A Generalized Frequency Division Multiplexing Transceiver Field

The present invention relates to multicarrier modulation techniques suitable for wireless communication systems. More particularly, the invention relates to a GFDM transceiver which is a candidate for the fifth generation (5G) of wireless communication systems. Background

Due to the demands of some emerging applications like machine type communications (MTC) and Internet of Things (loT) in the fifth generation of wireless communication systems (5G), new signaling techniques with a better time and frequency containment than that of orthogonal frequency division multiplexing (OFDM) have become necessary.

OFDM has been the technology of choice in many wired and wireless systems for many years. However, it has some limitations. These include a large out-of- band emission, which limits the utilization of non-contiguous spectrum known as carrier aggregation, and high sensitivity to synchronization errors, especially carrier frequency offset (CFO). As a result, in multiuser uplink scenarios where OFDMA is utilized, in order to avoid the large amount of interference caused by multiple CFOs, as well as timing offsets, stringent synchronization is required. It will be appreciated that this imposes a great amount of overhead to the network. Furthermore, the presence of multiple doppler shifts and propagation delays in the received uplink signal at the base station results in some residual synchronization errors, and hence multiuser interference (MUI). In order to tackle the MUI problem, some solutions exist. However, these existing solutions lead to an increased receiver computational complexity. Thus, OFDM loses one of its main advantages over other modulation techniques, namely its low complexity. The aforementioned requirements of 5G systems have stirred a great amount of interest among researchers, and thus led to the advent of a number of new signalling methods. All of these new signaling methods can be considered as filter bank multicarrier (FBMC) systems. Some of the FBMC systems have circular pulse shaping, while some FBMC systems have linear pulse shaping. FBMC signals with linear pulse shaping have attractive spectral properties. In addition, such systems are resilient to timing errors as well as frequency errors. However, the ramp-up and ramp-down of their signals as a result of the transient interval of the prototype filter causes additional latency issues. In con- trast, FBMC systems with circular pulse shaping remove the prototype filter transients due to their so called tail biting property.

Generalized Frequency Division Multiplexing (GFDM) is a generalized form of OFDM which preserves advantageous properties of OFDM while addressing its limitations. As GFDM can provide a very low out-of-band radiation, it removes the limitations of OFDM for carrier aggregation. Since GFDM uses only one cyclic prefix (CP) for a group of symbols, rather than a CP per symbol as is the case in OFDM, it is more bandwidth efficient than OFDM. Furthermore, through tail biting, GFDM removes the prototype filter transient intervals, and hence the latency. Additionally, its special block structure makes it an attractive choice for low latency applications like loT and MTC. Filtering the subcarriers using a well designed prototype filter will limit the intercarrier interference (ICI) to adjacent subcarriers only. This reduces the amount of leakage between subcarriers, and increases the resiliency of the system to CFO as well as to narrow band interference. In other words, GFDM has a robustness to synchronization errors. GFDM is also a good match for multiple input multiple output (MIMO) systems. As a result, GFDM is a candidate signalling method for 5G cellular systems.

However, the above-listed advantages of GFDM come at the expense of an increased bit error rate (BER) compared with OFDM. This degradation is due to the fact that GFDM is a non-orthogonal waveform. Consequently, non- orthogonality of the neighboring subcarriers and time slots results in self- interference. To tackle this self-interference, zero forcing (ZF) and minimum mean square error (MMSE) receivers can be derived. However, a ZF receiver incurs some BER performance loss due to its noise enhancement problem. Thus, the MMSE approach can be taken to reduce the noise enhancement effect.

ZF and MMSE receivers involve large matrix inversion operations. Therefore, they are found to be inefficient in practice, as they demand a high computational complexity. One proposed alternative solution is to take a time domain successive interference cancellation approach. This solution can completely remove the effect of the self-interference. However, that solution is a computationally exhaustive procedure. By taking advantage of the sparsity of the pulse shaping filter in the frequency domain to perform the interference cancellation in the frequency domain, the computational complexity of the receiver can be reduced.

