Carrier phase and amplitude estimation for phase shift keying using pilots and data

0.34 MB·English

by  Robby McKilliam

1
Carrier phase and amplitude estimation for phase
shift keying using pilots and data
Robby McKilliam, Andre´ Pollok, Bill Cowley, I. Vaughan L. Clarkson and Barry Quinn
Abstract—We consider least squares estimators of carrier of the data symbols d . The sets P and D are disjoint, i.e.,
i
phase and amplitude from a noisy communications signal that P D = where is the empty set, and we let L= P D
contains both pilot signals, known to the receiver, and data ∩ ∅ ∅ | ∪ |
be the total number of symbols transmitted.
signals, unknown to the receiver. We focus on signaling con-
3 We assume that time offset estimation has been performed
stellations that have symbols evenly distributed on the complex
1 unit circle, i.e., M-ary phase shift keying. We show, under andthatLnoisyM-PSKsymbolsareobservedbythereceiver.
0 reasonably mild conditions on the distributionof the noise, that The received signal after matched filtering is,
2 theleastsquaresestimatorofcarrierphaseisstronglyconsistent
n andasymptoticallynormallydistributed.However,theamplitude yi =a0si+wi, i P D, (1)
∈ ∪
a estimator is not consistent, but converges to a positive real
J number that is a function of the true carrier amplitude, the where wi is noise and a0 = ρ0ejθ0 is a complex number
noisedistributionandthesizeoftheconstellation.Ourtheoretical representing both carrier phase θ and amplitude ρ (by
9 0 0
reexsisutl,tsi.cea.,nnaolnsocobheereanptpldieedtetcotiotnh.eTcahseerwehsuelrtesnoof pMiloonttseymCbarollos definitionρ0 isapositiverealnumber).Ouraimistoestimate
] a from the noisy symbols y ,i P D . Complicating
T simulations are provided and these agree with the theoretical 0 { i ∈ ∪ }
matters is that the data symbols d ,i D are notknownto
I results. { i ∈ }
. the receiver and must also be estimated. Estimation problems
s Index Terms—Detection, phaseshift keying, asymptotic statis-
c of this type have undergone extensive prior study [2–10]. A
tics
[ practical approach is the least squares estimator, that is, the

minimisers of the sum of squares function
1
v I. INTRODUCTION
SS(a, d ,i D )= y as 2
0 In passband communication systems the transmitted signal { i ∈ } | i− i|
6 typicallyundergoestimeoffset(delay),phaseshiftandattenu- i∈XP∪D (2)
7 ation (amplitude change). These effects must be compensated = yi api 2+ yi adi 2,
1 | − | | − |
1. foratthereceiver.Inthispaperweassumethatthetimeoffset Xi∈P iX∈D
has been previously handled, and we focus on estimating the where x denotes the magnitude of the complex number x.
0
| |
3 phase shift and attenuation. We consider signalling constella- Theleastsquaresestimatorisalsothemaximumlikelihoodes-
1 tionsthathavesymbolsevenlydistributedonthecomplexunit timatorunderthe assumptionthatthe noisesequence w ,i
i
: circle such as binary phase shift keying (BPSK), quaternary Z is additive white and Gaussian. However, as we sh{ow, th∈e
v
}
i phaseshiftkeying(QPSK)andM-aryphaseshiftkeying(M- estimator works well under less stringent assumptions.
X
PSK). In this case, the transmitted symbols take the form, Theexistingliterature[2–8]mostlyconsiderswhatiscalled
r noncoherent detection where no pilot symbols exist (P = ).
a si =ejui, In the noncoherentsetting differential encoding is often use∅d,
where j =√−1 and ui is from the set {0,2Mπ,...,2π(MM−1)} and for this reason the estimation problem has been called
and M 2 is the size of the constellation. We assume that multiple symbol differential detection. A popular approach
≥
some of the transmitted symbols are pilot symbols known to is the so called non-data aided, sometimes also called non-
the receiver and the remainder are information carrying data decision directed, estimator based on the paperof Viterbi and
symbols with phase that is unknown to the receiver. So, Viterbi [2]. The idea is to ‘strip’ the modulation from the
receivedsignalbytakingy /y tothepowerofM.Afunction
i i
si =(dpii ii∈∈DP, iFs t:aRke7→n toRbiescM1ho∠seAnwanhdertehe|∠e|sdteimnoatteosrtohfethceomcaprlreixeraprhgausmeeθn0t
and
whereP isthesetofindicesdescribingthepositionofthepilot A= 1 F(y ) yi M. (3)
symbols pi, and D is a set of indices describing the position L | i| |yi|
i P D
∈X∪ (cid:0) (cid:1)
ApreliminaryversionofthispaperhasbeensubmittedtoICASSP’13[1]. Various choices for F are suggested in [2] and a statistical
Supported under the Australian Governments Australian Space Research
analysisispresented.Acaveatofthisestimatoristhatitisnot
Program. Robby McKilliam, Andre´ Pollok and Bill Cowley are with the
InstituteforTelecommunicationsResearch,TheUniversityofSouthAustralia, obvious how pilot symbols should be included. This problem
SA,5095.VaughanClarksoniswiththeSchoolofInformationTechnology& does not occur with the least square estimator.
ElectricalEngineering,TheUniversityofQueensland,QLD.,4072,Australia.
An important paper is by Mackenthun [7] who described
Barry Quinn is with the Department of Statistics, Macquarie University,
Sydney,NSW,2109,Australia. an algorithm to compute the least squares estimator requiring
2
onlyO(LlogL)arithmeticoperations.Sweldens[8]rediscov- be simplified. Section VIII presents the results of Monte-
ered Mackenthun’s algorithm in 2001. Both Mackenthun and Carlo simulations. These simulations agree with the derived
Sweldenconsideredonlythenoncoherentsetting,butweshow asymptotic properties.
in Section II that Mackenthun’salgorithm can be modified to
include pilot symbols. Our model includes the noncoherent II. MACKENTHUN’SALGORITHM WITHPILOTS
case by setting the number of pilot symbols to zero, that is,
InthissectionwederiveMackentun’salgorithmtocompute
putting P = .
