Toward the Optimal Bit Aspect Ratio in Magnetic Recording William - - PowerPoint PPT Presentation

Toward the Optimal Bit Aspect Ratio in Magnetic Recording

William E. Ryan

Associate Professor The University of Arizona

with contributions from

Fan Wang, Roger Wood, and Yan Li

March 25, 2004

Background Channel Model Approach for Shannon Codes Optimal Code Rates for Shannon Codes on the Lorentzian Channel Approach for LDPC Codes Optimal Code Rates for LDPC Codes on the Lorentzian Channel On the Optimal Bit Aspect Ratio Concluding Remarks

Outline

Coding on a magnetic recording channel: Lorentzian model due to ISI, the code rate loss is R2 -- on the AWGN channel it is R

• n the AWGN channel, performance improves with decreasing code

rate; on ISI channels such as the Lorentzian, it does not

Background

encoder (code rate R) data word record waveform playback waveform Tc = R Tu code word Tu

In [Ryan, Trans. Magn., Nov. 2000] we examined optimal code

rates empirically for specific parallel and serial turbo codes

Background (cont’d)

Su = PW50/Tu Performance of various PCCC's on the PR4-equalized Lorentzian channel with user density Su = 2.0.

Lorentzian model (in AWGN)

where s(t) = h(t) - h(t-Tc) is the dibit is AWGN with spectral density N0/2 and h(t) is the Lorentzian pulse

Ei = the energy per isolated Lorentzian pulse h(t) and pw50 is its

width measured at half its height

Channel Model

( )2

50 50

/ 2 1 1 4 ) ( pw t pw E t h

i

+ = π

) ( ) ( 2 1 ) ( t w kT t s a t r

c k k

+ − ∑ =

applying a whitened matched filter to r(t) leads to the discrete-time

equivalent model depicted below

where ⌧Edibit is the energy in s(t), ⌧f(D) is the minimum phase factor in the Tc-sampled auto-

correlation function of s(t), Rs(D)

⌧

Channel Model (cont’d)

+

nk ~ ak = Xk Yk

( )

2 / , N η

1 ±

) ( 2 1 D f Edibit

∑ =

k k

f 1

2

Approach Approach for Shannon Codes

Our goal is to determine optimal code rates for this channel for

both Shannon codes and LDPC codes.

high SNR low SNR medium SNR capacity measure Sc (channel density) capacity low in this region since Sc (and hence Su) is low capacity low in this region since Sc is high so that ISI is severe Sc = PW50/Tc

Approach for Shannon Codes (cont’d)

possibly better is data such as that in the figure below

capacity measure high SNR low SNR medium SNR R (code rate) coding overhead high large Sc and ISI becomes too severe to overcome with coding coding overhead low to provide sufficient coding gain, Su and Sc must be reduced

1

can now use the result of Arnold-Loeliger (ICC'01) (also, Pfister-

Siegel, GC'01) to compute the achievable information rate of the binary-input ISI channel assuming iid inputs

Note by computing the information rate for , we do

not assume PR equalization. Rather, optimal (ML) detection is assumed.

Note also that we use as our SNR measure

Approach for Shannon Codes (cont’d)

) ( 2 1 D f Edibit ) ( 2 1 D f Edibit

/ N Ei

Results for Shannon Codes

0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Information rate Ixy (info bits/channel bit) Channel density Sc Ei/No = 3dB Ei/No = 8dB Ei/No = 13dB Ei/No = 18dB Ei/No = 5.5dB Ei/No = 10.5dB Ei/No = 15.5dB

Information rate of Lorentzian channel versus channel density Sc.

Results for Shannon Codes (cont’d)

Note I xy is in units of information bits/channel bit we would like a capacity measure relative to a physical parameter

• f the channel, such as info bits/inch (along a track)

info bits/pw50 is particularly convenient:

note Sc = pw50/Tc may be regarded as channel bits/pw50 (Example: Sc = 3 3 channel bits/pw50)

define a new information rate

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′

50 50

bits channel bit channel bits info bits info pw S I pw I

c xy xy

Results for Shannon Codes (cont’d)

Normalized Information rate of Lorentzian channel versus channel density Sc.

