Two-Dimensional Signaling in Ricean Fading with Imperfect Channel by murplelake81


									    Two-Dimensional Signaling in Ricean Fading with Imperfect Channel Estimation

                           Xiaodai Dong Member, IEEE, and Lei Xiao Student Member, IEEE
                                     Department of Electrical and Computer Engineering,
                                  University of Alberta, Edmonton, Alberta, Canada T6G 2V4
                                             Email: {xdong, lxiao}
                                            Tel: 780 492-6989 Fax: 780 492-1811

   Abstract— In this paper, the analytical framework reported           In this paper, we present a general analysis of the SER of
in [1] for calculating the symbol error rate (SER) of two-           arbitrary 2-D signaling in Ricean fading with channel estima-
dimensional (2-D) signaling in Rayleigh fading with channel          tion errors, as an extension to [1]. Our analysis is applicable to
estimation errors is further developed to address the more
general case of frequency-flat Ricean fading. We show that in the     any channel estimation scheme where the estimated fading is
presence of channel estimation errors, the SER of arbitrary 2-D      jointly Gaussian with the actual fading. The Doppler frequency
signaling with polygonal decision regions in Ricean fading can       shift in the line-of-sight component is included in the Ricean
be expressed as a two-fold proper integral with finite integration    channel modeling and its effect on the error performance of
limits, which is suitable for numerical evaluation. Moreover,        arbitrary 2-D signaling is easily studied with our general SER
this new analysis is general in the sense that it is applicable
to any channel estimation scheme where the estimated and the         expression. The generality and flexibility of our work make it
actual channel gains are jointly complex-Gaussian. The effect of     a useful tool in the design and analysis of practical coherent
static channel estimation errors and dynamic channel estimation      receivers in microcellular and indoor wireless environments.
errors introduced by pilot symbol assisted modulation (PSAM)            The remainder of this paper is organized as follows. We
and minimum mean square error (MMSE) channel estimations             first discuss the system and channel models in Section II.
are studied using the newly derived SER formula. The effect
of Doppler frequency shift in the line-of-sight (LOS) component      Section III presents the derivation for the PDF’s of the decision
of the channel on the error performance is investigated in our       statistics and the equivalent noise of the system. The PDF’s
analysis. The analytical and numerical results presented in this     are then used in Section IV to further develop the symbol error
paper provide a useful tool on choosing suitable signaling formats   probability of coherent 2-D signaling with channel estimation
and optimizing parameters in the communication system design.        errors. In Section V, the SER formula is applied to three
                                                                     types of channel estimation errors: static estimation errors,
                                                                     pilot symbol assisted modulation and MMSE estimation errors.
                      I. I NTRODUCTION
                                                                     Finally, in Section VI and Section VII, we discuss some
   Extensive research has been conducted on the performance          numerical results and present our conclusions.
analysis of coherent signaling in fading channels. Ready-to-
evaluate probability of error expressions can be found in [2]                             II. S YSTEM M ODEL
for arbitrary 2-D signaling over various fading channels. These         Consider transmission of a carrier modulated signal s(t)
results, however, are acquired with the assumption of perfect        over a slowly time-varying frequency flat Ricean fading chan-
knowledge of channel fading gains, which is not possible in          nel. The channel gain is assumed to be almost constant during
practical communication systems. In [1], a new method of             one symbol interval T , but changes from symbol to symbol.
analysis was presented for determining the SER of arbitrary 2-       Further assuming perfect carrier frequency synchronization
D signaling in Rayleigh fading with channel estimation errors.       and symbol timing in the receiver, as well as intersymbol-
Performance studies on Ricean fading channels, however, have         interference free transmission, the sampled output of the
been less reported in the literature and are largely confined         receiver matched filter at t = i T can be expressed as
to perfect coherent and non-coherent detection [2], [3]. In
[3], the SER’s of MDPSK, coherent MPSK and noncoherent                                     2Es KR j2πfp (iT )
                                                                             zi   =               e           + αi     si + n i
MFSK were presented for Ricean fading channels. In [4],                                    1 + KR
performance analysis of linearly modulated signals was carried                       g
                                                                                  = (¯i + αi )si + ni
out in Ricean fading with minimum mean square estimation                          = g i si + n i                                   (1)
errors. And in [5], the probability density functions (PDF’s)
to describe the behavior of data demodulated with imperfect          where the transmitted signal si takes value from a set of M
channel estimation were derived and the SER and BER of               symbols in a constellation normalized to yield unit average
16 rectangular-QAM were presented. Their study, however,             energy, and gi is the multiplicative complex fading random
is only concerned with the decision statistics and cannot be         variable with mean gi which represents the line-of-sight (LOS)
applied to arbitrary 2-D signaling easily. To the best of the        component or the specular component. The zero-mean com-
authors’ knowledge, there is no general result on the SER            plex Gaussian random variable (R.V.) αi denotes the diffused
performance of arbitrary 2-D signaling over Ricean fading            or multipath component of the channel fading with variance
channel with channel estimation errors.                              2 E[|αi | ] = Es /(1 + KR ), where Es = 2 E[|gi | ] is the
                                                                     1        2                                    1       2
average energy of the received signal since the transmitted          Y respectively. Matrix L is the Hermitian covariance matrix
signal si is normalized to have unit average energy, and KR =        given by
  |¯i |2
E[|αi |2 ] denotes the Ricean K factor which by definition is
                                                                                                mxx mxy
                                                                                       L=                                    (7)
the ratio of the LOS power to the power in the multipath                                        m∗xy  myy
component of the fading. The additive white Gaussian noise n i                               ¯                  ¯
                                                                     where mxx = E[|X − X|2 ], myy = E[|Y − Y |2 ] and mxy =
has zero-mean and variance N0 in both its real and imaginary                 ¯        ¯
                                                                     E[(X − X)(Y − Y )∗ ] are the second central moments of the
parts, and is assumed to be independent of the channel fading        nonzero-mean complex Gaussian R.V.s X and Y . Eqn. (6) can
gi . Parameter fp is the Doppler frequency shift of the LOS
                                                                     be expressed in an alternative form as
component at the i-th symbol. It is reasonable to assume the
LOS Doppler shift fp approximately constant over several data
                      i                                                                                                ¯
                                                                        fX,Y (X, Y ) = π −2 |L|−1 exp{−|L|−1 (myy |X − X|2
frames, and therefore we drop the superscript notation i to                  +mxx |Y − Y                      ¯      ¯
                                                                                         ¯ |2 − 2 [m∗ (X − X)(Y − Y )∗ ])}. (8)
arrive at fp . Furthermore, the LOS component and the diffused
component have the following temporal property                       And in polar coordinates X = rx ejθx , Y = ry ejθy , the PDF
                                                                     (8) can be further expressed as
               gi /¯k
               ¯ g      =   ej2πfp (i−k)T                    (2a)
           1                  Es                                                         |L|−1
             E[αi α∗ ] =             J0 (2πfD |i − k|T )     (2b)     f (rx , θx , ry , θy ) = rx ry exp(−P ) exp − |L|−1
           2       k
                            1 + KR                                                         π2
                                                                             × myy rx − m∗ rx ry ej(θx −θy ) − mxy rx ry e−j(θx −θy )
where fD is the maximum Doppler frequency and J0 (·) is                                  xy

