Low Complexity Transceiver Design for GFDM

0.21 MB·English

by  Arman Farhang

1
Low Complexity Transceiver Design for GFDM
Arman Farhang, Nicola Marchetti and Linda E. Doyle
CTVR / The TelecommunicationsResearch Centre, Trinity College Dublin, Ireland,
Email: farhanga, marchetn, ledoyle @tcd.ie
{ }
Abstract—Due to its attractive properties, generalized fre- the base station (BS) results in some residual synchronization
quencydivisionmultiplexing(GFDM)isrecentlybeingdiscussed errors and hence multiuser interference (MUI), [4]. The MUI
as a candidate waveform for the fifth generation of wireless
problemcanbetackledwitharangeofdifferentsolutionsthat
communication systems (5G). GFDM is introduced as a gen-
are proposed in [5]–[7]. However, these lead to an increased
eralized form of the widely used orthogonal frequency division
5 multiplexing (OFDM) modulation scheme and since it uses only receiver computational complexity. Thus, one of the main
1 onecyclicprefix(CP)foragroupofsymbolsratherthanaCPper advantages of OFDM, i.e., its low complexity, is lost. The
0 symbol,itismorebandwidthefficientthanOFDM.Inthispaper, challengethereforeistoprovidewaveformswithmorerelaxed
2 we propose novel transceiver structures for GFDM by taking
synchronization requirements and more localized signals in
advantage of the particular structure in the modulation matrix.
n time and frequencyto suit future5G applications,withoutthe
Ourproposedtransmitterisbasedonmodulationmatrixsparsifi-
a cationthroughapplicationoffastFouriertransform(FFT)tore- penalty of a more complex transceiver.
J
ducetheimplementationcomplexity.Aunifiedreceiverstructure There are many suggestions on the table as candidate
3
for matched filter (MF), zero forcing (ZF) and minimum mean waveforms[8]–[12].Ingeneral,allofthesesignalingmethods
1
square error (MMSE) receivers is also derived. The proposed
canbeconsideredasfilterbankmulticarrier(FBMC)systems.

receiver techniques harness the special block circulant property
] They can be broadly broken into two categories, those with
of the matrices involved in the demodulation stage to reduce
T
the computational cost of the system implementation. We have linear pulse shaping [11], [12] and those with circular pulse
I
. derived the closed forms for the ZF and MMSE receiver filters. shaping,[8]–[10].Theformersignalswithlinearpulseshaping
s Additionally, our algorithms do not incur any performance loss have attractive spectral properties, [13]. In addition, these
c
as they maintain the optimal performance. The computational
[ systemsareresilientto thetimingaswell asfrequencyerrors.
costs of our proposed techniques are analyzed in detail and are
However, the ramp-up and ramp-down of their signal which
1 compared with the existing solutions that are known to have
v the lowest complexity. It is shown that through application of are due to the transient interval of the prototype filter result
0 ourtransceiver structureasubstantial amount of computational in additional latency issues. In contrast, FBMC systems with
4 complexity reduction can be achieved. circular pulse shaping remove the prototype filter transients
9 thankstotheirsocalledtailbitingproperty,[8].Thewaveform
2
of interest in this paper is known as generalized frequency
0 I. INTRODUCTION
divisionmultiplexing(GFDM)anditcanbecategorizedasan
.
1 OFDM has been the technology of choice in wired and FBMC system with circular pulse shaping. The focus of the
0
wireless systems for years, [1]–[3]. The advent of the paper, more specifically, is on the design of low complexity
5
fifth generation of wireless communication systems (5G) and transceivers for GFDM.
1
: the associated focus on a wide range of applications from GFDM has attractivepropertiesand as a resulthas recently
v
thoseinvolvingburstymachine-to-machine(M2M)like traffic received a great deal of attention. One of the main attractions
i
X to media-rich high bandwidth applications has led to a re- of GFDM is that it is a generalized form of OFDM which
r quirement for new signaling techniques with better time and preserves most of the advantageous properties of OFDM
a
frequencycontainmentthan that of OFDM. Hence, a plethora while addressing its limitations. As Datta and Fettweis have
of waveforms are coming under the microscope for analysis pointed out in [14], GFDM can provide a very low out-of-
and investigation. band radiation which removes the limitations of OFDM for
The limitations of OFDM are well documented. OFDM carrier aggregation. It is also more bandwidth efficient than
suffersfrom large out-of-bandemissions which not only have OFDM since it uses only one cyclic prefix (CP) for a group
interference implications but it also can reduce the potential of symbols in its block rather than a CP per symbol as is the
for exploiting non-contiguous spectrum chunks through such case in OFDM. Through circular filtering, GFDM removes
techniques as carrier aggregation. For future high bandwidth the prototype filter transient intervals and hence the latency.
applications this can be a major drawback. OFDM also has Additionally, its special block structure makes it an attractive
high sensitivity to synchronization errors especially carrier choice for the low latency applications like IoT and M2M,
frequencyoffset(CFO).Asacaseinpoint,inmultiuseruplink [15]. Filteringthe subcarriersusinga well-designedprototype
scenarios where OFDMA is utilized, in order to avoid the filter limits the intercarrier interference (ICI) only to adjacent
large amount of interference caused by multiple CFOs as subcarriers which reduces the amount of leakage between
well as timing offsets, stringent synchronization is required subcarriers and increases the resiliency of the system to CFO
which in turn imposes a great amount of overhead to the as well as narrow band interference. In other words, GFDM
network.ThisoverheadisnotacceptableforlightweightM2M has robustness to synchronization errors. As Michailow et al
applications for example. The presence of multiple Doppler reportin [15], GFDM is also a goodmatch for multiple input
shifts and propagation delays in the received uplink signal at multiple output (MIMO) systems.
2
The advantages of GFDM come at the expense of an and hence the computations can run in parallel which can
increased bit error rate (BER) compared with OFDM. This in turn reduce the overall processing delay of the system. As
degradationis due to the fact that GFDM is a non-orthogonal ourproposedtransceiverstructureisbasedonsparsificationof
waveform. Consequently, non-orthogonality of the neighbor- the matrices that are involved, it also provides savings in the
ing subcarriers and time slots results in self-interference. To memory requirements of the system.
tackle this self-interference, matched filter (MF), zero forcing The rest of the paper is organized as follows. Section II
(ZF) or minimum mean square error (MMSE) receivers can presentstheGFDMsystemmodel.SectionsIIIandIVinclude
be derived [16]. Since, the MF receiver cannot completely the design and implementation of our proposed GFDM trans-
remove the ICI, ZF receiver can be utilized. However, due to mitterandreceiverstructures,respectively.Thecomputational
its noiseenhancementproblem,ZF receiverincurssome BER complexity of our transceiver pair is analyzed in Section V.
performanceloss.Thus,theMMSEapproachcanbechosento Finally, the conclusions are drawn in Section VI.
reducethenoiseenhancementeffectandmaximizethesignal- Notations: Matrices, vectors and scalar quantities are de-
to-interferenceplusnoiseratio(SINR).AsMF,ZFandMMSE noted by boldface uppercase, boldface lowercase and normal
receivers involve large matrix inversion and multiplication letters, respectively. [A] and [a] represent the element in
m,n n
operations,theydemandalargecomputationalcomplexitythat the mth row and nth column of A and the nth element of
makes them inefficient for practical implementations. As an a, respectively and A 1 signifies the inverse of A. I and
− M
alternative solution, Datta et al, [17], take a time domain suc- 0 are the identity and zero matrices of the size M M,
M
×
cessive interference cancellation approach. This solution can respectively.D=diag(a)isadiagonalmatrixwhosediagonal
completelyremovetheeffectoftheself-interference.However, elements are formed by the elements of the vector a and
that solution is a computationally exhaustive procedure. In a C = circ(a) is a circulant matrix whose first column is
morerecentworkfromthesame researchgroup,Gasparetal, a. The round-down operator , rounds the value inside to
⌊·⌋
[18], take advantageof the sparsity of the pulse shaping filter the nearest integer towards minus infinity. The superscripts
in frequency domain to perform the interference cancellation ()T, ()H and () indicate transpose, conjugate transpose
∗
· · ·
in the frequencydomainand hencefurtherreducethe compu- and conjugate operations, respectively. Finally, δ(), M and
· (cid:13)
tational complexity of the receiver. Even though the solutions mod N represent the Dirac delta function, M-point circular
that are based on the results of [17] and [18] successive convolution and modulo-N operations, respectively.
interferencecancellationcanremovetheself-interference,they
can incur error propagation problems. Recently, Matthe´ et al, II. SYSTEMMODELFORGFDM
[19], have proposed a fast algorithm to calculate the ZF and
We consider a GFDM system with the total number of
MMSE receiver filters. Their approach is based on the Gabor
N subcarriers that includes M symbols in each block. In a
transform structure of GFDM. Although matrix inversion is
GFDM block,M symbolsoverlapin time. Therefore,we call
circumvented multiplication of the ZF and MMSE matrices
M, overlapping factor of the GFDM system. The MN 1
to the received signal is a bottle-neck in this approach as the ×
vector d = [dT,...,dT ]T contains the complex data
matrixto vectormultiplicationis a computationallyexpensive 0 N 1
symbols of the GFDM blo−ck where the M 1 data vector
operation. ×
d =[d (0),...,d (M 1)]T containsthe data symbolsto be
In this paper, we design a low complexity transceiver i i i −
transmittedontheith subcarrier.Toputitdifferently,d (m)is
structure for GFDM and therefore improve on the existing i
the data symbol to be transmitted at the mth time slot on the
approaches. The special structure of the modulation matrix is
ith subcarrier. The data symbols are taken from a zero mean
utilizedtoreducethecomplexityofthetransmitter.Compared
independentand identicallydistributed(i.i.d)processwith the
with the existing GFDM transmitter [20], so far known to
variance of unity. In GFDM modulation, the data symbols to
havethelowestcomplexity,ourproposedtransmitterstructure
be transmitted on the ith subcarrier are first up-sampled by
is more computationally efficient. Based on the lessons that
the factor of N to form an impulse train
we learned from ICI cancellation in uplink OFDMA systems
with interleaved subcarrier allocation, [6], we are able to M 1
−
substantially reduce the complexity of the ZF and MMSE si(n)= X di(k)δ(n−kN), n=0,...,NM −1. (1)
receiverscomparedwiththelowcomplexityreceiverstructure k=0
that is proposed in [18]. We propose a unified structure Then, s =[s (0),...,s (MN 1)]T is circularly convolved
i i i
for the MF, ZF and MMSE receivers. This unified receiver −
withtheprototypefilterandup-convertedtoitscorresponding
structure is beneficial as only the filter coefficients need to
subcarrierfrequency.Afterperformingthesameprocedurefor
be changed for implementation of different receivers. These
allthesubcarriers,theresultingsignalsaresummeduptoform
coefficients can be saved on memory and be used if needed
the GFDM signal x(n), [16].
in different scenarios. For instance, ZF receiver can be used
iAnsstoeuadrteocfhMniMquSeEsaorenediaretchtiagnhdsnigonaapl-ptroo-xniomisaetiroantiiossin(vSoNlvResd)., x(n)=NX−1MX−1di(m)g{(n−mN) mod MN}ej2πNin, (2)
our proposed receivers do not incur any performance loss i=0 m=0
compared with the optimal MF, ZF and MMSE receivers. where g is the ℓth coefficient of the prototype filter.
ℓ
Another advantage of our receiver structure with respect to Putting together all the transmitter output samples in an
interference cancellation receivers is that it is not iterative MN 1 vector x = [x(0),...,x(MN 1)]T, the GFDM
× −
3
Synthesisfilterbank Analysisfilterbank
Circ.Conv. Circ.Conv.
d0 N gn ej2Nπn e−j2Nπn g˘n N dˆ0
Circ.Conv. Circ.Conv.
d1 N gn g˘n N dˆ1
. . . . . . . .
CP CP
. . . . Addition Channel removal FDE . . . .
. . . . . . . .
Channel
ej2Nπn(N−1) estimation e−j2Nπn(N−1)
Circ.Conv. Circ.Conv.
dN−1 N gn g˘n N dˆN−1
Fig.1. Basebandblockdiagram ofaGFDMtransceiver system.
signal can be represented as multiplication of a modulation the assumptionof havingperfectsynchronizationand channel
matrix A of size MN MN to the data vector d, [16]. estimates, the equalized signal can be obtained as
×
x=Ad. (3) y=FHMNH−1FMNr, (7)
Modulation matrix A encompasses all signal processing where F is MN-point normalized discrete Fourier trans-
MN
stepsinvolvedinmodulation.Letg=[g ,...,g ]T hold form (DFT) matrix and H 1 is a diagonal matrix whose
0 MN 1 −
−
all the coefficients of the pulse shaping/prototype filter with diagonalelementsarereciprocalsoftheelementsofthevector
the length MN, the elements of A can be represented as, obtained from taking MN-point DFT of the zero padded
version of h, viz., h˜. The vector y = [y ,...,y ]T is
[A]nm =g (n mN) mod MN ej2Nπn⌊Mm⌋. (4) the output of the FDE block. 0 MN−1
{ − }
Basedontheequations(2)to(4),thematrixAcanbewritten In order to suppress or remove the ICI due to non-
as orthogonality of the subcarriers and estimate the transmitted
A= G E G ... E G , (5) data vector d from the equalized signal vector, three linear
1 N 1
(cid:2) − (cid:3) GFDM receivers; namely, MF, ZF and MMSE detectors are
where G is an MN M matrix whose first column contains considered in this paper.
×
the samples of the prototype filter g and its consecutive As it was discussed in, [16], the transmitted symbols can
columns are the copies of the previous column circularly be recovered through match filtering
shifted by N samples. E = diag [eT,...,eT]T is an
MN MN diagonal matirix whose{diaigonal eilem}ents are dˆMF =AHy. (8)
×
comprised of M concatenated copies of the vector e =
i However, MF receiver cannot completely remove the ICI.
[1,ej2Nπi,...,ej2Nπi(N−1)]T. Hence, ZF solution can be utilized to completely eliminate
GFDM systems use frequency domain equalization (FDE)
the ICI that is caused by non-orthogonalityof the subcarriers.
to tackle the wireless channel impairments and reduce the
The ZF estimate of the transmitted data vector can be found
channelequalizationcomplexity.Inthosesystems,aCPwhich
as
is longer than the channel delay spread is added to the dˆZF =(AHA)−1AHy. (9)
beginning of the GFDM block to accommodate the channel
transient period. If N is the CP length, the last N Since (AHA) 1AH can have large values, its multiplication
CP CP −
elements of the vector x are appended to its beginning in toy canresultinnoiseenhancement.Thisnoiseamplification
order to form the transmitted signal vector x¯ whose length problemcan be taken care of by utilizing the MMSE receiver
is MN +N . Let h = [h ,...,h ]T be the channel
impulserespoCnPse.Thus,theC0PlengthNNch−1needstobelonger dˆMMSE =(AHA+σν2IMN)−1AHy. (10)
CP
than the channel length Nch. The received signal which has Fig. 1, depicts the baseband block diagram of a GFDM
gone through the channel, after CP removal can be shown as transceiver when we have perfect synchronization in time
and frequency between the transmitter and receiver. Fig. 1
r=Hx+ν, (6)
summarizes the modulation and demodulation process that
where ν is the complex additive white Gaussian noise is discussed above. It is worth mentioning that g ’s for
n
(AWGN) vector, i.e, ν (0,σ 2I ), σ 2 is the noise n = 0,...,MN 1 are the prototype filter coefficients and
ν MN ν
variance, H = circ h˜∼aCndNh˜ is the zero padded version g˘ ’s are the rece−iver filter coefficients which can be taken
n
{ }
of h to have the same length as x. Due to the fact that H from the coefficients of MF, ZF or MMSE receiver filter. As
is a circulant matrix, an FDE procedure can be performed it was mentioned in Section I, GFDM is a type of filter bank
to compensate for the multipath channel impairments. With multicarrier system with circular pulse shaping. Therefore,
4
N-point CircularConvolution
dn M DFT P/S g¯0
z 1
− CircularConvolution
M N-point P/S g¯1
DFT
. . .
. . . d¯n d¯n . . . xn
. . .
. . .
Thiscommutator
. . .
turnsbyoneposition
z−1 aftereveryMsamples
N-point CircularConvolution
M P/S
DFT g¯N−1
(a) (b)
Fig.2. Concatenation of(a)and(b)showtheimplementation oftheproposedGFDMtransmitter.
GFDM transmitter and receiver can be thought of as a pair definition of F , F d can be implementedby performingM
b b
of synthesis and analysis filter banks, respectively. DFT operations of size N on the data samples, i.e., one per
From equations (3) and (8) to (10), one realizes that direct GFDM symbol. Let d¯ =F d=[d¯T,...,d¯T ]T where the
b 0 N 1
matrix multiplications and inversions that are involved, de- M 1 vector d¯ = [d¯(0),...,d¯(M 1)]T c−ontains the ith
i i i
× −
mandaverylargecomputationalcomplexityasallthematrices output of each DFT block, then (11) can be rearranged as
are of the size MN MN, with N being usually large, and
such complexity may×not be affordable for practical systems. x=ΓHd¯ =N−1ΓHd¯ , (12)
Therefore, in the remainder of this paper, low complexity X i κ
i=0
techniques will be proposed that can substantially reduce
where κ = (N i) mod N. As discussed in Appendix B,
the computational cost of the synthesis and analysis filter −
the M MN matrices Γ ’s have only M non-zero columns
banksthat are shown in Fig. 1, while maintainingthe optimal × i
and the sets of those column indices are mutually exclusive
performance.
with respect to each other.As a result, ΓHd¯ will be a sparse
i κ
vectorwithonlyM non-zeroelementslocatedonthepositions
III. PROPOSED GFDM TRANSMITTER
κ,κ+N,...,κ+(M 1)N.Onthebasisofthederivationsthat
This section presents our proposed low complexity GFDM are presentedin App−endixA, the non-zeroelementsof ΓHd¯
i κ
transmitter design and implementation. In the following sub- canbeobtainedfromM-pointcircularconvolutionofd¯ with
κ
sections, we will show how the synthesis filter bank of Fig. 1 the κth polyphasecomponentof the prototypefilter g thatis
κ
can be simplified to have a very low computational load. scaled by √N. Therefore, defining the non-zero elements of
ΓHd¯ as the vector x = [x ,x ,...,x ]T, we
i κ κ κ κ+N κ+(M 1)N
A. GFDM transmitter design get −
Starting from (3), one can realize that direct multiplication xκ =g¯κMd¯κ, (13)
(cid:13)
of the matrix A to the data vector d is a complex operation where g¯ =√Ng .
κ κ
which demands (MN)2 complex multiplications. Therefore,
complexity will be an issue for practical systems as the
B. GFDM transmitter implementation
number of subcarriers and/or the parameter M increases.
Accordingly, a low complexity implementation technique for In this subsection, implementation of the designed GFDM
GFDM transmitter has to be sought. To this end, equation(3) transmitter in Section III-A is discussed. From the equations
can be written as (11) to (13), GFDM modulation,based on our design, can be
summarized into two steps.
x=Ad=AFHF d, (11)
b b 1) M number of N-pointDFT operations, i.e., application
where F is the MN MN normalized block DFT matrix of N-point DFT to each individual GFDM symbol
b
nth,ait=inc0l,u.d.e.s,NM×1M. Vsa×ulibdmityatroifceesquΩantiio=n (1√11N)eis−jb2aπNsneidIMonatnhde wimhpiclehmiennctleuddebsyNtaksiunbgcaardrivearsn.taTgheisofcatnhebefasetffiFcoieunrtileyr
fact that FHF −=I . As it is derived in Appendix A, the transform (FFT) algorithm.
b b MN
resulting matrix from multiplication of the block DFT matrix 2) N number of M-point circular convolution operations.
F into AH is sparse and it is comprised of the prototype Therefore, the first and second steps of our GFDM trans-
b
filter coefficients scaled by √N. From equation (11), it can mitter can be implemented by cascading the block diagrams
be inferred that ΓH = AFH is also sparse since it is the shown in Fig. 2 (a) and (b), respectively. The blocks P/S
b
conjugate transpose of F AH. Hence, our strategy allows convert the parallel FFT outputs to serial streams. All the
b
us to make the matrix A sparse and real as the prototype commutators shown in Fig. 2 turn counter clockwise. Both
filter is usually chosen as a real filter. Due to (11) and the commutators located on the right hand side of the Fig. 2 (a)
5
and (b) turn after one sample collection. However, the one Appendix A, multiplication of AH by the block DFT matrix
located on the left hand side of (b) turns by one position resultsinasparsematrix.DuetothefactthatFHF =I ,
b b MN
aftersendingM samplestoeachM-pointcircularconvolution similar to the transmitter (equation(11)), equation (8) can be
block. written as
dˆ = FHF AHy
IV. PROPOSED GFDM RECEIVER MF b b
= FHΓy, (18)
In this section, we derive low complexity ZF and MMSE b
receivers for GFDM systems. It is worth mentioning that whereΓisasparsematrixwithonlyNM2 non-zeroelements
our solutions are direct and hence lower complexity of these that are the scaled version of the prototype filter coefficients.
receivers comes for free as they do not result in any per- Closed form of Γ = [ΓT,...,ΓT ]T is derived in Ap-
0 N 1
formance loss, thanks to the special structure of the matrix pendix A and it is shown that the m−atrix is real valued and
AHA. The characteristics of AHA will be discussed in the comprised of the prototypefilter elements. Non-zero columns
nextsubsectionandthenwewillderiveourproposedreceivers of the M MN block matrices Γ ’s are circularly shifted
i
on the basis of those traits. copies of e×ach other. Hence, multiplication of Γ and y is
i
equivalent to M-point circular convolution of M equidistant
A. Block-diagonalization of the matrix AHA elements of y starting from the κth position and circularly
folded version of the κth polyphase component of g scaled
The key idea behind our proposed GFDM receiver tech-
by √N, viz., √Ng˜ . Usually, the prototypefilter coefficients
niques is to take advantage of the particular structure of the κ
are real-valued. Thus, Γ is real-valued. Multiplication of FH
matrixAHAwhichispresentinbothZFandMMSEreceiver b
to the vector Γy can be implementedby applyingM number
formulations. Using (5), one can calculate AHA and find out
ofN-pointIDFToperations.Lety¯ =Γy=[y¯T,...,y¯T ]T
that it has the following structure 0 N 1
and y =[y ,y ,...,y ]T. Therefore, we h−ave
κ κ κ+N κ+(M 1)N
GHG GHE G GHE G −
1 N 1
 GHEHG GHG ··· GHE − G y¯i =Γiy=vκMyκ, (19)
AHA= ...1 ... ·.·.·. N... −2 . where vκ = √Ng˜κ. Finally, the M(cid:13)F estimates of d can be
GHEH G GHEH G GHG  obtained as
N−1 N−2 ··· (14) dˆ =FHy¯. (20)
MF b
From the definition of vector e , it can be straightforwardly
i
perceived that eH = e and hence EH = E . Therefore,
N i i N i i C. Low complexity ZF receiver
the columns of A−HA as shown in (14) a−re circularly shifted
with respect to each other. Accordingly, AHA is a block- Inserting (15) into (9), we get
circulant matrix with blocks of size M M. Following a
similar line of derivations as in [21] and×[6], AHA can be dˆZF =FHbD−1FbAHy. (21)
expanded as follows Multiplication of matrix AH to the vector y is the first
source of computational burden in ZF receiver which has
AHA=FHDF , (15)
b b computational cost of (MN)2. However, this complexity can
where D is an MN MN block-diagonal matrix, D = be reduced by taking advantage of the sparsity of the matrix
×
diag D ,...,D and D ’s are M M block matrices. Γ = F AH as it was suggested in the previous subsection.
0 N 1 i b
From{(15), D can−be}derived as × Equation(19) can be written as y¯ =Γ y=√Ncirc g˜ y .
i i κ κ
D=F (AHA)FH. (16) Let y˜ =D−1y¯ =[y˜0T,...,y˜NT 1]T where { }
b b −
As it is explained in Appendix B, Di’s can be derived from y˜i =√ND−i 1circ{g˜κ}yκ. (22)
polyphase components of the prototype filter. Therefore, from rearranging equation (17) as D =
i
Di =Ncirc{gκ(cid:13)Mg˜κ}, (17) Ncirc{g˜κ}circ{gκ} and inserting it into (22), we have
1
pwohneernetκof=g(Nan−d ig˜) m=od[Ng,,ggi is the i,t.h..p,oglypha]Tse icsomits- y˜i = √N(circ{g˜κ}circ{gκ})−1circ{g˜κ}yκ
i i i+(M 1)N i+N
circularly folded version. As (17) hig−hlights, Di’s are all real = 1 (circ g ) 1y
κ − κ
and circulant matrices. √N { }
= qκMyκ, (23)
(cid:13)
B. Low complexity MF receiver where q includes the first column of the circulant matrix
κ
Based on equation (8), direct implementation of MF re- (circ g ) 1 scaled by 1 . Due to the fact that the the
{ κ} − √N
ceiver involves a matrix to vector multiplication which has coefficientsof the prototypefilter are known,the vectorsq ’s
κ
the computational cost of (MN)2 complex multiplications. can be calculated offline. Additionally, since the prototype
This procedure becomes highly complex for large values of filter coefficients are real, q ’s are also real. From (23), one
κ
N and/or M which is usually the case. As discussed in may realize that calculation of the vector y˜ needs N number
6
CircularConvolution N-point
γ y¯n/y˜n/y˘n M P/S
0 IDFT
z 1
CircularConvolution −
γ N-point
1 M P/S
IDFT
. . .
yn . . . y¯n/y˜n/y˘n . . . dˆn
. . .
. . .
Thiscommutator . . .
turnsbyoneposition z 1
aftercollectingMsamples −
CircularConvolution N-point
γ M P/S
N 1 IDFT
−
(a) (b)
Fig.3. Unifiedimplementation ofourproposedMF,ZFandMMSE-basedGFDMreceivers fromcascading theblockdiagrams(a)and(b).
of M-point circular convolutions. After acquiring y˜, the ZF right hand side of Fig. 3 (a) will turn by one position after
estimates of the transmitted symbols can be obtained as collecting M samples from the ith branch, i.e., M 1
×
vector y¯ /y˜ /y˘ , in the clockwise direction. In the MF and
dˆ =FHy˜. (24) i i i
ZF b ZF receivers, the vectors γ are replaced by v ’s and q ’s,
i i i
Ascan beinferredfrom(24),findingdˆ fromy˜ requiresM respectively, and in MMSE receiver, they will be replaced by
ZF
number of N-point inverse DFT (IDFT) operations. pi’s. Due to the fact that in the MF and ZF receivers, the
vectors v and q are fixed and only depend on the prototype
i i
filter coefficients, they can be calculated offline and hence
D. Low complexity MMSE receiver
there is no need for their real-time calculation. However, in
Using (15) in (10) we get MMSEreceivers,thevectorsp dependonthesignalto noise
i
dˆ = (FHDF +σ 2I ) 1AHy ratio and hence they should be calculated in real-time. As
MMSE b b ν MN − mentionedearlierinSectionIV-D,circularconvolutionsinour
= FHbD¯−1FbAHy, (25) MMSE receiver need to be performedby taking advantage of
where D¯ = D + σ 2I = diag D¯ ,...,D¯ and fast convolution to keep the complexity low.
ν MN 0 N 1
D¯ = D + σ 2I . Recalling circ{ulant property−o}f D
i i ν M i
from (17), it can be understood that D¯ is also circulant and V. COMPUTATIONALCOMPLEXITY
i
can be expanded as D¯i = FHM(Φ∗κΦκ +σν2IM)FM where In this section, the computational complexity of our pro-
DΦ¯κ−1F=bAMHNyd,iwage{cFanMwgκri}te1. Let y˘ = [y˘0T,...,y˘NT−1]T = apnodsecdoGmFpDarMedttroantshmeietxteirstainngdorenceesivtheratstarruectkunroeswanretodhisacvuessthede
lowest complexity, [18], [20]. In both cases, total number of
y˘i = FHM(Φ∗κΦκ+σν2IM)−1Φ∗κFMyκ N subcarriers and overlapping factor of M are considered.
= pκMyκ, (26)
(cid:13)
where p includes the first column of the circulant matrix A. Transmitter complexity
κ
tFhHMe {m(aΦtr∗κixΦDκ¯+−1σνd2eIpMen)d−s1Φon∗κ}σFνM2 .anSdintchee, irnecMeiMveSrEcarnenceoitvebre, GFTDabMletIrapnrsemseitntetsr tihmepcleommepnuttaattiioonnsalbacsoemdpolenxitthyeonfudmifbfeerreonft
simplified as in (19) or (23), circular convolution of (26) complex multiplications (CMs).
needs to be calculated in the frequency domain, known as As discussed in Section III-B, our proposed GFDM trans-
fastconvolution,in ordertohavethelowestcomplexity.After mitterinvolvestwosteps.ThefirststepincludesM numberof
obtaining y˘, the MMSE estimates of the transmitted symbols N-point FFT operations that requires MN log N CMs. The
2 2
can be found as secondstepneedsN numberofM-pointcircularconvolutions.
dˆ =FHy˘. (27) Recallingequation(13),sinceg ’sarereal-valuedvectors,one
MMSE b κ
may realize that each M-point circular convolution demands
E. Receiver implementation M22 number of CMs. If M is a power of two, the complexity
canbefurtherreducedbyperformingthecircularconvolutions
In this subsection, we present a unified implementation
infrequencydomain.Thisisduetothefactthatcircularconvo-
of the MF, ZF and MMSE receivers that we proposed in
lutionintimeismultiplicationinthefrequencydomain.Thus,
SectionsIV-B,IV-CandIV-D.AsFig.3depicts,theproposed
to perform each circular convolution, a pair of M-point FFT
GFDM receivers can be implemented by cascading Fig. 3 (a)
and IFFT blocks together with M complex multiplications to
and (b). It is worth mentioning that the commutator on the
the filter coefficients in frequency domain are required.
1Since, g˜κ is a real vector and circularly folded version of gκ, Φ∗κ = The complexity relationships that are presented in Table I
MNdiag{FMg˜κ}. are calculated and plotted in Fig. 4 for N =1024 subcarriers
7
TABLEI
From Fig. 3, it can be understood that our proposed
COMPUTATIONALCOMPLEXITYOFDIFFERENTGFDMTRANSMITTER
IMPLEMENTATIONS receivers involve N and M numbers of M-point circular
convolutionsandN-pointIDFToperations,respectively.IDFT
Technique Number ofComplex Multiplications operationscanbeefficientlyimplementedusingN-pointIFFT
Directmatrixmultiplication (MN)2 algorithm which requires N2 log2N CMs. As mentioned ear-
lier,intheproposedMFandZFreceivers,thevectorsγ have
Proposedtransmitterin[20] MN(log2N+2log2M+L) fixed values and hence can be calculated and stored oiffline.
Ourproposed transmitter M2N(M+log2N) Furthermore, γ ’s are real-valued vectors. Thus, the number
i
of complex multiplications needed for N number of M-point
106 circular convolutions is NM2 2.
In contrast to the MF and ZF receivers, in the MMSE
receiver, the vectors γ ’s are not fixed and depend on the
i
signal-to-noiseratio(SNR).Hence,theyneedtobecalculated
ons in real-time. To this end, as highlighted in Section IV-D,
ati
plic thoseoperationscanbeperformedbyusingM-pointDFTand
ulti105 IDFT operations. Due to the fact that (Φ∗κΦκ+σν2IM) is a
m
x real-valued diagonal matrix, its inversion and multiplication
e
mpl to Φ∗κ only needs M2 CMs. The resulting diagonal matrix
of co (Φ∗κΦκ +σν2IM)−1Φ∗κ is multiplied into an M ×1 vector
er which needs M CMs. Since, M is not necessarily a power
b
m of 2, complexityof M-pointDFT and IDFT operationsin the
u
N
implementation of the circular convolutions is considered as
104 Proposed transmitter in [20]
M2. Obviously, if M is a power of 2, a further complexity
Proposed GFDM transmitter
OFDM transmitter reductionby taking advantageof FFT and IFFT algorithmsis
possible. Therefore, the complexity of our proposed MMSE
2 4 6 8 10 12 14 16 18 20
Block length M receiver only differs from the MF and ZF ones in the imple-
mentation of the circular convolution operations.
Fig.4. ComputationalcomplexitycomparisonofdifferentGFDMtransmitter
techniques andtheOFDMtransmitter technique forN =1024. Table II also presents the complexity of the direct MF, ZF
andMMSEdetectiontechniques,i.e.,directmatrixmultiplica-
tions and solutionsto the equations(9) and (10), respectively.
with respect to different values of overlapping factor M. As
Those solutions involve direct inversion of an MN MN
the authors of [20] suggest, L = 2 is chosen for calculating
×
matrixwhichhasthecomplexityof (M3N3)andtwovector
theirGFDMtransmittercomplexity.Duetothefactthatdirect
O
by matrix multiplications with the computational burden of
multiplication of A to the data vector d demands a large
2(MN)2 CMs.
number of CMs and is impractical, we do not present it in
The complexity formulas that are presented in Table II
Fig. 4. To give a quantitative indication of the complexity re-
are evaluated and plotted in Fig. 5 for different values of
ductionthat our proposedtransmitter providescomparedwith
overlapping factor M [1,21], N = 1024 and I = 8 for
thedirectcomputationoftheequation(3),inthesame system
∈
the receiver that is proposed in [18]. Based on the results of
setting as used for our other comparisons, i.e., N = 1024
[18], I =8 andL=2areconsidered.Duetothe factthatthe
and M [1,21], complexity reduction of around three orders
∈ complexityofMF,ZFandMMSEreceiverswithdirectmatrix
of magnitude can be achieved. According to Fig. 4, for the
inversion and multiplications is prohibitively high compared
small values of M our proposed transmitter structure has a
with other techniques (the difference is in the level of orders
complexity very close to that of OFDM. However, as M
of magnitude), they are not presented in Fig. 5. However, to
increases the complexity of our transmitter increases with a
higherpace than OFDM. Thisis dueto the overheadof NM2 quantifytheamountofcomplexityreductionthatourproposed
2 techniques provide, in the case of N = 1024 and M = 7,
number of CMs compared with OFDM. Compared with the
our proposed MF/ZF receiver is three orders of magnitude
transmitter structure that we are proposing in this paper, for
and the proposed MMSE receiver is six orders of magnitudes
small values of M up to 11, the transmitter proposed in [20]
simpler than the direct ones, respectively, in terms of the
demands about two times higher number of CMs. As M
required number of CMs. As Fig. 5 depicts, our proposed
increases, complexity of our technique gets close to that of
ZF receiver is around an order of magnitude simpler than the
the one proposed in [20]. GFDM transmitter of [20] is about
proposedreceiver with SIC in [18]. In addition, our proposed
3 to 4 times more complex than OFDM.
MMSE receiver has 2 to 3 times lower complexity than the
one in [18]. Apart from lower computational cost compared
B. Receiver complexity
with the existing receiver structures, our techniques maintain
Table II summarizes the computational complexity of dif- the optimal ZF and MMSE performance as they are direct.
ferent GFDM receivers in terms of the number of complex Finally,theZFandMMSEreceiversthatweareproposingare
multiplications. The parameter I is the number of iterations closer in complexityto OFDM as comparedto the receiverin
in the algorithm with interference cancellation. [18] which is over an order of magnitude more complex than
8
TABLEII
transceiver structures attractive for hardware implementation
COMPUTATIONALCOMPLEXITYOFDIFFERENTGFDMRECEIVER
TECHNIQUES of the real time GFDM systems.
Technique NumberofComplexMultiplications APPENDIX A
DirectZF 2(MN)2 DERIVATION OFFbAH
DirectMMSE 1(MN)3+2(MN)2
3 The key idea in the derivation of F AH is based on the
Matchedfilter+SIC,[18] MN(log2MN+log2M+L+I(2log2M+1)) b
fact that inner product of two complex exponential signals
ProposedMF/ZF M2N(M+log2N) with different frequencies is zero.
ProposedMMSE M2N(4M+log2N+3)
N 1
107 X− ej2Nπℓ(i−k) =Nδik. (A.1)
ℓ=0
From the definitions of F and A, Γ = F AH can be
b b
ations106 bolbotcakinemdaatrsicΓes=th[aΓtT0c,a.n.b.,eΓmTNa−th1e]TmawtihcearlelyΓsih’oswarneaMs ×MN
c
pli
multi Γ = 1 GHN−1 iℓEH, (A.2)
plex 105 i √N Xℓ=0 W ℓ
m
er of co wwehehreavWe iℓ =e−j2πNiℓ.BasedonthedefinitionofEℓ and(A.1)
b
um104 Matched filter+SIC [18] N−1
N MPraotpcohseedd fMiltMer S[1E8 r]eceiver X WiℓEHℓ =NΨκ, (A.3)
ℓ=0
Proposed MF/ZF receiver
OFDM receiver where κ=(N i) mod N, Ψ =diag [ ψT,...,ψT ]T ,
103 − κ { κ κ }
2 4 6 8 10 12 14 16 18 20
Block length M M block vectors
| {z }
Fig. 5. Computational complexity comparison ofdifferent GFDMreceiver ψκ =[0,...,1,...,0]T,
techniqueswithrespecttoeachotherandthatofOFDMreceiverwhenN =
1024andI=8for[18]. ↑
κth position
OFDM.
ψ ’s are N 1 vectors and Ψ is a diagonal matrix whose
κ × κ
VI. CONCLUSION maindiagonalelementsaremadeupofM concatenatedcopies
of the vector ψ . From (A.3) and (A.1), Γ ’s can be obtained
In this paper, we proposed low complexity transceiver κ i
as
techniques for GFDM systems. The proposed transceiver
Γ =√NGHΨ . (A.4)
techniques exploit the special structure of the modulation i κ
matrix to reduce the computational cost without incurring
Accordingly, it can be perceived that the block matrices Γ ’s
i
any performance loss penalty. In our proposed transmitter,
and hence the matrix Γ are sparse. The matrix Γ has only
i
block DFT and IDFT matrices were used to make the mod- M2 non-zero elements which are located on the circularly
ulation matrix sparse and hence reduce the computational
equidistantcolumnsκ,κ+N,...,κ+(M 1)N.Theelements
burden. We designed low complexity MF, ZF and MMSE −
of two consecutive non-zero columns of Γ are circularly
i
receivers by block diagonalization of the matrices involved
shiftedcopiesofeachother.Forinstance,thesecondnon-zero
in demodulation. It was shown that through this block di-
column of Γ is a circularly shifted version of the first non-
i
agonalization, a substantial amount of complexity reduction
zeroonebyonesample.From(A.4),thefirstnon-zerocolumn
in the matrix inversion and multiplication operations can be of Γ can be derived as √N[g ,g ,...,g ]T
i κ κ+(M 1)N κ+N
achieved. A unified receiver structure based on MF, ZF and which is the circularly folded version of t−he κth polyphase
MMSE criteria was derived. The closed form expressions for
componentoftheprototypefilter. Onecanfurtherdeducethat
the ZF and MMSE receiver filters were also obtained. We
the matrix Γ is a real one consisted of the prototype filter
also analyzed and compared the computational complexities
coefficients.
of our techniqueswith the existing onesknownso far to have
the lowest complexity. We have shown that all the proposed
APPENDIX B
techniques in this paper involve lower computational cost
CLOSED FORM DERIVATION OFD
than the existing low complexity techniques [18], [20]. For
instance, over an order of magnitude complexity reduction The polyphase components of the prototype filter g can
can be achieved through our ZF receiver compared with the be defined as the vectors g ,g ,...,g where g =
0 1 N 1 i
proposedtechniquein[18].Suchasubstantialreductioninthe [g ,g ,...,g ]T. As it is shown−in Appendix A,
i i+N i+(M 1)N
amountofcomputationsthatareinvolvedmakesourproposed Γ=F AH isaspar−sematrixwithonlyM non-zeroelements
b
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{ (cid:13) }
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