Equalization and Diversity

The minimum mean square error (MSE) criterion chooses the equalizer tap weights that make the average squared error between the desired symbol and the equalizer output as small as possible. The optimum set is called the Wiener solution.

Structure: adaptive linear combiner (transversal equalizer)

 y_k ─┬─[z⁻¹]─┬─[z⁻¹]─┬─ ... ─[z⁻¹]─┐
      │       │       │              │
     (x)w₀   (x)w₁   (x)w₂   ...    (x)w_N
      │       │       │              │
      └───────┴──────(Σ)─────────────┘
                      │ d̂_k (output)
                      v
           x_k ──►(+)─(−)──► e_k (error)
                      │
            [Adaptive algorithm] → updates w

Derivation

Let the input vector and weight vector be:

y_k = [y_k  y_(k−1)  ...  y_(k−N)]ᵀ
w_k = [w₀  w₁  ...  w_N]ᵀ

Equalizer output and error (x_k = desired/transmitted symbol):

d̂_k = y_kᵀ w_k = w_kᵀ y_k
e_k  = x_k − y_kᵀ w_k

Square the error:

|e_k|² = x_k² + w_kᵀ y_k y_kᵀ w_k − 2 x_k y_kᵀ w_k

Take expectation (w is treated as fixed):

ξ = E[|e_k|²]
  = E[x_k²] + wᵀ E[y_k y_kᵀ] w − 2 E[x_k y_kᵀ] w

Define:

R = E[y_k y_kᵀ]   (input correlation matrix, (N+1)×(N+1))
p = E[x_k y_k]    (cross-correlation vector)

Then the MSE is a quadratic ("bowl-shaped") function of w:

ξ = E[x_k²] + wᵀ R w − 2 pᵀ w

Set the gradient to zero to find the minimum:

∇ξ = ∂ξ/∂w = 2 R w − 2 p = 0
R ŵ = p
ŵ = R⁻¹ p          ← optimum (Wiener) weights

Minimum MSE, by substituting ŵ:

ξ_min = E[x_k²] − pᵀ R⁻¹ p = E[x_k²] − pᵀ ŵ

Error surface

 ξ
 |\           /
 | \         /
 |  \       /     quadratic bowl
 |   \_____/      single global minimum
 |      •  ξ_min
 +------ŵ--------> w

Since R is positive definite, the surface has a single global minimum, so any gradient search (steepest descent, LMS) converges to ŵ.

Practical note

Computing R⁻¹ directly needs about (N+1)³ operations and knowledge of the statistics. Adaptive algorithms find ŵ iteratively:

• LMS: w_(k+1) = w_k + α e_k y_k (simple, slow convergence).
• RLS: faster convergence, more computation.

RAKE receiver

A RAKE receiver is a CDMA receiver that collects the energy of several multipath components of the same signal and combines them, instead of treating them as interference. It has several correlators, called fingers, like the teeth of a garden rake. Each finger is locked to one multipath copy that is delayed by at least one chip, so the copies are nearly uncorrelated. Combining them gives multipath (time) diversity, which improves the SNR and reduces fading.

Working of an M-branch RAKE receiver

       +->[Corr. 1, τ1]->Z1->(x)α1-+
       |                           |
r(t) --+->[Corr. 2, τ2]->Z2->(x)α2-(Σ)-> Z'
       |                           |    |
       +->[Corr. M, τM]->ZM->(x)αM-+    v
               ^                  [∫ + decide]
     [PN code + path searcher]          |
                                        v  m'(t)

1. Path searching: a searcher finds the M strongest multipath delays τ₁ … τ_M from the channel's power delay profile using the PN code's sharp autocorrelation.
2. Correlation: correlator m multiplies r(t) by the PN code delayed by τ_m and integrates over the bit. Its output Z_m is the despread data from path m. Other paths, being more than one chip apart, appear as low-level noise.
3. Weighting: each output is multiplied by a weight α_m proportional to the signal strength (SNR) of that path, so strong paths count more:

α_m = Z_m² / Σ_(m=1..M) Z_m²

4. Combining: the weighted outputs are co-phased and summed:

Z' = Σ_(m=1..M) α_m Z_m

5. Decision: the combined output is integrated and compared with a threshold to give the data bit.

Why it works

If one path fades, others are likely strong, because fading on paths separated by more than one chip is independent. With maximal ratio weights the output SNR approaches the sum of the finger SNRs. IS-95 uses 3 fingers in the mobile and 4 in the base station; WCDMA also uses RAKE receivers.

Signal processing to minimize ISI

Equalization is the signal processing operation that minimizes ISI. An equalizer is a filter at the receiver whose response is roughly the inverse of the channel, so the combined channel + equalizer response is flat. Because the mobile channel changes with time, the equalizer is adaptive and tracks the channel using training sequences and decision feedback.

RAKE receiver

A RAKE receiver is a multi-correlator CDMA receiver. Each correlator ("finger") locks to one multipath component delayed by at least one chip, and the finger outputs are weighted and combined. It turns multipath into diversity gain instead of ISI.

