communications link analysis - hacettepe universitytoker/ele492/6. demodulation.pdf · spring 2017...

Demodulation(BER analysis under fading)

Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 1

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


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:

The basis functions constitute the axes of the signal-space and the weights define the constellationpoints in that space.

Each group of weight can be represented by a vector, sm, i.e. each constellation point is given by


φ1

φ2

64-QAM

φ1

φ22-FSK

φ1

BPSK

s1s2s1

s2

s1

s64

Valid for both LP and also BP representations.For BP φn(t) contains the carrier.

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.


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


MAP/ML Detector Using the signal space approach, ML rule becomes

For Gaussian noise, the ML detector reduces to

which is equivalent to

where . (For a constant modulus constellation (e.g. PSK, FSK) Em = E.)


Projection of rLP on to sLP,m

Pairwise Error Probability (PEP) The probability that symbol sj is mistaken for symbol sk that has Euclidean distance djk from sj is

djk is determined by i. Signal energy εs,i

ii. Channel gain α


φ1

φ1

φ2 φ2

φ1

Anti-podal, e.g. BPSK

Orthogonal, e.g. FSK

djk

djk

djk

Bi-orthogonal, e.g. QPSK

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:


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)

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


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 thepdf of the power gain of the channel,

3. Average the BER over the distribution of SNRs.


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 fadingchannel 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 - ∞ dBhalf 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,


Error Probability for Flat-Fading Channels Signal strength: Rayleigh pdf

To calculate SNR, we need signal power (its pdf). The mean power is . Use the Jacobian , to find the pdf of the received power:

Then the pdf of the SNR (per bit) is ( : mean SNR/bit)

For Rician-fading (Kr: Rice factor)

Then, the average BER is calculated by


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:


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


Error Probability for Frequency Selective Channels

That is another story !!!!


communications link analysis - hacettepe universitytoker/ele492/6. demodulation.pdf · spring 2017...

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