Understanding Error Types in Data Transmission

In the previous page, we explored single-bit errors—isolated corruptions where one bit flips independently of its neighbors. While elegant mathematical models like the Binary Symmetric Channel assume this independence, reality often behaves differently.

When a lightning strike induces voltage on a communication cable, it doesn't flip just one bit and politely stop. When a wireless signal fades due to multipath interference, multiple symbols are affected simultaneously. When a scratch appears on a disk surface, it corrupts a contiguous region of data.

These are burst errors—and understanding them is crucial because they occur far more frequently than isolated single-bit errors in many communication scenarios, yet require fundamentally different detection and correction strategies.

A burst error (also called a cluster error or error burst) occurs when multiple consecutive or nearly-consecutive bits are corrupted. The defining characteristic is the spatial or temporal correlation between errors—they cluster together rather than occurring independently.

Formal Definition:

A burst of length L is defined as a contiguous sequence of L bits where:

1. The first bit is erroneous
2. The last bit is erroneous
3. Any bits in between may or may not be erroneous

Importantly, the burst length is measured from the first to the last error, not by counting only the corrupted bits. Between the first and last corrupted bits, there may be correct bits mixed with incorrect ones.

Alternative Definition (Guard Band):

Some sources define burst errors using a guard band approach: two bursts are considered separate if at least g correct bits appear between them. This definition is useful for analyzing burst error statistics in real channels.

Why the Definition Matters:

The burst length definition is critical for error detection and correction:

CRC Capability: A CRC with an n-bit check value can detect:

• All bursts of length ≤ n
• Most (but not all) bursts of length > n

If the burst length is measured incorrectly, protection guarantees become invalid.

Interleaving Design: Burst error correction often uses interleaving, where the interleaving depth must exceed the maximum expected burst length. Underestimating burst length leads to uncorrectable errors.

Burst errors arise from physical phenomena that cause sustained signal degradation—disturbances that persist longer than the transmission of a single bit. Understanding these causes reveals why burst errors dominate in certain environments.

The Temporal Correlation Principle:

When an interfering phenomenon has a duration τ and the bit period is T, the minimum burst length L is approximately:

$$L \approx \lceil \tau / T \rceil$$

This explains why high-speed links tend to have longer burst lengths for the same physical disturbance—more bits fit into the same interference window.

The Binary Symmetric Channel model, which assumes independent errors, fails to capture burst error behavior. More sophisticated models are required to characterize channels where errors cluster.

The Gilbert-Elliott Model:

The most widely used burst error model is the Gilbert-Elliott channel, which uses a two-state Markov model:

State G (Good): Low error probability (typically 0 or very small) State B (Bad): High error probability (typically 0.5 or higher)

The channel transitions between states with probabilities:

• p: Probability of transitioning from G to B (entering a burst)
• r: Probability of transitioning from B to G (leaving a burst)

This elegantly captures the correlated nature of burst errors: once in the bad state, errors are likely, and the channel tends to stay in that state for some duration.

Key Parameters:

Average burst length (when in bad state): $$\bar{L} = \frac{1}{r}$$

Average gap between bursts (when in good state): $$\bar{G} = \frac{1}{p}$$

Fraction of time in bad state: $$\pi_B = \frac{p}{p + r}$$

Overall bit error probability: $$P_e = \pi_G \cdot P_{e|G} + \pi_B \cdot P_{e|B}$$

Example Parameterization:

For a channel with average burst length of 10 bits and bursts occurring every 1000 bits on average:

• r = 0.1 (average burst length = 10)
• p = 0.001 (average gap = 1000)
• π_B = 0.001 / (0.001 + 0.1) ≈ 0.01 (1% of time in bad state)

Extended Models:

Fritchman Model: Extends Gilbert-Elliott to multiple good states and bad states, enabling more precise modeling of channels with variable error-free gaps between bursts.

Hidden Markov Models (HMM): For channels with complex burst characteristics, HMMs can be trained on observed error sequences to create empirically-accurate models without assuming simple state structures.

Burst Length Distribution:

In the basic Gilbert model, burst lengths follow a geometric distribution: $$P(L = k) = (1-r)^{k-1} \cdot r$$

This means short bursts are most common, with probability decreasing exponentially for longer bursts. Real channels often exhibit heavier tails, requiring extended models.