the zero-th order Bessel function of the first kind. The LOS                                    ¯                   ¯
                                                                             +mxx ry − 2 (myy X ∗ rx ejθx − m∗ Y ∗ rx ejθx
Doppler shift fp is related to the maximum Doppler frequency                      ¯                 ¯
                                                                             −mxy X ∗ ry ejθy + mxx Y ∗ ry ejθy )                 (9a)
fD by fp = fD cos θ0 where θ0 is the angle of arrival of the
LOS component.                                                       where
   In coherent detection, the receiver generates a channel gain
                                                                                                 ¯          ¯
                                                                                  P = |L|−1 myy |X|2 + mxx |Y |2
estimation gi , and the decision variable D is given by
                                                                                                 ∗ ¯ ¯∗          ¯ ¯
                 zi          gi si + n                                                       −mxy X Y − mxy X ∗ Y .               (9b)
            D=      = si + (           − si ) = si + t,       (3)
                 gi              ˆ
                                 gi                                  To obtain the PDF of the decision variable D = rd ejθd , we
and then                                                             define another variable F = rf ejθf = Y . With the relation
                        gi si + n                                    rd = rx , θd = θx − θy , rf = ry , θf = θy , it is easy to verify
                  t=              − si = D − s i              (4)           ry
                                                                     that the Jacobian is |J| = r1 = r1 . Therefore, the joint PDF
                                                                                                 y      f
where t is the equivalent noise term that includes the effects of    of D and F is given by
both the channel estimation error and the AWGN. In most of
the practical channel estimation schemes such as the prevailing                                    |L|−1 −P      3
                                                                               fD,F (rd , θd , rf , θf ) =e rd rf
PSAM and MMSE, the channel estimate gi and the actual
                                               ˆ                                                     π2
channel gain gi are jointly complex Gaussian.                                        × exp − Brf + 2rA rf cos(θf + θA )          (10a)
  We can see from (3) that it is the noise term t that causes        where
a decision error. The SER can be obtained by integrating the
PDF of t outside si ’s decision region. In Section III, we present      A = rA ejθA = ard ejθd + b                               (10b)
our method to find the PDF of t.                                                                  ¯         ¯
                                                                        a = ra ejθa = |L|−1 (myy X ∗ − m∗ Y ∗ )                  (10c)
     III. T HE PDF      OF THE   E QUIVALENT N OISE T ERM               b = rb e = |L| (mxx Y − mxy X ∗ )
                                                                                jθb     −1       ¯ ∗       ¯                     (10d)
                                                                         2           2 2     2
  Recognizing the fact that the decision variable D in (3) is           rA    =     ra rd + rb + 2ra rb rd cos(θa − θb + θd )    (10e)
the ratio of two complex-valued nonzero-mean Gaussian R.V.s,            B     =     |L|−1 (myy rd − 2 {m∗ ejθd }rd + mxx ).
                                                                                                            xy                   (10f)
we rewrite variable D in a general form as
                                                                     Then, we can integrate (10a) with respect to rf and θf to
                                 X                                   obtain the PDF of the complex R.V. D. That is,
                             D=                          (5)
where X and Y are correlated nonzero-mean complex Gaus-                                                 e−P rd rA + B rA
                                                                                                                2      2