∅ the least squares estimator of the carrier phase and ampli-
In the literature it has been common to assume that the
tude [7]. Mackenthun specifically considered the noncoherent
data symbols d ,i D are of primary interest and that the
{ i ∈ } setting, so we modify the algorithm to include the pilot sym-
complex amplitude a is a nuisance parameter. The metric of
0 bols. For the purpose of analysing computational complexity,
performance is correspondingly the symbol error rate, or bit
we will assume that the number of data symbols D is
error rate. While estimating the symbols (or more precisely | |
proportional to the total number of symbols L, so that, for
thetransmittedbits)isultimatelythegoal,wetaketheopposite
example,O(L)=O(D ).InthiscaseMackentun’salgorithm
point of view here. Our aim is to estimate a0, and we treat requires O(LlogL) |ari|thmetic operations. This complexity
the unknown data symbols as nuisance parameters. This is
arises from the need to sort a list of D elements.
motivated by the fact that in many modern communication | |
Define the sum of squares function
systems the data symbols are coded. For this reason raw
symbol error rate is not of interest at this stage. Instead, we SS(a, d ,i D )= y as 2
i i i
desire an accurate estimator aˆ of a , so that the compensated { ∈ } | − |
0 i P D
arenceaidvdeditivseymnbooisles caˆh−a1nynielc.aTnhbeeadacdcituivraetenlyoismeocdheallnendelusisinga = |yi|2−∈Xa∪siyi∗−a∗s∗iyi+aa∗ , (4)
i P D
common assumption for subsequent receiver operations, such ∈X∪ (cid:0) (cid:1)
as decoding. The estimator aˆ is also used in the computation where denotes complex conjugate. The minimiser of SS
∗
with respect to a as a function of d ,i D is
of decoder metrics for modern decoders, and for interference i
{ ∈ }
cancellation in multiuser systems. Consequently, our metric 1 1
aˆ( d ,i D )= y s = Y (5)
of performance will not be symbol or bit error rate, but { i ∈ } L i ∗i L
aˆ a0 . It will be informative to consider the carrier phase i∈XP∪D
a|nd−amp|litudeestimatorsseparately,thatis,if aˆ=ρˆejθˆwhere where L= P D is the total number of symbols transmit-
| ∪ |
ρˆis a positive real number, then we consider θˆ θ and ted, and to simplify our notation we have put
0 π
|h − i |
ρˆ ρ .Thefunction denotesitsargumenttaken‘modulo
2|π−’ in0to| the interval [h·iππ,π). It will become apparent why Y = yis∗i = yip∗i + yid∗i.
θˆ θ0 π rather than −θˆ θ0 is the appropriate measure of i∈XP∪D Xi∈P iX∈D
h − i −
error for the phase parameter. NotethatY isafunctionoftheunknowndatasymbols di,i
{ ∈
It is possible to generalise the results we present here to D and we could write Y( d ,i D ), but have chosen to
i
} { ∈ }
allowdatasymbolswithvaryingconstellationsize,i.e.varying suppress the argument ( d ,i D ) for notational brevity.
i
{ ∈ }
M. For example, one might give more importance to certain Substituting 1Y for a into (4) we obtain SS minimised with
L
data symbols and use BPSK (M = 2) for these, but QPSK respect to a,
(M = 4) for other less important symbols. This is related to 1
what is called unequal error protection in the literature [11, SS( di,i D )=A Y 2, (6)
{ ∈ } − L| |
12].Tokeepourideasandnotationfocusedwedon’tconsider
this further here. where A = i P D|yi|2 does not depend on the di. The
The paper is organised in the following way. Section II leastsquaresesti∈ma∪torsofthedatasymbolsaretheminimisers
P
extends Mackenthun’s algorithm for the coherent case, when of (6). Observe that given candidate values for the data
both pilot symbols and data symbols exist. Section III de- symbols,we can computethe correspondingSS( di,i D )
{ ∈ }
scribes properties of complex random variables that we need. in O(L) arithmetic operations. It turns out that there are at
Section IV states two theorems that describe the statistical most M D candidate values of the least squares estimator of
| |
properties of the least squares estimator of carrier phase θˆ the data symbols [7, 8].
and amplitude ρˆ. We show, under some reasonably general To see this, let a = ρejθ where ρ is a nonnegative real.
assumptions about the distribution of the noise w ,...,w , Now,
1 L
t√haLt θˆhθˆθ− θ0iisπasycomnpvteortgiceasllyalnmoorsmtalsluyredliystritboutzeedroasaLnd that. SS(ρ,θ,{di,i∈D})= yi−ρejθsi 2
0 π
h − i →∞ i P D
However, ρˆis not a consistent estimator of the amplitude ρ . ∈X∪ (cid:12) (cid:12)
0 = y ρejθp 2+(cid:12) y ρ(cid:12)ejθd 2. (7)
The asymptotic bias of ρˆ is small when the signal to noise i i i i
− −
ratio (SNR) is large, but the asymptotic bias is significant Xi∈P(cid:12) (cid:12) iX∈D(cid:12) (cid:12)
when the SNR is small. Sections V and VI provide proofs We have slightly(cid:12)abused not(cid:12)ation he(cid:12)re by reus(cid:12)ing SS.
of the theorems stated in Section IV. In Section VII we This should not cause confusion as SS(a, d ,i D ),
i
{ ∈ }
consider the special case when the noise is Gaussian. In SS(ρ,θ, d ,i D ), and SS( d ,i D ) are easily
i i
{ ∈ } { ∈ }
this case, our expressions for the asymptotic distribution can told apart by their arguments. For given θ, the least squares
3
estimator of the ith data symbol d is given by minimising whenever i < k where i,k 0,1,..., D 1 . It is
i
y ρejθd 2, that is, convenient to define the indices∈in{to σ to b|e t|ak−en }modulo
i i
−
D , that is, if m is an integer not from 0,1,..., D 1
(cid:12)(cid:12) dˆi(θ)=e(cid:12)(cid:12)juˆi(θ) where uˆi(θ)= ∠(e−jθyi) , (8) |the|n we define σ(m) = σ(k) where k { m mod| D|−an}d
≡ | |
k 0,1,..., D 1 . The first sequence in S is
where ∠() denotesthe complexargument(cid:4)(orphase),(cid:7)and ∈{ | |− }
rounds its·argumentto the nearest multiple of 2π. A word⌊o·⌉f f =f(0)= dˆ(0),i D = b ,i D .
