For discrete modulations, with increasing SNR we should increase the constellation size M (or use a less powerful channel encoder) to increase the rate and hence approach capacity: Adaptive modulation and coding (more on this later) Fading Channels: Capacity, BER and Diversity 9/48 Introduction Capacity BER Diversity Conclusions Capacity fading channels 1 I Let us consider the following example Fading channel AWGN C = 1 bps/Hz Channel Random Encoder Switch AWGN C = 2 bps/Hz I The channel encoder is fixed I The switch takes both positions with equal probability I We wish to transmit a codeword formed by a long sequence of coded bits, which are then mapped to symbols I What is the capacity for this channel? 1Taken from E. Biglieri, Coding for Wireless Communications, Springer, 2005 Fading Channels: Capacity, BER and Diversity 10/48 Introduction Capacity BER Diversity Conclusions The answer depends on the rate of change of the fading 1. If the switch changes position every symbol period, the codeword experiences both channels with equal probabilities so we could use a fixed channel encoder, and transmit at a maximum rate 1 1 C = C + C = 1:5 bps=Hz 2 1 2 2 2. If the switch remains fixed at the same (unknown) position during the transmission of the whole codeword, then we should use a fixed channel encoder, and transmit at a maximum rate C = C1 = 1 bps=Hz or otherwise half of the codewords would be lost Fading Channels: Capacity, BER and Diversity 11/48 Introduction Capacity BER Diversity Conclusions What would be the capacity if the switch chooses from a 2 continuum of AWGN channels whose SNR, γ = Pjhj , 0 γ < , WN0 follows an exponential distribution (Rayleigh channel) ≤ 1 C1 = log(1+ γ 1) Channel Encoder C2 = log(1+ γ 2 ) Random Switch C3 = log(1+ γ 3 ) Fading Channels: Capacity, BER and Diversity 12/48 Introduction Capacity BER Diversity Conclusions Fast fading (ergodic) channel I If the switch changes position every symbol period, and the codeword is long enough so that the transmitted symbols experience all states of the channel (fast fading or ergodic channel) Z 1 C = E [log(1 + γ)] = log(1 + γ)f (γ)dγ (1) 0 1 −γ=γ where f (γ) = γ e , with γ being the average SNR I Eq. (1) is the ergodic capacity of the fading channel I For Rayleigh channels the integral in (1) is given by 1 1 1 C = exp E 2 ln(2) γ 1 γ R 1 e−t 2 where E1(x) = x t dt is the Exponential integral function 2y=expint(x) in Matlab Fading Channels: Capacity, BER and Diversity 13/48 Introduction Capacity BER Diversity Conclusions I The ergodic capacity is in general hard to compute in closed form (we can always resort to numerical integration) I We can apply Jensen's inequality to gain some qualitative insight into the effect of fast fading on capacity Jensen's inequality If f (x) is a convex function and X is a random variable E [f (X )] f (E [X ]) ≥ If f (x) is a concave function and X is a random variable E [f (X )] f (E [X ]) ≤ I log( ) is a concave function, therefore · C = E [log(1 + γ)] log (1 + E[γ]) = log (1 + γ) ≤ Fading Channels: Capacity, BER and Diversity 14/48 ' $ IntroductionSISO FadingCapacity Channel:BER ErgodicDiversity Capacity Conclusions The capacity of a fading channel2 with2 receiver CSI is2 less than the Assume iid Rayleigh fading with σ = E[ h ] = 1, and σz = 1. capacity of an AWGN channelh with| the| same average SNR 7 6 5 AWGN Channel Capacity Fading Channel Ergodic Capacity 4 3 Capacity (bps/Hz) 2 1 0 −10 −5 0 5 10 15 20 SNR (dB) 17 & Fading Channels: Capacity, BER and Diversity 15/48 % Introduction Capacity BER Diversity Conclusions Block fading channel I If the switch remains fixed at the same (unknown) position during the transmission of the whole codeword, there is no nonzero rate at which long codewords can be transmitted with vanishingly small probability or error I Strictly, the capacity for this channel model would be zero I In this situation, it is more useful to define the outage capacity Cout = r Pr(log(1 + γ) < r) = Pr(C(h) < r) = Pout ) Suppose we transmit at a rate Cout with probability of outage Pout = 0:01 this means that with probability 0:99 the instantaneous capacity of the channel (a realization of a r.v.) will be larger than rate and the transmission will be successful; whereas with probability 0:01 the instantaneous capacity will be lower than the rate and the all bits in the codeword will be decoded incorrectly Fading Channels: Capacity, BER and Diversity 16/48 Introduction Capacity BER Diversity Conclusions I Pout is
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