Ferkans — Interactive Telecom Tutor

Fading Changes Everything

In AWGN, BER decays exponentially with SNR: $P_b \sim e^{-c \cdot E_b/N_0}$ . In fading, the channel gain $|h|^2$ is random, so the instantaneous SNR $\gamma = |h|^2 E_s/N_0$ fluctuates. Even a single deep fade ( $|h| \approx 0$ ) causes a burst of errors. The result is a dramatic degradation: BER decays only algebraically $P_b \sim \bar{\gamma}^{-d}$ at high SNR, where $d$ is the diversity order. Understanding this transition and how to combat it (diversity, coding) is central to wireless system design.

Definition:
Coherent Detection

Coherent detection assumes the receiver has perfect knowledge of the channel state information (CSI), i.e., the complex channel gain $h$ is known. The received signal is

$r = h\, s_m + w$

and the ML detector with known $h$ selects

$\hat{m} = \arg\min_m |r - h\, s_m|^2 = \arg\max_m \operatorname{Re}\{h^* r\, s_m^*\} - \tfrac{1}{2}|h|^2|s_m|^2$

For equal-energy signals, this simplifies to

$\hat{m} = \arg\max_m \operatorname{Re}\{h^* r\, s_m^*\}$

The instantaneous BER (conditioned on $h$ ) is the AWGN BER evaluated at the instantaneous SNR $\gamma = |h|^2 E_s / N_0$ .

The average BER is obtained by averaging over the fading distribution:

$\bar{P}_b = \int_0^\infty P_b(\gamma)\, p_\gamma(\gamma)\, d\gamma$

Coherent detection requires channel estimation (Section 9.5). Estimation errors degrade performance, and the analysis assumes perfect CSI unless stated otherwise.

Theorem: Average BER for BPSK over Rayleigh Fading

For BPSK with coherent detection over a Rayleigh fading channel ( $|h|^2 \sim \text{Exp}(1)$ , so $\gamma \sim \text{Exp}(\bar{\gamma})$ ), the average BER is

$\bar{P}_b = \frac{1}{2}\left(1 - \sqrt{\frac{\bar{\gamma}}{1 + \bar{\gamma}}}\right)$

At high SNR ( $\bar{\gamma} \gg 1$ ):

$\bar{P}_b \approx \frac{1}{4\bar{\gamma}}$

This shows first-order diversity ( $d = 1$ ): the BER decays as $1/\bar{\gamma}$ , i.e., only 10 dB/decade, compared to the exponential decay in AWGN.

In Rayleigh fading, there is a significant probability of a deep fade ( $\gamma \approx 0$ ), where detection is unreliable. These deep fades dominate the average BER even at high average SNR, changing the decay from exponential to algebraic.

Proof

Conditional BER

For BPSK in AWGN: $P_b(\gamma) = Q(\sqrt{2\gamma})$ .

Craig formula substitution

Using Craig's formula:

$P_b(\gamma) = \frac{1}{\pi}\int_0^{\pi/2} \exp\!\left(-\frac{\gamma}{\sin^2\phi}\right) d\phi$

Average over Rayleigh fading

With $p_\gamma(\gamma) = \frac{1}{\bar{\gamma}} e^{-\gamma/\bar{\gamma}}$ :

$\bar{P}_b = \frac{1}{\pi}\int_0^{\pi/2} \underbrace{\int_0^\infty e^{-\gamma/\sin^2\phi} \cdot \frac{1}{\bar{\gamma}} e^{-\gamma/\bar{\gamma}}\, d\gamma}_{M_\gamma(-1/\sin^2\phi)}\, d\phi$

The inner integral is the MGF of the exponential distribution:

$M_\gamma(s) = \frac{1}{1 - s\bar{\gamma}} \implies M_\gamma\!\left(-\frac{1}{\sin^2\phi}\right) = \frac{\sin^2\phi}{\sin^2\phi + \bar{\gamma}}$

Closed-form evaluation

$\bar{P}_b = \frac{1}{\pi}\int_0^{\pi/2} \frac{\sin^2\phi}{\sin^2\phi + \bar{\gamma}}\, d\phi = \frac{1}{2}\left(1 - \sqrt{\frac{\bar{\gamma}}{1+\bar{\gamma}}}\right)$ $The last step uses the standard integral$ \frac{1}{\pi}\int_0^{\pi/2} \frac{\sin^2\phi}{\sin^2\phi + c}, d\phi = \frac{1}{2}(1 - \sqrt{c/(1+c)}) $.$ \blacksquare$

,

Definition:
MGF Approach for Averaging BER Over Fading

The moment generating function (MGF) approach exploits Craig's formula to express the average BER as

$\bar{P}_b = \frac{1}{\pi}\int_0^{\pi/2} M_\gamma\!\left(-\frac{g}{\sin^2\phi}\right) d\phi$

where $g$ is a modulation-dependent constant ( $g = 1$ for BPSK, $g = 1/2$ for orthogonal BFSK) and $M_\gamma(s) = E[e^{s\gamma}]$ is the MGF of the instantaneous SNR.