Linear transversal equalizer (LTE)

The most common equalizer structure is a tapped delay line (FIR) filter. The received signal passes through a chain of delay elements of one symbol period T; each tap output is multiplied by an adjustable complex weight c_n, and all products are summed.

y_k ─┬─[T]─┬─[T]─┬─[T]─┬─ ... ─[T]─┐
     │     │     │     │           │
   (x)c₋N₁ (x)  (x)   (x)  ...   (x)c_N₂
     │     │     │     │           │
     └─────┴─────┴──(Σ)┴───────────┘
                     │
                     v d̂_k ─►[Decision]─► x̂_k
                     │
        x_k (training) ─►(−)─► e_k ─► [Adapt taps]

Output:

d̂_k = Σ_(n = −N₁)^(N₂) c_n* y_(k−n)

• N₁ = number of taps for future (precursor) samples, N₂ for past (postcursor) samples.
• The error e_k = x_k − d̂_k (or decision − d̂_k in tracking mode) is used by an algorithm such as LMS to adjust the taps.
• The minimum MSE of an LTE is:

E[|e|²] = (N₀T/2π) ∫ dω / (|F(e^jωT)|² + N₀)

Merits: simple, stable, easy to adapt. Limitation: in channels with deep spectral nulls it boosts noise a lot, so non-linear equalizers (DFE, MLSE) are preferred for severe multipath.

Signal processing to minimize ISI

Equalization minimizes ISI. An equalizer is a receiver filter with roughly the inverse frequency response of the channel, so that channel + equalizer gives a flat, linear-phase response. Since the mobile channel is random and time-varying, adaptive equalizers are used: a known training sequence first sets the tap weights, and then the equalizer tracks the channel using its own decisions.

RAKE receiver

A RAKE receiver is a CDMA receiver with several correlators (fingers), each locked to a different multipath copy of the signal. Since the PN code's autocorrelation is very small beyond one chip, copies delayed by more than one chip can be separated and then combined. This gives multipath diversity instead of ISI.

Working of M-branch RAKE receiver

       +->[Corr. 1, τ1]->Z1->(x)α1-+
       |                           |
r(t) --+->[Corr. 2, τ2]->Z2->(x)α2-(Σ)-> Z'
       |                           |    |
       +->[Corr. M, τM]->ZM->(x)αM-+    v
                                  [∫ + decide]-> m'

1. A searcher finds the delays of the M strongest paths.
2. Each correlator multiplies the input by the PN code shifted to its path delay τ_m and integrates, giving output Z_m.
3. Each output is weighted by α_m, proportional to that path's strength, e.g. α_m = Z_m² / Σ Z_m².
4. The outputs are co-phased and summed: Z' = Σ α_m Z_m.
5. Z' is integrated over the bit and compared with a threshold to decide the bit.

If one path fades deeply, the others usually do not, so the combined output is much more reliable. IS-95 mobiles use 3 fingers; base stations use 4.

Interleaving

Interleaving rearranges the order of coded bits before transmission so that a burst of errors caused by a deep fade is spread out into isolated single errors after de-interleaving. Error-correcting codes can correct scattered errors easily but fail on long bursts, so interleaving gives time diversity without adding redundancy.

Block interleaver (m rows × n columns): bits are written into the array row by row and read out column by column. The receiver de-interleaver does the reverse.

 Write by rows →         Read by columns ↓
  1   2   3   4          out: 1 5 9 13 2 6 10 14
  5   6   7   8               3 7 11 15 4 8 12 16
  9  10  11  12
 13  14  15  16

Suppose a fade wipes out 4 consecutive transmitted bits, e.g. 1, 5, 9, 13. After de-interleaving they appear in different rows (code words), each with only one error, which the decoder can correct.

 Channel burst: [1 5 9 13]   (4 bits in a row)
 After de-interleave:
  X  2  3  4
  X  6  7  8      one error per code word
  X 10 11 12
  X 14 15 16

The interleaving depth must exceed the burst length, but larger depth adds delay; speech systems therefore limit delay to around 40 ms (GSM interleaves over 8 bursts).

Maximum Likelihood Sequence Estimation (MLSE)

MLSE is a non-linear equalizer that, instead of deciding symbols one by one, chooses the whole data sequence that is most likely to have produced the received signal. It treats the ISI channel as a finite-state machine and uses the Viterbi algorithm to search efficiently.

r(t)->[Matched ]->[ MLSE (Viterbi) ]-> data
      [ filter ]        ^
           |            |
           +->[Channel estimator]
               (from training)

1. A training sequence lets the channel estimator find the channel impulse response (L taps).
2. The channel has M^L states (M = alphabet size). For each received sample the Viterbi algorithm computes path metrics (squared distance between the received sample and the expected noise-free output) for every state transition.
3. Only the best ("survivor") path into each state is kept.
4. After enough samples the survivor with the minimum total metric gives the most likely transmitted sequence.

MLSE is optimal in the sense of minimizing sequence error probability and handles severe ISI, but complexity grows exponentially with channel length. GSM uses a Viterbi (MLSE) equalizer covering about 4 bit periods (≈ 15 µs) of delay spread.

Interleaving

Interleaving is the process of rearranging coded bits in time before transmission so that burst errors from deep fades are spread into random, isolated errors after de-interleaving, where the channel decoder can correct them.

How equalization offsets ISI

In a multipath channel, delayed copies of a symbol overlap with later symbols, causing inter-symbol interference (ISI). If the channel impulse response is f(t), the received signal is y(t) = x(t) ⊛ f(t) + n(t). An equalizer h_eq(t) is chosen so that the combined response is an impulse:

f(t) ⊛ h_eq(t) = δ(t)
F(f) · H_eq(f) = 1   →  H_eq(f) = 1 / F(f)

So the equalizer is an inverse filter of the channel. Since the mobile channel changes with time and position, the equalizer must be adaptive.

