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31 December 2008
Signal Processing for Bit-Patterned Media (BPM) Channels
Sheida Nabavi & Vijayakumar Bhagavatula
2
DSSC Channels Research Strategy
Low SNR will be one of the main challenges for future high-density storage systems
LDPC codes Iterative soft decoders
Channel models and signal processing approaches must be adapted to the new recording paradigms
Bit-patterned media (BPM)Two-dimensional magnetic recording (TDMR)Heat-assisted magnetic recording (HAMR)Multi-level cell flash memory & MRAM
Optimization of channels algorithms for hardware power/area/performance tradeoffs
3 3
LDPC Codes: Contributions
FPGA platform for LDPC decoder achieved 10-12 BERDesign, evaluation and encoder/decoder architecture of hardware-friendly progressive edge growth (PEG) quasi-cyclic LDPC codesEvaluation of LDPC codes under realistic magnetic recording channel models (not just AWGN)Improved reverse concatenation of LDPC codes and RLL codes using bit-flipping & interleavingSemi-analytical method for error floor estimation by identifying trapping sets using FPGA platform; estimated error floors down to 10-13 frame error ratesInvestigation of the concatenation of LDPC codes and outer RS codes
4
Error-pattern Flipping Chase2-type Decoder
Reed-Solomon (RS) codes are still of much use and interestSoft-decoding methods (e.g., Koetter-Vardy algorithm) proposed for error correction beyond half the minimum distance, but their complexity too highSoowoong Lee proposed a new Error-pattern flipping Chase2-type decoder , with parameter p
Flip dominant error-patterns before RLL decoder instead of bit-flipping or symbol-flipping Repeat error-only decoding with 2p modified received sequences while flipping p most probable error-patterns
6 6
TDMR Concept (INSIC)
20grains
40 channel bits from encoder
10 user bits 10 user bits
encode(¼-rate)
soft 2D decode
(shingled-) write
process
40 channel bitswith soft info.
(scanning)
establish timing & position of
bit-cells
hi-resln2D readprocess
can almost see grains but not grain-boundaries
A.
B.
C. D.
E.
F.
not all channel bits get written on grains
10 bits in 1 μin2
= 10 Terabits/in2
cornerwriter
TDMRToy example:
Use 2D image analysis methods to decode user bits from recovered channel bits & soft info., and grain statistics
READINGWRITING
Courtesy: Roger Wood
7
PLB IP Interface
Register Files
Write logic
Read logic
PowerPC405
DataBlockRAM
InstructionBlockRAM
RDG
RDG LDPCENC
Channel
SOVALDPCDecoder
Iteration Control
Analysis
LDPC Error Floor Estimation via FPGA Emulation (Yu Cai & Prof. Ken Mai)
PowerPC
SOVA_LDPCClock rate improved from 20Mhz to 100Mhz (5X)Two parallel SOVA_LDPC fit in one FPGA (2X)
Two PowerPC per FPGAFive FPGAs per boardHigh speed interconnect80GB on board DRAM
FPGA based SoC
Two parallel SOVA_LDPC per FPGA (2X speedup)Utilize all 5 FPGAs (5X speedup)50X speedup over previous FPGA implementation1000X speedup over workstation implementation
Future PlansResource UsageFPGA Floorplan
BEE2 Board
.5GB/s .
DRAM
XilinxVirtexII Pro 70
DRAM
XilinxVirtexII Pro 70
DRAM
XilinxVirtexII Pro 70
DRAM
XilinxVirtexII Pro 70
DRAM
XilinxVirtex II Pro 70
8
BPM Signal Processing: Outline
Motivation for BPMBPM 2D Pulse responseBER analysis of BPM Mitigating the effects of inter-track interferenceModeling the media noise
9
SNR is proportional to the number of grains per bitHigher density can be achieved by scaling
Size of bits should be reducedThe number of grains should
remain the same
For nano-scale magnetic grain, thermal energy is comparable to the energy of bit (superparamagnetic limit)
Why bit-patterned media?
