RLL 1,7 vs RLL 2,7 Encoding Differences

Run-Length Limited (RLL) encoding is a modulation technique used by magnetic disk controllers to store binary data efficiently and reliably on physical media. The two prominent variants, RLL (1,7) and RLL (2,7), differ fundamentally in their code rates, minimum and maximum run lengths of zeros, data density, and timing jitter tolerance. While RLL (2,7) achieves a higher theoretical data density by enforcing larger gaps between magnetic flux transitions, RLL (1,7) offers a wider detection window that provides superior tolerance to timing jitter, making it widely adopted in later high-density drive technologies.

Understanding RLL (d,k) Notation

In RLL \((d,k)\) schemes, user data bits are translated into channel bits (code bits) that dictate where magnetic flux transitions occur:

• \(d\) (Minimum constraint): The minimum number of consecutive 0s allowed between consecutive 1s. A higher \(d\) value spaces magnetic transitions further apart, reducing inter-symbol interference (pulse crowding).
• \(k\) (Maximum constraint): The maximum number of consecutive 0s allowed between consecutive 1s. This ensures frequent enough transitions for the drive’s phase-locked loop (PLL) clock to stay synchronized with the data stream.

RLL (1,7) Encoding

RLL (1,7) uses a code rate of 2/3, meaning 2 input data bits are mapped to 3 channel bits.

• Run-length constraints: \(d = 1\), \(k = 7\) (at least one 0 and at most seven 0s between 1s).
• Detection Window (\(T_w\)): \(T_w = \frac{2}{3} T \approx 0.67T\), where \(T\) is the user bit period.
• Minimum Transition Spacing (\(T_{\text{min}}\)): \((1 + d) \times T_w = (1 + 1) \times \frac{2}{3}T = 1.33T\).
• Maximum Transition Spacing (\(T_{\text{max}}\)): \((1 + k) \times T_w = (1 + 7) \times \frac{2}{3}T = 5.33T\).
• Density Ratio (\(DR\)): \((1 + d) \times \text{Rate} = 2 \times \frac{2}{3} \approx 1.33\).

RLL (2,7) Encoding

RLL (2,7) uses a code rate of 1/2, meaning 1 input data bit is mapped to 2 channel bits.

• Run-length constraints: \(d = 2\), \(k = 7\) (at least two 0s and at most seven 0s between 1s).
• Detection Window (\(T_w\)): \(T_w = \frac{1}{2} T = 0.50T\).
• Minimum Transition Spacing (\(T_{\text{min}}\)): \((1 + d) \times T_w = (2 + 1) \times \frac{1}{2}T = 1.50T\).
• Maximum Transition Spacing (\(T_{\text{max}}\)): \((1 + k) \times T_w = (1 + 7) \times \frac{1}{2}T = 4.00T\).
• Density Ratio (\(DR\)): \((1 + d) \times \text{Rate} = 3 \times \frac{1}{2} = 1.50\).

Key Technical Comparisons

1. Recording Density

RLL (2,7) provides a density ratio of 1.50, representing a 50% capacity improvement over standard Non-Return-to-Zero (NRZ) recording and an advantage over RLL (1,7)’s 1.33 ratio. Because \(T_{\text{min}}\) is \(1.50T\) in RLL (2,7) versus \(1.33T\) in RLL (1,7), magnetic transitions can be physically placed closer together on the disk surface for the same degree of pulse crowding.

2. Detection Window and Jitter Margin

The detection window (\(T_w\)) defines the timing margin the read channel has to resolve a transition. * RLL (1,7) has a window of \(0.67T\) (33% larger than RLL 2,7). * RLL (2,7) has a window of \(0.50T\).

Because RLL (1,7) has a broader detection window, it is less vulnerable to phase jitter, peak shift, and noise, allowing disk controllers to read data accurately at higher channel frequencies.

3. Maximum-to-Minimum Transition Ratio (\(T_{\text{max}} / T_{\text{min}}\))

• RLL (1,7): \(\frac{5.33T}{1.33T} = 4.0\)
• RLL (2,7): \(\frac{4.00T}{1.50T} \approx 2.67\)

A smaller ratio represents a narrower operational frequency range for the analog read/write circuits. RLL (2,7) has a tighter ratio, making automatic gain control (AGC) and equalizer design simpler in purely analog peak-detection channels.

Conclusion

RLL (2,7) was heavily used in early high-capacity hard drives because its \(1.50\) density ratio maximized capacity using simpler peak-detection read heads. As drive technology evolved toward Partial-Response Maximum-Likelihood (PRML) channels, RLL (1,7) became the preferred standard because its larger detection window (\(0.67T\)) and higher code rate (\(2/3\)) provided better noise tolerance and higher overall throughput at elevated recording frequencies.