Example: adaptive linear (transversal) equalizer

y_k ─┬─[T]─┬─[T]─┬─ ... ─[T]─┐
     │     │     │           │
   (x)w₀  (x)w₁ (x)w₂ ...  (x)w_N
     └─────┴────(Σ)──────────┘
                 │ d̂_k
                 ├──────►[Decision]──► x̂_k
                 v              │
     x_k ───►(+)(−)◄────────────┘ (tracking)
     (training)  │ e_k
                 v
          [LMS: w ← w + α e_k y_k]

Operation

1. Training mode: the transmitter sends a known training sequence (e.g. 26-bit midamble in GSM). The receiver compares the equalizer output d̂_k with the known symbol x_k to form the error e_k = x_k − d̂_k.
2. Weight update: an adaptive algorithm (e.g. LMS) adjusts the taps to reduce the mean square error:

w_(k+1) = w_k + α e_k y_k

The taps converge near the optimum ŵ = R⁻¹p, so the filter undoes the channel's ISI. 3. Tracking mode: during user data, the receiver's own decisions x̂_k replace the training symbols, so the equalizer keeps following slow channel changes. 4. Retraining: training is repeated periodically (every TDMA burst) because the channel changes faster than the equalizer can track over long periods.

Example: a two-path channel with ISI from one earlier symbol (f = [1, 0.5]) needs an inverse filter approximately [1, −0.5, 0.25]; the adaptive equalizer finds these taps automatically from the training sequence.

Need for diversity

In a fading channel the received signal often falls into deep fades (20–40 dB). Increasing transmit power to cover these fades is costly. Diversity sends or receives the signal over two or more independently fading paths; the chance that all paths fade at the same time is very small. It improves the average SNR and reduces the error rate, with low cost and no extra bandwidth (for space diversity).

Maximal ratio combining (MRC)

In MRC, the signals from all M antenna branches are co-phased, weighted in proportion to their own signal amplitude (SNR), and summed.

 Ant1 ->[x G1]-+
 Ant2 ->[x G2]-+->(Σ)-> output, SNR = Σ γ_i
 AntM ->[x GM]-+
   (co-phased, G_i ∝ r_i/N_i)

• Output SNR = sum of the branch SNRs: γ_M = Σ γ_i, the best possible for a linear combiner.
• Gives an acceptable SNR even when no single branch is good.
• Needs a separate receiver and phase tracking per branch, so it is the most complex method.

Interleaving is needed because mobile fading channels cause burst errors: when the mobile is in a deep fade, many consecutive bits are lost. Most error-correcting codes are designed for random, scattered errors and cannot correct long bursts. Interleaving spreads the bits of each code word over time so that one fade damages only a few bits of each code word. It provides time diversity without adding redundant bits.

Working (block interleaver)

 Write row-wise:        Read column-wise:
  a1 a2 a3 a4           a1 b1 c1 d1 a2 b2 ...
  b1 b2 b3 b4
  c1 c2 c3 c4           Burst hits a2 b2 c2 d2
  d1 d2 d3 d4

1. Coded bits are written into an m × n array row by row (each row = one code word).
2. They are read out column by column and transmitted.
3. The receiver's de-interleaver writes column by column and reads row by row, restoring the order.
4. A burst of up to m bits in the channel becomes at most one error per code word, which the decoder corrects.

Trade-off: deeper interleaving handles longer fades but adds delay. Example: GSM interleaves each 456-bit speech block over 8 TDMA bursts.

Need for equalization and diversity

• Equalization is needed to remove inter-symbol interference (ISI) caused by time-dispersive (frequency-selective) multipath. When the delay spread is a large part of the symbol period, earlier symbols overlap with later ones; an adaptive equalizer reverses the channel's effect.
• Diversity is needed to fight deep fading in flat-fading channels. Because the received signal randomly falls 20–40 dB, receiving multiple independently fading copies lets the receiver avoid or combine around deep fades, reducing BER without more transmit power.
• Both together raise link quality, allow higher data rates and lower outage probability.

Space diversity reception methods

1. Selection diversity
2. Feedback (scanning) diversity
3. Maximal ratio combining
4. Equal gain combining

Selection diversity (explained)

M antennas, spaced at least about λ/2 (at the mobile) apart, feed M receivers. The receiver continuously selects the branch with the highest instantaneous SNR and connects it to the demodulator.

 Ant1 ->[Rx1]-SNR1-+
 Ant2 ->[Rx2]-SNR2-+->[Select max]-> demod
 AntM ->[RxM]-SNRM-+

Probability that all M branches are below threshold γ (Rayleigh, mean SNR Γ):

P[γ₁..γ_M ≤ γ] = (1 − e^(−γ/Γ))^M

Mean output SNR improvement:

γ̄/Γ = Σ_(k=1)^(M) 1/k

Example: 2 branches give 1.5 times (≈ 1.76 dB) average SNR improvement, and much larger gain at low outage levels. Simple to build, but only one branch is used at a time.

A RAKE receiver is a receiver used in CDMA (DSSS) systems that collects and combines the energy of several multipath components of the same signal, providing multipath (time) diversity. It is named after a garden rake because it has several "fingers".

       +->[Finger 1: corr, τ1]-Z1-(xα1)-+
r(t) --+->[Finger 2: corr, τ2]-Z2-(xα2)-(Σ)-> decide
       +->[Finger M: corr, τM]-ZM-(xαM)-+

• Each finger is a correlator using the PN code delayed to match one multipath component. Paths more than one chip apart are uncorrelated, so each finger isolates one path.
• Outputs are weighted by path strength, α_m = Z_m²/ΣZ_m², co-phased and summed: Z' = Σ α_m Z_m.
• If one path fades, others still contribute, so fading is greatly reduced.
• Turns multipath, normally harmful, into a gain. Used in IS-95 (3 fingers in the mobile, 4 at the base station) and WCDMA.

An adaptive equalizer works in two modes.