10
Bits are stored in single domain magnetic islandsThe regions between the islands are nonmagnetic material Achieving densities over 1 Tb/in2
AdvantagesReducing transition and track edge noiseReducing or eliminating non-linear bit shiftSimplifying tracking
Bit-patterned media (BPM)
+ magnetization- magnetization
11
Inter-track interferenceBits or magnetic islands are very closeBesides ISI, there is ITI
Media noisemedia noise is caused by the fluctuations in the islands
LocationSizeThicknessMagnetizationShape
Write synchronizationFor more realistic channel model (modeling ITI and media noise) 2D pulse response needs to be used
Along track
Main Track
BPM challenges (signal processing)
...
...
.........
...
...... ......... ...
... .........
...
......... ...
...
...
...
12
2D pulse response
Readback voltage of the MR head
: signal flux injected into the MR element at the ABS
3D Reciprocity principle
My : media magnetizationi : imaginary write currentΨ : magnetic head potential
( , ) ( , )MRV x z C x zφ=
( , ) ( , 0, )x z x y zφ φ= =
0 ( , , )( , ) ( , , )
d y
d
M x x y z zx z dx dy dz x y z
i yδμφ
∞ + ∞
−∞ −∞
′ ′ ′∂ − −⎡ ⎤′ ′ ′ ′ ′ ′= Ψ ⎢ ⎥′∂⎣ ⎦
∫ ∫ ∫
32 2 2 2
( , )( , , )2 ( ) ( )
s x z dx dzyx y zx x y z zπ
∞ ∞
−∞ −∞
′ ′ ′ ′ΨΨ =
′ ′⎡ ⎤− + + −⎣ ⎦∫ ∫
My
d
δ xy
z
13
Applying the head surface magnetic potential approximation from Wiesen et al.*
Example MR head surface magnetic potential
*K. Wiesen, and B. Cross, “GMR Head Side-Reading and Bit Aspect Ratio,” IEEE Trans. Magn., Vol. 39, No. 5, Sep. 2003
gW
tCro
ss-tr
ack
Along-track
12
12
2 2 4 2 22 1 2exp cos exp 1 exp cos1( , ) 1 arctan
2 2 41 2exp cos exp
s
z x z z xg g g g g
x zz x z
g g g
π π π π π
ππ π π
⎛ ⎞⎡ ⎤⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
− + − +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎣ ⎦⎜ ⎟Ψ = − ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠
MR head (W = 20 nm, t = 4 nm, g = 10 nm)
14
xyz
d
δa
t
g
W
Example 2D Pulse Response
Numerical integration to obtain the magnetic potential and pulse response
MR head t = 4 nm, W = 20 nm, gap = 10 nm
Magnetic islanda=11 nm, δ= 10 nm, d=10 nm
Along-track PW50=22.3 nm, Cross-track PW50=30.1 nm
15
Readback signal
Using 2D linear superposition to obtain readback signalBits can be represented by their locations Length of ITI assumed to be 3
Main track and two adjacent tracksRectangular grid
1
1( ) ( , ) ( , )
N
z xm n N