                                                                                      fD (rd , θd ) =                eB.          (11)
sian variables. The joint PDF of X and Y can be obtained via                                             π|L|     B3
slightly modifying the joint PDF of two zero-mean Gaussian           Via a change of variables, it can be shown that (11) is
R.V.s given by Wooding [6] as                                        equivalent to [5, eqn. (15)], which was derived using a different
           fX,Y (X, Y ) = π −2 |L|−1 exp (−VH L−1 V)          (6)    method. Note that it is difficult to apply the fading estimate
                                                                     model given by [5, eqn. (1)] to PSAM in Ricean fading.
                    ¯        ¯
where V = [(X − X)∗ (Y − Y )∗ ]H with ∗ representing                    Our expression for the decision variable D is further devel-
complex conjugation and A the Hermitian transpose of
                                                                     oped to obtain the PDF of the combined noise term t, which
          ¯      ¯
matrix A. X and Y are the complex mean of R.V.s X and                is required for arbitrary 2-D signaling. Recall that t and D
have the relation t = rejθ = D − si given by (4). After some                 with well-behaved integrand that can be evaluated efficiently
mathematical manipulations, we have                                                                          ¯          ¯
                                                                             with numerical methods. With X = 0 and Y = 0 as a special
                                                                             case representing the Rayleigh fading scenario, (14) can be
                                      e−P r G + H G
                    ft (r, θ) =                  eH                  (12a)   further simplified to yield the same single integral expression
                                      π|L| H 3
                                                                             as given in [1].
where                                                                          The symbol error probability of an arbitrary coherent two
                                                                             dimensional constellation with channel estimation errors is the
   G(r)        2
            = ra r2 + 2f r + g                                       (12b)
                                                                             weighted sum of the probability of error in each erroneous
  H(r)      = hr + 2cr + d                                           (12c)   subregion [2], i.e.,
    ra                   ¯              ¯
            = |L|−2 [m2 |X|2 + |mxy |2 |Y |2
                      yy                                                                                    N
                     −2myy {mxy X    ¯
                                  ¯ Y ∗ }]                           (12d)                          Pe =         wj Pe,j                (15)
        f   =   ra {ejθ s∗ } + ra rb cos(θa − θb + θ)
                            i                                        (12e)                                 j=1