The present invention seeks to provide a GFDM transceiver which overcomes at least some of the above mentioned problems. Summary

The present invention provides, as set out in the appended claims, a method of transmitting data in a Generalized Frequency Division Multiplexing ,GFDM, communication system having N subcarriers with M symbols in each GFDM block, the method comprising the steps of:

downsampling the signal vector for transmission of size MN by a factor

M;

performing M number of discrete Fourier transform, DFT, operations of size N on selected down sampled data samples to provide DFT output data; and

performing N number of M point circular convolution operations on the

DFT output data with a pulse shaping filter to provide the GFDM output signal for transmission. The selected data samples are based on the equations and block diagrams of the transmitter, which are described in detail in the paragraphs below.

By using block DFT and IDFT matrices, the modulation matrix is made sparse, and thus the invention provides a low complexity GFDM transmitter.

Preferably, the method further comprises delaying the signal vector for transmission prior to down sampling. In one embodiment, the method further comprises converting the DFT output data from parallel to serial data prior to performing the circular convolution operations.

The step of performing M number of DFT operations of size N on the selected down sampled data samples may comprise performing a fast Fourier transform on the data samples.

The present invention also provides a method of demodulating received GFDM signals in a GFDM communication system having N subcarriers with M symbols in each GFDM block, the method comprising the steps of:

performing N number of M point circular convolution operations on selected samples of the received GFDM signal;

down sampling the circular convolved samples with a factor of M; and performing M number of inverse discrete Fourier transform ,IDFT, operations of size N on the down sampled data.

The selected samples are based on the equations and block diagrams of the receiver, which are described in detail in the paragraphs below. Through the block diagonalization of the matrices involved in demodulation, the invention provides a low complexity GFDM receiver.

Preferably, the method further comprises delaying the circularly convolved samples prior to down sampling. In one embodiment the method further comprises converting the output data from the I DFT operations from parallel to serial data to provide the demodulated data.

The step of performing M number of inverse discrete Fourier transform, I DFT, operations of size N on the down sampled data may comprise performing a fast Fourier transform on the data samples. In one embodiment the method comprises Zero Forcing detection, and wherein the circulant matrix is derived from a pulse shaping filter.

In one embodiment the method comprises Minimum Mean Square Error, MMSE, detection, and wherein the circulant matrix is derived from a pulse shaping filter and the signal to noise ratio.

The circular convolution operation may comprise fast convolution performed in the frequency domain.

The method may comprise matched filter, MF, detection.

The present invention also provides a transmitter in a Generalized Frequency Division Multiplexing ,GFDM, communication system having N subcarriers with M symbols in each GFDM block, comprising :

means for down sampling the signal vector for transmission of size MN by a factor M ;

means for performing M number of discrete Fourier transform, DFT, operations of size N on selected down sampled data samples to provide DFT output data; and

means for performing N number of M point circular convolution operations on the DFT output data with a pulse shaping filter to provide the GFDM output signal for transmission. The present invention also provides a receiver in a GFDM communication system having N subcarriers with M symbols in each GFDM block, the method comprising the steps of:

means for performing N number of M point circular convolution operations on selected samples of the received GFDM signal;

means for down sampling the circular convolved samples with a factor of M; and means for performing M number of inverse discrete Fourier transform ,IDFT, operations of size N on the down sampled data.

The present invention also provides a GFDM transceiver structure comprising: the transmitter and the receiver.

The present invention also provides a computer program comprising program instructions for causing a computer to perform a method of transmitting data in a Generalized Frequency Division Multiplexing ,GFDM, communication system having N subcarriers with M symbols in each GFDM block, the method comprising the steps of:

down sampling the signal vector for transmission of size MN by a factor M;

performing M number of discrete Fourier transform, DFT, operations of size N on selected down sampled data samples to provide DFT output data; and

performing N number of M point circular convolution operations on the DFT output data with a pulse shaping filter to provide the GFDM output signal for transmission.

The present invention also provides a computer program comprising program instructions for causing a computer to perform a method of demodulating received GFDM signals in a GFDM communication system having N subcarriers with M symbols in each GFDM block, the method comprising the steps of:

performing N number of M point circular convolution operations on selected samples of the received GFDM signal; down sampling the circular convolved samples with a factor of M; and performing M number of inverse discrete Fourier transform, IDFT, operations of size N on the down sampled data.