M 0 { i ∈ } { i ∈ }
caution, the notation is often used to denote rounding to
⌊·⌉ The next sequence f is given by replacing the element b
the nearest integer. This is not the case here. If the function 1 σ(0)
in f with b e j2π/M. Given a sequence x we use xe to
round() takes its argument to the nearest integer then, 0 σ(0) − i
· denote x with the ith element replaced by x e j2π/M. Using
i −
x = 2π round Mx . this notation,
⌊ ⌉ M 2π
f =f e .
Note that dˆ(θ) does not depend o(cid:0)n ρ. (cid:1)As defined, uˆ (θ) is 1 0 σ(0)
i i
not strictly inside the set 0,2π,...,2π(M−1) , but this is The next sequence in S is correspondingly
{ M M }
not of consequence, as we intend its value to be considered
f =f e e =f e ,
2 0 σ(0) σ(1) 1 σ(1)
equivalent modulo 2π. With this in mind,
and the kth sequence is
uˆ (θ)= ∠y θ
i i
⌊ − ⌉ f =f e . (11)
k+1 k σ(k)
which is equivalent to the definition from (8) modulo 2π.
In this way, all M D sequences in S can be recursively
We only require to consider θ in the interval [0,2π). | |
enumerated.
Consider how dˆ(θ) changes as θ varies from 0 to 2π. Let
i We wantto findthe f S correspondingto the minimiser
b =dˆ(0) and let k ∈
i i of (6). A na¨ıve approach would be to compute SS(f ) for
k
zi =∠yi−uˆi(0)=∠yi−⌊∠yi⌉ epaacrthickula∈rk{0r,e1q,u.ir.e.s,OM(|LD)|a−rit1h}m.eCticomoppeurtaintigonSs.SS(of,kt)hifsonraa¨ınvye
be the phase difference between the received symbol yi and approach would require O(LM D ) = O(L2) operations in
the hard decision resulting when θ =0, i.e. ⌊∠yi⌉. Then, total. Following Mackenthun [7]|, w|e show how SS(fk) can
be computed recursively.
b , 0 θ <z + π
i ≤ i M Let,
dˆi(θ)=b...b..iiee−−jj22ππ/kM/M,, zzii++ Mππ(2Mk≤−1θ)<≤zθi+<3zMπi+ π(2Mk+1) where, Yk =YSS(f(kf)k)==A−yipL1∗i|+Yk|2, yifk∗i (12)
.
Let bie−j2π =bi, zi+ π(2MM−1) ≤θ <2π. (9) =BXi∈P+iX∈DgkiiX∈,D
where B = y p is independent of the data symbols,
f(θ)= dˆ(θ),i D i P i ∗i
{ i ∈ } and fki denotes∈the ith symbol in fk, and for convenience,
P
be a function mapping the interval [0,2π) to a sequence of we put gki = yifk∗i. Letting gk be the sequence {gik,i ∈D}
M-PSK symbols indexedby the elements ofD. Observe that we have, from (11), that gk satisfies the recursive equation
f(θ) is piecewisecontinuous.Thesubintervalsof [0,2π)over
g =g e ,
which f(θ) remains constant are determined by the values of k+1 k ∗σ(k)
zi,i D . Let where gke∗σ(k) indicates the sequence gk with the σ(k)th
{ ∈ }
element replaced by g ej2π/M. Now,
kσ(k)
S = f(θ) θ [0,2π)
{ | ∈ }
Y =B+ g
be the set of all sequencesf(θ) as θ varies from 0 to 2π. If θˆ 0 0i
i D
is the least squaresestimatorof the phase then S containsthe X∈
sequence dˆ(θˆ),i D corresponding to the least squares can be computed in O(L) operations, and
i
{ ∈ }
estimator of the data symbols, i.e., S contains the minimiser
Y =B+ g
1 1i
of(6).Observefrom(9)thatthereareatmostM D sequences
in S, because there are M distinct values of d| (θ|) for each iX∈D
i =B+(ej2π/M 1)g + g
i D as θ varies from 0 to 2π. − 0σ(0) 0i
∈ i D
The sequences in S can be enumerated as follows. Let σ X∈
=Y +ηg ,
denote the permutationof the indices in D such that z are 0 0σ(0)
σ(i)
in ascending order, that is, where η =ej2π/M 1. In general,
−
z z (10) Y =Y +ηg .
σ(i) σ(k) k+1 k kσ(k)
≤
4
Input: yi,i P D EW =EZ is finite, then W has zero mean because
{ ∈ ∪ } | |
1 for i D do
2 φ∈=∠yi EW = 2π ∞zejθfZ,Θ(z,θ)dzdθ
34 uzi==⌊φφ⌉−u =Z10 Z20πejθ ∞zf (z)dzdθ
5 gi =yie−ju 2π Z0 Z0 Z
6 Y = i P yip∗i + i Dgi = 1 EZ 2πejθdθ =0.
7 aˆ= 1Y∈ ∈ 2π
8 Qˆ =LPL1|Y|2 P If X and Y are real randoZm0 variables equal to the real and
9 η =ej2π/M 1 imaginary parts of W =X+jY then the joint pdf of X and
−
10 σ =sortindices(z) Y is
11 for k =0 to M D 1 do f ( x2+y2)
| |− f (x,y)= Z .
12 Y =Y +ηgσ(k) X,Y 2π x2+y2
p
13 gσ(k) =(η+1)gσ(k)
14 Q= 1 Y 2 Wewillhaveparticularuseofcpomplexrandomvariablesof
15 if Q>L|Qˆ|then the form 1+W where W is circularly symmetric. Let R≥0
16 Qˆ =Q and Φ∈[0,2π) be real random variables satisfying,
17 aˆ= 1Y RejΦ =1+W.