        g   =   ra |si |2 + rb + 2ra rb {si ej(θa −θb ) }
                 2            2
                                                                     (12f)   where N is the total number of erroneous subregions and wj
        h =     myy |L|−1                                            (12g)   is the weighting coefficient related to the a priori probability
                                                                             of each symbol [2]. Eqns. (14) and (15) are a new general
        c =     |L|−1 (myy {s∗ ejθ } − {m∗ ejθ })
                                 i              xy                   (12h)   SER expression for arbitrary 2-D signaling in Ricean fading
        d =     |L|−1 (myy |si |2 − 2 {si m∗ } + mxx ).
                                             xy                      (12i)   with channel estimation errors.
The PDF of the equivalent noise term t given by (12) is new.                                        V. A PPLICATIONS
  IV. S YMBOL E RROR P ROBABILITY                  OF   2-D S IGNALING          To apply our general analysis to various channel estimation
                                                                             methods, the only effort needed is to specify second central
   To provide a general analysis applicable to arbitrary polyg-
                                                                             moments and the means of R.V.s X = gi si + ni and Y = gi ,  ˆ
onal 2-D constellations, we adopt the idea of Craig’s method
                                                                             which describe the statistical characteristics of the received
(see [7] for details in AWGN, and [2] for analysis in fading
                                                                             signal and the channel estimation. Now, we analyze three
scenarios). Instead of dealing with the corrupted received
                                                                             different kinds of channel estimation errors employing the
signal, we work directly with the noise term t superimposed
                                                                             derived formula in Section IV.
on the transmitted signal. Therefore, the probability of error
can be calculated exactly by integrating the PDF of t over                   A. Static Estimation Error
the erroneous decision region. Furthermore, the erroneous                       By the term static estimation error, we mean the estimated
decision regions can be partitioned into subregions so that each             channel gain gi is related to actual channel gain gi by a
subregion can be conveniently expressed in polar coordinates                 constant factor. Thus, the estimated fading gi = qejφ is
                                                                                                                           ˆ         gi
[2]. With the PDF of t available, the probability of error for               also Gaussian with nonzero mean, where q and φ denote the
the j-th erroneous subregion of symbol si is given by                        estimation error in amplitude and phase, respectively. In this
                 θ2 ,j     ∞                                                 case, the central moments and means are given by
      Pe,j =                     ft (r, θ)drdθ,
                θ1 ,j    R(θ)                                                                              2Es |si |2
                −P       θ2,j      ∞                                                          mxx    =                + 2N0            (16a)
                e                            G    1          G                                             1 + KR
            =                            r      + 2        e H drdθ (13a)                                       2Es
                π|L|     θ1,j     R(θ)       H3  H                                            myy    =                                 (16b)
                                                                                                           (1 + KR )q 2
where G, H and P are given in (12) and (9b) and R(θ) has
                                                                                                              2Es si
the form                                                                                      mxy    =                  ejφ            (16c)
                           xj sin ψj                                                                       (1 + KR )q
              R(θ) =                      .        (13b)
                      sin(θ − θ1,j + ψj )                                                                     2Es KR
                                                                                                X    =                  si             (16d)
Parameters xj ’s, θ1,j ’s, θ2,j ’s and ψj ’s depend on the geometry                                           1 + KR
of the j-th subregion and they are defined in [2].                                               ¯          e−jφ 2Es KR
                                                                                                Y    =                      .          (16e)
  Further simplification of (13a) is not easy. However, we can                                                q       1 + KR
use a simple change of variable u = 1 to transform the semi-
                                            r                                When the static error is small enough so that every signal
infinite integral to an integral with finite integration limits.               point is still inside its own decision boundary, the analysis in
That is,
                                                                             [2] with a rotated and/or scaled decision boundary yields a
             1 −P         θ2,j        R(θ)         2
                                                  ra + 2f u + gu2            solution in the form of a sum of one-fold integral. However,
  Pe,j =         e                           u                               Craig’s method is no longer applicable for situations where
            π|L|         θ1,j     0              (h + 2cu + du2 )3
                                       ra +2f u+gu2
                                        2                                    the signal point is outside its decision boundary, which can be
                +                     e h+2cu+du2 dudθ.               (14)   the case for severe static errors. The approach in this paper,
                    (h + 2cu + du2 )2                                        given by (12)-(15) and (16), is more general since we do not
where ra , f , g, h, c and d have been defined in (12). The                   put any constraint on the severity of the static error in the
probability of error (14) is now a two-fold proper integral                  channel estimation.
B. Pilot Symbol Assisted Modulation                                     And the temporal property in (2) should be modified as
   In the transmitter of a PSAM system, known symbols                                     gl = g i
                                                                                          ¯     ¯                                            (21a)
termed pilot symbols are inserted periodically into the data                      1                Es
stream such that one frame of L symbols begins with a                               E[αl α∗ ] =
                                                                                          i              J0 (2πfD |l − i|T )
                                                                                  2             1 + KR
pilot symbol followed by L − 1 data symbols. The period of
pilot symbols should satisfy the Nyquist sampling theorem to                                  × e−j2πfp (l−i)T .                             (21b)
ensure channel state information at non-pilot positions can be          Accordingly, some quantities in (19) need to be modified as
accurately estimated. That is L ≤ 2fD T . The receiver, with the
priori knowledge of the position and value of pilot symbols,                                            2Es hk,l sl J0 (2πfD |kL − l|T )
                                                                              mxy      =
can obtain a channel estimation as                                                                                   1 + KR
                          zk,0          nkp
                    gp =           k
                               = gp +                       (17)                       × ej2πfp (kL−l)T                                      (22a)
                           pk           pk                                                                      K2
                                                                                ¯                 2Es KR
where gp and gp are the channel fading and fading estimate
               ˆk                                                               Y      =                               hk,l                  (22b)
                                                                                                  1 + KR
at the pilot position in the k-th frame, respectively. The first                                                k=−K1
received symbol of the k-th frame is denoted by zk,0 and pk                                     ∗       2Es J0 (2πfD |k − n|LT )
                                                                              Ckn      = E[gpk gpn ] =
is the pilot symbol transmitted. The channel gain of the l-th                                                    1 + KR