Training mode

• The transmitter sends a fixed-length, known training sequence (e.g. a pseudo-random binary pattern or the 26-bit midamble in GSM) before or within each data burst.
• The receiver knows this sequence, so it computes the error e_k = x_k − d̂_k exactly and adjusts the tap weights (e.g. by LMS) towards their optimum.
• The training sequence is designed so that the equalizer can converge even in the worst channel; at the end of training the taps are near optimal for the current channel.

Tracking mode

• When user data follows, there is no known reference, so the equalizer uses its own decisions x̂_k (decision-directed adaptation): e_k = x̂_k − d̂_k.
• It continuously follows slow changes in the channel while data is being received.
• If the channel changes too much, decisions become wrong, so training is repeated periodically. In TDMA systems this happens in every time slot.

 |Training|  User data (tracking)  |Training| Data ...
 <-known->                          <-known->

Feedback (scanning) diversity scans the antenna branches in a fixed order and stays with the first branch whose signal is above a threshold. When that signal falls below the threshold, it scans again. Maximal ratio combining (MRC) uses all branches at once: it co-phases each branch, weights it in proportion to its SNR, and adds them.

 Feedback/scanning            MRC
 Ant1 -+                      Ant1 ->(xG1)-+
 Ant2 -+-[Switch]->Rx         Ant2 ->(xG2)-(Σ)->Rx
 AntM -+   ^                  AntM ->(xGM)-+
     [Compare with threshold]

How diversity improves network quality

Diversity is a receiver technique that gives the receiver two or more independently faded copies of the same signal and then selects or combines them. Because deep fades on independent branches rarely happen at the same time, the chance that all copies are weak is very small.

• If one branch is in a fade with probability p, the probability that all M independent branches fade together is about pᴹ (e.g. 0.1 → 0.001 for M = 3).
• The combined signal has a higher and more stable average SNR, so the bit error rate falls sharply and fewer calls drop.
• Unlike equalization, diversity needs no training sequence and adds little delay.
• Result: better coverage at cell edges, lower required transmit power, higher capacity and better voice/data quality.

RAKE receiver

A RAKE receiver is a diversity receiver used in CDMA (IS-95, WCDMA). In CDMA the chip duration is very short, so multipath copies that arrive more than one chip (Tc) apart are resolvable. Instead of treating these delayed copies as interference, the RAKE receiver collects them, like the fingers of a garden rake gathering leaves.

             +------------+   +----+   w1
        +--->| Correlator |-->|Z1  |--(x)--+
        |    |  code c(t) |   +----+       |
        |    +------------+                |
 r(t)   |    +------------+   +----+   w2  v   +-------+
 ------>+--->| Correlator |-->|Z2  |--(x)-(+)->|Integr.|
        |    | c(t - τ2)  |   +----+       ^   | & dec.|
        |    +------------+                |   +-------+
        |    +------------+   +----+   wM  |      |
        +--->| Correlator |-->|ZM  |--(x)--+      v
             | c(t - τM)  |   +----+          data bits
             +------------+
      (one "finger" per resolvable multipath)

Operation

1. The received signal r(t) is fed to M correlators, called fingers.
2. Each finger multiplies r(t) by the user's PN code delayed by the delay τₖ of one multipath component, and integrates over one bit. This despreads that one path; other paths look like noise to it because the PN code has low autocorrelation for shifts ≥ Tc.
3. A searcher (path finder) estimates the delays τₖ and strengths of the strongest paths and assigns them to the fingers.
4. Finger outputs Z₁ … Z_M are weighted by w₁ … w_M, proportional to the SNR (amplitude) of each path, so strong paths count more. This is maximal-ratio style combining: Z′ = Σ wₖ·Zₖ
5. The combined output Z′ is integrated and passed to a decision device to recover the bits.

Benefit: multipath, which harms narrowband systems, becomes a source of multipath (time) diversity, improving SNR and reducing fading in CDMA systems.

Signal processing operation that minimizes ISI: equalization

Equalization is the signal processing used at the receiver to reduce inter-symbol interference (ISI) caused by multipath delay spread and band-limited channels. When the delay spread is larger than the symbol period, each symbol spreads into the next. An equalizer is a filter whose response is roughly the inverse of the channel response, so that

H_eq(f) · H_ch(f) ≈ constant (flat), i.e. h_ch(t) * h_eq(t) ≈ δ(t)

Because the mobile channel changes with time, the equalizer is adaptive: it is first trained with a known training sequence, then tracks the channel in decision-directed mode using algorithms such as LMS or RLS. Common types are linear transversal (FIR) equalizers, decision feedback equalizers (DFE) and MLSE (Viterbi) equalizers.

Space diversity techniques

Space (antenna) diversity uses two or more antennas spaced far enough apart that their fading is uncorrelated: about λ/2 at the mobile and 10λ or more at the base station. The branch signals are then used in one of these ways:

1. Selection diversity – The receiver monitors the SNR of all M branches and connects the branch with the highest instantaneous SNR to the demodulator. It is simple and needs one receiver after the switch.

2. Feedback (scanning) diversity – The branches are scanned in a fixed order until one is found above a preset threshold; that branch is used until it falls below the threshold, then scanning restarts. Only one receiver is needed, but it performs a little worse than selection.

3. Maximal ratio combining (MRC) – Every branch is co-phased, weighted in proportion to its signal amplitude (gain ∝ signal voltage / noise power), and summed. The output SNR is the sum of branch SNRs (γ_M = Σ γᵢ), which is the best possible. It needs a receiver and phase correction on every branch.

4. Equal gain combining (EGC) – Branches are co-phased and added with equal weights. It performs only slightly worse than MRC and is easier to build because no amplitude estimation is needed.

 Ant1   Ant2   ...  AntM
  |      |           |
 [G1]   [G2]  ...   [GM]   gains / co-phasing
  |      |           |
  +------+-----+-----+
               |
      Select / Sum (combiner)
               |
          Demodulator

Selection and feedback pick one branch; MRC and EGC add all branches.