L
18 return aˆ The joint pdf of R and Φ can be shown to be
Algorithm 1: Mackenthun’s algorithm with pilot symbols
rf ( r2 2rcosφ+1)
f(r,φ)= Z − . (14)
2πpr2 2rcosφ+1
−
So, each Yk can be computed from it predecessor Yk 1 Since cosφ has period 2πpand is even on [ π,π] it follows
in a O(1) arithmetic operations. Given Yk, the value −of thatf(r,φ) hasperiod2π andis evenon[ π−,π] with respect
SS(fk) can be computed in O(1) operations using (12). Let to φ. The mean of RejΦ is equal to one b−ecause the mean of
kˆ =argminSS(fk).Theleastsquaresestimatorof a0 isthen W is zero. So,
computed according to (5),
E (RejΦ)=ERcos(Φ)=1, (15)
1 ℜ
aˆ= Y . (13)
L kˆ where () denotes the real part, and
ℜ ·
Pseudocode is given in Algorithm 1. Line 10 contains the E (RejΦ)=ERsin(Φ)=0, (16)
ℑ
function sortindices that, given z = z ,i D , returns the
i
{ ∈ } where () denotes the imaginary part.
permutation σ as described in (10). The sortindicies function ℑ ·
requires sorting D elements. This requires O(LlogL) oper-
ations. The sorti|nd|icies function is the primary bottleneck in IV. STATISTICALPROPERTIES OF THELEASTSQUARES
this algorithm when L is large. The loops on lines 1 and 11 ESTIMATOR
and the operations on lines 6 to lines 8 all require O(L) or In this section we describe the asymptotic propertiesof the
less operations. leastsquaresestimator.Inwhatfollowsweuse x todenote
π
h i
x taken ‘modulo 2π’ into the interval [ π,π), that is
−
x
III. CIRCULARLY SYMMETRICCOMPLEX RANDOM x =x 2πround ,
h iπ − 2π
VARIABLES
(cid:16) (cid:17)
where round() takes its argument to the nearest integer. The
Before describing the statistical properties of the least ·
directionofroundingforhalf-integersisnotimportantso long
squaresestimator,wefirstrequiresomepropertiesofcomplex
as it is consistent. We have chosen to round up half-integers
valued random variables. A complex random variable W is
here.Similarlyweuse x todenotextaken‘modulo 2π’into
saidtobecircularlysymmetricifitsphase∠W isindependent h i M
the interval π , π , that is
ofitsmagnitude W andifthedistributionof∠W isuniform −M M
| |
on [0,2π). That is, if Z 0 and Θ [0,2π) are real random (cid:2)x =x (cid:1)2π round Mx =x x .
variablessuchthatZejΘ≥=W,then∈Θisuniformlydistributed h i − M 2π −⌊ ⌉
The next two theorems describe(cid:0)the a(cid:1)symptotic properties of
on [0,2π) and is independent of Z. If the probability density
the least squares estimator. These are the central results and
function(pdf)of Z is f (z), thenthe jointpdfof Θ andZ is
Z
the chief original contributions of this paper.
1
f (z,θ)= f (z). Theorem 1. (Almost sure convergence) Let w be a se-
Z,Θ 2π Z { i}
quence of independent and identically distributed, circularly
Observe that for any real number φ, the pdf of W and ejφW symmetric complex random variables with Ew1 2 finite, and
are the same, that is, the pdf is invariant to phase rotation. If let y ,i P D be given by (1). Let aˆ=ρˆ|ejθˆ|be the least
i
{ ∈ ∪ }
5
squares estimator of a = ρ ejθ0. Put L = P D and let the proof of Theorem 1 by use of Kolmogorov’s strong law
0 0
| ∪ |
P and D increase in such a way that of large numbers [13].
| | | | The theorems place conditions on θˆ θ rather than
|P| p and |D| d as L . directlyonθˆ θ .Thismakessensebechaus−eth0eiπphasesθ and
0 0
L → L → →∞ −
θ +2πk areequivalentforanyintegerk.So,forexample,we
0
Let Ri 0 and Φi [0,2π) be real random variables expectthe phases 0.99π and 0.99π to be close together,the
satisfying≥ ∈ differencebetweenthembein−g 0.99π 0.99π =0.02π,
R ejΦi =1+ wi , (17) and not 0.99π 0.99π =1|.h9−8π. − iπ|
i
a s |− − |
0 i Theorem2requiresthefunctiongtobecontinuousat 2πk+
M
and define the continuous function π foreachk =0,...,M 1.Thisplacesmildrestrictionson
M −
thedistributionofthenoisew .Forexample,therequirements
G(x)=ph (x)+dh (x) where i
1 2 are satisfied if the joint pdf of the real and imaginary parts
h1(x)=ER1cos(x+Φ1), h2(x)=ER1cos x+Φ1 . of wi is continuous, since in this case f(r,φ) is continuous.
h i Because f(r,φ) has period 2π and is even on [ π,π] with
If p>0 and if G(x) is uniquelymaximised at x=0 over the respect to φ it follows that g has period 2π and−is even on
interval [ π,π) then [ π,π].
1) θˆ −θ 0 almost surely as L , −A key assumption in Theorem 1 is that G(x) is uniquely
0 π
h − i → →∞
2) ρˆ ρ G(0) almost surely as L . maximised at x=0 for x [ π,π). This assumption asserts
0
→ →∞ ∈ −
that G(x) G(0) for all x [ π,π) and that if x is a
Theorem 2. (Asymptotic normality) Under the same condi- ≤ ∈ − { i}
sequence of numbers from [ π,π) such that G(x ) G(0)
tionsasTheorem1, letf(r,φ) be thejointprobabilitydensity − i →
as i then x 0 as i . Although we will not
function of R1 and Φ1, let prove→it ∞here, this ias→sumption i→s no∞t only sufficient, but also
∞ necessary, for if G(x) is uniquely maximised at some x = 0
g(φ)= rf(r,φ)dr then θˆ θ xalmostsurelyasL ,whileifG(x6)is
Z0 notuhniq−uel0yiπm→aximisedthen θˆ θ →wil∞lnotconverge.One
PanudtλˆaLssu=methθˆatθ|0PL|π==p+θ0o(Lθˆ−π1/a2n)damˆndL|=DL|ρˆ=ρd0+Go((0L).−I1f/t2h)e. scyamnmcheetrcikc tahnadt nthoirsmaaslsluymdpistitorhinb−uhteod0ldi.πsWwehwenillwn1otisacttiermcuplatrtloy
−h − i h − i −
functiong iscontinuousat 2πk+ π foreachk=0,...,M further classify those distributions for which the assumption
M M −
1, then the distribution of (√LλˆL,√LmˆL) converges to the holds here.
bivariate normal with zero mean and covariance matrix Theorem 1 defines real numbers p and d to represent the
pA1+dA2 0 proportionofpilotsymbolsanddatasymbolsinthelimitasL
(p+Hd)2 goes to infinity. For Theorem 2 we need the slightly stronger
0 ρ2(pB +dB )
(cid:18) 0 1 2 (cid:19) condition that
as L , where P D
→∞ | | =p+o(L−1/2) and | | =d+o(L−1/2).