data symbol in a frame is estimated by linear interpolation of                         × e −j2πfp (k−n)LT
                                                                                                          .                      (22c)
channel gains of K nearby pilot symbols
                                  K2                                    It will be shown in Section VI through numerical evaluation
                         gl =                 ˆk
                                         hk,l gp                 (18)   that the LOS tracking method leads to significantly degraded
                                k=−K1                                   error performance compared to perfect carrier synchroniza-
                                                                        tion, unless some Doppler shift compensation mechanism is
where K1 = (K − 1)/2 , K2 = K/2 and hk,l ’s are the
                                                                        implemented in the interpolation filter. Note that the moments
coefficients of the interpolator. Since gl is a linear combination
                                                                        and means in (19) or (22) as well as the SER’s are different
of jointly Gaussian R.V.s gp 1 · · · gp 2 , it is also a Gaussian
                              ˆ−K        ˆK
                                                                        from symbol to symbol in a frame. The overall SER, therefore,
R.V.. Several interpolators, including Cavers’ optimum Weiner
                                                                        needs to be averaged over l, where l = 1, . . . , L − 1.
interpolator [8] and sinc interpolator [9], have been proposed
in the literature. Our analysis applies to any linear interpolating     C. Minimum Mean Square Error Estimator
schemes.                                                                   For an MMSE estimator, the error e = gi − gi and the
   Assume perfect carrier frequency synchronization, the sec-
                                                                        estimation gi are orthogonal (and hence independent, since
ond central moments and means needed for calculating the
                                                                        they are Gaussian) and the estimation is unbiased [4]. That is,
SER of the l-th symbol in a frame of size L are given by
                                                                        E[gi ] = E[ˆi ] and E[eˆi ] = 0. Therefore,
                                                                                   g           g∗
       mxx =                 |sl |2 + 2N0                     (19a)                                       2Es |si |2
                    1 + KR                                                                     mxx      =            + 2N0                   (23a)
                                   2N0                                                                    1 + KR
       myy = Hl CHT +      l             Hl 2                (19b)                                          2Es
                                  |pk |2                                                       myy      =                2
                                                                                                                     − 2σe                   (23b)
                       K2                                                                                 1 + KR
                              2Es hk,l sl J0 (2πfD |kL − l|T )                                 mxy      = myy si                             (23c)
      mxy      =                                               (19c)
                                           1 + KR                                                                2Es KR
                                                                                                 X      =               si                   (23d)
                        2Es KR                                                                                   1 + KR
        X      =               sl                               (19d)
                        1 + KR                                                                   ¯               2Es KR
                                                                                                 Y      =                                    (23e)
                        2Es KR
                                    K2                                                                           1 + KR
         Y     =                          hk,l ej2πfp (kL−l)T   (19e)
                        1 + KR                                          where σe = E[|gi − gi |2 ] denotes the energy of the estimation
                                                                        error, which is a parameter indicating the accuracy of the esti-
       Hl      = [h−K1 ,l , . . . , hK2 ,l ]                   (19f)    mation. The orthogonality between e and gi can be exploited
                          ∗          2Es J0 (2πfD |k − n|LT )           to further simplify eqn. (12) as
      Ckn      = E[gpk gpn ] =                                (19g)
                                              1 + KR                                                              2
                                                                                                    G(r)       = rb                          (24a)
where Ckn      is the element with index (n, k) in covariance
                                                                                                    H(r)       = hr2 + d,                    (24b)
matrix C.
   In practice, when an automatic frequency control (AFC)               which make it possible to solve the inner integral in eqn. (14)
loop is used to track the line-of-sight component, the received         analytically as
signal sample is written as                                                                                                 ¯
                                                                                                |Y |2
                                                                                                 ¯                        |Y |2 myy V
                                                                                               − myy     θ2,j   V exp    m3 R2 (θ)+V
                       2Es KR                                               M
                                                                           Pe,j M SE   =                                   yy
                                                                                                                                        dθ    (25)
             zl = (           + αl e−j2πfp lT )sl + nl .         (20)                          2π                 myy R2 (θ) + V
                       1 + KR                                                                           θ1,j
where V = mxx − myy |si |2 . The SER of arbitrary 2-D              Fig. 4 shows how the SER of PSAM 16 rectangular-QAM
signaling with MMSE channel estimation can be calculated           depends on K in Ricean fading. We find that K = 25 is
using (25). Our result reconciles with the previous analysis       sufficient when the SNR is less than 30dB. For the case where
reported in [4].                                                   SNR is 40dB or 50dB, however, a larger K value of 30 or 32,
                                                                   respectively, is required.
                 VI. N UMERICAL R ESULTS                              Fig. 5 depicts the effect of the LOS Doppler frequency
   In this section, we present some numerical results for          shift on the error performance of pilot symbol assisted 2-
the symbol error probabilities of 16 star-QAM [2] and 16           D modulation, using 16 rectangular-QAM as an example.
rectangular-QAM in Ricean fading with channel estimation           One interesting observation is that the LOS Doppler shift
errors. We adopt KR = 7dB in our numerical calculation,            has minimal effect on the symbol error probability in the
which is a typical value of KR for practical microcellular         case of perfect carrier frequency synchronization for low to
channels.                                                          medium range of SNR values. When the synchronization is
   Fig. 1 indicates that the performance of 16 rectangular-        not ideal, i.e. the receiver is locked at the frequency of the
QAM is degraded more severely than that of 16 star-QAM             LOS component, a substantial performance degradation is
in the presence of static estimation errors. Furthermore, the      observed. This degradation is caused by the spectral shift of
rectangular-QAM is no longer usable when the actual channel        the multipath component. In the ideal synchronization case, the
gain is 2dB smaller in amplitude than the estimated channel        power spectral density (PSD) of the fading process is band-
gain and differs 10◦ in phase. This is because the amount          limited to [−fD , fD ]. With an AFC loop locked to the LOS
of estimation error has displaced some signal points of 16         component with a Doppler shift of fp , the non-zero region
rectangular-QAM outside their decision boundaries. Although        of the fading process PSD is now [−fD − fp , fD − fp ] and
not shown here, our study finds that in the low to medium SNR       −fD ≤ fp ≤ fD , which makes |f |max = fD + |fp |. This shift
range, both constellations are more sensitive to static channel    of PSD makes our choice of L according to L ≤ 2fD T no     1

