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Coded modulation : Final issues

• Rotational invariance

• Multidimensional constellations

• Turbo coded modulation*

• Noisy-Champs 2004

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Rotational invariance• Most constellations are phase symmetric: A phase shift of will

map one signal point into another, for some value of • For example, uncoded BPSK is 180 phase invariant; uncoded

QPSK 90 phase invariant

• Coded modulation is in general not phase invariant, unless special care is taken

• Loss of phase synchronization:

• Try to resynchronize (it helps if system is not phase invariant, but still very time consuming)

• Use phase invariant coded modulation scheme with differential encoding/decoding.

• Will cause brief error burst at phase loss but will maintain synchronization

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Rotational invariance: Example• Uncoded BPSK: 180 phase invariant

• Coded BPSK: The coded scheme is 180 phase invariant iff,

• for every code sequence v(D), it holds that v(D)1(D) is also a codeword

• This in turn is true iff 1(D) is a codeword

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Rotational invariance: Example• Uncoded QPSK: 90 phase invariant

• Coded QPSK: Depends on mapping

• Gray mapping

• Natural mapping

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Rotational invariance: ExampleCoded QPSK with Gray mapping:

Rate ½ code with

• G(D) = (h(1)(D), h(0)(D))

• H(D) = (h(0)(D), h(1)(D)) V(D) = (v(1)(D), v(0)(D))

Vr(D) = (vr (1)(D), vr (0)(D))

= (v(0)(D), v(1)(D) 1(D))

Vr(D) HT(D) =

u(D) ( (h(1)(D))2 (h(0)(D))2) h(0)(D)1(D) = f(D) + h(0)(1)(D)

...is in general not = 0(D) for any choice of linear code

No 90 phase invariance possible

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Rotational invariance: ExampleCoded QPSK with natural mapping:

Rate ½ code with

• G(D) = (h(1)(D), h(0)(D))

• H(D) = (h(0)(D), h(1)(D))

V(D) = (v(1)(D), v(0)(D))

Vr(D) = (vr (1)(D), vr (0)(D))

= (v(0)(D)v(1)(D), v(0)(D)1(D))

Vr(D) HT(D) =

u(D) (h(1)(D))2 h(0)(D)1(D)

...is in general not = 0(D) for any choice of linear code

No 90 phase invariance possible

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Nonlinear parity checks• QPSK: V(D) = (v(1)(D), v(0)(D))• v(D) = v(0)(D) + 2v(1)(D), polynomial over Z4

• vr(D) = v(D) + 1(D) (mod 4)• Parity check polynomials over Z4 : h(D) = h(1)(D) + 2h(0)(D)• Notation:

• For = 2(1)+ (0) Z4 , let []1 = (1)

• Let [(D)]1 = the sequence of most significant bits in (D)• Parity check equation: [h(D) v(D) (mod 4)]1 = 0(D) • h(D) vr (D) = h(D) ( v(D) + 1(D) ) (mod 4) =

h(D)v(D) + h(D)1(D)(mod 4) = h(D)v(D) + h(1)(D) (mod 4) • Satisfied iff h(1) (mod 4) = h(1)(1) + 2h(0)(1) (mod 4) = 0

• Search for parity check matrices on this form and with h(1)(D) of degree < and with no constant term

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Nonlinear parity checks: Example• Naturally mapped QPSK, code rate 1/2

• Parity check equation: [h(D) v(D) (mod 4)]1 = [h(1)(D) v(0)(D) + 2 h(1)(D) v(1)(D) + 2 h(0)(D) v(0)(D) + 4 h(0)(D) v(1)(D) (mod 4)]1 = [h(1)

(D) v(0)(D)+2(h(1)(D)v(1)(D) + h(0)(D) v(0)(D) ) (mod 4)]1 = 0(D)

• Select h(1)(D) = Db +Da, 0<a<b<.

• h(1)(1) = 2 so h(0)(D) must have odd weight

• Parity check equation: [ (Db +Da) v(0)(D) + 2((Db +Da)v(1)(D) + h(0)

(D) v(0)(D)) (mod 4)]1 = 0(D)

• For , Z4 , + (mod 4) = + 2 ( )

• PCE: [ (Db Da) v(0)(D) + 2(Db v(0)(D) Da v(0)(D) (Db Da) v(1)

(D) h(0)(D) v(0)(D)) (mod 4))]1 = Db v(0)(D) Da v(0)(D) (Db Da) v(1)(D) h(0)(D) v(0)(D)) (mod 4) = 0(D)

• Nonlinear term: Will trellis have only 2 states?

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Nonlinear parity checks: Example• Naturally mapped QPSK, code rate 1/2

• Select h(1)(D)=D2 +D, h(0)(D)=D3 +D+1, H(D) = [h(1)(D)/h(0)(D),1]

• Two different encoders with 8 states:

• Differential encoder

• Embedded DE

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Nonlinear parity checksSome codes:

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Nonlinear parity checksSome codes for QAM constellations:

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Multidimensional constellations• Instead of using only a 1-D or 2-D signal constellation S:

• Send L signals from S, and consider the block of signals to be one signal point in the signal constellation SL

• If S contains 2I signal points, then SL will contain 2IL points: Can send IL bits (per L symbols)

• TCM system: k+1 = IL, = k/L

• Advantages of multidimensional signal schemes

• Wider range of spectral efficiency • Can achieve rotational invariance with linear codes

• Can use shaping to obtain further power savings

• Smaller peak-to-average power ratio

• Possible to obtain higher decoding speed

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Turbo coded modulation*• Binary turbo coding achieves a considerable coding gain

compared with convolutional codes• Many of the techniques for TCM can not be applied directly

• Set partitioning does not make sense• Rotational invariance is difficult to achieve

• Usual approach: Bit interleaving and Gray mapping• Also achieves a significant coding gain over TCM• One approach (Rosnes and Ytrehus, ICC 2004)

• Good and bad bit positions in Gray mapped QAM• List of all low weight codewords• Critical positions in low weight codewords mapped to

”good” bit positions• Can achieve extremely low error floors (ML decoding)

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Suggested exercises• 18.19-18.25

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Suggested exercises

Please acknowledge by e-mail to oyvind@ii.uib.no, in order to discuss format (length, use of projector etc.)

Date Subject Responsible

Nov. 11 Chapter 1, Chapter 9, Chapter 10, Chapter 14, Chapter 15

Geir Jarle Ness, Irina Gancheva, Susanna Spinsante

Nov. 11 Chapter 11 Lars Erik Danielsen

Nov. 16 Chapter 12 Øystein H. Nyheim

Nov. 16 Chapter 16 Sondre Rønjum

Nov. 16 Chapter 17 Joachim Knudsen

Nov. 16 Chapter 18 Olaf Garnaas

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Code length versus

performance

2004

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Final results mandatory exercise INF244

Name Time BER*

Implemented

3680 5,40E-05 0,199 PCCC

1120 2,15E-04 0,241 PCCC

13 5,13E-02 0,667 None

109 1,76E-02 1,915 Viterbi

2240 1,55E-03 3,474 SCCC

231 1,57E-02 3,624 Viterbi

107 4,02E-02 4,297 Viterbi

139 4,17E-02 5,794 Viterbi

3320 2,02E-03 6,700 Viterbi

139 5,75E-02 7,994 Viterbi

1. Joakim Knudsen2. Sondre Rønjum

3. Uncoded

2004