## Differential Space–Frequency Modulation and Fast

## 2D–Multiple–Symbol Differential Detection for

## MIMO–OFDM

Volker Pauli, Lutz Lampe∗ and Johannes Huber

Lehrstuhl f¨ur Informations¨ubertragung, Universit¨at Erlangen–N¨urnberg, Germany Email: {pauli, huber}@LNT.de

∗_{Department of Electrical & Comp. Eng., University of British Columbia, Canada}
Email: Lampe@ece.ubc.ca

**Abstract**

The combination of differential space–frequency modulation (DSFM) with orthogonal frequency division mul-tiplexing (OFDM) is attractive for transmission over time– and frequency–selective multiple–input multiple–output (MIMO) channels and detection without the need for channel state information (CSI) at the receiver. It is well known that simple differential detection results in a high error floor already for moderate time and frequency selectivity of the channel. More sophisticated multiple–symbol differential detection (MSDD), which jointly processes multiple received symbols, overcomes this limitation, usually at the price of higher detection complexity. In this paper we consider DSFM for MIMO–OFDM transmission and MSDD at the receiver. Inspired by previous work presented in the literature, we devise a novel DSFM scheme, which makes use of spatial and/or spectral (multipath) diversity and is particularly suited for MIMO–OFDM and power–efficient, low–delay MSDD. We further investigate the application of a two–dimensional (2D) observation window to MSDD (2D–MSDD) in order to exploit channel correlations in both time and frequency direction. We develop a representation of the detection problem that is amenable to tree–search decoding, whose application leads to a tremendous reduction in MSDD complexity or “fast” MSDD. An analytical approximation of the symbol–error rate of 2D–MSDD for MIMO–OFDM under spatially correlated fading is derived, which enables quick and accurate performance evaluations. Numerical and simulation results corroborate the efficacy of our approach and show that power efficiency close to that of coherent detection with perfect CSI is feasible in all standard fading scenarios at reasonable decoder complexity.

**Index Terms**

Orthogonal frequency division multiplexing (OFDM), multiple–input multiple–output (MIMO) transmission, differential space–frequency modulation (DSFM), differential space–time modulation (DSTM), multiple–symbol differential detection (MSDD), tree–search decoding, sphere decoding.

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) is a widely adopted modulation technique for transmission over frequency–selective channels. Recently, the combination of OFDM with transmit and receive antenna arrays, which is usually referred to as multiple–input multiple–output (MIMO)–OFDM, has become popular as a means to further boost bandwidth and/or power efficiency, cf. e.g. [1] for an overview. For both single–antenna OFDM and MIMO–OFDM transmission, receiver designs often include a dedicated channel estimation unit, which extracts channel state information (CSI) based on transmitted pilot symbols or tones, and which is followed by a data detection unit performing coherent detection under the assumption of perfect CSI. Since, in general, OFDM subchannels are correlated in time and frequency, efficient pilot–aided channel estimators employ two–dimensional (2D) filtering to suppress estimation noise, cf. e.g. [1]–[4]. However, especially in strongly frequency–selective channels and mobile environments with relatively fast changes of channel conditions, acquisition of accurate CSI may be impracticable. In such scenarios, differential encoding and detection without the need for CSI becomes an attractive alternative.

In this paper, we consider transmitter and receiver design for MIMO–OFDM transmission with differ-ential encoding and detection without CSI. As it is customary in the literature, detection without CSI is also referred to as differential or noncoherent detection in the following.

Related Literature: For single–antenna OFDM various detection schemes have been proposed in the literature, e.g., [5]–[10]. While [5] considers differential encoding in frequency direction and conventional differential detection (CDD), which relies on negligible variations of the channel frequency response at adjacent subchannels, improved one–dimensional (1D) and 2D differential detectors are devised in [6]–[10]. Thereby, 2D detectors exploit both temporal and spectral channel correlations, cf. e.g. [7]–[9].

For MIMO–OFDM the concept of space–frequency or space–time–frequency coding has attracted con-siderable attention in the literature, cf. e.g. [11]–[13]. Space–frequency codes for noncoherent detection have been designed and analyzed in [14]. While utilizing full space and frequency (multipath) diversity of the frequency–selective MIMO channel, detection complexity is exponential in the number of OFDM subchannels for fixed rate. Another approach pursued in e.g. [15]–[22] is to employ differential modulation with matrix constellations developed for differential space–time modulation (DSTM) and single–carrier MIMO transmission (e.g. [23]–[25]). In [15]–[20], diagonal constellations ([24, 25]) are considered and, similar to space–frequency codes for coherent detection with CSI, signal components of the DSTM code matrices are allocated to different transmit antennas and OFDM subchannels to accomplish diversity gains.

This is referred to as differential modulation diversity (DMD) in [15]. Signal allocation can be further optimized for the case of a known channel power–delay profile (PDP) at the transmitter as discussed in [17]–[19] (see also [13]). Whereas diversity and optimal coding gain are considered in [18], performance improvement of CDD is achieved in [17, 19] by creating a smooth logical channel for differential encoding in frequency direction. Extending the work in [21], orthogonal constellations ([23]) in conjunction with CDD, decision–feedback differential detection (DFDD) [15, 26], and prediction–based Viterbi decoding for MIMO–OFDM are studied in [22]. New constellations tailored for differential MIMO–OFDM were designed in [27, 28] assuming time–invariant channels.

Contributions: In this paper, we study MIMO–OFDM transmission for the general case in which differential encoding can be performed in time or in frequency direction. To exploit frequency diversity while minimizing transmission delay and detrimental effects of temporal channel variations on performance, we consider differential space–frequency modulation (DSFM) [19] based on diagonal constellations, where all components of a matrix signal are allocated within one OFDM symbol. We devise a novel signal– allocation scheme for DSFM, which unifies DMD of [15] with ideas from block–differential modulation proposed for frequency–flat MIMO fading channels in [29]. Under the reasonable assumption that the PDP is not known at the transmitter, it optimally combines the goals of (i) minimally correlated subchannels used for different components of one space–frequency matrix signal, (ii) maximally correlated subchannels experienced by matrix signals successive in differential encoding, and (iii) maximal correlation between subchannels comprised by the 2D observation window (see below) for detection without CSI. Thereby,

full space–frequency diversity and optimal coding gain are not our primary concern, as (i) this usually requires some knowledge of the PDP at the transmitter, (ii) corresponding signal constellations may become extremely large, which increases detection complexity, and (iii) asymptotic performance gains through increasing diversity beyond a certain point are not noteworthy for practically relevant error rates.

Different from the aforementioned literature, we study maximum–likelihood multiple–symbol differential detection (ML–MSDD) for DSFM (cf. e.g. [30, 31] for ML–MSDD). Inspired by 2D channel estimation schemes in e.g. [1]–[4], we propose ML–MSDD with a 2D “observation window”, which, different from previous work for noncoherent MIMO–OFDM, is capable of exploiting channel correlation in both time and frequency direction. In order to break the exponential complexity of ML–MSDD with increasing observation window size, we develop a representation of the detection problem that is amenable to low– complexity or “fast” optimal and suboptimal tree–search methods. In this context, we borrow from [32] and adapt tree–search detection methods originally developed for single–carrier DSTM to DSFM considered

here. Finally, we provide expressions for the symbol–error rate (SER) of 2D ML–MSDD for DSFM under spatially correlated fading, which enable a quick and accurate performance evaluation, and we illustrate the efficacy of our approach by means of numerical examples.

Organization: The remainder of this paper is organized as follows. Section II introduces the MIMO– OFDM transmission model. DSFM and the novel signal–allocation scheme are presented in Section III. The general 2D ML–MSDD decoder is developed and tree–search implementations are discussed in Section IV. In Section V, analytical performance expressions are derived, and performance and complexity results are presented in Section VI. Finally, Section VII concludes this paper.

Notation: Bold upper case X and lower case x denote matrices and vectors, respectively. The element in the ith row and jth column of a matrix X is denoted by the respective lower case xi,j. Conversely, an (M ×N )–dimensional matrix X and an M –dimensional vector x are defined as X = [x△ i,j]i=1,...,M

j=1,...,N and x

△ = [xi]i=1,...,M.(·)H,(·)T,||·||, tr{·}, det{·}, ⊗, ◦, E{·} and F {·} denote Hermitian transposition, transposition, Frobenius norm, trace and determinant of a matrix, Kronecker and Hadamard product, expectation and Fourier transform, respectively. IL and 1L are the L × L identity and all–ones matrix, respectively, and diag{X1, . . . , XL} is an LM × LN block–diagonal matrix with the M × N matrices Xl on its main diagonal.

II. MIMO–OFDM TRANSMISSION MODEL

In this section, we first introduce the spatially correlated time– and frequency–selective MIMO channel model using a time–domain description in Section II-A. Then, in Section II-B we present the OFDM transmission as well as the relevant frequency–domain channel parameters.