M 1 L L
−
H =h2(0)−2sin(Mπ ) g(2Mπk+ Mπ ), This stronger condition is required to prove the asymptotic
k=0 normality of √Lmˆ .
X L
A =ER2sin2(Φ ), A =ER2sin2 Φ , The next two sections give proofs of Theorems 1 and 2.
1 1 1 2 1 h 1i The proofs make use of variouslemmas, which are provedin
B =ER2cos2(Φ ) 1, B =ER2cos2 Φ h2(0). the appendix.
1 1 1 − 2 1 h 1i− 2
The proof of Theorem 1 is in Section V and the proof
V. PROOF OF ALMOSTSURE CONVERGENCE(THEOREM1)
of Theorem 2 is in Section VI. Before giving the proofs
Substituting dˆ(θ),i D from (8) into (7) we obtain SS
we discuss the assumptions made by these theorems. The { i ∈ }
minimised with respect to the data symbols,
assumption that w ,...,w are circularly symmetric can be
1 L
relaxed, but this comes at the expense of making the theorem SS(ρ,θ)= y ρejθp 2+ y ρejθdˆ(θ) 2
statementsmorecomplicated.Ifw isnotcircularlysymmetric i− i i− i
i
tahlseonothnethdeisttrraibnusmtioitntedofsyRmibaonlsd Φsii,imayPdepDend. Aons aa0reasunldt =XiA∈P−(cid:12)(cid:12)ρZ(θ)−ρZ(cid:12)(cid:12)∗(θ)iX∈+DL(cid:12)(cid:12)(cid:12)ρ2, (cid:12)(cid:12)(cid:12)
{ ∈ ∪ }
the asymptotic variance described in Theorem 2 depends on where
a and s ,i P D , rather than just ρ . The circularly
0 { i ∈ ∪ } 0 Z(θ)= y e jθp + y e jθdˆ (θ),
symmetric assumption may not always hold in practice, but i − ∗i i − ∗i
wefeelitprovidesasensibletradeoffbetweensimplicityand Xi∈P iX∈D
generality. and Z∗(θ) is the conjugate of Z(θ). Differentiating with
The assumption that Ew 2 =Ew 2 is finite implies that respect to ρ and setting the resulting expression to zero gives
1 i
Rihasfinitevariancesinc|eE|Ri2 =1|+E||wi|2.Thisisrequired the least squares estimator of ρ0 as a function of θ,
iWneTwheilolraelmso2usseotthheatfatchtethcaotnRstanhtassAfi1n,itAe2v,arBia1ncaendtoBsi2mepxliisfty. ρˆ(θ)= Z(θ)+Z∗(θ) = 1 (Z(θ)), (18)
i 2L Lℜ
6
where () denotes the real part. Substituting this expression and since |P| p and |D| d as L ,
ℜ · L → L → →∞
into SS(ρ,θ) gives SS minimised with respect to ρ and the
lim EG (λ)=G(λ)=ph (λ)+dh (λ).
data symbols, L 1 2
L
→∞
1 As is customary, let Ω be the sample space on which the
SS(θ)=A (Z(θ))2.
− Lℜ random variables w are defined. Let A be the subset of
i
We again abuse notation by reusing SS, but this should not the sample space {Ω o}n which G(λˆL) G(0) as L .
→ → ∞
causeconfusionasSS(ρ,θ)andSS(θ)areeasilytoldapartby Lemma 1 shows that Pr A =1. Let A′ be the subset of the
ρtˆheairreinppousittsi.veB.yHdoewfienviteiro,nρˆ(thθe) a=mpl(iZtu(dθe))ρ0maanydtaitkseensteigmaatitvoer issamupnliequseplaycemaoxnimwihsiecdha{λˆtLx}→=00,asitLfo→llow∞s.thBaetcaGu(sλˆeLG)(x)
values for some θ [ π,π). Thℜe least square estimator θˆ G(0) only if λˆL 0 as L . So A A′ and therefo→re
of θ0 is the minimis∈er o−f SS(θ) under the constraint ρˆ(θ) = Pr{A′}≥Pr{A}→=1. Part 1→of∞Theorem⊆1 follows.
(Z(θ)) > 0. Equivalently θˆ is the maximiser of (Z(θ)) It remains to prove part 2 of the theorem regarding the
wℜith no constraints required. ℜ convergenceof the amplitude estimator ρˆ. From (18),
We are thus interested in analysing the behaviour of the 1
maximiser of ℜ(Z(θ)). Recalling the definition of Ri and Φi ρˆ= Lℜ(Z(θˆ))=ρ0GL(λˆL). (21)
from (17),
Lemma 8 in the appendix shows that G (λˆ ) converges
L L
yi =a0si+wi almost surely to G(0) as L , and ρˆ consequently
→ ∞
w converges almost surely to ρ G(0) as required. It remains to
i 0
=a s 1+
0 i a s proveLemmas 1 and 8. These are provedin Section A of the
(cid:18) 0 i(cid:19)
=a s R ejΦi appendix.
0 i i
=ρ R ej(Φi+θ0+∠si).
0 i VI. PROOF OF ASYMPTOTICNORMALITY(THEOREM2)
Recalling the definition of dˆi(θ) and uˆi(θ) from (8), We first prove the asymptotic normality of √LλˆL. Once
uˆ (θ)= ∠y θ thisisdonewe willbeable toprovethenormalityof √LmˆL.