s x x m n P mT x nT=− =−
= − −∑ ∑
Channelx(-1,k)x(0,k)x(1,k)
( )s x
( , )P z x
Along track x(0,k)
Track of interestxT
zT
16
Readback signal is sampled at the nominal bit period Tx
Discrete time readback signal
Equalizer outputAssuming symmetric channels
Discrete-time BPM read channel
H(m,n)=P(mTz, nTx)
1
1
( ) ( , ) ( , ) ( )N
m n N
r k x m n H m k n v k=− =−
= − − +∑ ∑
1
0
1
( )( )( )
h kh kh k−
← →⎡ ⎤⎢ ⎥= ← →⎢ ⎥⎢ ⎥← →⎣ ⎦
Hmatrix representation
Channelx(-1,k)x(0,k)x(1,k)
( )s x( , )P z x
r(k)+
v(k)
Equalizerf(k)
ViterbiDetector
z(k) ˆ(0, )x k
0 1
( ) ( ) ( )( ) ( ) (0, ) ( ) ( ) [ ( 1, ) (1, )] ( ) ( )
z k f k r kf k h k x k f k h k x k x k f k v k
= ∗= ∗ ∗ + ∗ ∗ − + + ∗
17
Viterbi detector input signal
Viterbi detector input signal
where, where g(k) (gT = [g-K, …, g0,…gK]) is the PR target
, where f(k) (fT = [f-L,…, f0,…fL]) is the equalizer
, assuming h1(k)=h-1(k)
Using vector representation
where
( ) (0, ) ( )y k x k g k= ∗
0( ) ( ) ( ) ( )ISIc k h k f k g k= ∗ −
( ) ( 1, ) (1, )b k x k x k= − +
1( ) ( ) ( )ITIc k h k f k= ∗
( ) ( ) (0, ) ( ) ( ) ( ) ( ) ( ),ISI ITIz k y k x k c k b k c k v k f k= + ∗ + ∗ + ∗
un-equalized ISI ITI colored noisedesired output
0,( ) ( )ISI ITIk k kz k y k= + + +T T Tc x c b f v
[ ( ) ( )]ISI ISI ISIc N L c N L= − − +Tc[ ( ) ( )]ITI ITI ITIc N L c N L= − − +Tc
n(k) : overall noise
18
For performance analysis of a partial response channel we need to first determine the error eventsFor the Viterbi detector, an error event ε is an erroneous estimate state sequence
Error event
Correct path Erroneous path
k1 k2
zk1 zk2-1
xk1-1xk1-2
-1 -1
-1 1
1 -1
1 1
-1 -1
-1 1
1 -1
1 1
-1 -1
-1 1
1 -1
1 1
-1 -1
-1 1
1 -1
1 1
19
Probability of an error event, ε
Probability of a particular error event ε for the Viterbi detector
wheremc is the path metric for the correct path
me the path metric for the erroneous path of a particular error event ε
wH(ε) : The number of input bit errors or the Hamming weight
For a specific error event ε
( )1Pr( ) Pr( )2
Hw
e cm mε
ε ⎛ ⎞≤ < ⎜ ⎟⎝ ⎠
2 2|| || ||c k kkm = − =z y n
2 2 2ˆ ˆ|| || || || ||e k k k k kkm = − = + − = +yz y y n y ε n
2 22 1Pr( ) Pr( ) Pr( )2e c k k y km m< = + < = < −T
y yn ε n ε n ε
20
pdf of
* : convolutionpdf of
vk : AWGN electronics noise
y kTε n
0,
0,
f ( ) f ( )
f ( )*f ( )y k ISI k ITI k f k
ISI k ITI k f k
= + +
= +
T T T T
T T T
ε n w x w b w v
w x w b w v
Gaussian(not Gaussian) ?