*A. Time– and Frequency–Selective MIMO Channel*

We consider transmission with NT transmit andNR receive antennas over a both time– and frequency– selective channel. The spatial subchannel between transmit antenna i and receive antenna j is described by its continuous–time input delay–spread function [33]

hi,j(τ, t) = Lh

X

l=1

h(l)_{i,j}(t)γ(τ − τl), 1 ≤ i ≤ NT, 1 ≤ j ≤ NR, (1)
where theLh propagation paths are specified by their complex amplitudesh(l)i,j(t) and delays τl,1 ≤ l ≤ Lh,
and the pulse γ(t) accounts for transmitter pulse shaping and receiver matched filtering.

To describe the spatial correlation of the MIMO channel we use the correlation model of [34, 35] where channel matrices are modelled as

h

h(l)_{i,j}(t)i_{i}

=1,...,NT
j_{=1,...,NR}

= AW(l)(t)BH, (2)

with constant (NT × NT)–dimensional A and (NR × NR)–dimensional B modelling the transmit and receive correlation, respectively. The Lh mutually independent (NT× NR)–dimensional matrices W(l)(t), 1 ≤ l ≤ Lh, contain i.i.d. circularly symmetric complex Gaussian random processes with mean zero, variance σ2

l, and identical temporal correlation functions.

*B. OFDM Transmission*

OFDM transmission with D active subcarriers and subcarrier frequency spacing ∆f is assumed. One
OFDM frame is of durationTf = DT +Tg, whereT = 1/(∆f D) is the modulation interval and Tg denotes
the guard interval. With respect to the channel model (1) we make the following standard assumptions:
(i) the coefficients h(l)_{i,j}(t) change only negligibly during transmission of one OFDM frame, i.e., h(l)_{i,j}(t) ≈
h(l)_{i,j}(t + τ ), 0 ≤ τ ≤ Tf, (ii) the frequency responses of the transmitter and receiver filters are flat in the
frequency range of interest, i.e., |F{γ(t)}| ≈ 1, and (iii) the guard interval is sufficiently large to avoid
interference between OFDM frames, i.e., Tg > τLh. After symbol–rate sampling with 1/T , the effective

discrete–time channel between transmit antenna i and receive antenna j for the mth subcarrier of the kth OFDM frame is given by

gi,j[m, k] = Lh

X

l=1

h(l)_{i,j}(kTf)e−j2πm∆f τl, 1 ≤ m ≤ D, k ∈ IN. (3)
It can be shown that the autocorrelation ofgi,j[m, k] is separable into a spectral, a temporal, and two spatial
correlation functions:

ψgg[µ, κ, i, j, p, q] △

= Egi,j[m + µ, k + κ]gp,q∗ [m, k] = ψf[µ]ψt[κ]ψTx[i, p]ψRx[j, q] . (4) The spectral correlation is given by

ψf[µ] = Lh

X

l=1

σ_{l}2e−j2πµ∆f τl _{,} _{(5)}

and the temporal correlation of all paths is modelled by [36]

where J0(·) and Bh denote the zeroth order Bessel function of the first kind and the maximum fading
bandwidth. The spatial correlation coefficients ψTx_{[i, j] and ψ}Rx_{[i, j] are found to equal the elements in}
the _{ith row and jth column of Ψ}Tx △= AAH and _{Ψ}Rx △= (BBH_{)}T_{, respectively. For numerical and}
simulation results we use the simple exponential correlation model [35]

ψTx[i, j] = e−α||xTxi −xTxj ||, _{(7)}

ψRx[i, j] = e−α||xRxi −xRxj ||, _{(8)}

where xTx_{i} and xRx_{j} denote the position vectors of transmit antenna i and receive antenna j in multiples
of the wavelength, respectively, and _{α ∈ IR}+.

As usual, additive spatially, temporally, and spectrally white Gaussian noise (AWGN) nj[m, k] with mean zero and common variance σ2

n △

= E{|nj[m, k]|2} is assumed.

III. DIFFERENTIALSPACE–FREQUENCYMODULATION(DSFM)

In this section, we describe DSFM to achieve spatial and spectral diversity and facilitate noncoherent detection (discussed in Section IV). To this end, we briefly introduce diagonal signal constellations [24, 25] in Section III-A and present differential encoding in time and frequency direction in Section III-B. The novel signal–allocation (SA) scheme, i.e., the allocation of elements of the diagonal–matrix signals to OFDM subchannels and transmit antennas, is developed in Section III-C.

*A. Diagonal Signal Constellations*

One motivation to use diagonal signal constellations is thatnon–diagonal constellations, such as orthog-onal designs [23] or Cayley codes [37], require the channel to be almost constant during the transmission of one matrix symbol. This carries over to DSFM in that performance of non–diagonal constellations would be affected by channel variations in frequency direction. Additionally, diagonal constellations facilitate joint spatial and spectral diversity, cf. Section III-C.

A diagonal constellation of L (NS× NS)–dimensional matrix signals is defined as [24, 25]
V _{=}△
diagnej2π_{L}c1_{, . . . , e}j2π_{L}c_{NS}ol
l ∈ {0, . . . , L − 1}
, (9)

where the integer coefficients [c1, . . . , cNS] are optimized in [24, 25]. We choose L

△

= 2NSR _{where} N

S △ = NTNB, i.e.,(NSR)–tuples of bits are mapped to elements of the set V, and R denotes the data rate in bits per OFDM subchannel use. The factor NB ∈ IN is discussed below.

*B. Differential Encoding*

We denote data matrices chosen from V by V[m, k], where m and k are discrete–frequency and – time indices as described in detail in Section III-C. In DSFM, diagonal transmit matrices S[m, k] can be obtained either by differential encoding in time direction, i.e., time–differential space–frequency modulation (T–DSFM), via

S_{[m, k + 1] = V [m, k]S[m, k],} S_{[m, 1] = I}_{N}

S , (10)

or by differential encoding in frequency direction, i.e. frequency–differential space–frequency modulation (F–DSFM), via

S_{[m + 1, k] = V [m, k]S[m, k],} S_{[1, k] = I}_{N}

S. (11)

F–DSFM is preferable for burst transmission or if detection delay is to be minimized. T–DSFM is advantageous for continuous transmission since (i) the share of reference symbols S[m, 1] tends to zero with increasing transmission time and (ii) OFDM is usually employed for channels that exhibit significant frequency but moderate time selectivity.

*C. Signal Allocation (SA) Scheme*

There are several possibilities to use the three dimensions space, frequency, and time to transmit the two–dimensional matrices S[m, k]. In general, frequency and time are interchangeable, but to minimize the transmission delay, space–frequency modulation, i.e., DSFM, is considered in this paper, and all elements of S[m, k] are allocated to subcarriers of the kth OFDM frame. Hence, assuming D active subcarriers and D/NS being integer, the D/NS symbols [S[1, k], S[2, k], . . . , S[D/NS, k]] are transmitted during the kth OFDM frame using NT transmit antennas. In order to exploit full spatial diversity, the non–zero elements si,i[m, k], 1 ≤ i ≤ NS, of each S[m, k] are transmitted from all NT antennas over NB= NS/NT subcarriers per antenna. To maximize spectral diversity under the constraint that the PDP is not known at the transmitter, these NB subcarriers are spread uniformly over the used frequency band, i.e., spaced by D/NB subcarriers. This is similar to the subchannel grouping proposed for space–time–frequency coding in [11] also adopted in [15, 18, 27, 28] (cf. also [38] for single–antenna differential frequency coding).

Finally, differential encoding needs to be taken into account. In the case of T–DSFM (see (10)), the
proposed scheme ensures that the time between receiving two consecutive transmit symbols S[m, k] and
S_{[m, k+1] is the minimum of one OFDM frame. As long as s}_{i,i}_{[m, k] is assigned to the same OFDM }
subcar-rier and antenna assi,i[m, k+1], the “effective” fading bandwidth attains its minimum of BhTf. In the case of

F–DSFM (see (11)), the spectral correlation between the subcarriers allocated to si,i[m, k] and si,i[m + 1, k] should be maximized for all 1 ≤ i ≤ NS. As, different from e.g. [19], the channel PDP is assumed to be unknown at the transmitter, the most reasonable choice is to transmit[si,i[1, k], . . . , si,i[D/NS, k]] over a set of D/NS contiguous subcarriers. This can be regarded as an application of block–differential modulation [29] to DSFM.

