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.

PRINCIPLE OF EQUALIZATION

Carrier

Original Base-band Message x (t)

Transmitter Channel

Receiver Front end

IF Stage

Detector

Equalizer Decision Maker

f(t)

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 }

MSE ERROR = E [ | e (t) | 2 ] Aim: To minimize error MSE.

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.

• + 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)

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

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 ]

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.

DIVERSITY TECHNIQUES

• Powerful communications receiver technique that

provides wireless. link improvement at relatively low

cost

• Unlike equalization, diversity requires no training

overhead.

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.

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

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

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)

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.

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 ) !

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 ]

= 0.9999998

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

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.

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

< >

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.