Need for diversity

In a mobile channel the received signal suffers small-scale (Rayleigh) fading: the envelope can drop 20–40 dB below its average when the mobile moves only a fraction of a wavelength. During such deep fades the bit error rate becomes very high and calls may drop. Raising transmit power to cover the deepest fades is wasteful and causes interference.

Diversity solves this by giving the receiver several independently faded copies (branches) of the same signal:

• The chance that all M branches are in a deep fade at once is roughly pᴹ, which is very small.
• Average SNR improves and, more importantly, the SNR becomes much more stable (fewer deep fades).
• It gives a large BER improvement with no extra bandwidth (space, polarization) and no training overhead.
• It works well together with equalization (for ISI) and channel coding (for remaining errors).

Diversity branches can be obtained by space (antenna separation), polarization, frequency (separation > coherence bandwidth) or time (separation > coherence time). Whatever the source, the branches are then combined by one of the methods below.

Diversity combining techniques

Once the branch signals are available they must be combined. Let γᵢ be the SNR of branch i.

1. Selection combining

• The branch with the largest instantaneous SNR is chosen: γ_out = max(γ₁, …, γ_M).
• Only one demodulator after the switch; simple.
• Mean SNR improvement = Σₖ₌₁ᴹ 1/k times the single-branch mean (e.g. 1.83 for M = 3).

2. Feedback (scanning / switched) combining

• Branches are scanned in order; the first one above a threshold is used until it drops below the threshold.
• Needs only one receiver; slightly poorer than selection.

3. Maximal ratio combining (MRC)

• Each branch is co-phased and weighted by Gᵢ ∝ (signal amplitude)/(noise power), then summed.
• γ_out = Σ γᵢ, so the mean output SNR is M times the single-branch mean; this is the optimum linear combiner.
• Needs channel estimates and a receiver on each branch; used in RAKE receivers and modern base stations.

4. Equal gain combining (EGC)

• Branches are co-phased and summed with equal (unity) weights.
• Performance close to MRC (within about 1 dB), but simpler because only phase must be estimated.

  r1 --[co-phase, ×G1]--+
  r2 --[co-phase, ×G2]--+--(Σ)--> detector
  ...                   |
  rM --[co-phase, ×GM]--+
  MRC: Gi ∝ amplitude/noise   EGC: Gi = 1
  Selection: switch to max SNR branch only

Example: with three branches each having mean SNR Γ, selection combining gives a mean output SNR of Γ(1 + 1/2 + 1/3) = 1.83Γ (about 2.6 dB), while MRC gives 3Γ (about 4.8 dB). The real benefit is larger at low outage probabilities, where diversity can save 10 dB or more of fade margin.

Diversity techniques in wireless communication

Diversity means sending or receiving the same information over two or more independently fading paths and combining them, so that the probability of a deep fade on all paths together is small. The main types are:

1. Space (antenna) diversity – Several receive antennas spaced so their signals fade independently (≈ λ/2 at the mobile, 10λ or more at a base station).
2. Polarization diversity – Two antennas with orthogonal polarizations (vertical/horizontal or ±45°) at the same location; scattering makes the two polarizations fade differently. Saves space at base stations.
3. Frequency diversity – Same data sent on carriers separated by more than the coherence bandwidth, so they fade independently. Used in OFDM with coding and in frequency hopping.
4. Time diversity – Same data repeated at intervals greater than the coherence time, or spread in time using channel coding with interleaving. RAKE receivers use multipath time diversity.
5. Angle/pattern diversity – Directional antennas pointing in different directions receive differently faded components.

Diversity can be microscopic (against small-scale fading, as above) or macroscopic (against shadowing, by receiving from two base stations at different sites).

Combining methods used for space diversity

• Selection diversity – The branch with the highest instantaneous SNR is connected to the receiver. Simple; mean SNR gain = Σₖ₌₁ᴹ 1/k.
• Feedback (scanning) diversity – Branches are checked one by one; the first one above a threshold is used until it falls below it. Uses one receiver; slightly worse than selection.
• Maximal ratio combining (MRC) – All branches are co-phased, weighted by their signal-to-noise ratio, and summed; output SNR = sum of branch SNRs. Best performance, highest cost.
• Equal gain combining (EGC) – All branches are co-phased and summed with equal weights. Nearly as good as MRC, easier to build.

 Ant1 --+                      Ant1 --[w1]--+
        |--[switch to          Ant2 --[w2]--+--(Σ)--> Rx
 Ant2 --+   max SNR]--> Rx     AntM --[wM]--+
 Selection / feedback          MRC (w ∝ SNR), EGC (w = 1)

Adaptive equalization is equalization in which the filter coefficients are adjusted automatically to track a time-varying channel, so that inter-symbol interference (ISI) is removed even as the mobile moves.

Why adaptive: the mobile radio channel is unknown and changes with time (Doppler, moving scatterers), so a fixed equalizer cannot stay matched to it.

 x(k)   +-------------------+ d^(k)   +----------+
 ------>| Transversal filter|-------->| Decision |--> d~(k)
        | taps w0..wN       |    |    +----------+   |
        +-------------------+    |                   |
               ^                 v                   |
               |     e(k)=d(k)-d^(k)  <-- d(k) ------+
               +---- adaptive algorithm (LMS/RLS)
     d(k): training sequence, or decided symbol later

Two modes of operation

1. Training mode – The transmitter sends a known, fixed-length training sequence (e.g. a PN sequence or the GSM midamble). The receiver knows d(k), computes the error e(k) = d(k) − d̂(k), and the algorithm sets the tap weights to near optimum.
2. Tracking (decision-directed) mode – After training, user data follows. The detector's own decisions d̃(k) are used as the reference to keep updating the taps as the channel slowly changes.