In summary, according to the proposed SA scheme the diagonal elements si,i[m, k], 1 ≤ i ≤ NS, of
S_{[m, k], 1 ≤ m ≤ D/N}_{S}_{,} _{k ∈ IN, are transmitted from antenna [mod (i − 1, N}_{T}_{) + 1] over subcarrier}
[(i − 1)D/NS+ m] of the kth OFDM frame, while all other antennas do not transmit over this subcarrier.
Fig. 1 illustrates the SA scheme for the example of NT = 2 and NB = 2, NS = NTNB= 4, and NR = 1.
Compared to related SA schemes based on diagonal–matrix signals this scheme is advantageous in that
neither the effective fading bandwidth ([16, 27]) nor the spectral spacing between subcarriers allocated to
si,i[m, k] and si,i[m + 1, k] ([15]) are increased beyond their respective minima BhTf and 1. These issues
will be discussed further in Section VI.

IV. MULTIPLE–SYMBOL DIFFERENTIALDETECTION(MSDD)

In MSDD, N received symbols are jointly processed. N is referred to as observation window size and, usually, the observation window extends in the direction of differential encoding, i.e., time for single–carrier transmission, cf. e.g. [30]–[32]. A straightforward adaptation of MSDD to multicarrier transmission is to apply a 1D observation window which extends either in time or in frequency depending on the direction of differential encoding. Such an approach has been chosen for DFDD and Viterbi detection in [15, 22].

Inspired by 2D channel estimation schemes for OFDM and MIMO–OFDM, cf. e.g. [1]–[4], we present in this section MSDD with a 2D observation window. We first derive the ML metric for 2D MSDD in Section A and then transform it into a form amenable to tree–search detection algorithms in Section IV-B. Implementation details and variants of 2D MSDD are then discussed in Section IV-C.

*A. Maximum–Likelihood Metric for 2D MSDD*

From the channel model (3) and according to the SA scheme devised in Section III-C, we define the NS× NR MIMO–OFDM channel matrix

G_{[m, k]}_{=}△ h_{g}_{mod(i−1,N}
T)+1,j[(i − 1)D/NS+ m, k]
i
i_{=1,...,NS}
j_{=1,...,NR}
, 1 ≤ m ≤ D/NS, k ∈ IN. (12)
The NS× NR receive matrix R[m, k] corresponding to transmission of a symbol S[m, k] is given by

where N[m, k] is an NS × NR AWGN matrix. For illustration see Fig. 1. The 2D MSDD observation window extends over

N = Nf · Nt (14)

received matrices R[mi, kj], 1 ≤ i ≤ Nf, 1 ≤ j ≤ Nt, i.e., Nf andNt are the dimensions of the window in frequency and time, respectively. To allow for detection of the data symbols V[m, k], we need kj+1= kj+1, 1 ≤ j ≤ Nt− 1, for T–DSFM and mi+1= mi+ 1, 1 ≤ i ≤ Nf− 1, for F–DSFM. At the same time, there are no restrictions on the indices mi for T–DSFM and kj for F–DSFM and appropriate choices will be discussed in Section IV-C.3. The corresponding received symbols can be collected in an NfNS× NtNR matrix1

R △_{=}hR_{[m}_{i}_{, k}_{j}_{]}i

i_{=1,...,Nf}
j=1,...,Nt

, (15)

where “” emphasizes the rectangular extension of the 2D observation window in time and frequency. In
order to formulate the ML–MSDD decision rule it is convenient to block–vectorize the received matrix
R _{and therewith the transmission model. To this end, we define matrices}

Xk △_{=}XT_{[m}_{1}_{, k}_{1}_{] , X}T_{[m}_{2}_{, k}_{1}_{] , . . . , X}T_{[m}_{N}

f, kNt]

T

, X _{∈ {R, G, N , S} ,} _{(16)}

such that Rk, Gk, Nkand Sk are of dimensionN NS×NR andN NS×NS, respectively, and anN NS×N NS block–diagonal matrix

Sk D

△

= diag{S[m1, k1], S[m2, k1], . . . , S[mNf, kNt]} , (17)

where “k” indicates that matrices X[mi, kj] are stacked first according to the frequency index mi and then according to the time index kj. We can then rewrite (13) to match the 2D–MSDD observation window as

Rk _{= S}k
DG

k

+ Nk . (18)

Since given Sk, Rk is zero–mean complex Gaussian distributed with autocorrelation matrix

ΨkRR|S
△
= E
RkRkH
Sk
= Sk_{D}ΨkGG
Sk
D
H
+ σ_{n}2NRIN NS , (19)
where _{Ψ}kGG
△

= EnGkGkHo_{, the 2D ML–MSDD decision rule reads}
ˆ
Sk _{= argmin}
Sk∈VN
tr
RkH_{Ψ}k
RR|S
−1
Rk
. (20)

1_{For the sake of readability, we omit the dependence of 2D–MSDD block matrices of index sets}_{{m}

It should be clear from the above that this decision rule applies to both T–DSFM and F–DSFM. From ˆ

Sk _{the} _{N}_{f}_{(N}_{t}_{− 1) and (N}_{f} _{− 1)N}_{t} _{data–matrix estimates ˆ}V_{[m, k] are determined using (10) and (11),}
respectively.

*B. Tree–Search Algorithms for 2D MSDD*

As the brute–force approach to solving (20) by searching through all LN −1_{= 2}NSR(NtNf−1) _{relevant S}k

is clearly intractable for observation window sizes of interest, we apply tree–search algorithms, which have recently attracted renewed attention to solve search problems in high–dimensional spaces, cf. e.g. [39].

For a representation of (20) amenable to the application of tree–search algorithms, we make the as-sumption that the channel coefficients collected in G[m, k] defined in (12) are mutually uncorrelated. We note that this is a fairly mild assumption, since it only requires that (i) NB subchannels spaced by (D/NB)∆f are uncorrelated, i.e., the channel coherence bandwidth is significantly less than (D/NB)∆f , and (ii) the minimum distance between different elements of the transmit or receive antenna array is greater than half of the wavelength [34]. The effect of subcarrier and spatial correlation on performance and detection complexity will be discussed in Section VI. Applying this assumption, it can be shown that the autocorrelation matrix of the (assumed) fading–plus–noise process Gk+ Nk can be written as

ΨkGG+ σ
2
nNRIN NS = NR Ψ
t
⊗ Ψf + σ_{n}2I_{N} ⊗ I_{N}
S
△
= C ⊗ INS , (21)

with time and frequency correlation matrices
Ψt △=ψt_{[k}
i− kj]
i=1,...,Nt
j=1,...,Nt and Ψ
f △_{=}_{ψ}f_{[m}
i− mj]
i_{=1,...,Nf}
j_{=1,...,Nf}
. (22)

Following the derivations for single–carrier DSTM in [40, Section V], we arrive at
ˆ
Sk _{= argmin}
Sk∈VN
N
X
i=1
N
X
j=i
˜
RH
i,jSj
2
, (23)

where ˜R_{i,j} _{= u}△ _{i,j}R_{j} _{and R}_{j} _{and S}_{j} _{denote the} _{jth N}_{S} _{× N}_{R} _{and} _{N}_{S} _{× N}_{S} _{submatrices of R}k _{and}
Sk_{, respectively. The coefficients} _{u}_{i,j} _{are the elements of an upper triangular matrix U , which can be}
determined through the Cholesky factorization UHU _{= C}−1 _{(cf. [40, Section V],[32, Section III-A]), or}
adaptively using the recursive least–squares algorithm, cf. e.g. [41].

With the ML–MSDD metric in the form of (23) efficient tree–search algorithms devised for single–carrier DSTM and 1D MSDD in [32, 42, 43] can now be readily applied to DSFM and 2D MSDD.

*C. Implementation Aspects*

*1) Tree–Search Algorithms: Various variants of tree–search algorithms applicable to solve (23) have*

been discussed and compared in [32], including those of [42, 43]. For the sake of brevity, we do not repeat the description of these algorithms, but only mention those variants particularly suited for MSDD and considered for performance discussion in Section VI.

• For an exact solution of (23), i.e., 2D ML–MSDD, the sphere decoder using Schnorr–Euchner enumeration presented in [32, Section III-B.1] is a favorable choice. In the following this algorithm will —as in [32]— be referred to as multiple–symbol differential sphere decoder (MSDSD).

• A suboptimal variant of MSDSD, subsequently referred to as MSDSD–FM–LD, uses a Fano–type metric (cf. [32, Section B.2]) and a lattice–decoder (LD) based symbol search (cf. [32, Section III-C.1]). Consequently, it does not necessarily find the ML solution but has lower complexity especially for low signal–to–noise ratio (SNR).

*2) Sorting: There are several possibilities to sort the received matrices R*[mi, kj] collected in Rinto a
stackedN NTNR× NR dimensional matrix, like e.g. Rk in (18). While it is clear that the sorting does not
influence the performance of ML MSDD, we also found that it has only a small effect on the complexity
of the tree–search decoders. Therefore, we only consider the sorting strategy (16) in the following.