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

Channel density Sc (= PW50/Tc) Information rate Ixy’ (bits/PW50) Ei/No = 3dB Ei/No = 8dB Ei/No = 13dB Ei/No = 18dB Ei/No = 5.5dB Ei/No = 10.5dB Ei/No = 15.5dB

Results for Shannon Codes (cont’d)

Information rate of Lorentzian channel versus code rate R.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3

Information rate Ixy (= Code rate R) Information rate Ixy’ (bits/PW50) Ei/No = 3dB Ei/No = 8dB Ei/No = 13dB Ei/No = 18dB Ei/No = 5.5dB Ei/No = 10.5dB Ei/No = 15.5dB

Examination of deviation of I(X;Y) with Rs(D) truncation parameter κmax

Results for Shannon Codes (cont’d)

Approach Approach for LDPC Codes

Extrinsic information transfer (EXIT) chart

provides a simple way of determining the capacity limit (or decoding

threshold) for a specific coding scheme.

describes the flow of extrinsic information through SISO processors

(detectors/decoders) operating cooperatively and iteratively.

SNR: Ei/N0 Inner SISO Detector Outer SISO Decoder Channel input

Approach for LDPC Codes (cont’d)

possibly better is data such as that in the figure below

Channel SNR increasing Inner information transfer characteristic Outer information transfer characteristic Iin-1 (Iout-2) Iout-1 (Iin-2)

1 1

Approach for LDPC Codes (cont’d)

EXIT chart for channel density Sc=1/3 and LDPC code rate 0.61

Results for LDPC Codes

Information rate I(X;Y) for Lorentzian channel versus channel density - Shannon codes and LDPC codes.

Results for LDPC Codes (cont’d)

Scaled Information rate I’(X;Y) for Lorentzian channel versus channel density - Shannon codes and LDPC codes.

Results for LDPC Codes (cont’d)

Scaled Information rate I’(X;Y) for Lorentzian channel versus code rate - Shannon codes and LDPC codes.

Results for LDPC Codes (cont’d)

Decoding threshold vs. code rate for various user densities

The information-theoretic areal density may be computed via

Iareal (bits/nm2) = I'xy (bits/PW50) / [ L50 (nm/PW50) x TW-1 (tracks/nm)] where L50 is the length of PW50 in nm and TW is the track width.

It is well-known that the SNR along a track is proportional to the

bit-length2 under the Lorentzian model (Bergmans, Immink)

One may argue that at the optimal track density (which maximizes

areal density), SNR will be proportional to the bit-width2 as well (let bit-width = TW): SNR = αTW2

On the Optimal Bit Aspect Ratio On the Optimal Bit Aspect Ratio

Combining these two equations yields Since α and L50 are constants dependent on a specific hard disk

drive, we define a normalized areal density measure

We may plot Iareal,norm as a function of Sc (since I'xy is a function of

Sc) and the normalized track width (since in the previous equation may be replaced by ).

Optimal Bit Aspect Ratio (cont’d)

( )

SNR L I I

xy areal 50

/ ′ ⋅ = α

( )

SNR I L I I

xy areal norm areal

/ / /

50 ,

′ = = α

TW α SNR TW α

Optimal Bit Aspect Ratio (cont’d)

Iareal,norm is maximized at TWnorm = 3.4 and Sc = 2.3. We could convert Iareal,norm,max = 0.433 to a density in bits/in2 by scaling this value by the factor , if known. 50

/ L α

Optimal Bit Aspect Ratio (cont’d)

Even in the absence of knowledge of a density measure in bits/in2, this analysis yields the following operating values at the optimum: SNR: Ei/N0 = 10.5 dB Code rate: R = 0.62 Channel density: Sc = 2.35 User density: Su = 1.45 For comparison, today’s (approximate) values: SNR: Ei/N0 = 18 dB Code rate: R = 0.95 Channel density: Sc = 3.0 User density: Su = 2.85

These results serve as a guide to choosing the optimal operating

parameters (linear density, bit aspect ratio, code rate, etc.).

This work can be extended to include media noise and/or

perpendicular recording.

It can also be extended to codes which do not have iid inputs (e.g.,

Markovian codes).

One of the implications is that work toward increased areal

densities should target bit-width, not bit-length, leading to new challenges in track servo design.

Conclusion