There is also provided a computer program comprising program instructions for causing a computer program to carry out the above method which may be embodied on a record medium, carrier signal or read-only memory.

Brief Description of the Drawings

The invention will be more clearly understood from the following description of an embodiment thereof, given by way of example only, with reference to the accompanying drawings, in which :-

Figure 1 shows a block diagram of the main processing steps of the GFDM transmitter of the present invention;

Figure 2 shows a block diagram of the structure of the GFDM receiver of the present invention;

Figure 3 shows a block diagram of the main processing steps of the GFDM receiver of the present invention;

Figure 4 shows a table of the computation complexity of a GFDM transmitter implemented with direct matrix multiplication versus the GFDM transmitter of the present invention based on the number of complex multiplications; and

Figure 5 shows a table of the computational complexity of a direct ZF and MMSE GFDM receiver implementation versus the GFDM receivers of the present invention based on the number of complex multiplications.

Detailed Description of the Drawings

The transceiver techniques of the present invention exploit the special structure of the modulation matrix to reduce the computational cost without incurring any performance loss penalty. This is achieved by providing a GFDM transmitter where block DFT and IDFT matrices are used to make the modulation matrix sparse, and hence reduce the computational burden. Furthermore, a low complexity ZF receiver and a MMSE receiver are achieved by block diagonalization of the matrices involved in demodulation, while a low complexity MF receiver is achieved by sparsification of the matrices involved in demodulation. This enables a substantial amount of complexity reduction in the matrix inversion and multiplication operations to be achieved.

The typical steps involved in processing data in a GFDM transceiver are firstly described below. This is followed by a description of how the GFDM transceiver of the present invention implements these steps in order to achieve the above mentioned processing advantages.

Consider a GFDM system with a total number of N subcarriers that includes M symbols in each block. In a GFDM block, M symbols overlap in time. Therefore, M is called the overlapping factor of the GFDM system. The MN ^χ 1 vector d =

[do ^T ,...,d ^T _n -i] ^T contains the complex data symbols of the GFDM block, where the M x 1 data vector di =[ di (0), ... ,di(M- 1 )] ^T contains the data symbols to be transmitted on the i ^th subcarrier. To put it differently, di (m) is the data symbol to be transmitted at the m ^th time slot on the i ^th subcarrier. The data symbols are taken from a zero mean independent and identically distributed (i.i.d) process with the variance of unity.

In GFDM modulation, the data symbols to be transmitted on the i ^th subcarrier are first up-sampled by the factor of N to form an impulse train :

T

where δ is Dirac delta function. Then, Si =[si (0), ... ,Si (MN - 1 )] is circularly convolved with the prototype filter and up-converted to its corresponding subcarrier frequency. After performing the same procedure for all the subcarriers, the resulting signals are summed up to form the GFDM signal x(n) :

N- l M -i

(«) =∑ ∑ Μ™)ΰ _{ (»- Ν) _{mmi MN }e} i (2) where gi is the l prototype filter coefficient.

Putting together all the transmitter output samples in an MN ^χ 1 vector x=[x(0),...,x(MN - 1 )] ^T , the GFDM signal can be represented as multiplication of a modulation matrix A of size MN ^χ MN and the data vector d: x = Ad (3)

It will be understood that the modulation matrix A encompasses all signal processing steps involved in modulation.

Let g =[go, ... ,gMN-i] hold all the coefficients of the pulse shaping/prototype filter and g be the } ^h prototype filter coefficient, A can be represented as,

,2i L M J where

where ξί = N((M - i) mod M) and mod N is modulo-N operation. Based on (4) and (5), the matrix A can be written as

A = β £ι§ . .. . €N- : G (6) where G is an MN ^χ M matrix whose first column contains the samples of the prototype filter g and its other columns are circularly shifted copies of g by N samples.

