Ferkans — Interactive Telecom Tutor

Definition:
Selection Combining (SC)

Selection combining chooses the branch with the highest instantaneous SNR and discards the others:

$\gamma_{\text{SC}} = \max_{l=1,\ldots,L} \gamma_l$

For i.i.d. Rayleigh branches with CDF $F_{\gamma}(x) = 1 - e^{-x/\bar{\gamma}}$ , the CDF of $\gamma_{\text{SC}}$ is

$F_{\gamma_{\text{SC}}}(x) = \left(1 - e^{-x/\bar{\gamma}}\right)^L$

Selection combining is the simplest diversity technique: it requires monitoring all branches but processing only one. However, it wastes the signal energy received on the non-selected branches.

SC only needs to measure the SNR (or signal strength) on each branch; it does not require knowledge of the channel phase. This makes it suitable for non-coherent systems.

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Definition:
Equal-Gain Combining (EGC)

Equal-gain combining co-phases all branches and adds them with equal weights:

$r_{\text{EGC}} = \sum_{l=1}^{L} e^{-j\angle h_l}\, r_l = \sum_{l=1}^{L} |h_l|\, s + \tilde{w}$

where $r_l = h_l s + w_l$ is the received signal on branch $l$ . The output SNR is

$\gamma_{\text{EGC}} = \frac{\left(\sum_{l=1}^{L} |h_l|\right)^2} {\sum_{l=1}^{L} N_0} = \frac{\left(\sum_{l=1}^{L} |h_l|\right)^2}{L\, N_0}$

EGC requires knowledge of the channel phases but not the amplitudes, making it simpler than MRC while still capturing energy from all branches.

By the Cauchy-Schwarz inequality, EGC is always suboptimal compared to MRC. The performance gap is small for branches with similar average SNR, typically less than 1 dB for i.i.d. branches.

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Definition:
Maximal-Ratio Combining (MRC)

Maximal-ratio combining weights each branch proportionally to its channel gain and inversely proportionally to its noise power:

$r_{\text{MRC}} = \sum_{l=1}^{L} \frac{h_l^*}{\sigma_l^2}\, r_l$

For equal noise power $\sigma_l^2 = N_0$ on all branches, the combining weights are $w_l = h_l^*$ , and the output SNR is

$\gamma_{\text{MRC}} = \sum_{l=1}^{L} \frac{|h_l|^2}{N_0} = \sum_{l=1}^{L} \gamma_l$

MRC is the optimal linear combining strategy: it maximises the output SNR among all linear combiners. It requires full CSI (both amplitude and phase) of every branch.

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Theorem: MRC Maximises Output SNR

Among all linear combiners $r = \sum_{l=1}^{L} w_l\, r_l$ , the weight vector $\mathbf{w} = \mathbf{h}^* / N_0$ (i.e., the matched filter to the channel vector) maximises the output SNR:

$\gamma_{\text{MRC}} = \frac{|\mathbf{w}^H \mathbf{h}|^2\, E_s} {\mathbf{w}^H \mathbf{R}_{w} \mathbf{w}} = \frac{\|\mathbf{h}\|^2 E_s}{N_0} = \sum_{l=1}^{L} \gamma_l$

where $\mathbf{R}_{w} = N_0 \mathbf{I}$ for i.i.d. noise. No other linear combiner can achieve a higher SNR.

MRC is the spatial matched filter. Just as the matched filter in time maximises SNR by aligning with the signal waveform, MRC aligns the combining weights with the channel vector $\mathbf{h}$ to maximise the signal component while keeping the noise normalised.

Proof

Output SNR expression

The combined output is $r = \mathbf{w}^H \mathbf{r}$ where $\mathbf{r} = \mathbf{h} s + \mathbf{w}$ . The signal power is $|\mathbf{w}^H \mathbf{h}|^2 E_s$ and the noise power is $\mathbf{w}^H (N_0 \mathbf{I}) \mathbf{w} = N_0 \|\mathbf{w}\|^2$ .

$\gamma = \frac{|\mathbf{w}^H \mathbf{h}|^2 E_s}{N_0 \|\mathbf{w}\|^2}$

Cauchy-Schwarz bound

By the Cauchy-Schwarz inequality, $|\mathbf{w}^H \mathbf{h}|^2 \leq \|\mathbf{w}\|^2 \|\mathbf{h}\|^2$ , with equality iff $\mathbf{w} = \alpha \mathbf{h}$ for some scalar $\alpha$ .

