BSC Channel

The Binary Symmetric Channel (BSC) is the simplest and most widely analyzed discrete memoryless channel model. It takes a binary input, flips the bit with a fixed crossover probability p, and delivers the (possibly flipped) bit as output. The BSC forms the foundation for understanding how noise degrades binary communication and for computing channel capacity in closed form. It appears in nearly every GATE information theory section.

Core Concept Explanation

The Binary Symmetric Channel is characterized by one parameter: the crossover probability p, where 0 < p < 0.5. With probability 1-p, the input bit is transmitted correctly. With probability p, it is flipped (0 becomes 1 or 1 becomes 0). The channel is called symmetric because the crossover probability is the same for both input values.

The channel transition matrix is fully described by: P(Y=0|X=0) = P(Y=1|X=1) = 1-p and P(Y=1|X=0) = P(Y=0|X=1) = p. This matrix has a symmetric structure that makes capacity analysis particularly clean. The symmetry ensures that the capacity-achieving input distribution is always the uniform distribution P(X=0) = P(X=1) = 0.5.

When p = 0, the channel is noiseless and C = 1 bit per use. When p = 0.5, every input bit is equally likely to be correct or flipped, making the output completely independent of the input, so C = 0. When p = 1, every bit is always flipped, which is also a perfect channel (just invert the output), so C = 1. Therefore, the BSC is useless only when p = 0.5, not when p = 1.

Mathematical Expression

The capacity of the BSC is derived as the maximum of I(X;Y) over input distributions. Due to channel symmetry, the maximum is achieved by the uniform input distribution. The derivation proceeds as:

I(X;Y) = H(Y) - H(Y|X)

H(Y|X) = H(p) because given the input, the output is determined entirely by whether a flip occurred, which has probability p. With uniform input, H(Y) = 1 (output is also uniform). Therefore:

C = 1 - H(p) bits per channel use

where H(p) = -p log2(p) - (1-p) log2(1-p) is the binary entropy function. H(p) ranges from 0 (when p=0 or p=1) to 1 (when p=0.5). Therefore C ranges from 0 (when p=0.5) to 1 (when p=0 or p=1). The capacity is symmetric about p=0.5.

Practical Understanding

The BSC is a good model for binary channels with random independent bit errors, such as hard-decision decoded wireless links, optical fiber with threshold detection, and memory cells with random read errors. In hard-decision decoding of BPSK over an AWGN channel, the resulting bit-flip channel is approximately a BSC with p = Q(sqrt(2 Eb/N0)).

An important practical insight is that for the BSC, moving to p > 0.5 is unusual in practice but theoretically equivalent to a channel with crossover probability 1-p by flipping all output bits. This means the BSC is effectively only analyzed for p in [0, 0.5], with capacity being a monotonically decreasing function of p in that range.

Solved Numerical Example

BSC capacity calculation requires computing the binary entropy function H(p) and subtracting it from 1. The numerical values of H(p) for common crossover probabilities are worth memorizing: H(0.1) = 0.469, H(0.11) = 0.5, H(0.2) = 0.722, H(0.25) = 0.811.

Given:
BSC with crossover probability p = 0.1

Why this formula applies:
BSC capacity with uniform input achieves maximum I(X;Y) = 1 - H(p).

Formula:
C = 1 - H(p)
H(p) = -p log2(p) - (1-p) log2(1-p)

Substitution:
H(0.1) = -0.1 * log2(0.1) - 0.9 * log2(0.9)
log2(0.1) = log10(0.1)/log10(2) = -1/0.301 = -3.3219
log2(0.9) = log10(0.9)/0.301 = -0.0458/0.301 = -0.1520

Calculation:
H(0.1) = -0.1*(-3.3219) - 0.9*(-0.1520)
       = 0.3322 + 0.1368 = 0.4690 bits
C = 1 - 0.4690

Final Answer: C = 0.531 bits per channel use

Exam Tip: For BSC, C = 1 - H(p) is valid ONLY because the channel is symmetric and the optimal input is uniform. For non-symmetric binary channels (like Z-channel or S-channel), the capacity-achieving input is NOT uniform and you must optimize P(X=1) numerically or using calculus. GATE sometimes tests whether students blindly apply 1 - H(p) to non-symmetric channels.

• C = 1 - H(p) where H(p) is the binary entropy function. This closed-form result applies because the channel is symmetric.
• The capacity-achieving input distribution is always uniform (P(X=0) = P(X=1) = 0.5) for any crossover probability p.
• When p = 0.5, H(p) = 1 and C = 0. The channel is completely useless, not because it corrupts all bits but because it does so randomly.
• When p = 1, every bit is always flipped, which is still a perfect channel: just invert all outputs. C = 1 - H(1) = 1 - 0 = 1 bit.
• For p in [0, 0.5], capacity decreases monotonically from 1 to 0. For p in [0.5, 1], capacity increases symmetrically back to 1.

Quick Revision

• BSC is defined by crossover probability p: both 0 and 1 flip with same probability p.
• C_BSC = 1 - H(p) where H(p) = -p log2(p) - (1-p) log2(1-p).
• Capacity-achieving input is always uniform for BSC. This is due to the channel's output symmetry.
• H(p) = H(1-p), so C(p) = C(1-p). Capacity is symmetric about p = 0.5.
• p = 0: C = 1 (noiseless). p = 0.5: C = 0 (useless). p = 1: C = 1 (always flip, perfect after inversion).
• Exam trap: For Z-channel or asymmetric binary channels, C = 1 - H(p) does not apply. Symmetry is the prerequisite for this formula.
• Common values: H(0.1) = 0.469, H(0.2) = 0.722, H(0.25) = 0.811, H(0.5) = 1.