estimation errors in Ricean fading than in Rayleigh fading.        longer sufficient in providing zero mean square error recovery
   The effect of dynamic estimation errors on the performance      from the sinc interpolation. If an accurate estimate of fp is
of 2-D signaling is investigated using the pilot symbol assisted   available to the receiver, we can compensate this undesirable
modulation scheme. Although our analysis is valid for any lin-     shift by either adjusting the pass-band of the interpolation filter
ear interpolator, we choose the sinc interpolator [9] because of   to [−fD − fp , fD − fp ] or shifting the estimation from pilots
its simple implementation and close-to-optimum performance.        back to [−fD , fD ]. If the value of fp is not available to the
For a sinc interpolator, we have                                   receiver, we observe that |f |max = fD + |fp | ≤ 2fD . So the
                                                                   design criteria for the frame size L should be L ≤ 4fD T in 1
                                    l                              this case, which requires double the number of pilots inserted
                     hk,l = sinc      − k).                (26)
                                   L                               and thus leads to a reduced channel throughput.
To smooth the abrupt truncation in the time domain, a window
                                                                                            VII. C ONCLUSION
should be applied. Our numerical study shows that Hamming
window gives better performance than rectangular window,              In this paper, we have presented a new, general analysis
Blackman window and Hanning window. All the numerical              of the symbol error probability of arbitrary 2-D signaling in
results for PSAM in this paper, therefore, are evaluated using     Ricean fading with imperfect channel estimation. The SER
the sinc interpolator with Hamming window.                         of arbitrary 2-D signaling in Ricean fading with channel
   In Fig. 2, we plot the SER’s of PSAM 16 rectangular-QAM         estimation errors has been expressed as a two-fold proper
and 16 star-QAM in Ricean fading with KR = 7dB. The frame          integral with finite integration limits, which can be evaluated
size L is fixed as 15 for a fading rate of fD T = 0.03. Our         efficiently. Applications of the newly derived formula to
numerical results agree with the simulations. In the high SNR      three types of channel estimation errors have been studied.
region, the error floor of 16 star-QAM is less significant than      Numerical examples of 16 rectangular-QAM and 16 star-QAM
that of 16 rectangular-QAM. Another advantage of 16 star-          with static and dynamic estimation errors in Ricean fading
QAM is that the ring ratio of the constellation is adjustable      have been presented. The effect of Doppler frequency shift on
for systems operating in different SNR’s.                          the error performance has also been studied and it has been
   Fig. 3 plots the optimum ring ratios of various 16-ary          shown that we need to double the number of pilot symbols
circular constellations in terms of minimizing the SER of          to achieve acceptable performance when a receiver is locked
PSAM in Ricean fading. The signal space diagrams of these          to the LOS component and the Doppler frequency shift is
constellations were presented in [2]. We find that the optimum      unknown. It has been shown that key parameters in PSAM
ring ratios for different constellations roughly have the same     and the optimum ring ratios for circular 2-D signaling can be
tendency as SNR increases and the more signaling points on         easily and accurately determined by our analytical results.
the outer ring, the larger the optimum ring ratio.                                              R EFERENCES
   An important parameter in PSAM is the interpolation order
                                                                    [1] X. Dong and N.C. Beaulieu, “SER of two-dimensional signalings in
K. While a small K value worsens performance, increasing                Rayleigh fading with channel estimation error”, Proc. ICC’03, pp. 2763-
K blindly implies huge buffer size and long detection delay.            2767, May 2003.