Therefore $\gamma \leq \|\mathbf{h}\|^2 E_s / N_0 = \sum_{l=1}^{L} \gamma_l$ .

Optimal weights

Equality is achieved when $\mathbf{w} = \mathbf{h}^*$ (or any scalar multiple). This is the MRC weight vector. The maximum output SNR equals the sum of the branch SNRs. $\blacksquare$

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Theorem: MRC BER for BPSK in Rayleigh Fading

For BPSK with $L$ -branch MRC over i.i.d. Rayleigh fading, the exact average BER is

$P_b = \left(\frac{1 - \mu}{2}\right)^L \sum_{k=0}^{L-1} \binom{L-1+k}{k} \left(\frac{1+\mu}{2}\right)^k$

where $\mu = \sqrt{\bar{\gamma}/(1+\bar{\gamma})}$ and $\bar{\gamma}$ is the average SNR per branch.

At high SNR ( $\bar{\gamma} \gg 1$ ), this simplifies to

$P_b \approx \binom{2L-1}{L} \frac{1}{(4\bar{\gamma})^L}$

confirming diversity order $d = L$ .

The formula extends the single-branch result (which uses $\mu = \sqrt{\bar{\gamma}/(1+\bar{\gamma})}$ ) to $L$ branches. As $\bar{\gamma}$ increases, $\mu \to 1$ and the dominant term is $(1-\mu)^L/2^L \approx (1/(2\bar{\gamma}))^L \cdot 2^{-L}$ .

Proof

Conditional BER

Given the MRC output SNR $\gamma_{\text{MRC}} = \sum_l \gamma_l$ , the conditional BER is $P_b(\gamma) = Q(\sqrt{2\gamma})$ .

Averaging over Erlang distribution

Since $\gamma_{\text{MRC}} \sim \text{Erlang}(L, \bar{\gamma})$ , we compute

$P_b = \int_0^\infty Q(\sqrt{2\gamma})\, f_{\text{Erlang}}(\gamma; L, \bar{\gamma})\, d\gamma$

Using the Craig representation $Q(x) = \frac{1}{\pi}\int_0^{\pi/2} \exp(-x^2/(2\sin^2\theta))\,d\theta$ and integrating over $\gamma$ yields the closed-form expression involving $\mu$ and the binomial sum.

High-SNR approximation

For $\bar{\gamma} \gg 1$ : $\mu \approx 1 - 1/(2\bar{\gamma})$ , $(1-\mu)/2 \approx 1/(4\bar{\gamma})$ , and $(1+\mu)/2 \approx 1$ . The sum $\sum_{k=0}^{L-1} \binom{L-1+k}{k} = \binom{2L-1}{L}$ , giving $P_b \approx \binom{2L-1}{L}/(4\bar{\gamma})^L$ . $\blacksquare$

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MRC Combining Procedure

Complexity:

O(L)

per symbol

Input: Received signals

r_1, \ldots, r_L

; channel

estimates

\hat{h}_1, \ldots, \hat{h}_L

; noise variance

N_0

.

Output: Combined signal

r_{\text{MRC}}

and detected symbol

\hat{s}

.

1. for

l = 1, \ldots, L

do

2.

\qquad w_l \leftarrow \hat{h}_l^* / N_0

\qquad

(conjugate of estimated channel gain)

3. end for

4.

r_{\text{MRC}} \leftarrow \sum_{l=1}^{L} w_l\, r_l

(weighted combination)

5.

\gamma_{\text{MRC}} \leftarrow \sum_{l=1}^{L} |\hat{h}_l|^2 / N_0

(combined SNR for soft decisions)

6.

\hat{s} \leftarrow \text{demod}(r_{\text{MRC}})

(apply standard AWGN detector to combined signal)

7. return

\hat{s}

,

\gamma_{\text{MRC}}

MRC converts the $L$ -branch fading channel into an equivalent AWGN channel with SNR $\gamma_{\text{MRC}} = \sum_l \gamma_l$ . After combining, standard AWGN detection (e.g., nearest-neighbour for QAM) is applied to the combined signal.

Example: MRC with Two Branches

A 2-branch MRC receiver in Rayleigh fading has $h_1 = 0.3 e^{j\pi/6}$ and $h_2 = 1.2 e^{-j\pi/3}$ with $N_0 = 0.01$ and transmitted symbol $s = 1$ (BPSK).

(a) Compute the branch SNRs $\gamma_1$ and $\gamma_2$ .

