BEC Channel

The Binary Erasure Channel (BEC) is a discrete memoryless channel where each transmitted bit is either received correctly or erased (declared unknown) with probability epsilon. Unlike the BSC which corrupts bits, the BEC never delivers incorrect bits: the receiver always knows exactly which bits were erased. The BEC is the correct model for packet networks, optical burst switches, and scenarios where loss rather than corruption is the dominant impairment.

Core Concept Explanation

The Binary Erasure Channel has a three-symbol output alphabet: {0, e, 1} where e denotes erasure. The transition probabilities are P(Y=0|X=0) = 1-epsilon, P(Y=e|X=0) = epsilon, P(Y=1|X=0) = 0 and similarly P(Y=1|X=1) = 1-epsilon, P(Y=e|X=1) = epsilon, P(Y=0|X=1) = 0.

The key distinction from the BSC is that in the BEC, there are no bit errors: when a bit arrives, it is always correct. The receiver knows which positions were erased and must use coding redundancy to recover the lost bits. This is fundamentally different from the BSC where the receiver cannot distinguish a corrupted bit from a correct one without additional information.

The erasure probability epsilon characterizes channel quality. When epsilon = 0, every bit is received: noiseless channel with C = 1. When epsilon = 1, every bit is erased: useless channel with C = 0. Unlike the BSC which has symmetric capacity about p=0.5, the BEC capacity decreases monotonically with epsilon from 1 to 0 over [0,1].

Mathematical Expression

The capacity of the BEC is derived cleanly using I(X;Y) = H(Y) - H(Y|X). With uniform input:

H(Y|X) = H(epsilon) — this is the binary entropy of the erasure probability because given X, Y is either correct (prob 1-epsilon) or erased (prob epsilon), making it a binary uncertainty.

H(Y): The output alphabet is {0, e, 1}. P(Y=0) = 0.5*(1-epsilon), P(Y=e) = epsilon, P(Y=1) = 0.5*(1-epsilon). Therefore H(Y) = -2*(0.5*(1-epsilon)) * log2(0.5*(1-epsilon)) - epsilon * log2(epsilon).

After simplification: H(Y) = (1-epsilon) + H(epsilon) where the (1-epsilon) term comes from the uniform distribution of non-erased outputs. Substituting:

C = H(Y) - H(Y|X) = [(1-epsilon) + H(epsilon)] - H(epsilon) = 1 - epsilon bits per channel use

This remarkably simple result C = 1 - epsilon says that the fraction of channel uses that deliver useful information equals the fraction of non-erased bits, which is the intuitive answer. The BEC is unique in that its capacity equals the fraction of received bits regardless of coding.

Practical Understanding

The BEC is the natural model for packet erasure networks, where each IP packet either arrives intact (receiver can check with CRC) or is dropped by a router. The capacity C = 1 - epsilon gives the maximum fraction of the transmitted data that can be reliably recovered. Forward error correction codes for packet networks are designed to approach this limit.

The BEC also has deep theoretical importance: it was the first channel for which LDPC codes were proven to achieve capacity using belief propagation decoding. Richardson and Urbanke's density evolution analysis was developed specifically on the BEC before being extended to other channels. The BEC thus served as a proving ground for modern iterative coding theory.

In wireless communications, the BEC arises when deep fading causes entire symbols to be unreliable. The receiver marks these positions as erasures rather than passing uncertain hard decisions to the decoder. Hybrid ARQ protocols use erasure correction as one component of their retransmission strategy.

Solved Numerical Example

BEC capacity calculation is more direct than BSC: since C = 1 - epsilon, only the erasure probability is needed. GATE problems on BEC often also test comparison with BSC capacity and the derivation of H(Y).

Given:
BEC with erasure probability epsilon = 0.25
Uniform input P(X=0) = P(X=1) = 0.5

Why this formula applies:
BEC capacity with uniform input = 1 - epsilon (shown by H(Y) - H(Y|X) derivation).

Formula:
C = 1 - epsilon

Substitution:
C = 1 - 0.25 = 0.75

Verification using I(X;Y) = H(Y) - H(Y|X):
P(Y=0) = P(Y=1) = 0.5*(1-0.25) = 0.375
P(Y=e) = 0.25
H(Y) = -2*0.375*log2(0.375) - 0.25*log2(0.25)
      = -0.75*log2(0.375) - 0.25*(-2)
      = -0.75*(-1.415) + 0.5 = 1.061 + 0.5 = 1.561 bits
H(Y|X) = H(epsilon) = H(0.25) = -0.25*log2(0.25) - 0.75*log2(0.75)
        = 0.5 + 0.75*0.415 = 0.5 + 0.311 = 0.811 bits
I(X;Y) = 1.561 - 0.811 = 0.75 bits

Final Answer: C = 0.75 bits per channel use (both methods agree)

Exam Tip: BEC capacity C = 1 - epsilon is linear in epsilon, while BSC capacity C = 1 - H(p) is a concave function of p. For the same numerical value (p = epsilon = 0.1), BEC has higher capacity (0.9) than BSC (0.531). This is because erasures are less harmful than bit flips: the receiver knows exactly where the uncertainty lies in the BEC.

• BEC output alphabet is {0, e, 1} where e denotes erasure. No bit flips occur: received bits are always correct.
• C_BEC = 1 - epsilon. This is a linear function of erasure probability, derived from H(Y) - H(Y|X) = (1-epsilon) + H(epsilon) - H(epsilon).
• For the same parameter value, BEC capacity exceeds BSC capacity because the receiver has more information (it knows which positions were erased).
• LDPC codes were first proven to achieve BEC capacity using belief propagation, making the BEC the foundational channel for modern iterative coding theory.
• The BEC models packet erasure networks, ARQ retransmission systems, and fading channels where deep fades make symbols unreliable.

Quick Revision

• BEC parameter: epsilon = erasure probability. Output alphabet: {0, e, 1}. No crossover: received bits are always correct.
• C_BEC = 1 - epsilon. Linear, simple, and intuitive: fraction of non-erased bits equals capacity.
• BEC capacity derivation: C = H(Y) - H(Y|X) = [(1-epsilon) + H(epsilon)] - H(epsilon) = 1 - epsilon.
• Comparison: for same parameter value, C_BEC = 1 - epsilon > C_BSC = 1 - H(epsilon). Since H(epsilon) > epsilon for 0 < epsilon < 1, BEC always has higher capacity.
• epsilon = 0: C = 1 (noiseless). epsilon = 1: C = 0 (every bit erased). No symmetry-based jump at epsilon = 1.
• Exam trap: BEC capacity is NOT 1 - H(epsilon). That would be BSC capacity. BEC is 1 - epsilon (no binary entropy function needed).
• LDPC codes achieve BEC capacity under belief propagation decoding. This was the breakthrough that launched modern iterative coding theory.