Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Caution: The Complex Normal Distribution !

Evripidis Karseras

Imperial College London

13 May 2014

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Complex Signals

Quote by Carl Gauss

j =√−1, “shadow of shadows”

• Complex signals are also called quadrature signals.

• As engineers we know that many applications involve this notion.

• The choice of the word “imaginary” is rather unfortunate.

• Mathematicians: The definitions are consistent so all is good !

• Engineering: Euler proved the relationship with real trigonometricfunctions.

• The rest is history.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Visualise1

1 Image from DSPrelated.com

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Real World Complex Signals2

• Generate two sinusoids with a phase difference of 90 degrees with agenerator.

• Plug the output of these two generators on the vertical andhorizontal inputs of an oscilloscope.

• Physical representation of a complex signal.

• Everything is real. We treat these numbers in a special way.

2 Example attributed to Richard Lyons (DSPrelated.com)

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Second Order Statistics

• Consider a random vector z ∈ Cn×1 where z = x+ jy.

• The p.d.f of z is that of the real vector z̄ = [x,y]T ∈ R2n×1.

• For real Gaussian random variables, this p.d.f can be exactlycharacterised from the covariance matrix Kz̄ = E[z̄z̄∗] (assumezero mean).

• The following relationships can be easily proven:

E[xxT ] =1

2Re{Kz +Mz} E[yyT ] =

1

2Re{Kz −Mz}

E[xyT ] =1

2Im{−Kz +Mz} E[yxT ] =

1

2Im{Kz −Mz}

where Mz = E[zzT ] is called the pseudo-covariance matrix.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Circular Symmetry

• We need both matrices Kz and Mz to fully characterise z.

Definition - Circular Symmetry

• The p.d.f of a random variable z is the same as that of ejφz.

• Based on the above in order to have Circular Symmetry:

Kejφz = Kz, Mejφz = Mz.

• The first condition holds for any zero-mean complex z.

• For the second: Mejφz = ej2φMz.

• We have circular symmetry iff Mz vanishes !

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

The Gaussian Case

• The Circularly-symmetric complex normal is written as CN (0,K)(since we do not need matrix M).

• Alternatively, N(

0, 12

[Re{K} −Im{K}Im{K} Re{K}

])• The p.d.f can be proven to be p(z) = 1

2π|Kz |e−z∗K−1

z z.

• The univariate case with unit variance: p(z) = 1πe−|z|2 explains

better the term circular.

• The magnitude of z has the Rayleigh distribution which helps inmodelling multipath fading communications channels.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Some Properties

Theorem

Complex vector z is C.S.N iff it can be written as z = Aw whereA ∈ Cn×m and w ∼ CN (0, I). Also z ∼ CN (0,AA∗).

• In simple words, any linear transformation of unit-variance i.i.dcircularly-symmetric complex normal random variables results in acircularly-symmetric normal random variable.

Shift Invariance

Let z ∼ CN (0,Kz). Suppose that there exists y = Bz. Theny ∼ CN (0,BKzB

∗).

• Proof sketch: Use the theorem above.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

• Basically everyone just uses the CN straight ahead.

• In most engineering applications it is the easiest thing to do.

• Does not mean that it is actually correct.

• In many publications the Hermitian transpose is used in place of thetranspose operator.

• Hardly any mention as to why the pseudo-covariance matrixvanishes.

• The use of complex distributions in engineering problems is stillbeing studied, especially in cases where the pseudo-covariancematrix does not vanish.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Complex Multitask Bayesian Compressive Sensing

• What is Multitask BCS:

yl = Φlwl + εl l = 1 · · ·L

• We have to solve several BCS problems at once.

• The paper considers complex datasets.

• Motivation: A proper treatment of this issue.

• The hyper-prior is shared:

wlm ∼ CN (0, αm).

• Hence the name “multitask”.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Some quotes...

• “group sparsity of these two components...jointly treats the real andimaginary components”

• “exploiting the fact that the real and imaginary parts are likely toshare the same sparsity pattern.”

• “the real and imaginary components have been treated as twoseparate variables in the literature ”

• “handles complex-value problem by dividing a complex weight intoindependent real and imaginary components”

• “reduces the sparsity by a factor of two”

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Proposed Recovery Algorithm

• Main Contribution: Equation (9).

• For the parameter estimation an iterative procedure is followed.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Experimental Results

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

References

Robert G. Gallager.Circularly-Symmetric Gaussian random vectors. Book Chapter.

Qisong Wu, Yimin D. Zhang, Moeness G. Amin and Braham HimedComplex Multitask Bayesian Compressive Sensing. ICASSP 2014.

Preliminaries Random Complex Numbers Complex Normal Distribution Example paper References

Thanks !