(b) Compute the MRC output SNR.

(c) Compare with selection combining.

Solution

Branch SNRs

$\gamma_1 = |h_1|^2 / N_0 = 0.09 / 0.01 = 9$ (9.5 dB)

$\gamma_2 = |h_2|^2 / N_0 = 1.44 / 0.01 = 144$ (21.6 dB)

MRC output SNR

$\gamma_{\text{MRC}} = \gamma_1 + \gamma_2 = 9 + 144 = 153$ (21.8 dB)

The MRC output SNR is the sum of the branch SNRs.

Comparison with SC

$\gamma_{\text{SC}} = \max(9, 144) = 144$ (21.6 dB)

MRC provides 0.2 dB gain over SC in this case. The gain is small because one branch dominates. When branches have comparable SNR, the MRC advantage is larger (up to 10 $\log_{10} L$ dB for equal branches). $\blacksquare$

Example: Comparing SC, EGC, and MRC at L = 4

Four i.i.d. Rayleigh fading branches have average SNR $\bar{\gamma} = 10$ dB per branch. Compute the average output SNR for SC, EGC, and MRC.

Solution

MRC average output SNR

$E[\gamma_{\text{MRC}}] = L\, \bar{\gamma} = 4 \times 10 = 40 = 16.0 \text{ dB}$ $

SC average output SNR

For i.i.d. branches: $E[\gamma_{\text{SC}}] = \bar{\gamma} \sum_{l=1}^{L} \frac{1}{l} = 10 \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}\right) = 10 \times 2.083 = 20.83 = 13.2 \text{ dB}$

EGC average output SNR

EGC performance lies between SC and MRC. For Rayleigh fading with $L = 4$ branches:

$E[\gamma_{\text{EGC}}] \approx \bar{\gamma}\left(1 + \frac{\pi}{4}(L-1)\right) = 10\left(1 + \frac{3\pi}{4}\right) = 10 \times 3.356 = 33.56 = 15.3 \text{ dB}$

Summary: MRC (16.0 dB) > EGC (15.3 dB) > SC (13.2 dB). The EGC loss relative to MRC is only 0.7 dB — a remarkably small price for not needing amplitude estimates. $\blacksquare$

SC vs EGC vs MRC Comparison

Compare the BER performance of the three combining techniques as a function of average SNR per branch. All three achieve the same diversity order $L$ , but MRC provides the best coding gain (leftmost curve), followed by EGC, then SC.

Parameters

Number of branches4

Modulation scheme

Show no-diversity (L=1) reference

BER with MRC

Exact BER curves for MRC with varying number of branches and modulation schemes. Observe how each additional branch steepens the BER curve by one decade per 10 dB.

Parameters

Number of MRC branches2

Modulation scheme

Adding Diversity Branches

Watch the BER curve evolve as diversity branches are added one by one. Each frame adds a new branch to the MRC receiver, steepening the high-SNR slope and shifting the curve leftward.

Parameters

Modulation scheme

Maximum number of branches6

Quick Check

Why is MRC optimal among all linear combiners?

It selects the best branch, discarding weak ones

It weights branches proportionally to their SNR, which is the spatial matched filter

It uses equal weights, which averages out the fading

It requires no channel knowledge

Correction:

It weights branches proportionally to their SNR, which is the spatial matched filter

MRC applies the Cauchy-Schwarz-optimal weight vector $\mathbf{w} = \mathbf{h}^*$ , which is the matched filter to the spatial channel vector. This maximises the output SNR to $\gamma_{\text{MRC}} = \sum_l \gamma_l$ .

Common Mistake: EGC Requires Phase Knowledge

Mistake:

Implementing EGC without estimating the channel phases, assuming that equal-gain means no channel knowledge is needed.

Correction:

Equal-gain combining co-phases the branches before adding them: $r_{\text{EGC}} = \sum_l e^{-j\angle h_l} r_l$ . Without accurate phase alignment, the signals would add destructively and EGC would perform worse than selection combining.

The "equal-gain" refers to equal amplitude weights (all branches weighted by 1), not equal or zero channel knowledge. EGC needs phase estimates but not amplitude estimates — it is simpler than MRC but more demanding than SC.