2(0, )k vv N σ→
f kTw v
22(0, )f v fN σ→Tw v w
The PDF of the overall noise and the output error sequence inner product
-1 1
1/2 1/2
x
f(x)
-2 0 2
1/21/41/4
b
f (b)
21
Distribution of ITI and un-equalized ISI
pdf ofAssuming uniform binary input data is a train of impulses
Monic constraint target (gT = [g-1, 1, g1]),Density of 2 Tbit/in2 , The minimum distance error event
0,ISI k ITI k+T Tw x w b0,f ( )ISI k ITI k+T Tw x w b
23
Most probable error events
Pr(ε) Pr(ε)++-+++0-+0++-++0+0-+0-0-+0+0+++-+--+-0++0-++-0-
4.037.138.988.038.08
10.2512.0812.0812.1312.0512.0511.1311.1311.18
1.66e-032.46e-051.48e-064.77e-066.33e-062.89e-071.13e-081.14e-081.51e-081.97e-082.02e-085.19e-085.19e-087.03e-08
+0+-++++-+-+0--+-+0++0++++0++0-0++-0+-+0-+-+-+0-+0+-++-+0++-0-+
11.1813.9613.3012.9812.9813.0313.0312.0314.2314.2514.2514.3114.3014.28
7.14e-081.07e-093.28e-093.34e-093.36e-094.33e-094.34e-098.50e-095.28e-105.29e-105.41e-107.36e-107.42e-107.72e-10
xεxε2
yε2
yε
Using computer search to obtain most probable error eventsMonic constraint target (gT = [g-1, 1, g1])Density of 2 Tbit/in2
σ2v= 0.01, The minimum distance error event, (dmin )2= 4.03
25
Conventional PR4 and EPR4 targets do not perform well
Vp=1σv : noise standard deviation
PR channels for BPM
e 10SNR 20 log ( / )p vV σ=
(db)
26
Analytical and simulation BER vs SNR
Monic constraint target gT = [g-1, 1, g1]
Density of 2 Tbit/in2
Tx and Tz 18 nmMR head
t = 4 nm, W = 20 nm, g = 10 nmMagnetic island
a=11 nm, δ= 10 nm, d=10 nm
0.021 0.213 0.0210.101 1 0.1010.021 0.213 0.021
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
H
29
GPR equalizerTarget length : 3g = [g-1 1 g1]
Density of 2 Tbit/in2
Tx and Tz 18 nmMR head
t = 4 nm, W = 18 nm, g = 6 nmMagnetic island
a=11 nm, δ= 10 nm, d=10 nm
ITI significantly degrades the performance of the channel
Effect of Inter-track Interference (ITI)
0.038 0.294 0.0380.113 1 0.1130.038 0.294 0.038
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
H
30
2D-GPR equalizer
Using 2D approach to mitigate the effect of ITI
Assuming the parallel readback of the data in three adjacent tracks
Using minimum mean squared error (MMSE) approach to optimize the 2D equalizer and target
Viterbi algorithm does not exist for the true 2D motion
G( m,n)
H( m,n) F( m,n)+v(j,k)
y(j,k) z(j,k)equalizer
target+
_ e(j,k)d(j,k)
x(j,k) x(j,k)Viterbi Detector
^
channel
31
j=0 is representing the main track
Mean square errorwhere
Vector representation for 2D GPR
, 0,0 ,
, 0,0 ,
, 0, ,
, 0, ,
T
M N M N
T
L L L L
T
k M k N k M k N
T
k L k L k L k L
f f f
g g g
y y y
x x x
− −
− −
+ − −
+ − −
⎡ ⎤= ⎣ ⎦
⎡ ⎤= ⎣ ⎦
⎡ ⎤= ⎣ ⎦
⎡ ⎤= ⎣ ⎦
f
g
y
x
… …
… …
… …
… …
( ) k ke k = T Tf y - g x
n(j,k)
G( m,n)
H( m,n) F( m,n)+y(j,k) z(0,k)=zk
equalizer
target +_e(k)
d(0,k)=dk
x(j,k)
{ }2( ) 2E e k = − +T T Tf Rf f Tg g Ag
{ } { } { }, ,k k k k k kE E E= = =T T TR y y T y x A x x
32
Minimizing the E{e(k)2} with considering the constraint
Lagrange functional to be minimized
where
2D GPR zero-ITI forcing equalizer (ZIFE)
2 2J = − + −T T T T Tf Rf f Tg g Ag λ (E g - c)
[ ]1 0 0 0 0 0 0 T=c
T T -1 -1 -1
T -1 -1
-1
λ = (E (A - T R T) E) cg = (A - T R T) Eλf = R Tg
Taking derivative respect to f, g and λ
0, 1 0,1
0 0 01
0 0 0g g−
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
G or 0, 1 0,10 0 0 1 0 0 0T
g g−⎡ ⎤= ⎣ ⎦g
0 0 0 0 1 0 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
TE
33
Results of 2D GPR ZIFE
2D GPR ZIFE3×3 targetObtained for σv
2=0.01
1D GPR (multi-track)Obtained for σv
2=0.01
2D GPR ZIFE improves the BER
0 0 00.132 1 0.130
0 0 0
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
G
[ ]0.076 1 0.077 T=g
34
2D-GPR equalizer
Pseudo 2D Viterbi algorithm
For a 3×3 target, tracing the history of two consecutive data in three tracks
64 states 8 branches
1, 1 1,0 1,1
0, 1 0,1
1, 1 1,0 1,1
1g g gg gg g g
− − − −
−
−
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
G
-1 -1-1 -1-1 -1
.