Training is repeated periodically (for TDMA, in every burst/time slot).

Algorithms

• Zero forcing (ZF) – forces ISI to zero; amplifies noise at spectral nulls.
• LMS – w(k+1) = w(k) + μ·e(k)·x(k); simple (2N+1 operations per iteration), slow convergence.
• RLS – faster convergence, more computation.

Structures: linear transversal equalizer, decision feedback equalizer (DFE), and MLSE (Viterbi) equalizer.

Performance factors: rate of convergence, misadjustment, computational complexity and numerical stability.

An adaptive equalization algorithm updates the tap weights w of the equalizer so that the error e(k) = d(k) − d̂(k) between the desired and actual output becomes small. Two common ones:

1. Least Mean Square (LMS) algorithm

• Minimises the mean square error E[e²(k)] using a stochastic gradient (steepest descent) step.
• Update equations:

 d^(k)  = wᵀ(k)·x(k)          (filter output)
 e(k)   = d(k) - d^(k)        (error)
 w(k+1) = w(k) + μ·e(k)·x(k)  (weight update)

• μ is the step size: larger μ gives faster convergence but more misadjustment; stability needs 0 < μ < 2/λ_max.
• Needs only about 2N+1 operations per iteration (N = taps), so it is simple and robust.
• Disadvantage: slow convergence when the eigenvalue spread of the input correlation matrix is large.

2. Recursive Least Squares (RLS) algorithm

• Minimises the weighted sum of past squared errors Σ λⁿ⁻ⁱ e²(i), where λ (0 < λ ≤ 1) is a forgetting factor.
• Recursively updates the inverse correlation matrix P(k) and a gain vector k(k):

 k(n)   = P(n-1)x(n) / [λ + xᵀ(n)P(n-1)x(n)]
 e(n)   = d(n) - wᵀ(n-1)x(n)
 w(n)   = w(n-1) + k(n)·e(n)
 P(n)   = [P(n-1) - k(n)xᵀ(n)P(n-1)] / λ

• Converges about an order of magnitude faster than LMS and is insensitive to eigenvalue spread.
• Needs about 2.5N² + 4.5N operations per iteration and can be numerically unstable.

(Zero forcing is a third simple algorithm that makes the combined channel–equalizer response have zero ISI.)

Space diversity uses M receiving antennas separated enough (about λ/2 at the mobile, 10λ or more at the base station) that each receives an independently faded copy of the signal. The copies are then used in one of four ways.

1. Selection diversity

 Ant1-[Rx1]-+
 Ant2-[Rx2]-+--> [select max SNR] --> demodulator
 AntM-[RxM]-+

The branch with the highest instantaneous SNR is connected to the output. Simple; output SNR = max(γᵢ).

2. Feedback (scanning) diversity

 Ant1-+
 Ant2-+--[scan switch]--[Rx]--> output
 AntM-+        ^          |
               +--[SNR < threshold?]

Branches are scanned in sequence; the first branch above a threshold is held until it falls below it. Needs only one receiver.

3. Maximal ratio combining (MRC)

 Ant1-[co-phase]-[×G1]-+
 Ant2-[co-phase]-[×G2]-+--(Σ)--> detector
 AntM-[co-phase]-[×GM]-+
       Gi ∝ amplitude / noise

Each branch is co-phased and weighted by its SNR, then summed. Output SNR = Σ γᵢ (the best result).

4. Equal gain combining (EGC)

Same as MRC but all gains Gᵢ = 1 after co-phasing. Performance is close to MRC and the circuit is simpler.

Maximum Likelihood Sequence Estimation (MLSE)

MLSE equalization is a non-linear equalizer that, instead of deciding each symbol separately, chooses the whole sequence of symbols that most probably produced the received samples. It is the optimum detector for a channel with ISI and additive white Gaussian noise.

 r(t)  +---------+  y(k)  +-----------------+   a^(k)
 ----->| Matched |------->| MLSE (Viterbi   |-------->
       | filter  |        | algorithm)      |  data
       +---------+        +-----------------+
            ^                    ^
            |                    |
       +------------------------------+
       | Channel estimator (CIR from  |
       | training sequence)           |
       +------------------------------+

Working

1. The received signal passes through a matched filter and is sampled once per symbol, giving y(k).
2. A channel estimator finds the discrete channel impulse response (CIR) h₀, h₁, …, h_L, usually by correlating with a known training sequence (e.g. the GSM midamble).
3. The channel with memory L is a finite-state machine with Mᴸ states (M = alphabet size). The possible sequences form a trellis.
4. For each candidate sequence the receiver predicts the noiseless output ŷ(k) = Σ hⱼ·a(k−j) and computes a metric, the squared Euclidean distance Σ |y(k) − ŷ(k)|².
5. The Viterbi algorithm searches the trellis efficiently: at each state it keeps only the survivor path with the smallest metric, so complexity grows linearly with sequence length rather than exponentially.
6. The survivor with minimum metric is the maximum-likelihood sequence; its symbols are output.

Simple illustration: for BPSK (M = 2) and a two-tap channel h = [h₀, h₁] (L = 1), the trellis has 2¹ = 2 states (previous symbol +1 or −1). Each new symbol gives two branches per state, and at each step only the better of the two paths entering each state survives. After the whole burst, one survivor remains, and that is the decided sequence.

Comparison with other equalizers

Features: minimum probability of sequence error; complexity grows as Mᴸ, so it is practical only for short channel memory. GSM receivers commonly use MLSE (Viterbi) equalizers.

