Chapter 6_Mobile Basics
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Transcript of Chapter 6_Mobile Basics
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EQUALIZATION AND DIVERSITY
EQUALIZATION
• Compensates for inter-symbol interference (ISI) created by
multi-path within time dispersive channels.
• An equalizer within a receiver compensates for average
range of expected channel amplitude and the delay
characteristics.
• Equalizers are generally adaptive since channel is unknown
and time varying.
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PRINCIPLE OF EQUALIZATION
Carrier
Original Base-band Message x (t)
Transmitter Channel
Receiver Front end
IF Stage
Detector
Equalizer Decision Maker
f(t)
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Reconstructed y(t)
ηb(t)
Message d(t) ∧
Equivalent heq(t) Noise
SYSTEM EQUATIONS
Ø y (t) = x (t) ⊗ f * (t) Y (f) = Y (f) F* (f) Ø Output of Equalizer is:
∧ => X (f) F* (f) Heq (f) = X (f) => F* (f) Heq (f) = 1 ∧ ERROR e (t) = d (t) - d (t)
+
d(t)
D (f) = Y (f) Heq (f) = X (f) ………. { Desired }
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MSE ERROR = E [ | e (t) | 2 ] Aim: To minimize error MSE.
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EQUALIZER OPERATING MODES
• Training.
• Tracking.
• The Training sequence is a known pseudo-random signal
or a fixed bit pattern sent by the transmitter. The user data
is sent immediately after the training sequence.
• The equalizer uses training sequence to adjust its frequency
response Heq (f) to satisfy eq.(1) and is optimally ready for
data sequence. the adjustment goes on dynamically, it is
adaptable equalizer.
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• + TDMA system are well suited for equalization since data
is sent in time blocks.
DIGITAL COMMUNICATIONS EQUALIZERS
• In discrete form, we sample signals at interval of ‘T’ seconds : t = k T; ∧ d (k) = y (k) * heq (k)
∧ e (k) = d (k) - d (k) • The Output of Equalizer is:
∧ d (k) = y (k) * heq (k) N = Σ Wnk y (k – n)
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n=0 = W0k y (k) + W1k y (k –1) + ……….+ WNK y (k - N)
BLOCK DIAGRAM OF DIGITAL EQUALIZER
y (k-1) y (k) y (k-2) y (k-N)
d(k)
-
e(k) d (k)
Z-1 Z-1 Z-1
Σ
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N delays.
N + 1 Taps.
N + 1 Tunable Complex Multipliers or weights.
OPERATING MODE OF DIGITAL EQUALIZER
• The weights are updated continuously by the adaptive algorithms,
• The adaptive algorithm is controlled by the error signal, ∧ e (k) = d (k) - d (k) and the equalizer weights are updated to minimize the cost function Min E [ e(k ) e(k)* ] = Min E [ | e (k) |2 ]
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Wnk+1 = Wnk + K ek-1 * yn ∧ e k-1 = d k-1 - d k-1 • The equalizer weights are varied until convergence is
reached. TYPES OF EQUALIZERS
• Linear Equalizers.
• Non Linear Equalizers.
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DIVERSITY TECHNIQUES
• Powerful communications receiver technique that
provides wireless. link improvement at relatively low
cost
• Unlike equalization, diversity requires no training
overhead.
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PRINCIPLE OF DIVERSITY
• Small Scale fading causes deep and rapid amplitude
fluctuations as mobile moves over a very small distances.
• If we space 2 antennas at 0.5 m, one may receive a null
while the other receives a strong signal. By selecting the
best signal at all times, a receiver can mitigate or reduce
small-scale fading. This implies Antenna Diversity.