Burst errors pose unique challenges for error detection. Schemes optimized for independent single-bit errors can fail dramatically when errors cluster.

Parity Failure:

A single parity bit detects any odd number of errors. However, a burst that flips an even number of bits produces correct parity despite significant corruption:

• Transmitted: 10110110 + parity 0 = 101101100
• Burst flips bits 3-6: 101011000
• Receiver calculates parity: even (same as transmitted)
• Result: 4-bit corruption goes undetected

Statistically, a burst of length L has approximately 50% probability of containing an even number of errors, making simple parity useless against bursts.

Hamming Code Vulnerability:

Standard Hamming codes are designed to correct single-bit errors and detect double-bit errors. Against bursts, they fare poorly:

• A 2-bit burst may be incorrectly "corrected" to wrong data
• A 3+-bit burst may be completely missed
• The carefully designed check bit positions assume independent errors

Checksum Weakness:

Internet checksums (1's complement addition) can miss certain burst patterns:

• Two errors that cancel in addition (e.g., +256 and -256 in different bytes)
• Burst errors that swap bytes or flip complementary bit patterns

Studies show approximately 1 in 65,536 arbitrary error patterns evades 16-bit checksum detection—acceptable for TCP/IP with underlying CRC protection, but dangerous as a sole defense.

When burst errors exceed simple detection capabilities, specialized techniques transform the problem into one that simpler codes can handle.

Interleaving: Spreading Bursts Across Codewords

The key insight of interleaving is that if we spread data across multiple codewords before transmission, a burst error affecting consecutive transmitted bits will be distributed across multiple codewords at the receiver—converting one long burst into multiple short errors that simpler codes can correct.

Block Interleaving Example:

Consider 4 codewords of 8 bits each. Instead of transmitting them sequentially:

• Normal order: C1(8 bits), C2(8 bits), C3(8 bits), C4(8 bits)

We interleave by transmitting the first bit of each codeword, then the second bit of each, etc.:

• Interleaved: C1[1], C2[1], C3[1], C4[1], C1[2], C2[2], ...

A 4-bit burst now affects one bit in each of four codewords rather than four consecutive bits in one codeword. If each codeword can correct single-bit errors, the burst is fully corrected.

Interleaving Parameters:

The interleaving depth d determines burst tolerance:

• Maximum correctable burst = d × (single-error correction capability)
• For Hamming codes correcting 1 error: depth d handles bursts up to d bits
• For Reed-Solomon correcting t errors: depth d handles bursts up to d × t symbols

Trade-offs:

• Deeper interleaving → longer latency (must buffer more data)
• Deeper interleaving → larger memory requirements
• Interleaving adds no redundancy overhead—same efficiency, better burst tolerance

Convolutional Interleaving:

Block interleaving has constant latency but requires large buffers. Convolutional (or helical) interleaving uses variable delays that average to a lower latency while achieving similar burst spreading. This is the preferred technique for streaming applications.

Understanding the fundamental differences between single-bit and burst errors is essential for selecting appropriate error control strategies.

Choosing the Right Strategy:

For channels dominated by single-bit errors:

• Hamming SEC-DED codes are efficient and effective
• Simple CRC + ARQ (retransmission) works well
• Low redundancy overhead is achievable

For channels with significant burst errors:

• Interleaving is essential—without it, even sophisticated codes fail
• Reed-Solomon codes excel (they correct symbol errors, making interleaving naturally effective)
• Higher redundancy is necessary, but reliability improves dramatically

For mixed channels:

• Concatenated codes combine inner codes for random errors with outer codes for bursts
• Modern iterative codes (Turbo, LDPC) with interleaving handle both
• Adaptive systems measure channel conditions and adjust protection dynamically

We have explored burst errors in depth—understanding how they differ from single-bit errors and why they require specialized treatment. Let's consolidate the key insights:

What's Next:

Having understood both single-bit and burst errors, we'll next investigate what causes these errors. The upcoming page on Error Sources examines the physical and environmental factors that introduce errors into communication systems—from fundamental physics to practical engineering challenges.