*3) Observation Window Construction (OWC): Let us introduce the variables* p, 1 ≤ p ≤ D/(NSNf),
and q ∈ IN to identify the position of the Nf × Nt rectangular observation window in the frequency–time
plane. The corresponding sets of frequency and time indices of symbols inside the observation window are
denoted as M(p) △_{= {m}(p)
1 , m
(p)
2 , . . . , m
(p)
Nf} ⊂ {1, 2, . . . , D/NS} and K
(q) △_{= {k}(q)
1 , k
(q)
2 , . . . , k
(q)
Nt} ⊂ IN. As

stated in Section IV-A, the index sets in the direction of differential encoding, i.e., M(p) _{for F–DSFM and}
K(q) _{for T–DSFM, must be sets of consecutive integers, and adjacent observation windows must overlap}
by one index, i.e., m(p+1)_{1} = m(p)_{N}_{f} for F–DSFM and k(q+1)_{1} = k_{N}(q)_{t} for T–DSFM.

On the other hand, it is not immediately clear how the index sets perpendicular to the direction of
differential encoding, i.e., M(p) _{for T–DSFM and} _{K}(q) _{for F–DSFM, should be chosen. We note that for}
T–DSFM the sets M(p) _{are disjoint and that the union} _{∪}D/(NSNf)

p=1 M(p) is the set of integers from 1 to
D/NS. Similarly, K(q) are disjoint and ∪q∈INK(q) = IN in case of F–DSFM. While an optimization with
respect to error rate appears to be intractable, it is intuitively clear that the channel matrices G[m(p)_{i} , k_{j}(q)]
captured in the observation window should be correlated to the largest possible degree, since strong channel
correlations facilitate channel estimation based on received samples within the observation window, which

is done implicitly in MSDD (cf. e.g. [32, 40] for the relation of (20) to linear prediction).

In the case of F–DSFM, minimizing detection delay would suggest to chose K(q) _{as sets of consecutive}
integers as well. In practically relevant scenarios, this also coincides with the choice for maximal correlation,
as can be seen from Fig. 2, which shows the temporal correlation ψt_{[κ] [cf. (6)] for normalized fading}
bandwidths up to BhTf = 0.01. The same is true considering M(p) for T–DSFM and certain commonly
used channel models such as the typical urban (TU) model [44] and the model with exponential PDP
with decay parameter d = 1 (τl = lT , σl2 ∝ e−l/d, l ∈ IN). This can be seen from Fig. 3, where the
spectral correlation ψf_{[µ] [cf. (5)] is plotted assuming D = 192 and ∆f = 8 kHz as used for digital audio}
broadcasting (DAB) systems [45]. For channels such as the hilly terrain (HT) channel model [44] or the
two–ray (2–Ray) channel with τ2 − τ1 = 20 µs we observe a “strongly non–concave” behavior of ψf[µ],
which suggests that partitioning the D/NS frequency indices into sets M(p) of non–consecutive integers
is likely to improve the performance of T–DSFM with 2D MSDD. A meaningful figure of merit which is
to be maximized for each set M(p) _{is the sum of spectral correlations within the observation window,}

ρ M(p) △
=
Nf
X
i=2
i−1
X
l=1
ψ
fh_{m}(p)
i − m
(p)
l
i
. (24)

We therefore propose the following greedy–type algorithm to construct the sets M(p)_{.}
Initialize I(0) _{= {1, . . . , D/N}
S}
for p = 1, . . . , D/(NSNf)
Pick m(p)_{1} ∈ I(p−1)
n
m(p)_{2} , . . . , m(p)_{N}_{f}o= argmax
n
m(p)_{2} ,...,m(p)_{Nf}o⊆I(p−1)_{\m}(p)
1
n
ρm(p)_{1} , . . . , m(p)_{N}_{f}o
M(p) _{=}n_{m}(p)
1 , . . . , m
(p)
Nf
o
I(p) _{= I}(p−1)_{\M}(p)
end

To illustrate the effectiveness of this algorithm, we consider the example of D = 192, ∆f = 8 kHz,
NT = 3, 2–Ray PDP with τ2 − τ1 = 20 µs, and Nf = 2. In this case, the algorithm groups the indeces in
sets M(p) _{as} _{{p, p + 25} for 1 ≤ p ≤ 25, {p + 25, p + 31} for 26 ≤ p ≤ 31, and {63, 64} for p = 32, with}
1.0, 0.99, and 0.88 as respective values of ρ M(p)_{, cf. also Fig. 3. Hence, the optimal ρ M}(p)_{ from}
(24) is practically attained and 2D–MSDD effectively sees a frequency–flat channel for 1 ≤ p ≤ 31, and
only for p = 32 the dispersive channel as for 2D–MSDD without OWC is effective.

V. PERFORMANCEANALYSIS

In this section, we derive expressions for the SER performance of DSFM with 2D ML–MSDD as pre-sented above, and we illustrate their usefulness to guide the design of appropriate transmission parameters.

*A. Derivation*

The derivation of the SER expressions is done in two steps. First, the pairwise error probabilityPEP(Sk → ˆ

Sk_{) that ˆ}Sk _{is detected while S}k _{6= ˆ}Sk _{was transmitted is considered. Thereby, we can directly apply well–}
known techniques for evaluating the PEP for quadratic receivers, cf. e.g. [46]. In the second step, we identify
the dominant error events and sum the corresponding PEPs to tightly approximate the average SER.

Using the results of [46, Section III] the PEP(Sk → ˆSk_{) can be written as}
PEPSk _{→ ˆ}Sk_{= −} X
R H poles
Res 1
s
N NSNR
Y
i=1
1+sλi Ψrr_{|S}F
−1
!
, (25)

where the summation is taken over all residues corresponding to poles located in the right–hand (RH) side of the complex s–plane, λi(X) denotes the ith eigenvalue of X, and

Ψrr_{|S}
△
= I_{N}
R⊗ S
k
D ˜C
I_{N}
R⊗ S
k
D
_{H}
(26)
F _{= ˆ}△ Sk
D C−1⊗ INS
ˆ_{S}k
D
H
− Sk_{D} C−1_{⊗ I}_{N}
S
_{S}k
D
_{H}
(27)
˜
C _{= Ψ}△ Rx_{⊗ Ψ}t_{⊗} ˜_{Ψ}f_{◦ 1}_{N}
BNf ⊗ Ψ
Tx
+ σ_{n}2I_{N N}
RNS (28)
˜
Ψf[mx−my]
△
= hψfhmx−my+(i−j)_{N}D_{S}
ii
i_{=1,...,NS}
j_{=1,...,NS}
(29)
˜
Ψf =△ h ˜Ψf[mi − mj]
i
i_{=1,...,Nf}
j_{=1,...,Nf}
. (30)

ΨTx, _{Ψ}Rx, and _{Ψ}t are defined in Section II-B and (22), respectively. Evaluation of PEP(Sk → ˆSk_{) is}
analytically easy for simple eigenvalues in λi Ψrr|SF. Otherwise, numerical methods can be used [47].
It is important to note that while the correlation matrix C [cf. (21)] assuming uncorrelated elements in
G_{[m, k] is used in the decision rule (23) [cf. also (27)], the true fading–plus–noise correlation matrix ˜}C
appears in (26). Thus, the effect of metric mismatch due to the decoder’s assumption of no correlation
within G[m, k] is captured by the PEP analysis.

Next, we apply a truncated union bound over the dominant error events to approximate the average
SER. As argued in [32] for single–carrier transmission the dominant error events are single transmit–
symbol errors ˆS_{[m}_{i}_{, k}_{j}_{] with maximum correlation}

ζi,j △

1 ≤ i ≤ Nf, 1 ≤ j ≤ Nt, with the true transmit symbol S[mi, kj]. Thus, we define sets ˆS k

i,j of all matrices ˆ

Sk _{that differ from S}k _{only in ˆ}S_{[m}_{i}_{, k}_{j}_{] 6= S[m}_{i}_{, k}_{j}_{] and that maximize ζ}_{i,j}_{. Note, that at this S}k _{can be}
fixed. Taking into account that a data–symbol error occurs whenever the preceding or succeeding transmit
symbol is erroneously detected the SER for the data–symbol V[mi, kj] in the ith row and jth column of
the observation window can be approximated via

SERTD_{i,j}=X
ˆ
Sk∈ ˆSk_{i,j}
PEPSk _{→ ˆ}Sk_{+} X
ˆ
Sk∈ ˆSk_{i,j+1}
PEPSk _{→ ˆ}Sk _{(32)}
SERFDi,j=
X
ˆ
Sk∈ ˆSk_{i,j}
PEPSk _{→ ˆ}Sk_{+} X
ˆ
Sk∈ ˆSk_{i+1,j}
PEPSk _{→ ˆ}Sk _{(33)}

and the average SER over the entire 2D observation window can be approximated via

SERTD = 1
Nf(Nt− 1)
Nf
X
i=1
Nt−1
X
j=1
SERTDi,j , (34)
SERFD = 1
(Nf − 1)Nt
Nf−1
X
i=1
Nt
X
j=1
SERFD_{i,j} (35)

for TD– and FD–OFDM, respectively.