The key advantage is that the MGF is known in closed form for all standard fading distributions:

Rayleigh: $M_\gamma(s) = (1 - s\bar{\gamma})^{-1}$
Ricean ( $K$ factor): $M_\gamma(s) = \frac{1+K}{1+K-s\bar{\gamma}} \exp\!\left(\frac{Ks\bar{\gamma}}{1+K-s\bar{\gamma}}\right)$
Nakagami- $m$ : $M_\gamma(s) = (1 - s\bar{\gamma}/m)^{-m}$

The remaining integral over $\phi$ has finite limits ( $0$ to $\pi/2$ ) and can be evaluated analytically or by simple numerical quadrature.

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Theorem: Diversity Order

The diversity order of a communication system is defined as

$d = -\lim_{\bar{\gamma} \to \infty} \frac{\log \bar{P}_e}{\log \bar{\gamma}}$

At high SNR, the average error probability behaves as

$\bar{P}_e \approx (G_c \cdot \bar{\gamma})^{-d}$

where $G_c$ is the coding gain (a constant that shifts the BER curve horizontally).

For BPSK over:

AWGN: $d = \infty$ (exponential decay, faster than any polynomial)
Rayleigh fading (no diversity): $d = 1$
Rayleigh with $L$ -branch MRC diversity: $d = L$
Ricean fading: $d = 1$ (same as Rayleigh at high SNR)
Nakagami- $m$ : $d = m$

The diversity order counts the number of independent copies of the signal available to the receiver. Each independent copy reduces the probability of a simultaneous deep fade, changing the BER slope by one decade per 10 dB of SNR.

Proof

Rayleigh example

From Theorem 9.3: $\bar{P}_b \approx 1/(4\bar{\gamma})$ at high SNR.

$d = -\lim_{\bar{\gamma}\to\infty} \frac{\log(1/(4\bar{\gamma}))}{\log\bar{\gamma}} = -\lim_{\bar{\gamma}\to\infty} \frac{-\log 4 - \log\bar{\gamma}}{\log\bar{\gamma}} = 1$ .

MRC diversity

With $L$ -branch MRC, $\gamma_{\text{total}} = \sum_{\ell=1}^L \gamma_\ell$ where $\gamma_\ell$ are i.i.d. exponential. Then $\gamma_{\text{total}} \sim \text{Gamma}(L, \bar{\gamma})$ , and at high SNR:

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

giving diversity order $d = L$ . $\blacksquare$

Example: BER of BPSK in Rayleigh vs AWGN

Compare the BER of BPSK at $E_b/N_0 = 20$ dB in: (a) AWGN, (b) Rayleigh fading, (c) Rayleigh with $L = 2$ MRC diversity. Quantify the SNR penalty of fading.

Solution

AWGN

$P_b = Q(\sqrt{2 \times 100}) = Q(14.14) \approx 10^{-45}$

Essentially error-free.

Rayleigh fading (no diversity)

$\bar{P}_b = \frac{1}{2}\left(1 - \sqrt{\frac{100}{101}}\right) \approx \frac{1}{4 \times 100} = 2.5 \times 10^{-3}$

At $E_b/N_0 = 20$ dB, Rayleigh fading gives BER $\sim 10^{-3}$ while AWGN gives BER $\sim 10^{-45}$ . A catastrophic gap.

Rayleigh with $L = 2$ MRC

$\bar{P}_b \approx \binom{3}{2} \frac{1}{(4 \times 100)^2} = \frac{3}{160000} \approx 1.9 \times 10^{-5}$

Two-branch MRC reduces BER from $10^{-3}$ to $10^{-5}$ . The improvement is dramatic: each additional diversity branch steepens the BER slope by 10 dB/decade.