                                                                                                                                                    16 Star−QAM
                                  −2                                                                                           3.6
                             10                                                                                                                     Rotated (8,8)
            Symbol Error Rate


                                                                                                          Optimum Ring Ratio
                                                                                                                               3.2                  (4,12)
                                          Perfect, 16 rect.−QAM
                                          q=2dB, φ=10° 16 rect.−QAM                                                            2.8
                             10           q=−2dB, φ=10° 16 rect.−QAM
                                          Perfect, 16 star−QAM
                                          q=2dB, φ=10° 16 star−QAM                                                             2.4
                                          q=−2dB, φ=10° 16 star−QAM
                             10                                                                                                2.2
                                      0        10           20        30         40        50
                                                         Average E /N (dB)
                                                                  b   0                                                         2
                                                                                                                                 0           10               20       30      40           50        60
                                                                                                                                                               Average E /N (dB)
                                                                                                                                                                       b   0
Fig. 1. SER of 16 rectangular-QAM and 16 star-QAM in a Ricean fading
channel (KR = 7dB) with static estimation error q = ±2dB in amplitude
and φ = 10◦ in phase.                                                                           Fig. 3. Optimum ring ratios of PSAM 16-ary constellations in Ricean fading
                                                                                                with KR = 7dB, fD T = 0.03, fp T = 0, L = 15, K = 36 and Hamming