i ⌊ i− ⌉ Recall that λˆ is the maximiser of the function G defined
= θ +Φ +∠s θ L L
⌊ 0 i i− ⌉ in (20). The proof is complicated by the fact that GL is not
≡⌊hθ0−θiπ+Φi+∠si⌉ (mod 2π) differentiable everywhere due to the function h·i not being
= λ+Φ +∠s , differentiable at multiples of π . This prevents the use of
i i M
⌊ ⌉ “standard approaches” to proving normality that are based
wwe−hejeuˆcrieo(nθw)s,iediteprfuouˆtllioλ(wθ)=sethqhauθti0,v−awlehθneitnπmiaonddDulw,oh2eπre.,BaescainusSeedcˆ∗iti(oθn) I=I, aoshtnoλˆwthse.thSmaitmeatihnlaervdapelrruiovepaettirhvtieeeosGreh′Lmavde[o1e4bs–ee1exn9i]s.tu,sHaendodwbiesyveesrq,oumLaleemtoomfzeatrho9e,
∈ L
y e jθdˆ (θ)=ρ R ej(λ+Φi+∠si λ+Φi+∠si ) present authors to analyse the behaviourof polynomial-phase
i − ∗i 0 i −⌊ ⌉ estimators [20]. Define the function
=ρ R ej(λ+Φi λ+Φi )
0 i −⌊ ⌉
1 1
=ρ0Riejhλ+Φii (19) RL(λ)= L Risin(λ+Φi)+L Risinhλ+Φii. (22)
i P i D
since x+∠s = x +∠s for all x R as a result of ∠s X∈ X∈
i i i
being⌊a multipl⌉e of⌊2π⌉. Otherwise, whe∈n i P, Whenever GL(λ) is differentiable G′L(λ) = RL(λ), and
M ∈ so R (λˆ ) = G (λˆ ) = 0 by Lemma 9. Let Q (λ) =
L L ′L L L
yie−jθp∗i =ρ0Riej(λ+Φi). ERL(λ)−ERL(0) and write
Now, 0=R (λˆ ) Q (λˆ )+Q (λˆ )
L L L L L L
−
Z(θ)=ρ0 Riej(λ+Φi)+ρ0 Riejhλ+Φii. =√L RL(λˆL)−QL(λˆL) +√LQL(λˆL).
Xi∈P iX∈D Lemma 11 show(cid:0)s that (cid:1)
Let
1 √LQ (λˆ )=√Lλˆ p+Hd+o (1)
G (λ)= (Z(θ)) (20) L L L P
L
ρ Lℜ
0
where o (1) denotes a sequenc(cid:0)e of random var(cid:1)iables con-
P
andputλˆL = θˆ θ0 π = θ0 θˆ π.Sinceθˆisthemaximiser verging in probability to zero as L , and p,d and H
of (Z(θ)) it−fohllo−wsithatλˆhL is−thie maximiserof GL(λ). We are defined in the statement of Theore→ms∞1 and 2. Lemma 16
wilℜlshowthatλˆL convergesalmostsurelytozeroas L . shows that
→∞
The proof of part 1 of Theorem 1 follows from this.
Recall the functions G, h1 and h2 defined in the statement √L RL(λˆL)−QL(λˆL) =oP(1)+√LRL(0).
of Theorem 1. Observe that
It follows(cid:0)from the three equat(cid:1)ions above that,
P D
EGL(λ)= |L|h1(λ)+ |L|h2(λ) 0=oP(1)+√LRL(0)+√LλˆL p+Hd+oP(1)
(cid:0) (cid:1)
7
and rearranging gives, VIII. SIMULATIONS
√LR (0) We present the results of Monte-Carlo simulations with
√Lλˆ =o (1) L .
L P the least squares estimator. In all simulations the noise sam-
− p+Hd+o (1)
P
ples w ,...,w are independent and identically distributed
1 L
Lemma17showsthatthedistributionof √LR (0)converges circularly symmetric and Gaussian with real and imaginary
L
to the normalwith zero mean and variance pA +dA where parts having variance σ2. Under these conditions the least
1 2
A and A are defined in the statement of Theorem 2. It squares estimator is also the maximum likelihood estimator.
1 2
followsthatthedistributionof√Lλˆ convergestothenormal Simulations are run with M =2,4,8 (BPSK, QPSK,8-PSK)
L
with zero mean and variance and with signal to noise ratio SNR = ρ20 between -20dB
2σ2
and 20dB in steps of 1dB. The amplitude ρ = 1 and θ
pA +dA 0 0
1 2
. is uniformly distributed on [ π,π). For each value of SNR,
(p+Hd)2 −
T = 5000 replications are performed to obtain T estimates
We now analyse the asymptotic distribution of √LmˆL. Let ρˆ1,...,ρˆT and θˆ1,...,θˆT.
T (λ)=EG (λ). Using (21), Figures 1, 2 and 3 show the sample mean square error
L L
(MSE) of the phase estimator when M = 2,4,8 with
√LmˆL =√Lρ0 GL(λˆL) G(0) L = 4096 and for varying proportions of pilots symbols
−
=√Lρ0(cid:0)GL(λˆL) TL(λˆ(cid:1)L)+TL(λˆL) G(0) . |P| = 0,3L2,L8,L2,L. When |P| 6= 0 (i.e. coherent detection)
− − the mean square error is computed as 1 T θˆ θ 2.
Lemma 18 shows t(cid:0)hat (cid:1) Otherwise, when P = 0 the mean squareTerroi=r1ihs cio−mpu0tieπd
as 1 T θˆ |θ |2 as in [1]. The dotsP, squares, circles
√L G (λˆ ) T (λˆ ) =o (1)+X , T i=1h i − 0i
L L L L P L and crosses are the results of Monte-Carlo simulations with
−
P
the least square estimator. The solid lines are the estimator
where X =√(cid:0)L G (0) T (0)(cid:1). Lemma 19 shows that
L L − L MSEs predicted by Theorem 2. The prediction is made by
√L(cid:0) T (λˆ ) G(0(cid:1)) =o (1). dividing the asymptotic covariancematrix by L. The theorem
L L P
− accuratelypredictsthe behaviourof the phase estimatorwhen
Itfollowsthat√Lmˆ(cid:0) =ρ X +o (cid:1)(1).Lemma20showsthat L is sufficiently large. As the SNR decreases the variance of
L 0 L P
thedistributionofX convergestothenormalwithzeromean thephaseestimatorapproachesthatoftheuniformdistribution
L
and variance pB +dB as L where B and B are on [ π,π) when P = 0 and the uniform distribution on
definedinthestat1emento2fTheor→em∞2.Thus,thed1istributi2onof [ π−, π ) when P| |=6 0 [1]. Theorem 2 does not model
−M M | |
√Lmˆ convergesto the normalwith zero mean and variance this behaviour in the sense that for any fixed L there exist
L
ρ2(pB +dB ) as required. Because X does not depend on sufficiently small values of SNR for which Theorem 2 does
0 1 2 L
λˆ , it follows that cov(X ,√Lλˆ )=0, and so, not produce accurate predictions of the MSE. As the SNR
L L L
increases the variance of the estimators converge to that of
cov(√Lmˆ ,√Lλˆ ) cov(ρ X ,√Lλˆ )=0 the estimator where all symbols are pilots, i.e. P =L.