.
.+1 +1+1 +1+1 +1
-1 -1-1 -1-1 -1
-1 -1-1 -1-1 +1
.
.
.
-1 -1-1 -1-1 +1
+1 +1+1 +1+1 +1
-1-1 -1-1-1 +1
State 1
State 2
State 64
G(m,n )
H(m,n) F(m,n )+ z(k)equalizer
target
+_ e(k)d(k)
Pseudo 2D Viterbi detector
channelˆ(0, )x k
++
( 1, )x k−( 2, )x k−
(0 , )x k(1, )x k(2, )x k
( 1, )v k− (0, )v k (1, )v k
(0, )r k( 1, )r k−
(1, )r k
35
2D GPR equalizers
2D GPR targetObtained for σv
2=0.01
0.056 0.310 0.0770.183 1 0.1800.051 0.315 0.068
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
G
37
Modified Viterbi algorithm (VA)
The locations of bits (islands) are fixedOne track is readAssuming symetric channel
Along track x0,k Main Track
1
0
1
( )( )( )
h kh kh k
← →⎡ ⎤⎢ ⎥= ← →⎢ ⎥⎢ ⎥← →⎣ ⎦
H
0 1
( ) ( ) ( )( ) ( ) (0, ) ( ) ( ) ( ( 1, ) (1, )) ( ) ( )
z k f k r kf k h k x k f k h k x k x k f k v k
= ∗= ∗ ∗ + ∗ ∗ − + + ∗
r(k)ChannelH(m,n)
Equalizerf(k)
ViterbiDetector+
v(k)z(k)( 1, )x k−
(0, )x k(1, )x k
ˆ(0, )x k
38
Considering ITI in VA
ITI is mainly due to directly adjacent bits i.e. x-1,k , x1,k
Assuming no track mis-registration Desired equalizer output
Where
( ) (0, ) [ (1, ) ( 1, )] ( )kd g k x k x k x k n kα≅ ∗ + + − +
0( ) ( ) ( )g k h k f k= ∗
( ) ( ) ( )n k v k f k= ∗
( , ) {1, 1} [ (1, ) ( 1, )] { 2,0,2}x j k x k x k∈ − ⇒ + − ∈ −
1 0( ) ( )
kh k f kα
=∗
39
Modified VA trellis
Trellis with ITI
( ) (0, ) 2 ( )( ) (0, ) ( )( ) (0, ) 2 ( )
k
g k x k n kd g k x k n k
g k x k n k
α
α
∗ + +⎧⎪= ∗ +⎨⎪ ∗ − +⎩
trellis
40
Results of the modified VA
Target : g = [0.1 1 0.1]α = 0.2
Modified VA improves the performance of the channel
41
Modeling the Media Noise
Considering location fluctuations and size fluctuations
.........
...
......... ...
...
...
...
......... ......
a...
TxTz ......
...
...
...... ...