Time diversity

Time diversity is obtained by transmitting the same information repeatedly at time intervals greater than the coherence time of the channel, so that the copies experience independent fading. (Coherence time T_c ≈ 0.423/f_d, where f_d is the maximum Doppler shift.)

Two implementations of time diversity

1. Channel coding with interleaving

• Redundancy is added by an error-correcting code (convolutional, block, turbo).
• An interleaver rearranges coded bits so that bits that are adjacent in the code are sent far apart in time (more than T_c).
• A deep fade then produces scattered single-bit errors after de-interleaving, which the decoder corrects. Used in GSM, IS-95 and LTE.

 data->[encoder]->[interleaver]->channel
     ->[de-interleaver]->[decoder]->data

2. RAKE receiver (multipath time diversity)

• In spread-spectrum CDMA, multipath copies arriving more than one chip apart are separate in time.
• A RAKE receiver uses several correlators ("fingers"), each locked to one delayed path, and combines their outputs with maximal-ratio weights.
• Thus time-separated echoes of the same symbol provide diversity instead of interference.

Limitations of time diversity: it adds delay (interleaver depth must exceed T_c), and it gives little benefit when the mobile is slow or stationary because the coherence time becomes very long. Repetition coding and ARQ retransmission are simpler forms of the same idea.

Need for equalization

In mobile channels the transmitted signal arrives over many paths with different delays. When the delay spread is comparable to or larger than the symbol period (or the signal bandwidth exceeds the coherence bandwidth), each symbol overlaps the following symbols. This is inter-symbol interference (ISI). ISI causes:

• High, irreducible bit error rate that cannot be cured by raising power.
• A limit on the maximum data rate of the link.

Equalization is needed to compensate for this ISI (and for band-limited, frequency-selective channels) so that high data rates can be sent reliably.

Basic equalization technique

The equalizer is a filter at the receiver whose response is the inverse of the channel. If f(t) is the combined impulse response of transmitter, channel and receiver RF/IF, and h_eq(t) is the equalizer's, the target is

f(t) * h_eq(t) = δ(t), i.e. F(f) · H_eq(f) = 1

so the overall response is flat and ISI-free.

 x(t)->[Modulator]->[Channel f(t)]->(+)->[Rx RF/IF]
                                     ^        |
                                   n(t)       v y(t)
            +-----------------+       +-------------+
 d~(k) <----| Decision device |<------| Equalizer   |
    |       +-----------------+ d^(k) | h_eq(t)     |
    |                                 +-------------+
    +--> e(k) = d(k) - d^(k) --> adjust taps --^

Steps

1. Training: a known training sequence is sent first. The equalizer compares its output with the known symbols and adjusts its taps (e.g. by LMS) to minimise the error.
2. Tracking: user data follows; the equalizer uses its own decisions to keep adjusting as the channel changes.
3. The training is repeated periodically, e.g. each TDMA time slot.

Structure: the most basic form is a linear transversal (FIR) equalizer: a tapped delay line with delays T and tap weights wₖ:

d̂(k) = Σₙ wₙ · y(k − n)

Non-linear types (decision feedback equalizer, MLSE) are used when the channel has deep spectral nulls.

Importance of diversity

The mobile channel causes multipath (Rayleigh) fading: the received signal can drop by 20–40 dB in a fraction of a wavelength of movement. Diversity is important because:

• It gives several independently faded copies, so the chance of all being in a deep fade together (≈ pᴹ) is very small.
• It greatly reduces BER and call drops without extra transmit power.
• Many forms (space, polarization) need no extra bandwidth.
• It needs no training overhead and is cheap compared with the improvement it gives (often 10–20 dB gain at low BER).

Types of diversity techniques

1. Space (antenna) diversity

• Two or more antennas separated so that fading is uncorrelated (≈ λ/2 at mobile, ≥ 10λ at base station).
• Combining: selection, feedback, maximal ratio and equal gain combining.

2. Polarization diversity

• Horizontally and vertically (or ±45°) polarized antennas at one location. Scattering de-correlates the two polarizations.
• Compact; widely used at base stations; gain is a little less than space diversity.

3. Frequency diversity

• The same information is sent on carriers separated by more than the coherence bandwidth (B_c ≈ 1/(5σ_τ)).
• Disadvantage: needs extra spectrum and a receiver per frequency. Frequency hopping and OFDM use this idea.

4. Time diversity

• Repeats the information at intervals greater than the coherence time (T_c ≈ 0.423/f_d).
• Implemented with channel coding plus interleaving, or with a RAKE receiver in CDMA.

5. Angle (pattern) diversity

• Directional antennas pointing in different directions pick up differently faded multipath components.

Antenna diversity uses more than one antenna to obtain independently faded copies of the signal. The main kinds are:

1. Space diversity – identical antennas physically separated (≈ λ/2 at the mobile, 10λ or more at the base station) so fading is uncorrelated.
2. Polarization diversity – two co-located antennas with orthogonal polarizations (V/H or ±45°); multipath scattering makes the two polarizations fade differently.
3. Angle (pattern) diversity – directional antennas or beams pointing in different directions, receiving different groups of multipath waves.

Space diversity outputs are combined by selection, feedback, maximal ratio or equal gain combining.

Summary: space diversity gives the best gain but needs room; polarization diversity gives most of the gain with a single compact antenna, so it is common on modern base stations.