L L 0 L L
→ | |
Figures 1, 2 and 3 also display the sample MSE of the
as L . The lemmas that we have used are proved in
noncoherent phase estimator of Viterbi and Viterbi [2] de-
→ ∞
Section B of the appendix.
scribed by (3). This estimator requires selection of a function
F that transforms the amplitude of each sample prior to
VII. THEGAUSSIAN NOISECASE the final estimation step. Viterbi and Viterbi propose several
viable alternatives, from which we have chosen F(x) = 1.
Let the noise sequence wi be complex Gaussian with The Viterbi and Viterbi estimator is only applicable in the
{ }
independent real and imaginary parts having zero mean and
noncoherent setting, i.e. when P = 0. The sample MSE of
variance σ2. The joint density function of the real and imagi- the least squares estimator (wh|en| P = 0) and the Viterbi
nary parts is | |
and Viterbi estimator is similar. The least squares estimator
1 e−2σ12(x2+y2). appears slightly more accurate for some values of SNR.
2πσ2 Figures 4, 5 and 6 show the variance of the amplitude esti-
Theorems 1 and 2 hold, and since the distribution of w is matorwhenM =2,4,8andwithL=32,256,2048andwhen
1
circularly symmetric, the distribution of R ejΦ1 is identical thenumberofpilotssymbolsis P =0,L,L.Thesolidlines
1 | | 2
to the distribution of 1+ 1 w . It can be shown that arethevariancepredictedbyTheorem2.Thedotsandcrosses
ρ0 1
show the results of Monte-Carlo simulations. Each point is
g(φ)= co2sπφe−κ+ Ψ(√√2κπcκosφ)e−κsin2φ 2+κcos2φ creoqmupiruetseGd(a0s)thtoebuenbkinaoswedn.eIrnroprraT1cticeTi=G1(0ρˆ)im−aρy0nGo(t0b)e2k.nTowhins
P (cid:0) (cid:1)
(cid:0) (cid:1) at the receiver, so Figures 4, 5 and 6 serve to validate the
where κ= ρ20 and Ψ(t)= 1 t e x2/2dx is the cumu- correctness of our asymptotic theory, rather than to suggest
2σ2 √2π −
lative density function of the stand−a∞rd normal. The value of the practical performance of the amplitude estimator. When
R
A ,A ,B and B can be efficiently computed by numerical SNR is large G(0) is close to 1 and the bias of the amplitude
1 2 1 2
integration using this formula. estimatorissmall.However,G(0)growswithoutboundasthe
8
varianceof thenoise increases,so the biasissignificantwhen
SNR is small.
Figure 7 shows the MSE of the phase estimator when
0.1
M = 4 and L = 32,256,2048 and the number of pilots is
P = L,L. The figure depicts an interesting phenomenon.
|Wh|en L8 = 2048 and P = L = 256 the number of 0.01
| | 8
pilots symbols is the same as when L = P = 256. When
| | ˆθ
the SNR is small (approximately less than 0dB) the least 0.001 SE
squares estimator using the 256 pilots symbols and also the M
2048 256 = 1792 data symbols performs worse than the
estima−tor that uses only the 256 pilots symbols. A similar 10−4
Theory
phenomenon occurs when L = 256 and |P| = L8 = 32. This ViterbiViterbi
behaviour suggests modifying the objective function to give 10−5 |PP|==L0
thepilotssymbolsmoreimportancewhentheSNRislow.For | |
P =L/8
example, rather than minimise (2) we could instead minimise |P|=L/2 SNR(dB)
a weighted version of it, 10−6 | |
20 10 0 10 20
− −
SSβ(a,{di,i∈D})= |yi−asi|2+β |yi−adi|2, Fig.1. PhaseerrorversusSNRforBPSK withL=4096.
i P i D
X∈ X∈
where the weight β would be small when SNR is small and
near 1 when SNR is large. Computing the aˆ that minimises
SS can be achieved with only a minormodification to algo-
β 0.1
rithm1.Line5ismodifiedtog =βy e ju andlines7and17
i i −
are modified to aˆ = 1 Y. For fixed β the asymptotic
P +βD
properties of this weig|h|ted|es|timator could be derived using 0.01
the techniques we have developed in Sections IV, V and VI.
ˆθ
This would enable a rigorous theory for selection of β at the 0.001 SE
receiver. One caveat is that the receiverwould require knowl- M
edge about the noise distribution in order to advantageously
choose β. We do not investigate this further here. 10−4 Theory
ViterbiViterbi
P =L
| |
IX. CONCLUSION 10−5 |PP|==L0/32
| |
P =L/8
We consideredleast squaresestimatorsof carrierphaseand |P|=L/2 SNR(dB)
10−6 | |
amplitude from noisy communications signals that contain
20 10 0 10 20
both pilot signals, known to the receiver, and data signals, − −
unknown to the receiver. We focused on M-ary phase shift Fig.2. PhaseerrorversusSNRforQPSKwith L=4096.
keying constellations. The least squares estimator can be
computed in O(LlogL) operations using a modification of
an algorithm due to Mackenthun [7], and is the maximum
likelihoodestimatorinthecasethatthenoiseisadditivewhite
0.1
and Gaussian.
We showed, under some reasonably general conditions on
the distribution of the noise, that the phase estimator θˆ is 0.01
strongly consistent and asymptotically normally distributed.
However,the amplitude estimator ρˆ is biased, and converges ˆθ
to G(0)ρ0. This bias is large when0the signal to noise ratio 0.001 MSE
is small. It would be interesting to investigate methods for
correcting this bias. A method for estimating G(0) at the 10−4 Theory
receiver appears to be required. ViterbiViterbi
P =L
maMncoentoefCtahrelolesaimstuslaqtuioanressweesrteimuasteodr atondasaslessos ttohevpaelirdfaotre- 10−5 ||PP||==L0/32
| |
P =L/8
our asymptotic theory. Interestingly, when the SNR is small, |P|=L/2 SNR(dB)
it is counterproductive to use the data symbols to estimate 10−6 | |
the phase (Figure 7). This suggests the use of a weighted 20 10 0 10 20
− −
objective function, which would be an interesting topic for
Fig.3. PhaseerrorversusSNRfor 8-PSK withL=4096.
future research.
9
1
1
0.01
ˆρ
e
c
n
a
ri
a
v 0.01
10−4 L=32 ˆθ
E
Theory L=256 MS
P =L
| |
P =0
|P|=L/2 SNR(dB) L=2048
10−6 | |
20 10 0 10 20
− − 10−4 L=32
Fig.4. Unbiasedamplitude errorversusSNRforBPSK.
L=256
1
Theory
P =L L=2048
|P|=L/8 SNR(dB)
10−6 | |
20 10 0 10 20
− −
0.01
ˆρ Fig.7. PhaseerrorversusSNRforQPSK.
e
c
n
a
ri
a
v
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|P|=0 Conf. Acoust. Spe. Sig. Process., Nov. 2012.
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an [7] K. M. Mackenthun, “A fast algorithm for multiple-symbol
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Fig.6. Unbiasedamplitude errorversusSNRfor 8-PSK.
[11] M. Aydinlik and M. Salehi, “Turbo coded modulation for
10
unequal error protection,” IEEE Trans. Commun., vol. 56, no. Since
4, pp. 555–564, Apr. 2008. h (λ) = ER cos(λ+Φ ) ER ,
1 1 1 1
[12] S. Sandberg and N. Von Deetzen, “Design of bandwidth- | | | |≤
efficient unequal error protection LDPC codes,” IEEE Trans. and
Commun., vol. 58, no. 3, pp. 802–811, Mar. 2010. h (λ) = ER cos λ+Φ ER
[13] P. Billingsley, Probability and measure, John Wiley & Sons, | 2 | | 1 h 1i|≤ 1
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[14] R.vonMises, “Ontheasymptoticdistributionof differentiable ∈ −
statisticalfunctions,”AnnalsofMathematicalStatistics,vol.18, sup T (λ) G(λ) o(1)ER 0
L 1
no. 3, pp. 309–348, Sep. 1947. λ [ π,π)| − |≤ →
[15] A. W. van der Vaart, Asymptotic Statistics, Cambridge ∈−
University Press, 1998. as L . Lemma 3 shows that
→∞
[16] D. Pollard, “New ways to prove central limit theorems,”
sup G (λ) T (λ) 0
Econometric Theory, vol. 1, no. 3, pp. 295–313, Dec. 1985. L L
| − |→
[17] D. Pollard, Convergence of Stochastic Processes, Springer- λ∈[−π,π)
Verlag, New York, 1984. almost surely as L .
[18] D. Pollard, “Asymptotics by empirical processes,” Statistical →∞
Science, vol. 4, no. 4, pp. 341–325, 1989. Lemma 3. Put T (λ)=EG (λ). Then
L L
[19] S.A. van de Geer, Empirical Processes in M-Estimation,
Cambridge Series in Statistical and Probabilistic Mathematics. sup GL(λ) TL(λ) 0
| − |→
Cambridge University Press, 2009. λ [ π,π)
∈−
[20] R. G. McKilliam, B. G. Quinn, I. V. L. Clarkson, B. Moran,
almost surely as L .
and B. Vellambi, “Polynomial phase estimation by phase →∞
unwrapping,” arxiv.org/abs/1211.0374, Nov 2012. Proof: Put D (λ)=G (λ) T (λ) and let
L L L
[21] P.Billingsley,Convergenceofprobabilitymeasures,JohnWiley −
& Sons, 2nd edition, 1999. λ = 2π(n 1) π, n=1,...,N
n N − −
be N points uniformly spaced on the interval [ π,π). Let
APPENDIX −
L = [λ ,λ + 2π) and observe that L ,...,L partition
A. Lemmas required for the proofof almost sure convergence n n n N 1 N
[ π,π). Now
(Theorem 1) −
Lemma 1. G(λˆL)→G(0) almost surely as L→∞. λ s[uπp,π)|DL(λ)|
∈−
Proof: Since G(x) is uniquely maximised at x=0, = sup sup D (λ) D (λ )+D (λ )
L L n L n
| − |
0 G(0) G(λˆ ), n=1,...,Nλ∈Ln
≤ − L UL+VL,
and since λˆ is the maximiser of G (x), ≤
L L where
0≤GL(λˆL)−GL(0). UL = sup |DL(λn)| and
n=1,...,N
Thus,
V = sup sup D (λ) D (λ ).
L L L n
0≤G(0)−G(λˆL) n=1,...,Nλ∈Ln| − |
G(0) G(λˆ )+G (λˆ ) G (0) Lemma 4 shows that for any N and ǫ>0,
L L L L
≤ − −
G(0) G (0) + G (λˆ ) G(λˆ )
L L L L Pr lim U >ǫ =0,
≤| − | | − | L
≤2λ∈s[−uπp,π)|GL(λ)−G(λ)|, that is, UL → 0 almonsLt→su∞rely as Lo→ ∞. Lemma 5 shows
andthelastlineconvergesalmostsurelytozerobyLemma2. that for any ǫ>0,
Pr lim V >ǫ+ 4πER =0.
aLsemLma 2.. supλ∈[−π,π)|GL(λ)−G(λ)| → 0 almost surely If we choose N nlaLrg→e∞enoLugh thatN4πERioi <ǫN then
→∞
Proof: Put T (λ)=EG (λ) and write
L L Pr lim sup D (λ) >3ǫ
L
sup |GL(λ)−G(λ)| (L→∞λ∈[−π,π)| | )
λ [ π,π)
∈− Pr lim (U +V )>3ǫ
L L
=λ∈ss[−uuπpp,π)|GGL((λλ))−TTL((λλ))++TL(sλu)p−GT(λ()λ|) G(λ). ≤≤PrnLLl→im∞(UL+VL)>ǫ+oǫ+ 4NπERi
≤ | L − L | | L − | n →∞ o
Now,λ∈[−π,π) λ∈[−π,π) ≤=P0.rnLl→im∞UL >ǫo+PrnLl→im∞VL >ǫ+ 4NπERio
TL(λ)−G(λ)= |PL| −p h1(λ)+ |DL| −d h2(λ) Thus sup D (λ) 0 almost surely as L .
=(cid:0)o(1)h1(λ(cid:1))+o(1)h(cid:0)2(λ) (cid:1) λ∈[−π,π)| L |→ →∞