*B. Evaluation*

For illustration purposes, we evaluate SERFD_{i,j} from (33) for F–DSFM with D = 192, ∆f = 8 kHz,
NT = 3, NB = 1, NR = 1, R = 1, and a time– and frequency–selective channel with HT PDP and
BhTf = 0.01. The 2D–MSDD observation window extends over Nf = 10 subcarriers and different numbers
Nt of OFDM frames. Fig. 4 shows the SNR in terms of 10 log10(Eb/N0) required to achieve individual
error rates of SERFD_{i,j} = 10−5 _{as function of the position} _{[i, j], 1 ≤ i ≤ N}

f − 1, 1 ≤ j ≤ Nt, for Nt = [1, 2, 4].2 Simulation results for Nt = 1 and Nt= 2 are included, as well.

We observe a kind of “hammock behavior” with larger values for the required SNR towards the edges of the 2D observation window. This can be explained by regarding MSDD as detection with inherent prediction–based channel estimation (cf. [40, 32]), which yields estimation with lower estimation error for symbols located in the center positions of the observation window than for symbols at its edges. However, as Nt increases this effect diminishes, which means that the application of the 2D window is

2_{E}

bandN0denote the average received energy per information bit, the two–sided noise power spectral density of the complex equivalent–

highly beneficial for achieving uniform individual error rates. It can also be seen that increasing Nt leads to significant improvements in power efficiency of the detector even in this relatively fast fading scenario. Quite remarkably, for Nt = 4 the gap to coherent detection with perfect CSI (not shown in the figure) is only approximately 0.6 dB on average and 0.4 dB in the center positions of the observation window. Finally, we note that simulated SERs closely match the results obtained from (33), which nicely confirms the accuracy of the proposed SER approximation.

VI. RESULTS ANDDISCUSSION

In this section, we present numerical and simulation results to illustrate the performance of MIMO– OFDM with the proposed DSFM SA scheme and fast 2D MSDD. We continue to exemplarily consider the digital–audio–broadcasting system OFDM parameters D = 192 and ∆f = 8 kHz. The coefficients [c1, . . . , cNS] for the diagonal constellations (see (9)) were taken from [25, Table I]. To limit the number

of parameters, we fix the data rate to R = 1 bit per OFDM subchannel use and assume a single receive
antenna, i.e.,NR = 1. Fast 2D MSDD is implemented using MSDSD and MSDSD–FM–LD, respectively,
as outlined in Section IV-C.1. We recall that only MSDSD is optimal in that it finds the exact ML–MSDD
solution. As benchmark detectors we consider conventional differential detection (CDD) (N = 2) and
decision–feedback differential detection (DFDD) [15, 26] with 1D observation window of length N , their
respective lattice–decoder (LD) based counterparts differential lattice decoding (DLD) [48, 49] and DFDD–
LD [49], and coherent detection with perfect CSI at the receiver. Numerical results for these detectors are
obtained based on the same mathematical foundation as for MSDD using a truncated union bound over the
dominant single–symbol error events (cf. e.g. [26, 46]). In the case of F–DSFM, DFDD was implemented
making use of _{N − 1 pilot symbols S[m, k], 1 ≤ m ≤ N − 1, k ∈ IN, at the beginning of each OFDM}
frame in order to avoid a high error floor due to (i) a growing observation window at the start of DFDD
and (ii) subsequent excessive error propagation. The corresponding rate–loss is accounted for in Eb/N0.
Unless specified otherwise, we assume a spatially uncorrelated channel, i.e. _{Ψ}Tx = INT.

*A. SER Performance*

*1) F–DSFM and ML–MSDD: To illustrate the benefits of the new SA scheme and the application of*

a 2D observation window we consider the example of F–DSFM with different values of NS = NT, i.e., NB = 1, a channel with HT PDP and BhTf = 0.003, and MSDD with different observation window parameters [Nf, Nt]. Table I compares the SNR required to achieve SER = 10−5 for (i) the proposed

SA scheme, (ii) a scheme, where si,i[m, k] and si,i[m + 1, k] are separated by NT subcarriers while the
effective fading bandwidth is not increased, e.g. [15], and (iii) a scheme, wheresi,i[m, k] and si,i[m + 1, k]
are allocated to neighboring subcarriers but the effective fading bandwidth is increased by a factor NT,
e.g. [16, 27]. SA schemes such as [17]–[19] that exploit knowledge of the PDP at the transmitter are
not included in the comparison. The SA variant proposed in [20, Section V] is not considered either, as
the pseudo–random subcarrier assignment seriously complicates MSDD (and DFDD) due to the fact that
Ψf changes as the observation window slides across the received data. The respective SNR values for
coherent detection with perfect CSI are also included for comparison. The results are obtained from the
SER approximation (35), except for the values for SA according to [15] and [Nf, Nt] = [5, 2], which are
simulation results (“—” indicates the ocurrence of an error floor above SER = 10−5_{). Here,} _{N}

f = 5 is not sufficient for tracking the channel variations in frequency direction and performance is significantly degraded by multiple–symbol errors due to fading. These are not accounted for in (35), which therefore is too optimistic in this case.

The new SA method consistently achieves the best power efficiency. The gains over the scheme of [15] are more pronounced than those when comparing with [16, 27], since the correlations of the channel gains captured within the observation window are lowest for the scheme of [15] (see also Figs. 2 and 3). In particular, in the case of 1D MSDD the SA schemes of [16, 27] achieve the same power efficiency as our SA scheme, as the increase of the effective fading bandwidth has no effect here. With larger total window size N = NfNt the gains are smaller as the performance approaches that of coherent detection. Similarly, comparing 2D MSDD and 1D MSDD, the largest gains are achieved for relatively small N . For example, for the new SA scheme 2D MSDD with [Nf, Nt] = [5, 2] improves performance by 0.6 − 0.9 dB over 1D MSDD with Nf = N = 10. 2D MSDD with [Nf, Nt] = [5, 2] even achieves practically the same power efficiency as 1D MSDD with Nf = N = 20, i.e., double the window size, which corresponds to significant savings in decoder complexity especially in low SNR (see Section VI-B). We finally observe that the new SA scheme combined with 2D MSDD with [Nf, Nt] = [10, 2] reduces the performance gap to coherent detection to no more than 1 dB.

*2) T–DSFM and ML–MSDD: Next, we consider T–DSFM and (i) compare 1D and 2D ML–MSDD,*

(ii) illustrate the performance improvements due to the observation window construction (OWC) proposed
in Section IV-C.3, and (iii) further demonstrate the benefits of the new SA scheme. Fig. 5 shows the
SNR required to achieve SER = 10−5 _{as a function of the normalized fading bandwidth} _{B}

hTf for various channel PDPs. NT = 3 transmit antennas and NB = 1 are assumed. 1D MSDD with Nt = N = 5 and

Nt = N = 10 and 2D MSDD with [Nf, Nt] = [2, 5] are compared. Note that for the considered case of NB = 1 the results for Nt = N are independent of the PDP. The curves for the SA schemes of [16, 27] and Nf = 1 are also included, as well as the curve for coherent detection with perfect CSI. Again, the results are obtained from (34).

From the results for the 2–Ray channel we observe that extending the observation window over Nf = 2

neighboring subcarriers is only beneficial for relatively fast fading with BhTf &0.002. In contrast to this, application of the proposed OWC scheme leads to consistent improvements when applying[Nf, Nt] = [2, 5] compared to [Nf, Nt] = [1, 5] for all fading bandwidths. Extension of the observation window in frequency direction is also beneficial for the TU and exponential PDP. For the case of the HT PDP, where the maximal channel frequency correlation is smaller than for 2–Ray, TU, and exponential PDP (see Fig. 3), 2D MSDD with [Nf, Nt] = [2, 5] noticeably outperforms 1D MSDD with [Nf, Nt] = [1, 5] only for BhTf &0.002.

Comparing the respective curves for 2D MSDD and [Nf, Nt] = [2, 5] with that for 1D MSDD and N = Nt = 10, we conclude that it is not per se clear that 2D MSDD is advantageous over 1D MSDD for identical total window size N = NfNt (see also Table I for DMD [15]). In particular, Nf and Nt are not simply interchangeable for fixed N and the same channel correlation, but the direction of differential encoding needs to be taken into account too. Whereas increasing the observation window in direction of differential encoding lowers the error of predicting the channel variation, increasing the other window dimension improves the suppression of estimation noise prior to prediction. This also emphasizes the usefulness of the SER approximations developed in Section V to guide the choice of the parameters [Nf, Nt] for different channel scenarios and DSFM parameters.