^{1 1 *} ' ^{1 J J} is an MNx MN diagonal matrix whose diagonal elements are comprised of M concatenated copies of the vector e. _i = [1 , e ^J w _? . , . , e ³ w ^{ ^ ^1} \ ¹ , Typical GFDM systems use frequency domain equalization (FDE) to tackle the wireless channel impairments and reduce the channel equalization complexity. Accordingly, a cyclic prefix (CP) which is longer than the channel delay spread is added to the beginning of the GFDM block to accommodate the channel transient period. If NCP is the CP length, the last NCP elements of the vector x are appended to its beginning in order to form the transmitted signal vector * whose length is MN+NCP.

Let h be the channel impulse response. Thus, the CP length NCP needs to be longer than the channel length Q. After CP removal, the received signal which has gone through the channel, can be shown as r = Hx 4- v , (7)

where fis the complex additive white Gaussian noise (AWGN) vector, i.e, v ~ CA (0. <J _V IM N ) _* σν is the noise variance, H =circ{h} and ^ is the zero padded version of h to have the same size as x.

Due to the fact that H is a circulant matrix, an FDE procedure can be performed to compensate for the multipath channel impairments. With the assumption of having perfect synchronization and channel estimates, the equalized signal can be obtained as where FMN is MN-point normalized discrete Fourier transform (DFT) matrix and H ¹ is a diagonal matrix whose diagonal elements are reciprocals of the elements of the vector obtained from taking MN-point DFT of the zero padded version of h, viz., h . The vector y =[yo, ... ,yMN-i] ^T is the output of the FDE block.

At a typical GFDM receiver, in order to remove/suppress inter-carrier interference (I CI) due to non-orthogonality of the subcarriers and estimate the transmitted data vector d from the equalized signal vector a number of different types of linear GFDM receivers can be used. These include zero forcing (ZF), minimum mean squared error (MMSE) and matched filter (MF) receivers.

The ZF estimate of the transmitted data vector from equalized signal vector using (3) can be found as d _ZF = (A ^H A) ^_1 A ^H y. (9)

The ZF receiver completely eliminates the ICI caused by non-orthogonality of

H — 1 H

the subcarriers. However, since (A A) A can have large values, its multiplication to y can result in noise enhancement. This noise amplification problem can be taken care of by utilizing a MMSE receiver ii.M SE = (A ^fl A + σ„ ² Ι _Μ M) ^{^ 1} A ^H y. (10)

The matched filter receiver estimate of the transmitted data vector from the equalized signal vector using (3) can be defined as: d _MF = A ^H y _{( 10a)}

It will be appreciated that direct matrix inversions and multiplications in (9), (10) and (10a) demand high computational complexity. This is due to the fact that all the matrices are of the size MN xMN which may not be affordable for practical communication systems. Furthermore, it will be understood from a review of equation (3) that direct multiplication of the matrix A to the data vector d is a complex operation which demands (MN) ² complex multiplications.

The present invention therefore provides an implementation of a GFDM transceiver which substantially reduces its computational cost, while at the same time maintaining optimal performance. This implementation is described in the paragraphs below.

With regard to the implementation of GFDM transmitter of the present invention, equation (3) can be rewritten as: x = Ad = A5¾¾d. _; (11) where Fb is the MN xMN normalized block DFT matrix that includes M xM submatrices

and n, /= 0,...,N -1. The Validity of equation (11) is based on the fact that

The matrix resulting from the multiplication of the block DFT matrix

|_|

Fb into A is sparse and it is comprised of the prototype filter coefficients scaled by VN. The fact that it is sparse will be understood from the following derivation

|_|

of FbA The key idea in this derivation is based on the fact that inner product of two complex exponential signals with different frequencies is zero.

ΛΓ-1

1=0 From the definitions of Fb and A, can be obtained as

I - L ^A o ¹ N - l J where H's are M χ MN block matrices that can be mathematically shown as

where ^{= e N} Based on the definition of ^£ and the second last equation above,

iV-l

1=0

where κ=(Ν - i) mod N,

¾ = iiag{[ φ !.T IT

, . ,, ,ψ^

M block ¥ec1iors

^ _K = [0,... _t l ₎ ...,0] ^T ,

†

K ^til position ψ _κ 'ε are N ^χ 1 vectors and Ψκ is a diagonal matrix whose main diagonal elements are made up of M concatenated copies of the vector ψ _κ . From the equations above, Ti's can be obtained as