Comparison of Combining Techniques

Property	SC	EGC	MRC
Output SNR	$\max_l \gamma_l$	$(\sum \|h_l\|)^2 / (L N_0)$	$\sum \gamma_l$
Diversity order	$L$	$L$	$L$
SNR loss vs MRC (L=4)	2.8 dB	0.7 dB	0 dB (optimal)
CSI requirement	SNR only	Phase only	Full (amplitude + phase)
Complexity	Low	Medium	Medium
Chains processed	1	$L$	$L$

Why This Matters: MRC in RAKE Receivers

In CDMA systems (IS-95, UMTS), the RAKE receiver implements MRC across multipath components. Each "finger" of the RAKE despreads and demodulates one resolvable multipath tap, and the outputs are combined using MRC weights.

If the channel has $L_p$ resolvable paths (each separated by at least one chip duration $T_c = 1/W$ ), the RAKE receiver achieves diversity order $L_p$ — turning the multipath channel from an enemy (ISI) into an ally (diversity).

In modern OFDM systems (LTE, 5G NR), the RAKE is replaced by frequency-domain equalisation, but the underlying principle of multipath diversity remains the same.

See full treatment in Channel Estimation in OFDM

Maximal-Ratio Combining (MRC)

The optimal linear diversity combining technique that weights each branch by the conjugate of its channel gain. The output SNR equals the sum of the branch SNRs: $\gamma_{\text{MRC}} = \sum_l \gamma_l$ .

Selection Combining (SC)

A diversity combining technique that selects the branch with the highest instantaneous SNR and discards the rest. Simplest to implement but wastes energy on non-selected branches.

⚠️Engineering Note

MRC Performance with Imperfect Channel Estimation

The MRC analysis assumes perfect channel state information at the receiver (CSIR). In practice, the channel must be estimated from pilot symbols, introducing estimation error $\hat{h}_l = h_l + e_l$ where $e_l$ is the estimation noise.

With imperfect CSI, the MRC weights become $w_l = \hat{h}_l^*$ , and the output contains a residual self-interference term. The effective SNR degrades to approximately

$\gamma_{\text{eff}} \approx \frac{\gamma_{\text{MRC}}}{1 + \gamma_{\text{MRC}} / \gamma_{\text{est}}}$

where $\gamma_{\text{est}}$ is the channel estimation SNR (depends on the number of pilot symbols and pilot power).

At high operating SNR, the BER floor caused by estimation error can dominate. In 5G NR, demodulation reference signals (DM-RS) typically achieve $\gamma_{\text{est}} \approx 20$ – $30$ dB, which is sufficient for up to $L = 4$ MRC branches before the estimation error becomes the bottleneck.

Practical Constraints

•
Pilot overhead scales linearly with $L$ branches (each branch needs channel estimation)
•
High-mobility scenarios degrade estimation quality due to channel ageing
•
In mm-wave bands, phase noise further degrades coherent combining

Diversity Combining Architectures — Block diagrams of the three classical diversity combining techniques: selection combining (top), equal-gain combining (middle), and maximal-ratio combining (bottom). Each receives $L$ independently faded copies $r_1, \ldots, r_L$ and produces a combined output. SC selects the best branch; EGC co-phases and adds; MRC weights by $h_l^*$ before adding.

Array Gain

The increase in average output SNR due to coherent combining of signals from multiple antennas. For MRC with $L$ branches, the array gain is $L$ (i.e., $10\log_{10} L$ dB). Array gain exists even without fading; diversity gain requires fading.

Receive Diversity

Definition: Selection Combining (SC)

Definition: Equal-Gain Combining (EGC)

Definition: Maximal-Ratio Combining (MRC)

Theorem: MRC Maximises Output SNR

Output SNR expression

Cauchy-Schwarz bound

Optimal weights

Theorem: MRC BER for BPSK in Rayleigh Fading

Conditional BER

Averaging over Erlang distribution

High-SNR approximation

MRC Combining Procedure

Example: MRC with Two Branches

Branch SNRs

MRC output SNR

Comparison with SC

Example: Comparing SC, EGC, and MRC at L = 4

MRC average output SNR

SC average output SNR

EGC average output SNR

SC vs EGC vs MRC Comparison

Parameters

BER with MRC

Parameters

Adding Diversity Branches

Parameters

Quick Check

Common Mistake: EGC Requires Phase Knowledge

Comparison of Combining Techniques

Why This Matters: MRC in RAKE Receivers

Maximal-Ratio Combining (MRC)

Selection Combining (SC)

MRC Performance with Imperfect Channel Estimation

Diversity Combining Architectures

Array Gain

Definition:
Selection Combining (SC)

Definition:
Equal-Gain Combining (EGC)

Definition:
Maximal-Ratio Combining (MRC)