1
1( ) ( , ) ( , ( ) , ) ( )
m n N
z m x n mnm n N
r k x m n P mT T k n T T a a v k= =
=− =−
= − + Δ − + Δ + Δ +∑ ∑
Channel S(x)x(-1,k)x(0,k)x(1,k)
+ r(k)v(k)
-
( , )P z x
42
Obtaining an analytical 2D pulse response
Along-track pulse response can be considered as subtraction of two transitionsFor a<<PW50 derivative of transition pulses can be interpreted as pulse responses
2D Gaussian pulseUsing erf(x) as a transition response
2D Lorentzian pulseUsing arctan(x) as a transition response
Wx : along- track PW50 Wz : cross-track PW50
a
a/2
2 2
2 2
1( , ) exp{ ( )}2 x z
x zP x z Aσ σ
= − +along-track 50
2.3548xPWσ =
cross-track 502.3548z
PWσ =
2 2( , )
1/ 2 / 2x z
AP x zx z
W W
=⎛ ⎞ ⎛ ⎞
+ +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
43
Comparison of the 2D Gaussian and numerical pulses
Channel density 2Tbit/inch2
GPR equalizerTarget: [g-1 1 g1]
along-track PW50
Cross-track PW50
2D Gaussian pulse is a good candidate to model the BPM pulse response
44
Modeling media noise using analytical pulse response
Location fluctuations Randomness at the sampling points of the 2D Gaussian pulse,
Size fluctuations Amplitude, along-track PW50 and cross-track PW50 fluctuations
where
1Wx aaΔ ≅ Δ
2Wz aaΔ ≅ Δ
3A aaΔ ≅ Δ
2 21( , ) ( )exp ,2 ( ) ( )
x zA
x Wx z Wz
x zP x z Ac W c W
⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞+Δ +Δ⎪ ⎪⎢ ⎥= +Δ − +⎨ ⎬⎜ ⎟ ⎜ ⎟+Δ +Δ⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭2.3548c =
45
Extracting size and location statistics, using image processing techniquesUsing an image of BPM nano-mask
Media noise characteristics
Courtesy of Chip Hogg and professor Sara Majetich
46
Location and size fluctuations
Location and size fluctuations can be modeled by Gaussian distributions
Along-track size fluctuationsAlong-track location fluctuation
47
Correlation coefficients
Location fluctuations are highly correlated
Size fluctuations are less correlated
Along-track location fluctuations
Along-track Size fluctuations
48
Modeling correlated noise
Passing white noise through a digital filterNoise auto-correlation function modeled by a decaying exponential
f(k)n(k) v(k)
R n(k
)
R v(k
)
( ) kvR k a=( ) ( )nR k kδ=
D
+
a
n(k) v(k)b
f(k)
2( ) ( 1) ( 1 ) ( )v k av k a n k= − + −
white noise colored noise
, |a|<1
49
Decaying exponential auto-correlation function is a good model for correlated media noise
Modeling correlated noise (cont’d)
Along-track location fluctuations, a=0.9
Along-track Size fluctuations, a=-0.15
50
Matched-pulse PR channelsMinimizing noise amplification of the equalizers
GPR channelsNoise whitening equalizers
NPML channelsNoise predicting detectorsPredictor filter is embedded in the VA
Read channel for BPM with media noise
v(k)r(k) PR
EqualizerNPML detector
z(k)
Predictor
y(k)
-+ +
ChannelH(m,n)
( 1, )x k−
(0, )x k(1, )x k
ˆ (0 , )x k
v(k)r(k) PR
EqualizerViterbi detector
y(k)+
ChannelH(m,n)
( 1, )x k−
(0, )x k(1, )x k
ˆ (0 , )x k
Media noise
Media noise
51
Channel density 2Tbit/inch2
PR equalizerg=[0.1 1 0.1]
GPR equalizerg=[g-1 1 g1]Obtained for σ 2=0.05and 6% fluctuations
NPML detectorg=[0.1 1 0.1]
Gaussian fluctuations
Highly correlated normalized location and size fluctuations
α =0.95
Error performance of the PBM channels with media noise
solid lines: no media noise, dashed lines: 8% size and location fluctuations and doted lines: 8%
normalized correlated size and location fluctuations.
21β α= −
52
Summary
2D pulse response of BPM obtained using the 3D reciprocity principleThe ITI has the dominant impact on the error performance of the BPM channels and taking ITI into account is important for BPM channelsThe 2D GPR equalizers and the modified VA improve the BER performance of the BPM channelsMedia noise is correlated and can be modeled by a Gaussian random process2D Gaussian pulse can be used for the BPM 2D pulse responseNPML channels improve the BER performance of the BPM channels in the presence of the correlated media noise