SNR penalty

To achieve $P_b = 10^{-5}$ :

AWGN: $E_b/N_0 \approx 9.6$ dB
Rayleigh: $\bar{\gamma} = 1/(4 \times 10^{-5}) = 25000 = 44$ dB
Rayleigh + MRC-2: $\bar{\gamma} \approx 24$ dB

Rayleigh fading penalty: $44 - 9.6 = 34.4$ dB. MRC-2 recovers about 20 dB of this penalty. $\blacksquare$

BER in Fading vs AWGN

Compare BER curves in AWGN and fading channels. The AWGN curve drops exponentially (steep waterfall), while the Rayleigh fading curve drops as $1/\bar{\gamma}^d$ (gentle algebraic slope). Increase the diversity order $L$ to steepen the fading BER curve. Switch to Ricean fading to see how a line-of-sight component improves performance.

Parameters

Modulation

Diversity order

L

1

Fading type

Ricean

K

factor (dB)6

AWGN vs Fading BER Slopes

A cinematic comparison of BER curves on a log scale. The AWGN curve drops steeply (exponential decay), while the Rayleigh fading curve follows a gentle

1/\bar{\gamma}

slope. Then diversity branches (

L = 2, 4

) are added, progressively steepening the fading BER curves.

Each additional diversity branch steepens the BER slope by one decade per 10 dB of SNR increase.

Monte Carlo BER Convergence

Watch a Monte Carlo BER simulation converge as the number of transmitted symbols increases. The estimated BER (with confidence interval) converges to the theoretical value. Observe that reliable estimation of $P_b = 10^{-k}$ requires approximately $100/P_b = 10^{k+2}$ symbols.

Parameters

Modulation

SNR (dB)10

Channel

Quick Check

A system with 4-branch MRC diversity over Rayleigh fading has diversity order $d = 4$ . If the average SNR increases by 10 dB, by approximately how many decades does the BER decrease?

4 decades (BER improves by a factor of $10^4$ )

1 decade

10 decades

Infinite (error-free at high SNR)

Correction:

4 decades (BER improves by a factor of

10^4

)

With diversity order $d = 4$ , BER $\sim \bar{\gamma}^{-4}$ . A 10 dB (factor 10) increase in $\bar{\gamma}$ gives BER reduction of $10^4$ (4 decades). This is the defining property of diversity order: each 10 dB of SNR yields $d$ decades of BER improvement.

Common Mistake: Error Floor in Fading Without Diversity

Mistake:

Expecting that increasing transmit power alone will achieve arbitrarily low BER in a Rayleigh fading channel without diversity.

Correction:

Without diversity ( $d = 1$ ), $\bar{P}_b \approx 1/(4\bar{\gamma})$ . Achieving $P_b = 10^{-6}$ requires $\bar{\gamma} = 250{,}000 = 54$ dB, which is impractical.

Diversity is essential in fading channels. With $L = 4$ MRC, the same $P_b = 10^{-6}$ requires only $\bar{\gamma} \approx 16$ dB — a 38 dB saving. Alternatively, coding provides time/frequency diversity, and MIMO provides spatial diversity.

The lesson: in fading channels, spending power on diversity is far more effective than simply increasing transmit power.

Non-Coherent Detection Trades 3 dB for No CSI

Non-coherent detection does not require knowledge of the channel phase, using only the signal envelope for detection. For orthogonal signaling (e.g., non-coherent BFSK), the BER in Rayleigh fading is

$\bar{P}_b = \frac{1}{2 + \bar{\gamma}}$

compared to $\bar{P}_b \approx 1/(4\bar{\gamma})$ for coherent BPSK. The penalty is approximately 3 dB (a factor of 2 in SNR).

Differentially coherent detection (DPSK, DQPSK) is intermediate: it uses the previous symbol as a phase reference, requiring no explicit channel estimation. The BER for DPSK in Rayleigh fading is

$\bar{P}_b = \frac{1}{2(1 + \bar{\gamma})}$

The 3 dB penalty is often acceptable in rapidly time-varying channels where channel estimation is unreliable.

Coherent vs Non-Coherent vs Differential Detection

Aspect	Coherent	Differential	Non-coherent
CSI required	Full (amplitude + phase)	Previous symbol phase	None
Example modulations	BPSK, QPSK, QAM	DPSK, DQPSK	Non-coherent FSK
BER (Rayleigh)	$\frac{1}{4\bar{\gamma}}$	$\frac{1}{2(1+\bar{\gamma})}$	$\frac{1}{2+\bar{\gamma}}$
SNR penalty vs coherent	Baseline	$\approx$ 1 dB (AWGN), 3 dB (fading)	$\approx$ 3 dB
Applicable to QAM?	Yes	Limited (DQPSK)	No (amplitude info lost)
Complexity	High (channel estimator)	Medium	Low