Accordingly, it can be perceived that the block matrices H's and hence the

2

matrix Γ are sparse. The matrix Π has only M non-zero elements, which are located on the circularly equidistant columns κ,κ+Ν,...,κ+(Μ-1 )Ν. The elements of two consecutive non-zero columns of Π are circularly shifted copies of each other. For instance, the second non-zero column of Π is circularly shifted version of the first non-zero one by one sample. From the last equation, the first non-zero column of H can be derived as v^[.f _KS .ff _K+ (Af -i>Af _s _- - ^■ - SK+ ! ¹ which is the circularly folded version of the K ^th polyphase component of the prototype filter. One can further deduce that the matrix Γ is a real one consisted of the prototype filter coefficients.

Now shown from the derivation above that the multiplication of the block DFT

H H H

matrix Fb into A is sparse, from equation (1 1 ), it can be inferred that Γ = AFb

|_|

is also sparse since it is the conjugate transpose of FbA . Hence, this enables the matrix A to be made sparse and real, as the prototype filter is usually chosen as a real filter. Due to equation (1 1 ) and the definition of Fb, Fbd can be implemented by performing M DFT operations of size N on the data samples, i.e., one per GFDM symbol.

If one lets d = J-tci = Jcl ₀ , . . . , d ^ .... , f ^" where the M x 1 vector d i=[ di(0), di(M-1 )] ^T contains the ^h output of each DFT block, then equation (1 1 ) can be rearranged as

N -l

x = r ^H d = V" fd _Ki (12) where ^{K = ~~} 0 ^m °d N

The following paragraphs set out a closed form derivation of D, from which it will be understood that the M xMN matrices H's have only M non-zero columns and the sets of those column indices are mutually exclusive with respect to each

-pi! _J

other. As a result, ^» will be a sparse vector with only M non-zero elements located on the positions ^K > ^K + ^N - · · ^» « * + ( ^M - 1)^·.

Closed form derivation of D

The polyphase components of the prototype filter g can be defined as the vectors go,gi , ... = [gi,gi+N , ... ,gi+(M-i )N ] ^T . As has been previously shown, ^r = ^& ¹¹ j _{s a} sparse matrix with only M non-zero elements in each column. The elements of Γ can be mathematicall represented as:

Where ^gn is circularly folded version of and k=(N-i) mod N. From the equation above, it can be deduced that each group of M consecutive rows of Γ, i.e., H's, whose non-zero elements are comprised of the elements of the vectors

8 ' ^s , is mutually orthogonal to the other ones. This is due to the fact that the sets of column indices of H's with non-zero elements are mutually exclusive with respect to each other. The block-diagonal matrix D (which will be derived in a

H H

subsequent paragraph), can be calculated as D = Ft>(A A)Fb which can be rearranged as:

Due to orthogonality of H's with respect to each other, i.e r _f r ^j ; O "M™,' . i " ≠ ^■* j ·'. , it can be discerned that D has a block-diagonal structure. Based on the last equation above, only equidistant columns of 17s with circular distance of N are non-zero and two consecutive and non-zero columns are circularly shifted copies of each other with one sample. Consider

Γο, then the elements [ ^Γ «> = ^* Λ¾ο]ο

JiJ. i -i)o = 3 feu.cJu-i) and

[FQ]ON = V^[go] ( -l) i [Γο](Μ-1)ίν = V^[go] (Af -2) illustrate that the consecutive and non-zero columns of Γο are circularly shifted versions of each other. One can conclude that the same property holds for the other non-zero columns of Γο and all the other 17s.

A closed form for D can be derived by:

D is an MNxMN matrix comprised of MxM submatrices which are all zero

|_|

except the submatrices located on the main diagonal, i.e., Di = ΠΓ . From a previous equation, it can be understood that the first non-zero columns of the

|_|

matrices Π and Γ i are equal to and , respectively, and the rest of their non-zero columns are circularly shifted version of their first non-zero column. Removing zero columns of I7s

where ¹ and ¹ are circulant matrices with the first columns equal to and are real and circulant, Di is

L real and circulant matrix which can be obtained as t = Wcirc{g _re @g _K }.

where is M-point circular convolution. On the basis of the derivation of FbA above, the non-zero elements of ^Γ » ^Η<1 can be obtained from M-point circular d th

convolution of ^K with the κ polyphase component of the prototype filter gk.