Maximum Likelihood Sequence Estimation (MLSE) equalization is a non-linear equalizer that decides on the most probable sequence of transmitted symbols, rather than one symbol at a time. It is optimum for channels with ISI and white Gaussian noise.

 r(t) +---------+ y(k) +----------------+  a^(k)
 ---->| Matched |----->| MLSE: Viterbi  |------->
      | filter  |      | trellis search |  data
      +---------+      +----------------+
           ^                   ^
           |   +-----------+   |
           +---| Channel   |---+
               | estimator |  (CIR h0..hL from
               +-----------+   training sequence)

Working

1. The matched filter maximises SNR and its output is sampled once per symbol to give y(k).
2. The channel estimator uses the known training sequence to estimate the discrete channel impulse response h₀ … h_L.
3. Because ISI comes from the last L symbols, the channel behaves like a finite-state machine with Mᴸ states, drawn as a trellis.
4. For every path in the trellis the expected output ŷ(k) = Σ hⱼ·a(k−j) is formed and the metric |y(k) − ŷ(k)|² is accumulated.
5. The Viterbi algorithm keeps only one survivor (lowest metric) per state at each step, so the search is efficient.
6. The surviving path with the least total metric is the maximum-likelihood sequence and is output.

Merits: minimum sequence error probability; handles deep spectral nulls better than linear equalizers. Limitation: complexity grows exponentially with channel memory (Mᴸ states). GSM receivers use a Viterbi (MLSE) equalizer.

Diversity

Diversity is a technique in which the receiver gets two or more copies of the same information over independently fading paths (branches) and selects or combines them. Since all branches are unlikely to fade deeply at the same time, the combined signal is much more reliable. It is a powerful, low-cost way to fight small-scale (Rayleigh) fading.

1. Space (antenna) diversity

• Uses M receiving antennas separated so their signals are uncorrelated: about λ/2 at the mobile and 10λ or more at the base station (where scatterers are far away).
• No extra bandwidth or transmit power is needed.
• Branch signals are combined by:
  • Selection – choose the branch with the highest SNR.
  • Feedback/scanning – stay on a branch until it falls below a threshold, then switch.
  • Maximal ratio combining – co-phase, weight each branch by its SNR and add; output SNR = Σ γᵢ (best).
  • Equal gain combining – co-phase and add with equal weights; close to MRC.

 Ant1 --[co-phase ×G1]--+
 Ant2 --[co-phase ×G2]--+--(Σ / select)--> detector
 AntM --[co-phase ×GM]--+

• Example: two antennas on a base-station mast several metres apart.
• With MRC, M branches give M times the mean SNR of one branch; with selection the gain is Σₖ₌₁ᴹ 1/k.

2. Time diversity

• The same information is sent at time intervals greater than the coherence time T_c (≈ 0.423/f_d), so the copies fade independently.
• Implementations:
  • Channel coding with interleaving – coded bits are spread in time, so a fade causes scattered errors that the decoder corrects (GSM, IS-95, LTE).
  • RAKE receiver – in CDMA, multipath copies separated by more than one chip are despread separately and combined.
  • Repetition coding and ARQ.
• Drawback: adds delay; useless when the mobile is stationary (T_c becomes very long).

 data->[coder]->[interleaver]->fading channel
     ->[de-interleaver]->[decoder]->data

(Other types: frequency diversity – carriers separated by more than the coherence bandwidth; polarization diversity – orthogonally polarized antennas.)

Fundamentals of equalization

Equalization is a receiver technique that removes inter-symbol interference (ISI) caused by multipath delay spread or band-limited channels. ISI appears when the delay spread is comparable to the symbol period, so symbols overlap and the error rate becomes high.

• The equalizer is a filter whose response is the inverse of the channel: F(f)·H_eq(f) = 1, i.e. f(t) * h_eq(t) = δ(t), giving an overall flat, ISI-free response.
• Since the mobile channel is unknown and time-varying, the equalizer is adaptive:
  1. Training mode: a known training sequence is sent; the error e(k) = d(k) − d̂(k) is used to set the taps.
  2. Tracking mode: detected data symbols are used as the reference to follow channel changes.
• Basic structure: a transversal (tapped delay line) filter, d̂(k) = Σ wₙ·y(k − n).
• Types: linear (transversal, lattice) and non-linear (decision feedback equalizer, MLSE).
• Algorithms: zero forcing, LMS, RLS.

 y(k)->[T]->[T]->...->[T]
   |    |    |          |
  w0   w1   w2   ...   wN   (adapted by LMS/RLS)
   +----+----+----------+--(Σ)--> d^(k) -> decision

RAKE receiver

A RAKE receiver is used in CDMA to collect the energy of several multipath copies instead of suffering from them. Paths that arrive more than one chip apart are resolvable because the PN code has low autocorrelation away from zero shift.

             +------------+   +----+   w1
        +--->| Correlator |-->|Z1  |--(x)--+
        |    |  code c(t) |   +----+       |
        |    +------------+                |
 r(t)   |    +------------+   +----+   w2  v   +-------+
 ------>+--->| Correlator |-->|Z2  |--(x)-(+)->|Integr.|
        |    | c(t - τ2)  |   +----+       ^   | & dec.|
        |    +------------+                |   +-------+
        |    +------------+   +----+   wM  |      |
        +--->| Correlator |-->|ZM  |--(x)--+      v
             | c(t - τM)  |   +----+          data bits
             +------------+
      (one "finger" per resolvable multipath)

Function

1. M correlators ("fingers") each despread the received signal using the PN code delayed to match one strong multipath delay τₖ.
2. A searcher finds the path delays and amplitudes and assigns fingers.
3. Finger outputs Zₖ are weighted by wₖ (proportional to path strength) and summed: Z′ = Σ wₖZₖ (maximal-ratio combining).
4. The sum is integrated over the bit period and a decision gives the data.

So multipath, which causes ISI in narrowband systems, is turned into time diversity, raising SNR and reducing fading in IS-95 and WCDMA.