Download BER Analysis Under Fading

Demodulation
(BER analysis under fading)
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Model for Channel and Noise
 For a narrowband channel, the received signal (in the baseband) is (Low-Pass (LP) representation)

: complex-valued channel gain
 n(t): zero mean Gaussian noise
i.e. n(t) has white spectrum.
 Autocorrelation of n(t) (and also the inphase and quadrature-phase components of n(t))
 which becomes (as B → ∞)
 Inphase and quadrature-phase components of n(t) are uncorrelated.
 Pass-band (band-pass) representation of n(t) is
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Signal-Space Representation
 Remember the signal-space representation of digitally modulated signals.
 Each modulation scheme can be represented by certain basis functions φn(t) and their weights:
Valid for both LP and also BP
representations.
For BP φn(t) contains the carrier.
 The basis functions constitute the axes of the signal-space and the weights define the constellation
points in that space.
 Each group of weight can be represented by a vector, sm, i.e. each constellation point is given by
BPSK
s2
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2-FSK
φ2
s1
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φ2
64-QAM
φ1
s2
φ1
s1
s1
φ1
s64
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Signal-Space Representation
 Received signal:
where nn represent the noise terms in the signal space and n’(t) represents noise in the orthogonal space.
 We may simply ignore the noise in the orthogonal space. Does not affect the decision.
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MAP/ML Detector
 All transmitted symbols are equally likely
 The modulation format does not have memory (e.g. ASK, PSK, FSK, QAM, but not CPM, GMSK etc.)
 The channel is AWGN. Both the channel gain α and phase rotation φ is known at the RX.
 Then, «if the signal r(t) was received, then which symbol sm(t) was most likely transmitted?»
 Which symbol maximizes
 Maximum A Posteriori (MAP) detector.
 With equiprobable symbols, the MAP detector becomes identical to Maximum Likelihood (ML) detector
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MAP/ML Detector
 Using the signal space approach, ML rule becomes
 For Gaussian noise, the ML detector reduces to
which is equivalent to
where
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Projection of rLP on to sLP,m
. (For a constant modulus constellation (e.g. PSK, FSK) Em = E.)
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Pairwise Error Probability (PEP)
 The probability that symbol sj is mistaken for symbol sk that has Euclidean distance djk from sj is
φ2
djk
φ1
Anti-podal, e.g. BPSK
φ2
djk
djk
φ1
φ1
Orthogonal, e.g. FSK
Bi-orthogonal, e.g. QPSK
 djk is determined by
 i. Signal energy εs,i
 ii. Channel gain α
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Error Probability
 Binary Orthogonal Signals (2-FSK, 2-PPM)
 Signal (bit) energy: εb, SNR/symbol (bit): γs = γb = εb/N0,
 Symbol (bit) error probability = PEP:
 Antipodal Signals (BPSK)
 Signal (bit) energy: εb, SNR/symbol (bit): γs = γb = εb/N0,
 Symbol (bit) error probability = PEP:
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Error Probability
 Bi-orthogonal Signals (QPSK, 4-QAM)
 Bit energy: εb, SNR/bit : γb = εb/N0,
 Signal energy: εs = 2εb, SNR/symbol : γs = εs/N0,
 Symbol (bit) error probability: can be complicated
 Upper-left figure: exact error region → difficult to calculate
 Calculate the PEP for each pair in error for sj
 Add the PEPs for each pair up
 → Pairwise error regions overlap
 → Sum-PEP gives an upper bound on symbol (bit) error probability.
 → This approach gives an approximation on exact error probability.
(left & bottom points)
(diagonal point)
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Error Probability
 A simplification: pairwise error with nearest neighbours cover the
whole error region, no need to calculate the last test.
 PEP:
 Symbol error probability (from union bound)
 If we use Gray coding to map pair-of-bits to symbols, bit error probability
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Error Probability for Flat-Fading Channels
 In a flat-fading channel, the channel gain is NOT static but varies with time !
 SNR is time varying !
 BER is time varying !
 Approach to calculate BER in flat-fading channels:
 1. Determine the BER for any arbitrary SNR,
 2. Determine the probability that a certain SNR occurs in the channel – in other words, determine the
pdf of the power gain of the channel,
 3. Average the BER over the distribution of SNRs.
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Error Probability for Flat-Fading Channels
 In a fading channel, sometimes we are on a «mountain» (high SNR) and at other times in a
«valley» (low SNR).
 BER-SNR relation is highly nonlinear (Q-function),
 Being on a «mountain» slightly improves BER as compared to the average conditions,
 Being in a «valley» drastically decreases BER as compared to the average conditions,
 Being in a «valley» dominates the behaviour → for the same average SNR, performance of a fading
channel is significantly worse than that of an AWGN channel.
 Example: A fading channel has an average SNR of 10 dB. Fading causes the SNR to be - ∞ dB
half of the time while it is 13 dB the rest of the time. Consider DBPSK.




SNR = - ∞ dB → BER = 0.5
SNR = 13 dB → BER = 10-9.
Average BER is
For an AWGN channel with 10 dB SNR,
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Error Probability for Flat-Fading Channels
 Signal strength: Rayleigh pdf
 To calculate SNR, we need signal power (its pdf). The mean power is
Jacobian
, to find the pdf of the received power:
 Then the pdf of the SNR (per bit) is (
. Use the
: mean SNR/bit)
 For Rician-fading (Kr: Rice factor)
 Then, the average BER is calculated by
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Error Probability for Flat-Fading Channels
 For a Rayleigh fading channel:
 Binary antipodal signals (BPSK):
 Binary orthogonal signals (2-FSK):
 Differential binary antipodal signals (DBPSK):
 Differential binary orthogonal signals:
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Error Probability for Flat-Fading Channels
 For a Rician fading channel:
Differential binary antipodal signals (DBPSK):
 Differential binary orthogonal signals:
 Example: Calculate the average BER of DBPSK with γB = 12 dB and Kr = -3 dB, 0 dB, 10 dB.
Rician fading, use the above expression:
For Kr = -3 dB:
For Kr = 0 dB:
BER = 2.3 x 10-2
For Kr = 10 dB:
BER = 5.6 x 10-4
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Error Probability for Frequency Selective Channels
That is another story !!!!
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