⚠️Engineering Note

Link Budget Fading Margins in Practice

The algebraic BER decay in fading ( $\bar{P}_b \sim \bar{\gamma}^{-d}$ ) has direct implications for link budget design. To achieve a target BER without diversity, the required fading margin — the extra SNR relative to AWGN — is enormous:

Target BER	AWGN $E_b/N_0$	Rayleigh ( $d=1$ )	Margin
$10^{-3}$	6.8 dB	24 dB	17.2 dB
$10^{-5}$	9.6 dB	44 dB	34.4 dB
$10^{-6}$	10.5 dB	54 dB	43.5 dB

In 5G NR and LTE, the standard approach is to use diversity (2-4 Rx antennas, transmit diversity via SFBC) to increase $d$ , combined with HARQ to handle residual errors. The link adaptation algorithm (AMC) selects MCS based on the effective post-diversity SNR, targeting a BLER of 10% for initial transmission (relying on HARQ retransmissions for reliability).

The 3GPP link budget methodology accounts for fading through the shadow fading margin (log-normal, typically 7-10 dB) and the fast fading margin (handled implicitly by HARQ and diversity).

Practical Constraints

•
Without diversity: 30-40 dB fading margin needed for BER < 1e-5
•
With 2-branch MRC: margin reduces by ~20 dB
•
5G NR target BLER: 10% for eMBB, 0.001% for URLLC

📋 Ref: 3GPP TR 38.901 (channel models), 3GPP TS 38.214 (link adaptation)

Key Takeaway

The core message of this section in three points:

Fading changes BER decay from exponential ( $e^{-c\bar{\gamma}}$ , AWGN) to algebraic ( $\bar{\gamma}^{-d}$ ), where $d$ is the diversity order. This is the most fundamental difference between AWGN and fading channels.
Diversity steepens the BER slope: each independent replica of the signal increases $d$ by 1, and $d$ additional decades of BER improvement are gained per 10 dB of SNR.
The MGF approach (Craig's formula + fading MGF) provides a unified analytical framework for computing average BER over any fading distribution, avoiding numerical integration of nested integrals.

Why This Matters: Spatial Diversity and MIMO Detection

The diversity order analysis in this section considers receive diversity (MRC) with $L$ independent branches. In practice, spatial diversity is achieved through MIMO antenna arrays. The MIMO book develops this in full depth:

Spatial diversity: Alamouti code, space-time block codes (Chapter 10 of this book provides an introduction)
MIMO detection: ML, ZF, MMSE, sphere decoding — extending the detection theory of Section 9.1 to vector channels
Diversity-multiplexing tradeoff (DMT): the fundamental trade-off between diversity gain and spatial multiplexing gain in MIMO systems (includes CommIT contributions by Caire et al.)
Massive MIMO: when $L \to \infty$ , the channel hardens and approaches AWGN behaviour — closing the gap between fading and AWGN performance analysed in this section

Diversity Order

The negative slope of the log-BER vs log-SNR curve at high SNR: $d = -\lim \log P_e / \log \bar{\gamma}$ . It equals the number of independent signal copies available to the receiver and determines the rate of BER improvement with increasing SNR.

Coherent Detection

A detection method that requires knowledge of the channel's complex gain (amplitude and phase) at the receiver. Coherent detection achieves the best performance but requires accurate channel estimation, which may be impractical in rapidly varying channels.

Detection in Fading Channels

Fading Changes Everything

Definition: Coherent Detection

Theorem: Average BER for BPSK over Rayleigh Fading

Conditional BER

Craig formula substitution

Average over Rayleigh fading

Closed-form evaluation

Definition: MGF Approach for Averaging BER Over Fading

Theorem: Diversity Order

Rayleigh example

MRC diversity

Example: BER of BPSK in Rayleigh vs AWGN

AWGN

Rayleigh fading (no diversity)

Rayleigh with $L = 2$ MRC

SNR penalty

BER in Fading vs AWGN

Parameters

AWGN vs Fading BER Slopes

Monte Carlo BER Convergence

Parameters

Quick Check

Common Mistake: Error Floor in Fading Without Diversity

Non-Coherent Detection Trades 3 dB for No CSI

Coherent vs Non-Coherent vs Differential Detection

Link Budget Fading Margins in Practice

Key Takeaway

Why This Matters: Spatial Diversity and MIMO Detection

Diversity Order

Coherent Detection

Definition:
Coherent Detection

Definition:
MGF Approach for Averaging BER Over Fading