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Complex and Quaternion-Valued Lattices for Digital Transmission Sebastian Stern Robert F.H. Fischer A NACHRICHTENTECHNIK Work supported by Deutsche Forschungsgemeinschaft (DFG) under grant FI 982/12-1 Complex and Quaternion-Valued Lattices

  1. Complex and Quaternion-Valued Lattices for Digital Transmission Sebastian Stern Robert F.H. Fischer A NACHRICHTENTECHNIK Work supported by Deutsche Forschungsgemeinschaft (DFG) under grant FI 982/12-1 Complex and Quaternion-Valued Lattices for Digital Transmission 1

  2. A NACHRICHTENTECHNIK Outline 1. Complex-Valued Lattices in Communications Complex-Valued Lattices in Communications � System Model and Motivation � � Introduction to Complex-Valued Lattices � � Coded-Modulation Schemes based on Complex-Valued Lattices � � Complex-Valued Lattice Reduction for Channel Equalization � 2. Quaternion-Valued Lattices in Communications Quaternion-Valued Lattices in Communications � � System Model and Motivation � � Introduction to Quaternion-Valued Lattices � � Coded-Modulation Schemes based on Quaternion-Valued Lattices � � Quaternion-Valued Lattice Reduction for Channel Equalization Complex and Quaternion-Valued Lattices for Digital Transmission 2

  3. A NACHRICHTENTECHNIK Digital Transmission: System Model (I) q [ κ ] x [ k ] s ( t ) s RF ( t ) Encoding RF Modulation Mapping Modulation F 2 A ⊂ C C R Digital Pulse-Amplitude Modulation: Transmitter Digital Pulse-Amplitude Modulation: Transmitter � transmission of sequence of bits q [ κ ] ∈ F 2 � � encoding and mapping � � � channel encoding over the fi nite fi eld (Hamming space) � mapping to complex-valued signal constellation A (Euclidean space) � � modulation � ∞ � s ( t ) = x [ k ] g ( t − kT ) k = −∞ � � transition from discrete-time to continuous-time domain � transmit fi lter g ( t ) (usually: band limitation) � � symbol period T � Complex and Quaternion-Valued Lattices for Digital Transmission 3

  4. A NACHRICHTENTECHNIK Digital Transmission: System Model (II) q [ κ ] x [ k ] s ( t ) s RF ( t ) Encoding RF Modulation Mapping Modulation F 2 A ⊂ C C R Digital Pulse-Amplitude Modulation: Transmitter Digital Pulse-Amplitude Modulation: Transmitter � radio-frequency (RF) modulation � s RF ( t ) = | s ( t ) | · cos(2 πf c t + arg { s ( t ) } ) � complex signal modulated onto the real-valued carrier (frequency f c ) � � amplitude of RF signal given by | s ( t ) | � � phase of RF signal given by arg { s ( t ) } � s RF ( t ) Example Example 1 � A = { 1 , − 1 , i , − i } � � g ( t ) = rect( t/ 1 s) � t/ s ⇒ | s ( t ) | = 1 − 1 1 � x [ k ] = [ . . . , − i , 1 , − 1 , . . . ] � − 1 � f c = 1 Hz � − 1 − i 1 Complex and Quaternion-Valued Lattices for Digital Transmission 4

  5. A NACHRICHTENTECHNIK Digital Transmission: System Model (III) r RF ( t ) r ( t ) y [ k ] q [ κ ] ˆ RF Filter Decoding Demod. Sampling Demapping R C C F 2 Digital Pulse-Amplitude Modulation: Receiver Digital Pulse-Amplitude Modulation: Receiver � RF demodulation � � � equivalent complex baseband signal obtained from RF receive signal � receive fi lter and sampling � � usually matched filter g ∗ ( − t ) � → maximization of signal-to-noise ratio (SNR) � � sampling → transition from continuous-time to discrete-time domain � decoding and demapping � � � channel decoding w.r.t. coded-modulation scheme → interaction between channel code and signal constellation � � demapping to estimated source bits Complex and Quaternion-Valued Lattices for Digital Transmission 5

  6. A NACHRICHTENTECHNIK Discrete-Time Equivalent-Complex-Baseband Domain q [ κ ] x [ k ] s ( t ) s RF ( t ) Encoding RF Modulation Mapping Modulation F 2 A ⊂ C C R r RF ( t ) r ( t ) y [ k ] ˆ q [ κ ] RF Filter Decoding Demod. Sampling Demapping R C C F 2 Discrete-Time Equivalent Complex Baseband (ECB) Discrete-Time Equivalent Complex Baseband (ECB) � digital signal processing usually performed in � � baseband domain (complex signals) � � discrete-time domain � � discrete-time ECB transmission model with � � transmit symbols x [ k ] � � receive symbols y [ k ] � � complex-valued channel model � → equivalent representation of distortions in ECB domain Complex and Quaternion-Valued Lattices for Digital Transmission 6

  7. A NACHRICHTENTECHNIK Complex-Valued Channel Models (I) Additive White Gaussian Noise (AWGN) Channel Additive White Gaussian Noise (AWGN) Channel y [ k ] = x [ k ] + n [ k ] ���� ���� ���� Re { y [ k ] } +Im { y [ k ] } i Re { x [ k ] } +Im { x [ k ] } i Re { n [ k ] } +Im { n [ k ] } i � discrete-time complex-valued noise n [ k ] � � noise samples usually zero-mean Gaussian with some variance σ 2 � n � samples are white over time (i.i.d.) � � transmission performance depends on SNR expressed as σ 2 x /σ 2 � n Block-Based Transmission over the AWGN Channel Block-Based Transmission over the AWGN Channel y = ¯ x + ¯ n ¯ � sequence of transmit/receive symbols and noise split into blocks � [ y 1 , y 2 , . . . , y N b ] = [ x 1 , x 2 , . . . , x N b ] + [ n 1 , n 2 , . . . , n N b ] � �� � � �� � � �� � y x n ¯ ¯ ¯ � for brevity, block index omitted � Complex and Quaternion-Valued Lattices for Digital Transmission 7

  8. A NACHRICHTENTECHNIK Complex-Valued Channel Models (II) Single-Input/Single-Output (SISO) Block-Fading Channel Single-Input/Single-Output (SISO) Block-Fading Channel · y = h x + n ¯ ¯ ���� ¯ ���� ���� ���� Re { y } +Im { y } i Re { h } +Im { h } i Re { ¯ x } +Im { ¯ x } i Re { ¯ n } +Im { ¯ n } i ¯ ¯ � equivalently described by real-valued matrix equation � � Re { � � Re { h } � � Re { ¯ � � Re { ¯ � y } − Im { h } x } n } = + ¯ Im { y } Im { h } Re { h } Im { ¯ x } Im { ¯ n } ¯ � �� �  h (d) − h (c)   h (c) h (d)  � complex-valued fading factor h = h (d) + h (c) i � → usually complex Gaussian h (d) Re { ¯ x } Re { y } h (d) direct link ¯ − h (c) x y h (c) cross link ¯ ¯ h (d) Im { ¯ x } Im { y } ¯ R 2 R 2 C C Complex and Quaternion-Valued Lattices for Digital Transmission 8

  9. A NACHRICHTENTECHNIK Complex-Valued Channel Models (III) Multiple-Input/Multiple-Output (MIMO) Block-Fading Channel Multiple-Input/Multiple-Output (MIMO) Block-Fading Channel N tx Transmit Antennas N rx Receive Antennas n 1 ¯ x 1 y 1 h 1 , 1 ¯ ¯ Antenna 1 Antenna 1 h j, 1 h 1 ,i h 1 ,N tx n j ¯ x i y j h j,i ¯ ¯ Antenna i Antenna j h N rx , 1 n N rx ¯ x N tx h j,N tx h N rx ,i y N rx ¯ ¯ Antenna N tx Antenna N rx h N rx ,N tx → wireless multi-antenna transmission (same time and frequency) Representation via MIMO System Equation: Representation via MIMO System Equation:         y 1 h 1 , 1 . . . h 1 ,N tx x 1 n 1 ¯ ¯ ¯  .   . .   .   .  = ... · + . . . . .         . . . . . y N rx h N rx , 1 . . . h N rx ,N tx x N tx n N rx ¯ ¯ ¯ � �� � � �� � � �� � � �� � Y H X N channel matrix transmit symbols noise receive symbols Complex and Quaternion-Valued Lattices for Digital Transmission 9

  10. A NACHRICHTENTECHNIK Lattices in Communications Fields of Application Fields of Application � � channel coding � coded modulation � signal constellations � � lattice-reduction-aided M I MO equalization � ⇒ design of coded-modulation schemes for AWGN or M I MO scenarios Problem Problem � transmit and receive signals are complex-valued � � lattice theory is most often considered over real numbers � ⇒ complex-valued lattices are required Complex and Quaternion-Valued Lattices for Digital Transmission 10

  11. A NACHRICHTENTECHNIK Real-Valued Lattices Definition of a Lattice [CS ’ 99, Fis ’ 02] � V � � Λ ( G ) = g v ζ v | ζ v ∈ Z v =1 � created by generator matrix G = [ g 1 , . . . , g V ] ∈ R U × V � � de fi ned over integer ring Z � � infinite set of points (vectors) over U -dimensional Euclidean space � � Abelian group w.r.t. addition � Integer ring Z Integer ring Z � Euclidean ring d min = 1 � � division with small remainder possible � − 2 − 1 0 1 2 � Euclidean algorithm well-de fi ned � � two nearest neighbors � � squared minimum distance d 2 min = 1 � ⇒ how can we extend the de fi nition to complex lattices? Complex and Quaternion-Valued Lattices for Digital Transmission 11

  12. A NACHRICHTENTECHNIK Complex-Valued Lattices Generalized Definition [Ste ’ 19] � V � � Λ ( G ) = g v ζ v | ζ v ∈ I v =1 � complex generator matrix G = [ g 1 , . . . , g V ] ∈ C U × V � � de fi ned over complex integer ring I � 2 Complex Integer Rings Complex Integer Rings i → 1 Im − � Gaussian integers G = Z + Z i 0 � − 1 � Euclidean ring � − 2 � four nearest neighbors � min = | i | 2 = 1 � squared minimum distance d 2 � − 2 − 1 0 1 2 � isomorphic to 2D real-valued integer lattice Z 2 � Re − → � Eisenstein integers E = Z + Z ω � 2 � ω = e i 2 π → ω 1 3 Eisenstein unit (sixth root of unity) � Im − 0 � Euclidean ring � − 1 � six nearest neighbors � min = | ω | 2 = 1 � squared minimum distance d 2 − 2 � � isomorphic to 2D hexagonal lattice A 2 � − 2 − 1 0 1 2 Complex and Quaternion-Valued Lattices for Digital Transmission 12

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