                                                                 16 rect.−QAM, PSAM
                                                                 16 star−QAM, PSAM
                                                                 16 star−QAM, perfect
                             10                                  16 rect.−QAM, perfect                                              0
         Symbol Error Rate

                                                                 simulation, PSAM

                             10                                                                                                     −2
                                                                                                         Symbol Error Rate

                             10                                                                                                     −4
                                                                                                                                            E /N = 10dB
                                                                                                                                             b 0
                                      0   10        20       30     40      50        60   70                                               Eb/N0= 20dB
                                                         Average Eb/N0 (dB)                                                                 Eb/N0= 30dB
                                                                                                                               10           E /N = 40dB
                                                                                                                                             b 0
Fig. 2. SER of PSAM 16 rectangular-QAM and 16 star-QAM in Ricean                                                                            E /N = 50dB
                                                                                                                                                b    0
fading with KR = 7dB, fD T = 0.03, L = 15 and K = 30. The ring ratio
                                                                                                                                        5   10            15      20      25      30        35        40
for 16 star-QAM is 2.40.                                                                                                                                    Interpolation order, K

                                                                                                Fig. 4. The effect of K on the SER of 16 rectangular-QAM in Ricean fading
 [2] X. Dong, N.C. Beaulieu and P.H. Wittke, “Signal constellations for                         with KR = 7dB, fD T = 0.03 and L = 15.
     fading channels”, IEEE Trans. Commun., pp. 703-714, May 1999.
 [3] J. Sun and I. S. Reed, “Performance of MDPSK, MPSK, and nonco-
     herent MFSK in wireless Rician channels”, IEEE Trans. Commun., pp.
     813-816, June 1999.
 [4] M. G. Shayesteh and A. Aghamohammadi, “On the Error Probability                                                         10

     of Linearly Modulated Signals on Frequency-Flat Ricean, Rayleigh,
     and AWGN Channels”, IEEE Trans. Commun., pp. 1454-1466, Febru-
     ary/March/April 1995.                                                                                                   10

 [5] S. K. Wilson and J. M. Cioffi, “Probability Density Function for Analyz-
     ing Multi-Amplitude Constellations in Rayleigh and Ricean Channel”,
                                                                                                         Symbol Error Rate

     IEEE Trans. Commun., pp. 380-386, March 1999.                                                                           10

 [6] R. A. Wooding, “The Multivariate Distribution of Complex Normal
     Variables”, Biometrika, Vol. 43, pp. 212-215, June 1956.
 [7] J. W. Craig, “A New Simple and Exact Result for Calculating the                                                         10
                                                                                                                                            fpT=0, Ideal Sync.
     Probability of Error for Two-Dimensional Signal Constellations”, Proc.
                                                                                                                                            fpT=0.005, Ideal Sync.
     IEEE Milit. Commun. Conf. MILCOM’91, Boston, MA, pp. 0571-0575
                                                                                                                                  −4        f T=0.01, Ideal sync.
     (25.5.1-25.5.5), 1991.                                                                                                                  p
                                                                                                                                            fpT=0.005, LOS Tracking
 [8] J.K. Cavers, “An Analysis of Pilot Symbol Assisted Modulation for
                                                                                                                                            fpT=0.01, LOS Tracking
     Rayleigh Fading Channels”, IEEE Trans. VT., Vol. 40, pp. 686-693,                                                            −5
     Nov. 1991.                                                                                                              10
                                                                                                                                        0   5            10    15    20     25         30        35   40
 [9] Y.S. Kim, C.J. Kim, G.Y. Jeong, Y.J. Bang, H.K. Park and S.S. Choi,                                                                                      Average Eb/No (dB)
     “New Rayleigh Fading Channel Estimator Based on PSAM Channel
     Sounding Technique”, Proc. ICC’97, pp. 1518-1520, June 1997.                               Fig. 5. The effect of Doppler shift on the SER of PSAM 16 rectangular-
                                                                                                QAM in Ricean fading with KR = 7dB, fD T = 0.03, L = 15, K = 36
                                                                                                and Hamming window.

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