Cyclic Coded Integer-Forcing Equalization Or Ordentlich Joint work - - PowerPoint PPT Presentation

Cyclic Coded Integer-Forcing Equalization

Or Ordentlich Joint work with Uri Erez EE-Systems, Tel Aviv University ACC workshop, February 15th, 2011

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

The Gaussian ISI channel

decoder CSI encoder

xk yk nk m ˆ m

H(D)

Σ

yk = xk +

• m=0

hmxk−m + nk = xk + ISIk + nk nk is AWGN with unit power. Mutual Info: I(Sx(·)) =

1 2π

• ω log
• 1 + Sx(ejω)|H(ejω)|2

dω Assume (for simplicity) white input: Sx(ejω) = const = σ2

x

CSI@Rx only

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Reliable communication over the ISI channel

We are interested in schemes where decoding is decoupled from equalization.

• Turbo equalization not considered.

Multi-carrier (frequency domain) - OFDM/DMT

Transforms the ISI channel into parallel AWGN subchannels - simplifies equalization Coding over a channel with varying SNR may incur an unbounded gap-to-capacity PAPR Non-applicable to channels with finite alphabet (magnetic etc.)

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Reliable communication over the ISI channel

Single carrier (time domain)

(Tomlinson-Harashima Precoding (THP) requires complete CSI@Tx - inapplicable...) Linear equalizers: ZF-LE, MMSE-LE Decision-feedback equalization (DFE)

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Closer look at DFE

encoder

decoder

postcursor

DFE equivalent channel: (FFE)

xk yk nk

−

m ˆ m

H(D) A(D)

Σ Σ

ˆ xk y ′

k

G(D)

• r equivalently:

encoder

decoder

xk zk

−

m ˆ m

G(D) Σ

ˆ xk y ′

k

G(D) − 1

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

“Optimality” of DFE

MMSE-DFE is known to be “optimal” assuming correct detection of past symbols (CDEF) 1 2 log (1 + SNRDFE−MMSE−U) = C But how can one get error-free decisions? Must replace slicer with decoder Possible solution: Guess-Varanasi interleaving We pursue different solution: Move decoder before feedback loop

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Preview of Suggested Approach

Equalize the channel to integer-valued impulse response Add Zero-Padding/Cyclic prefix (as in OFDM) so that Linear Convolution → Cyclic Convolution Use linear cyclic code ⇒ closed under integer-valued cyclic convolution Decode convolved codeword which is also a codeword Apply DFE

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Integer-Forcing Equalization

encoder decoder

xk yk nk m

Σ Σ −

H(D)

ˆ xk, ˆ m ˆ x′

k

y ′

k

I(D) − 1

I(D) H(D)

• J. Zhan, B. Nazer, U. Erez, M. Gastpar ISIT 2010:

Integer-Forcing Equalization proposed ⇒ FFE(D) = I(D)

H(D) such that I(D) = FFE(D)H(D) is a

monic polynomial with integer coefficients

More general than Zero-Forcing where I(D) = 1 Less general than DFE since coefficients have to be integers FFE part is reminiscent of partial response equalization by lattice reduction (R. Fischer & C. Siegl 2005)

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

DFE- IF Equalization

FFE

xn yn wn Σ H(D) y ′

n

Y ′(D) = FFE(D)H(D)

• I(D)

X(D) + FFE(D)W (D)

y ′

n = xn + L

• k=1

in−kxk + zn For DFE-IF choose I(D) so as to maximize SNRDFE−IF = σ2

x

σ2

z

= σ2

x 1 2π

1/π

−1/π |I(ejω)|2 |H(ejω)|2 dω

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Cyclic Codes

Let C = {xk}2NR

k=1 be a linear code over Zq

C is cyclic if any cyclic shift of codeword is also a codeword ⇒ Cyclic linear code is closed under integer-valued cyclic convolution with operations performed over Zq. x ∈ C ⇒ x′ = [x ⊗ i] ∈ C Examples of cyclic codes:

”Most” algebraic codes: BCH, RS over prime field,. . . Classes of LDPC codes: type-I EG, type-I PG (Kou, Lin, Fossorier 2001), codes by Shibuya and Sakaniwa (2003)

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Finding I(D)

We would like to maximize SNRDFE−IF = σ2

x

σ2

z

= σ2

x 1 2π

1/π

−1/π |I(ejω)|2 |H(ejω)|2 dω

σ2

z can be written in matrix form

σ2

z = i

     k0 k−1 k−2 . . . k−L k1 k0 k−1 . . . k−(L−1) . . . . . . . . . ... kL kL−1 kL−2 . . . k0      iT = i˜ KiT where km =

1 2π

π

−π 1 |H(ejω)|2e−jmwdω.

A shortest lattice vector problem Can use LLL as an approximate solution

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Performance

How much do we lose w.r.t. “ideal” DFE by integer forcing? Theorem The noise enhancement caused by the IF equalizer is upper bounded by σ2

z ≤ σ2 ZF−DFE · min n≥p+1

• n2 (1.4πn)

1 n

πe

• |z0z1 . . . zp−1|2p
• µ,ν |z∗

µzν − 1|

1

n

where z0, z1, . . . , zp−1 are the maximum-phase zeros of H(D)H∗(D−∗), and p + 1 is the channel’s length. Bound is based on Minkowski bound for shortest lattice vector Not tight in general

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Simulation results

8-PAM constellation is used in a TCM-like manner cyclic LDPC n=255, k=175 (R = 2.6862

bits cahnnel use) code for

IF-DFE and MMSE-LE uncoded transmission for MMSE-DFE Channel is 1 + 0.894D + 0.814D2 + 0.239D3 − 0.070D4 + 0.036D5 − 0.022D6

2 3 4 5 6 7 8 9 10 10

−4

10

−3

10

−2

10

−1

10 SNRnorm[dB] Block Error Rate IF−MMSE−DFE MMSE−LE DFE−MMSE

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization

Summary, Extensions and Open Questions

Integer-Forcing equalization allows channel decoding before applying the DFE loop. Gains:

No error propagation Channel coding is much more effective

Penalties:

DFE coefficients must be integers Code must be cyclic

Method is effective for channels of moderate lengths, and high SNR Extension to MMSE exists Explore specific channel models for which IF is advantageous

Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization