Discrete Signaling for Non-Coherent, Single-Antenna, Rayleigh Block-Fading ChannelsMarcin Pikus12, Gerhard Krammer2, and Georg Bocherer2

1Huawei Technologies Duesseldorf GmbH, Germany 2Institute for Communications Engineering, Technical University of Munich, Germany

Rayleigh Block Fading Channel (RBFC)

Used in modeling of wireless channels — it captures theuncertainty of wireless channels and fadings correlation-in-time

The fading coefficient H is constant within a block of Tsymbols and changes independently between blocksThe parameter T represents the channel coherence timeThe fading coefficient H is unknown to the transmitter andreceiverThe receiver has to estimate the channel state and the datafrom the received symbols

√γX × + Y

H ∼ NC (0, 1) N ∼ NC (0, I)

E[‖X‖2] = T

[Y1, . . . ,YT ]† = H·√γ·[X1, . . . ,XT ]† + [N1, . . . ,NT ]† (1)

Y = H·√γX + N (2)

Signaling in the Literature

CSI Capacity When the receiver knows H, Gaussian IID inputsignal X ∼ NC (0, I) achieves the capacity. This rateupper-bounds all non-coherent rates.

Gaussian IID When the receiver does not know H, Gaussian IIDinput signal X ∼ NC (0, I) in not capacity-achieving.However it performs well for large T [1].

Pilot schemes The transmitter inserts a fixed pilot symbol insideeach fading block. At the receiver the pilot symbol isused to estimate the fading H and then the estimate H isused to decode the data [2].

USTM Unitary Space-Time Modulation transmits in each fadingblock a vector Φ uniformly distributed on theT -dimensional complex sphere with radius

√T [3].

Experimental Signaling

Discrete Product Form is the proposed discrete signaling schemeinspired by the capacity achieving distribution for RBFC[4].

On-Off USTM is an extension of the USTM. The schemetransmits either a vector Φ as in USTM or a vector of0’s. We optimize the probability P0 of transmitting the0-symbol. This scheme is an isotropicaly (continuous)counterpart of DPF.

On-Off Gaussian is an extension of Gaussian IID signaling. Thescheme transmits either a vector of Gaussian IID RVs ora vector of 0’s. We optimize the probability P0 oftransmitting the 0-symbol.

The Capacity-Achieving Coding SchemeThe capacity-achieving signaling over non-coherent RBFC (alsocalled product form) reads as

X = V Φ (3)

V and Φ are independent RVsΦ is uniformly distributed on the complex T -dimensionalsphere with radius

√T

V is discrete real non-negative RV

v1Φ1

v1 Φ2

v 2Φ3

v2Φ4

Discretization

1 Choose spheres radius, i.e., choose RV V

V =

0, with probability 1− P1

v1 =√

1P1, with probability P1

(4)

2 Sample the spheres by choosing the following points

Θ = [Θ1, . . . ,ΘT ] (5)

where Θi are IID uniformly distributed RVs on the QPSKalphabet A4 = {1,−1, j ,−j}.

We get the Discrete Product Form scheme

X =

0, with probability 1− P1√1

P1Θ with probability P1,

(6)

and P1 is chosen to maximize the mutual information.

Computing the Mutual Information

The following expansion will be used to compute the mutualinformation

I(Y ; X ) = h(Y )− h(Y |X ) (7)

The RV Y |{X = x} is a zero-mean circular-symmetricGaussian RV. The covariance matrix equals to

E[Y Y †

∣∣∣X = x]

= InB + γxx † (8)

Using the log-det formula, the entropy h(Y |X ) is

h(Y |X ) = EX [h(Y |X = X )]

= T log(πe) + P1 log(

1 + γTP1

).

Results: Rate vs Eb/N0

Computing h(Y )

The following formula can be used to compute h(Y )

h(Y ) = EY [− log p(Y )]. (9)

The knowledge of p(y) is needed to evaluate (9). However, adirect computation fails due to exponential complexity growthin block length T

p(y) =∑

xP(x)p(y |x) (10)

Instead, we use the fact that Y conditioned on V and H is avector of IID RVs. First expand

p(y) = Pr(V=0) p(y |V=0)︸ ︷︷ ︸∼ NC (0, I)

+ Pr(V=v1)p(y |V=v1) (11)

Then the second term

p(y |v1) = EH[p(y |v1,H)

]= EH

[ T∏i=1

p(yi |v1,H)]

= EH

T∏i=1

∑θi∈A4

P(θi )p(yi |θi , v1,H)

= EH

T∏i=1

∑θi∈A4

14 ·

1π

e−|yi−H√γθi v1|2

= EH[f(y , v1,H

)](12)

The complexity of computing f(y , v1,H

)grows linearly with

respect to the block size T . To evaluate the expectationin (12) we apply MC averaging.

Discussion/Conclusions

Gap to continuous dist.: The proposed Discrete Product Form(light green) achieves similar performance to itsisotropical counterpart (On-Off USTM, blue) up to 0.4b/CU and saturates approx at 1 b/CU and 2 b/CU forblock lengths T = 2 and T = 50, respectively.

Using the 0-symbol: Compare the performance of thecorresponding schemes: USTM (dark green) and On-OffUSTM (blue) or Gaussian IID (red) and On-Off Gaussian(brown). Introducing a non-zero probability mass at the0-symbol significantly improves the performance. Also forhigher SNR, especially with small T .

Using on-a-sphere signal: For T = 2 observe the gap betweenOn-Off Gaussian (brown) and On-Off USTM (blue).Both signals are on-off schemes so the gain is due to theon-a-sphere structure of the On-Off USTM during thon-cycle. For low rates, also Discrete Product Form(light green) is better than On-Off Gaussian for the samereason.

References

[1] F. Rusek, A. Lozano, and N. Jindal, “Mutual information of IID complexGaussian signals on block Rayleigh-faded channels,” IEEE Trans. Inf. Theory,vol. 58, no. 1, pp. 331–340, Jan. 2012.

[2] B. Hassibi and B. Hochwald, “How much training is needed inmultiple-antenna wireless links?” IEEE Trans. Inf. Theory, vol. 49, no. 4, pp.951–963, Apr. 2003.

[3] B. Hochwald and T. Marzetta, “Unitary space-time modulation formultiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inf.Theory, vol. 46, no. 2, pp. 543–564, Mar. 2000.

[4] M. Pikus, G. Kramer, and G. Bocherer, “Discrete signaling for non-coherent,single-antenna, rayleigh block-fading channels,” IEEE CommunicationsLetters, vol. PP, no. 99, pp. 1–1, 2016.

Institute forCommunications Engineering Technische Universitat Munchen

0 2 4 6 8 100

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Eb/N0 [dB]

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[b/C

U]

CSI CapacityOn-Off USTMDisc. Prod. FormGaussian IIDOn-Off GaussianUSTMPilot Scheme

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U]

CSI CapacityOn-Off USTMDisc. Prod. FormGaussian IIDOn-Off GaussianUSTMPilot Scheme

Information rates for block length T = 2. Information rates for block length T = 50.