A comparison of the respective results for the different SA schemes demonstrates the superior perfor-mance of the propose SA method. Finally, we note that both 1D and 2D MSDD with N = 10 achieve a performance within 0.4 − 1.0 dB of that for detection with perfect CSI even for relatively fast fading channels.

*3) F–DSFM and Different Detection Algorithms: We now compare MSDSD, which accomplishes ML–*

MSDD, and suboptimal MSDSD–FM–LD for 2D MSDD with observation window[Nf, Nt] = [5, 2]. CDD, DLD, DFDD, DFDD–LD with 1D windows of size Nf = N = 2 and Nf = N = 10, respectively, and coherent detection with perfect CSI are considered as benchmark detectors. The SER is shown as function of the SNR for F–DSFM with NT = 3 and NB = 1 and TU, exponential (d = 1), and HT PDP in Figs. 6 a)–c), respectively. Lines and markers represent numerical and simulation results, respectively.

coherent detection with perfect CSI within about 1 dB for all channel scenarios and in the SER–range of interest. MSDSD–FM–LD only looses a fraction of a decibel compared to optimal MSDSD and still provides consistent improvements over DFDD and CDD. This is especially true in highly dispersive channels, cf. Fig. 6 c), where CDD and computationally efficient DFDD–LD even encounter relatively high error floors. It should be noted that these gains are due to both the application of MSDD instead of DFDD and the use of a 2D as opposed to an 1D observation window.

*4) Spatial vs. Spectral Diversity: Fig. 7 presents numerical (lines) and simulation (markers) results for*

the SER of F–DSFM with ML–MSDD at 10 log10(Eb/N0) = 15 dB as a function of the inter–transmit–
antenna spacing
xTx_{i} − xTx_{i+1}

, 1 ≤ i ≤ NT − 1, when assuming a linear equispaced transmit antenna array. As in [35] we choseα = 0.8 [cf. (7), (8)] such that spatial subchannels are fairly uncorrelated if the respective transmit antennas are separated by half of the wavelength, cf. [34]. 2–Ray (τ2− τ1 = 20 µs) and HT PDPs are considered. We compare SA according to Section III-C with NB = NS = 3 (dashed lines) and NT = NS = 3 (solid lines) (ignore the dash–dotted line for the moment), employing 2D MSDD with [Nf, Nt] = [5, 2] in both cases.

As can be seen, the performance for NT = NS converges to that for NT = 1 as the antenna spacing tends to zero. This is because for the new SA scheme the assignment of elements si,i[m, k] to subcarriers does not depend on NB. Hence, increasingNT while keeping NS fixed can only improve performance by reducing subchannel correlation, i.e., correlation among elements within G[m, k] (see Fig. 1). Since for the 2–Ray channel NS = 3 > Lh = 2, diversity order, which is min{NSNR, NTNRLh} (cf. e.g. [20]), can be increased from two forNB = NS = 3 to three for NT = NS = 3, and relatively large performance gains are observed with increasing antenna spacing. For the HT PDP, on the other hand, the same diversity order of three is accomplished in both cases and thus, performance is largely independent of the antenna spacing. To further illustrate the advantage of the proposed SA scheme, Fig. 7 also includes the SER curve for DMD [15] with NT = NS = 3 and the HT channel (dash–dotted line). We note that for NB = NS DMD and the new SA scheme are identical. Here it is advantageous to set[Nf, Nt] = [10, 1] due to the increased spectral spacing of symbols si,i[m, k] and si,i[m + 1, k]. Different from the proposed SA scheme, the performance with DMD significantly deteriorates for larger spatial correlation, as si,i[m, k], 1 ≤ i ≤ NT, are transmitted over neighboring subcarriers. Due to the increased spectral spacing of symbols captured in the observation window, the SER for NT > 1 is higher than that for NT = 1 even if there is no spatial correlation.

*B. Complexity*

The computational complexity of MSDD, i.e., the number of operations required by MSDSD or MSDSD–
FM–LD to find a solution ˆSk_{, essentially hinges on the SNR and on the overall size} _{N = N}_{f}_{N}_{t} _{of the}
observation window, cf. e.g. [32] for the single–carrier case. This means, that the gains of 2D MSDD
over 1D MSDD with equal window size N as observed especially for F–DSFM come at practically
no computational cost. Furthermore, since the MSDD metric assumes uncorrelated channel coefficients
collected in G[m, k] [cf. (12)], complexity tends to grow with increasing correlation due to a more
pronounced metric mismatch. Subsequently, we present simulation results for the computational complexity
in terms of the average number of real–valued floating–point operations (flops) per decoded symbol.

*1) F–DSFM and Different Detection Algorithms: In Fig. 8 computational complexities of MSDSD,*

MSDSD–FM–LD, DFDD, DFDD–LD, CDD, and DLD are compared for the same system and channel parameters as in Fig. 6. While the computational complexity of MSDSD is quite high at low SNR, MSDSD– FM–LD succeeds in keeping average complexity reasonably low over the entire range of relevant SNR. In particular, the complexity of MSDSD–FM–LD is very well comparable to that of DFDD–LD, which is quite remarkable considering the performance advantages of MSDSD–FM–LD over DFDD–LD and the small gap in performance compared to coherent detection with perfect CSI (see Fig. 6). Furthermore, it can be seen from a comparison of subplots a)–c) that the PDP hardly influences the decoder complexity.

*2) Spatial vs. Spectral Diversity: Fig. 9 illustrates the dependence of computational complexity of*

MSDSD on correlation in the equivalent (NS × NR)–dimensional MIMO channel, i.e., among elements within G[m, k]. The same transmission parameters as in Fig. 7 are applied. As expected, for reasonably large antenna spacing with spatially uncorrelated channels, the complexity with NT > 1 is lower than that for NT = 1. This effect is particularly pronounced for the 2–Ray channel, where NB > Lh for NT = 1 and thus, the assumption of uncorrelated channel coefficients in G[m, k] is grossly idealistic. Hence, given sufficient antenna spacing, the use of multiple antennas is advantageous also from a complexity–point of view.

VII. CONCLUSIONS

In this paper, we studied DSFM for MIMO–OFDM transmission without CSI at the receiver. We focussed on the application of MSDD for power–efficient transmission. We devised a novel SA scheme that allows to make use of both space and frequency diversity and is particularly suited for differential transmission in both time and frequency direction in conjunction with detection without CSI. We further proposed the

application of MSDD with a two–dimensional observation window, which exploits channel correlations in both time and frequency direction for detection. Transforming the 2D MSDD problem into the form of 1D MSDD, efficient optimal and suboptimal tree–search algorithms proposed in previous work for single– carrier MIMO systems become directly applicable. We presented an analytical approximation of the SER that allows us to quickly and accurately assess the performance of DSFM with 2D MSDD. Numerical and simulation results confirmed considerable performance improvements due to (i) the new SA scheme, (ii) the use of MSDD as compared to DFDD or CDD and (iii) the application of 2D MSDD as compared to 1D MSDD. Employing tree–search algorithms to implement 2D MSDD, these gains entail only moderate increases in computational complexity.

REFERENCES

[1] G. L. St¨uber, J. R. Barry, S. W. McLaughlin, Y. Li, M. A. Ingram, and T. G. Pratt. Broadband MIMO–OFDM Wireless Communications.
*Proceedings of the IEEE, 92(2):271–294, February 2004.*

*[2] P. H¨oher, S. Kaiser, and P. Robertson. Two–Dimensional Pilot–Symbol–Aided Channel Estimation By Wiener Filtering. Proc. of IEEE*
*Conference on Acoustic, Speech, and Signal Processing (ICASSP), pages 1845–1848, April 1997.*

[3] Y. Li, L. J. Cimini, and N. R. Sollenberger. Robust Channel Estimation for OFDM Systems with Rapid Dispersive Fading Channels.
*IEEE Trans. Commun., 46(7):902–915, July 1998.*

*[4] M.-X. Chang and Y. T. Su. Model–Based Channel Estimation for OFDM Signals in Rayleigh Fading. IEEE Trans. Commun., 50(4):540–*
544, April 2002.

*[5] V. Engels and H. Rohling. Multilevel Differential Modulation Techniques (64-DAPSK) for Multicarrier Transmission Systems. European*
*Transactions on Telecommunications, 6:633–640, Nov.-Dec. 1993.*

*[6] M. Luise, R. Reggiannini, and G. M. Vitetta. Blind Equalization/Detection for OFDM Signals over Frequency–Selective Channels. IEEE*
*J. Select. Areas in Commun., 16:1568–1578, October 1998.*

[7] P. K. Frenger and N. A. B. Svensson. Decision–Directed Coherent Detection in Multicarrier Systems on Rayleigh Fading Channels.
*IEEE Transactions on Vehicular Technology, 48(2):490–498, March 1999.*

*[8] E. Haas and S. Kaiser. Two–Dimensional Differential Demodulation for OFDM. IEEE Trans. Commun., 51(4):580–586, April 2003.*
*[9] M.-X. Chang and Y. T. Su. Blind and Semiblind Detections of OFDM Signals in Fading Channels. IEEE Trans. Commun., 52(5):744–754,*

May 2004.

*[10] T. Cui and C. Tellambura. Joint Channel Estimation and Data Detection for OFDM Systems via Sphere Decoding. In Proc. of IEEE*
*Global Telecommunications Conference (Globecom), Dallas, TX, November 2004.*

[11] Z. Liu, Y. Xin, and G. B. Giannakis. Space–Time–Frequency Coded OFDM over Frequency–Selective Fading Channels. *IEEE*
*Transactions on Signal Processing, 50:2465–2476, October 2002.*

[12] H. B¨olcskei, M. Borgmann, and A. J. Paulraj. Impact of Propagation Environment on the Performance of Space–Frequency Coded
*MIMO–OFDM. IEEE J. Select. Areas in Commun., 21:427–439, April 2003.*

*[13] W. Su, Z. Safar, and K. J. R. Liu. Full–Rate Full–Diversity Space–Frequency Codes with Optimum Coding Advantage. IEEE Trans.*
*Inform. Theory, 51:229–249, January 2005.*

*[14] M. Borgmann and H. B¨olcskei. Noncoherent Space–Frequency Coded MIMO–OFDM. IEEE J. Select. Areas in Commun., 23:1799–1810,*
September 2005.

*[15] L. Lampe and R. Schober. Differential Modulation Diversity for OFDM. In Proc. of 6th International OFDM–Workshop (InOWo), pages*
19–1–19–6, Hamburg, September 2001.

*[16] J. Wang and K. Yao. Differential Unitary Space–Time–Frequency Coding for MIMO OFDM Systems. In Proc. of 36th Asilomar Conf.*
*on Signals, Systems and Computers, volume 2, pages 1867–1871, Pacific Grove, November 2002.*

*[17] T. Himsoon, W. Su, and K. J. R. Liu. Multiband Differential Modulation for UWB Communication Systems. In Proc. of IEEE Global*
*Telecommunications Conference (Globecom), November 2005.*

[18] S. Liu, G. Huang, and J.-W. Chong. Performance of Differential Unitary Space–Time Modulation over MIMO–OFDM Systems. In
*Proc. of International Conference on Communications, Circuits and Systems, pages 140–144, Chengdu, June 2004.*

[19] W. Su and K. J. R. Liu. Differential Space–Frequency Modulation via Smooth Logical Channel for Broadband Wireless Communications.
*IEEE Trans. Commun., 53(12):2024–2028, December 2005.*

[20] T. Himsoon, W. Su, and K. J. R. Liu. Single–Block Differential Transmit Scheme for Frequency Selective MIMO–OFDM Systems. In
*Proc. of IEEE Wireless Communications and Networking Conference (WCNC), pages 532–537, March 2005.*

[21] S. N. Diggavi, N. Al-Dhahir, A. Stamoulis, and A. R. Calderbank. Differential Space–Time Coding for Frequency–Selective Channels.
*IEEE Communications Letters, 6:253–255, June 2002.*

[22] P.-H. Chiang, D.-B. Lin, and H.-J. Li. Performance of 2IMO Differentially Transmit Diversity Block Coded OFDM Systems in Doubly
*Selective Channels. In Proc. of IEEE Global Telecommunications Conference (Globecom), pages 3768–3773, Saint Louis, November*
2005.

*[23] V. Tarokh and H. Jafarkhani. A Differential Detection Scheme for Transmit Diversity. IEEE J. Select. Areas in Commun., 18(7):1168–*
1174, July 2000.

*[24] B. L. Hughes. Differential Space–Time Modulation. IEEE Trans. Inform. Theory, 46(7):2567–2578, November 2000.*

*[25] B. M. Hochwald and W. Sweldens. Differential Unitary Space–Time Modulation. IEEE Trans. Commun., 48(12):2041–2052, December*
2000.

*[26] R. Schober and L. Lampe. Noncoherent Receivers for Differential Space–Time Modulation. IEEE Trans. Commun., 50(5):768–777,*
May 2002.

*[27] Q. Ma, C. Tepedelenio˘glu, and Z. Liu. Differential Space–Time–Frequency Coded OFDM With Maximum Multipath Diversity. IEEE*
*Transactions on Wireless Communications, 4(5):2232–2243, September 2005.*

*[28] H. Li. Differential Space–Time–Frequency Modulation Over Frequency–Selective Fading Channels. IEEE Communications Letters,*
7(8):349–351, August 2003.

*[29] S. Lv, G. Wei, J. Zhu, and Z. Du. Differential Unitary Space–Time Modulation in Fast Fading Channel. In Proc. of IEEE Vehicular*
*Technology Conference (VTC), Los Angeles, September 2004.*

*[30] D. Divsalar and M. K. Simon. Multiple–Symbol Differential Detection of MPSK. IEEE Trans. Commun., 38(3):300–308, March 1990.*
[31] D. Makrakis, P. T. Mathiopoulos, and D. P. Bouras. Optimal Decoding of Coded PSK and QAM Signals in Correlated Fast Fading
*Channels and AWGN: A Combined Envelope, Multiple Differential and Coherent Detection Approach. IEEE Trans. Commun., 42(1):63–*
74, January 1994.

*[32] V. Pauli and L. Lampe. Tree–Search Multiple–Symbol Differential Decoding for Unitary Space–Time Modulation. Acc. for publ.: IEEE*
*Trans. Commun., preprint available from http://www.ece.ubc.ca/∼lampe/publicat.html, November 2005.*

*[33] R. Steele. Mobile Radio Communications. John Wiley & Sons, New York, 2nd edition, 1999.*

[34] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn. Fading Correlation and Its Effects on the Capacity of Multielement Antenna
*Systems. IEEE Trans. Commun., 48(3):502–513, March 2000.*

*[35] N. Chiurtu, B. Rimoldi, E. Telatar, and V. Pauli. Impact of Correlation and Coupling on the Capacity of MIMO Systems. In Proc. of the*
*3rd IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), pages 154–157, Darmstadt, December*
2003.

*[36] R. H. Clarke. A Statistical Theory of Mobile–Radio Reception. The Bell Systems Technical Journal, 47:957–1000, July 1968.*
*[37] B. Hassibi and B. M. Hochwald. Cayley Differential Unitary Space–Time Codes. IEEE Trans. Inform. Theory, 48(6):1485–1503, June*

2002.

*[38] Z. Liu, Y. Xin, and G. B. Giannakis. Block Differentially Encoded OFDM with Maximum Multipath Diversity. IEEE Transactions on*
*Wireless Communications, 2:420 – 423, May 2003.*

[39] A. D. Murugan, H. El Gamal, M. O. Damen, and G. Caire. A Unified Framework for Tree Search Decoding: Rediscovering the
*Sequential Decoder. IEEE Trans. Inform. Theory, 52(3):933–953, March 2006.*

*[40] E. Chiavaccini and G. M. Vitetta. Further Results on Differential Space–Time Modulations. IEEE Trans. Commun., 51(7):1093–1101,*
July 2003.

*[41] R. J. Young and J. H. Lodge. Detection of CPM Signals in Fast Rayleigh Flat–Fading Using Adaptive Channel Estimation. IEEE*
*Transactions on Vehicular Technology, pages 338–347, May 1995.*

*[42] P. Pun and P. Ho. Fano Space–Time Multiple Symbol Differential Detectors. In Proc. of IEEE Global Telecommunications Conference*
*(Globecom), pages 3169–3174, Saint Louis, November 2005.*

*[43] T. Cui and C. Tellambura. Multiple–Symbol Differential Unitary Space–Time Demodulation with Reduced–Complexity. In Proc. of*
*IEEE International Conference on Communications (ICC), Seoul, May 2005.*

[44] European Telecommunications Standards Institute (ETSI). Digital Cellular Telecommunications System (Phase 2+); Radio Transmission and Reception; (3GPP TS 05.05 version 8.9.0 Release 1999). ETSI TS 100 910, April 2001.

[45] European Telecommunications Standards Institute (ETSI). Radio Broadcasting Systems; Digital Audio Broadcasting (DAB) to Mobile, Portable and Fixed Receivers. ETSI ETS 300 401, February 1995.

[46] M. Brehler and M. K. Varanasi. Asymptotic Error Probability Analysis of Quadratic Receivers in Rayleigh–Fading Channels With
*Application to a Unified Analysis of Coherent and Noncoherent Space–Time Receivers. IEEE Trans. Inform. Theory, 47(6):2383–2399,*
September 2001.

