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RLL(2,7) and the Density Ratio
MFM doubled FM's density by noticing that FM wasted a transition on every bit. That was free money and it ran out. Getting more required giving something up, and the question was what.
Peter Franaszek at IBM answered it in 1972 by treating the problem as counting. A magnetic channel has a minimum spacing between transitions — write them closer and their fields overlap and both read back shifted and smaller. So the useful question is not how many BITS you can write per inch but how much DATA you can carry per transition. Forbid short runs and each transition covers more ground; the price is that you have forbidden sequences, so you need more channel bits per data bit.
Shannon had already worked out the exchange rate. The capacity of a run-length constraint is the log of the largest eigenvalue of its counting recurrence, and no code can beat it. RLL(2,7) — minimum run 2, maximum 7, rate one half — achieves a density ratio of 1.5 against MFM's 1.0, at 96.6 percent of the theoretical limit. Fifty percent more capacity on the same head, the same medium and the same transition spacing, bought entirely with arithmetic. It shipped in the IBM 3370 in 1979 and then in essentially every hard disk of the 1980s.
This is the sibling of the MFM blueprint and the two should be read together. Same rate, same detection window, minimum run raised from 1 to 2 — and the whole difference is what the decoder has to remember.
Hejuru
3 hours
Amabwiriza
1
1
Encode a message by hand and check the constraint
Encode a message by hand and check the constraint
Write out the IBM (2,7) table on paper: 11 goes to 1000, 10 to 0100, 011 to 001000, 010 to 100100, 000 to 000100, 0011 to 00001000, 0010 to 00100100.
Encode 110100110001011. Take the longest match from the front each time, emit its codeword, and move on. Then check the constraint: count the zeros between every pair of 1s. Every gap must be at least 2 and at most 7 — not by luck, but because the table was built so no legal concatenation can violate it.
NOW MEASURE THE GAIN, PHYSICALLY. Cut two strips of graph paper. The medium sets a minimum spacing between transitions, so mark every 1 in each stream at that SAME spacing — use 5 mm squares and put one transition per square, with the required empty squares between them.
Encode the same fifteen bits as MFM on the first strip and as RLL(2,7) on the second, and lay them side by side. Measure both with the steel rule. The RLL strip carries the same message in about two thirds of the length, and that ratio is the density ratio the notebook computes. You have just measured a 50 percent capacity gain with a ruler.
Ibikoresho by'iyi ntambwe:
Graph Paper2 impapuroIbikoresho bikenewe:
Steel Rule2
2
Which code, and why
Which code, and why
The decision tree an engineer actually works down, and the order matters: the medium sets the minimum transition spacing before any coding question is asked, and everything after that is choosing how much data to fit into a spacing you do not control.
Note branch 4. A magnetic disk with peak detection barely cares about DC content; an optical disc reading a reflected LEVEL cares enormously. That single difference is why EFM in blueprint 9 looks wasteful next to RLL(2,7) until you notice it is solving a problem RLL does not have.
Flow
CHOOSING A CHANNEL CODE — the decision an engineer actually makes.
Work down. Each step is a constraint the medium or the electronics imposes.
1. HOW CLOSE CAN TWO TRANSITIONS BE BEFORE THEY INTERFERE?
This is a MEDIUM question, not a coding one. Write two reversals closer than
about the head gap and their fields overlap and both read back smaller and
shifted -- 'bit shift', and it is the real limit.
-> the answer sets your minimum physical spacing. Everything else is chosen
to put as much DATA as possible into that spacing.
2. HOW GOOD IS YOUR CLOCK?
- spindle within 0.5 %, quartz reference, PLL available -> you can afford a
large k. Go to 3.
- cheap spindle, wide temperature range, no PLL -> keep k small. FM or MFM.
Stop here; this is why cassette-based storage used FM-like codes.
3. CAN YOUR DECODER AFFORD STATE?
- a handful of gates -> MFM. It is a one-bit-of-history rule and it fits in
almost nothing.
- a small state machine, ~8 states -> RLL(2,7). Variable-length, needs a
lookahead buffer, and in 1979 that WAS the expensive part.
- a real controller chip -> RLL(1,7) at rate 2/3, or better. Lower density
ratio than (2,7) but a wider window, which matters more once the medium is
the noisy part rather than the electronics.
4. IS THE CHANNEL AC-COUPLED, AND DOES BASELINE WANDER HURT?
- magnetic disk, peak detection -> mostly no. MFM and RLL do not bother
controlling the running digital sum.
- optical disc, threshold detection on a reflected level -> YES, badly. You
must control DC content. That is why EFM spends three merging bits per
symbol on it, and why it looks so wasteful next to RLL(2,7) until you
notice it is doing a different job. See blueprint 9.
5. WHAT IS THE ERROR MODEL?
- independent random bit errors -> the channel code is enough, plus a light
ECC.
- BURSTS (a scratch, a fingerprint, a media defect) -> the channel code
cannot help you at all and you need an interleaved burst code on top.
Blueprint 9.
THE MISTAKE TO AVOID: choosing the code with the best density ratio. Density
ratio is one of four numbers, and the other three -- window, k, and DC content --
are what actually decide whether the drive works at three in the morning in a
warm rack.3
3
Capacity, the real table, and the cost
Capacity, the real table, and the cost
Gupakira ikaye ya Jupyter…
Ibikoresho bikenewe:
Desktop Computer4
4
Compendium: the exchange rate is a theorem
Compendium: the exchange rate is a theorem
WHY (2,7) AND NOT DENSER. Push d higher and density rises, but the rate falls, codewords lengthen, and error propagation worsens — one bad channel bit corrupts several data bits because the decoder loses its place in a variable-length table. (2,7) sat where a small state machine could cope. RLL(1,7) later became preferred with a LOWER density ratio, 1.333 against 1.5, because its wider detection window was worth more once the MEDIUM rather than the electronics became the noisy part. That reversal is the most instructive thing here.
THE SIBLING, MEASURED, against MFM. MFM: rate 1/2, d=1, k=3, DR 1.0, decoder is one bit of history. RLL(2,7): rate 1/2, d=2, k=7, DR 1.5, decoder is an eight-state machine with lookahead. Fifty percent more capacity for a decoder that in 1979 was a chip and in 1972 was a board. Nothing about the head or medium changed.
WHAT CAME AFTER. PRML stopped detecting individual transitions at all: it accepts that neighbours interfere, models the interference, and uses a Viterbi decoder to find the most likely SEQUENCE. That is why the modern drive in blueprint 2's table beats the simple spacing-loss budget by more than an order of magnitude.
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Blueprint zijyanye
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CC0 Umurenge rusange
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Shyigikira Umuremyi ugura ibicuruzwa binyuze muri Blueprint ye Komisiyo y'Umuremyi byashyizweho n'Abacuruzi, cyangwa kora verisiyo nshya y'iyi Blueprint ukayinjiza nk'isano muri Blueprint yawe kugira ngo musangire inyungu.