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DIVERSITY IMPROVEMENT
• Consider a fading channel (Rayleigh)
Input s (t) Output r (t)
• Input-output relation
r (t) = α (t) e -j θ (t) s (t) + n (t)
• Average value of signal to noise ratio
SNR = Γ = (Eb / No) α 2 (t)
Channel
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AVERAGE SNR IMPROVEMENT USING DIVERSITY
• p.d.f. p( γi ) = (1 / Γ ) e – γi / Γ , where ( γi ≥ 0 )
γi = instantaneous SNR
γ Probability [γi ≤ γ] = ∫ p (γi ) d γi 0
• M diversity branches,
Probability [γi > γ] = 1 – ( 1 – e –γ / Γ )M
• Average SNR improvement using selection Diversity,
M γ / Γ = Σ 1 / k
k = 1
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Example : Assume that 5 antennas are used to provide
space diversity. If average SNR is 20 dB, determine the
probability that the SNR will be > 10 dB. Compare this
with the case of a single receiver.
Solution : Γ = 20 dB => 100. Threshold γ = 10 dB =
10.
Prob [γi > γ] = 1 – ( 1 – e –γ/ Γ ) M
For M = 5
Prob = 1 – (1 – e –0.1 )5 = 0.9999
For M = 1 (No diversity)
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Pr = 1 – (1 – e –0.1 ) = 0.905
MAXIMAL RATIO COMBINING (MRC)
• MRC uses each of the M branches in co-phased and weighted manner such that highest achievable SNR is available. If each branch has gain Gi,
M rM = total signal envelope = Σ Gi . ri i= 1 assuming each branch has some average noise power N, total noise power NT applied to the detector is, M NT = N Σ Gi 2 i = 1
• SNR = γM = rM 2 / 2 NT M
• Max [γM ] = ½ . Σ ( ri 2 ) / N = Σ ri when Gi = ri / N.
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i = 1
AVERAGE SNR IMPROVEMENT
Average SNR = γM = M Γ
M Probability (γM ≤ γ) = 1 - e -γ/ Γ . Σ ( γ / Γ) k-1 k =1 ( k –1 ) !
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EXAMPLE : Repeat earlier problem for MRC case
M Probability (γM > γ) = 1 - e -γ/ Γ . Σ ( γ/ Γ) k-1 k =1 ( k –1 ) !
γ = 10, Γ = 100, M = 5.
5 Pr ( γM > 10) = 1 - e - 0.1. Σ ( 0.1 )k-1 k =1 ( k –1 ) ! = e - 0.1 [ 1 + 0.1 / 1 + 0.12 / 2 ! + 0.13 / 3 ! + 0.14 / 4 ! ] = 0.905 [ 1.1051708 ]
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= 0.9999998
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TYPES OF DIVERSITY • Space Diversity
• Either at the mobile or base station. • At base station, separation on order of several tens of
wavelength are required.
• Polarization Diversity • Orthogonal Polarization to exploit diversity • High art of space diversity is avoided.
• Frequency Diversity • More than one carrier frequency is used
• Time Diversity : • Information is sent at time spacings • Greater than the coherence time of Channel, so
multiple repetitions can be resolved
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PRACTICAL DIVERSITY RECEIVER – RAKE RECEIVER
• CDMA system uses RAKE Receiver to improve the signal
to noise ratio at the receiver.
• Generally CDMA systems don’t require equalization due to
multi-path resolution.
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BLOCK DIAGRAM OF RAKE RECEIVER
M1 M2 M3 Z 1 α 1 m’(t)
r ( t) Z 2 α 2 z’ z
α M
Z M :
PINE AND GREEN (1958)
Correlator 1
Correlator 2
Correlator M
Σ
T ∫ (•)dt 0
< >
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PRINCIPLE OF OPERATION
• M Correlators – Correlator 1 is synchronized to strongest multipath M1. The correlator 2 is synchronized to next strongest multipath M2 and so on.
• The weights α1 , α2 ,……,αM are based on SNR from each correlator output. (α is proportional to SNR of correlator.)
M Z 1 = Σ αM ZM m =1 • Demodulation and bit decisions are then based on the
weighted Outputs of M Correlators.
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