[47] E. Biglieri, G. Caire, G. Taricco, and J. Ventura–Traveset. Computing Error Probabilities over Fading Channels: A Unified Approach.
*European Transactions on Telecommunications, 9:15–25, 1998.*

*[48] K. L. Clarkson, W. Sweldens, and A. Zheng. Fast Multiple Antenna Differential Decoding. IEEE Trans. Commun., 49(2):253–261,*
February 2001.

*[49] C. Ling, W. H. Mow, K. H. Li, and A. C. Kot. Multiple–Antenna Differential Lattice Decoding. IEEE J. Select. Areas in Commun.,*
23:1821–1829, September 2005.

..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
transmit ant. 1
transmit ant. 2
receive ant. 1
su
b
ca
rr
ie
r
−
→
su
b
ca
rr
ie
r
−
→
su
b
ca
rr
ie
r
−
→
S_{[1, k]}
S_{[}D
4, k]
0
0
0
0
0
0
0
0
s1,1[1, k]
s1,1[D4, k]
s2,2[1, k]
s2,2[D4, k]
s3,3[1, k]
s3,3[D4, k]
s4,4[1, k]
s4,4[D4, k]
g1,1[1, k]s1,1[1, k] + n1[1, k]
g1,1[D4, k]s1,1[
D
4, k] + n1[
D
4, k]
g2,1[D4+1, k]s2,2[1, k] + n1[
D
4+1, k]
g2,1[D2, k]s2,2[
D
4, k] + n1[D2, k]
g1,1[D2+1, k]s3,3[1, k] + n1[
D
2+1, k]
g1,1[
3D
4 , k]s3,3[
D
4, k] + n1[
3D
4 , k]
g2,1[
3D
4 +1, k]s4,4[1, k] + n1[
3D
4 +1, k]
g2,1[D, k]s4,4[D4, k] + n1[D, k]
R_{[1, k]}
R_{[}D
4, k]
1
1
1
D
4
D
4
D
4
D
2
D
2
D
2
3D
4
3D
4
3D
4
D
D
D

0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |ψ t [κ ]| − → κ −→ BhTf= 0.001 BhTf= 0.003 BhTf= 0.01

0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |ψ f [µ ]| − → µ −→ TU HT 2–Ray(τ2−τ1= 20 µs) exponential(d = 1)

1 2 3 4 5 6 7 8 1 2 3 4, 9 22 22.5 23 23.5 24 24.5 25 25.5 26 Nt = 1 Nt = 2 Nt = 4 time–pos. j −→ ←− frequency–pos. i re q u ir ed 10 lo g10 (E b /N 0 ) in d B − → Simulation results Numerical results

Fig. 4. Required10 log_{10}(E_{b}/N_{0}) to achieve SER = 10−5for individual positions of 2D observation window
of MSDD. Parameters: F–DSFM, NT = 3, NB = 1, HT PDP, BhTf = 0.01, Nf = 10 and Nt = [1, 2, 4].

TABLE I

REQUIRED10 log_{10}(E_{b}/N_{0})TO ACHIEVESER = 10−5 USING MSDDAND VARIOUSSA SCHEMES.
PARAMETERS: F–DSFM,NS = NT, NB = 1, HT PDP, BhTf = 0.003. FOR COMPARISON: COHERENT

DETECTION WITH PERFECT CSI. NUMERICAL RESULTS FROM(35),FIGURES WITH“∗”ARE SIMULATION RESULTS AND “—”INDICATES AN ERROR FLOOR ABOVESER = 10−5.

SA [Nf, Nt] NT= 2 NT = 3 NT = 4 NT = 6
[1
5
] [10, 1] 30.7 26.0 23.4 20.7
[5, 2] ∗_{35.0} ∗_{29.2} _{—} _{—}
[20, 1] 29.3 24.0 20.8 18.1
[10, 2] 28.8 23.3 20.1 17.2
[1
6
,
2
7
]
[10, 1] 29.8 24.0 20.4 17.4
[5, 2] 29.2 23.4 19.9 17.1
[20, 1] 28.7 23.1 19.7 16.8
[10, 2] 28.6 22.9 19.5 16.7
S
ec
tio
n
II
I-C
[10, 1] 29.8 24.0 20.4 17.4
[5, 2] 28.9 23.1 19.7 16.8
[20, 1] 28.7 23.1 19.7 16.8
[10, 2] 28.4 22.8 19.4 16.5
coherent: 27.4 21.9 18.5 15.8

10−4 10−3 10−2 21.5 22 22.5 23 23.5 24 24.5 [Nf, Nt] = [1, 5] [Nf, Nt] = [2, 5], 2–Ray (neighbor) [Nf, Nt] = [2, 5], 2–Ray (OWC) [Nf, Nt] = [2, 5], HT (OWC) [Nf, Nt] = [2, 5], TU (OWC) [Nf, Nt] = [2, 5], exponential (d = 1, OWC) [Nf, Nt] = [1, 10] [Nf, Nt] = [1, 5], SA according to [16, 27] [Nf, Nt] = [1, 10], SA according to [16, 27] BhTf −→ re q u ir ed 10 lo g10 (E b /N 0 ) in d B − →

coherent with perfect CSI

Fig. 5. Required 10 log_{10}(E_{b}/N_{0}) to achieve SER = 10−5 vs. B_{h}T_{f} for various PDPs and observation
windows of MSDD. Parameters: T–DSFM, NT = 3, NB = 1. For comparison: results for SA scheme of
[16, 27] and coherent detection with perfect CSI.

10 15 20 25 10−5 10−4 10−3 10−2 10−1 10 15 20 25 10−5 10−4 10−3 10−2 10−1 10 15 20 25 10−5 10−4 10−3 10−2 10−1 a) TU b) Exponential (d = 1) c) HT CDD DFDD DFDD–LD MSDSD MSDSD–FM–LD

coherent with perfect CSI Lines:_{Markers: simulation results}numerical results

S
E
R
−
→
10 log_{10}(Eb/N0) in dB −→

Fig. 6. SER vs. 10 log_{10}(E_{b}/N_{0}) for TU, exponential (d = 1), and HT PDPs. F–DSFM and MSDSD and
MSDSD–FM–LD with[Nf, Nt] = [5, 2], CDD and DLD, DFDD and DFDD–LD with N = 10 and coherent
detection with perfect CSI. Parameters: NT = 3, NB = 1, and BhTf = 0.001.

10−2 10−1 100 101 10−3 10−2 10−1 2–Ray,NT= 1, NB= 3 2–Ray,NT= 3, NB= 1 HT, NT= 1, NB= 3 HT, NT= 3, NB= 1 HT, NT= 3, NB= 1, [15] 2–Ray HT

Lines: numerical results Markers: simulation results

xTx_{i} − xTx_{i+1}
−→
S
E
R
−
→

Fig. 7. SER at 10 log_{10}(E_{b}/N_{0}) = 15 dB vs. antenna spacing for a linear equispaced transmit–antenna
array. SA scheme according to Section III-C for [NT, NB] = [1, 3] and [NT, NB] = [3, 1] and 2D MSDD
with [Nf, Nt] = [5, 2]. Also shown: SA according to [15] for [NT, NB] = [3, 1] and 1D MSDD with
[Nf, Nt] = [10, 1]. Parameters: F–DSFM, 2–Ray (τ2− τ1 = 20 µs) and HT PDP, BhTf = 0.001.

10 20 30
102
103
104
10 20 30
102
103
104
10 20 30
102
103
104
a) TU b) Exponential (d = 1) c) HT
CDD
DLD
DFDD
DFDD–LD
MSDSD
MSDSD–FM–LD
av
er
ag
e
#
o
f
fl
o
p
s
p
er
sy
m
b
o
l
−
→
10 log_{10}(Eb/N0) in dB −→

Fig. 8. Average computational complexity vs. 10 log_{10}(E_{b}/N_{0}) for various detection algorithms and PDPs.
Parameters: F–DSFM, NT = 3, NB = 1, BhTf = 0.001. MSDSD and MSDSD–FM–LD with [Nf, Nt] =
[5, 2], DFDD and DFDD–LD with Nf = N = 10, CDD and DLD with Nf = N = 2.

10−2 10−1 100 101
103
104
2–Ray,NT= 1, NB= 3
2–Ray,NT= 3, NB= 1
HT, NT= 1, NB= 3
HT, NT= 3, NB= 1
2–Ray
HT
xTx_{i} − xTx_{i+1}
−→
av
er
ag
e
#
o
f
fl
o
p
s
p
er
sy
m
b
o
l
−
→

Fig. 9. Average computational complexity of MSDSD at 10 log_{10}(E_{b}/N_{0}) = 15 dB vs. antenna spacing for
a linear equispaced transmit–antenna array. SA scheme according to Section III-C for [NT, NB] = [1, 3]
and [NT, NB] = [3, 1]. 2D MSDD with [Nf, Nt] = [5, 2]. Parameters: F–DSFM, 2–Ray (τ2 − τ1 = 20 µs)
and HT PDP, BhTf = 0.001.