^■■ pi! j

Therefore, defining the non-zero elements of ¹ » ^U|t as the vector

3c#s == [¾t XK+N I · ^■ · ' +(M—I }N] * one gets! x* = v¾ _K ®ct _K (13)

From the equations (1 1 ) to (13) above, the implementation of the GFDM transmitter of the present invention comprises the following two main steps:

1 ) M number of N-point DFT operations. This involves the application of an N- point DFT to each individual GFDM symbol which includes N subcarriers. This can be efficiently implemented by taking advantage of fast Fourier transform (FFT) algorithm; and

2) N number of M-point circular convolution operations.

Therefore, the first and second steps in the GFDM transmitter of the present invention can be implemented from cascading the block diagrams shown in Figures 1 (a) and (b), respectively. The blocks P/S convert the parallel FFT outputs to serial streams. All the commutators shown in Figure 1 turn counter clockwise. Both commutators located on the right hand side of the Figure 1 (a) and Figure 1 (b) turn after one sample collection. However, the one located on the left hand side of Figure 1 (b) turns by one position after sending M samples to each M-point circular convolution block. The present invention also provides a low complexity implementation of ZF, MMSE and MF receivers for GFDM systems. Figure 2 shows a block diagram of the structure of the GFDM receiver of the present invention, according to one embodiment. It comprises two main processing steps, namely performing circular convolution operations on selected samples of the received GFDM signal, followed by performing inverse discrete Fourier transform, I DFT, operations on the circular convolved data. This will be explained in more detail in the paragraphs below.

It should be noted that the receiver structure of the present invention does not

|_| result in any performance loss, thanks to the certain structure of the matrix A A.

|_|

The characteristics of A A are firstly discussed in the next subsection, and then the implementation of the receivers of the present invention will be derived on the basis of those characteristics.

The GFDM receiver implementation of the present invention takes advantage of

|_|

the particular structure of the matrix A A which is present in both ZF and MMSE

|_|

receiver formulations. Using equation (6), one can calculate A A and find out that it has the following structure:

From the definition of the vector it can be straightforwardly perceived that

H _ _ £ _| _ _|

ejv_j - ^{~ f} ii _an d hence *"N-i ^ Therefore, the columns of A A as shown

|_| in (14) are circularly shifted with respect to each other. Accordingly, A A is a

|_|

blockcirculant matrix with blocks of size MxM. A A can be expanded as follows A ^H A = (15) where D is an MN ^χ MN block-diagonal matrix, D = diag {DO, ... , DN-I} and Di's are M xM block matrices. From equation (15), D can be derived as

As was previously explained, Di's can be derived from the polyphase components of the prototype filter.

V _i = Ncxc{g _K ®g _K }, (17) where κ =(N -/)mod N, is the i ^th polyphase component of g and

Ei = L¾ 9i+(M - l)Ni ^{■ ■ ■} i 9i+NI is its circularly folded version. It can be seen from equation (17) that Di's are all real and circulant matrices.

1. ZF receiver

To implement the ZF receiver of the present invention, inserting equation (15) into equation (9): dzp = ^V- ^x 3= _h A^y. (18)

|_|

The multiplication of matrix A to the vector y is the first source of computational

2

burden in a ZF receiver, which has computational cost of (MN) . However, this complexity can be reduced by taking advantage of the sparsity of the matrix

1 = J- _b A . The closed form of L - μ o » . . . » A JV_I I previously derived shows that the matrix is real valued and comprised of the prototype filter elements. Non-zero columns of the M xMN block matrices H's are circularly shifted copies of each other. Hence, the multiplication of H and y is equivalent to an M-point circular convolution of M equidistant elements of y starting from the K ^th position and circularly folded version of the K ^th polyphase

component of g scaled by VN, viz. Consequently,

y _t = Tiy_ = v^circ{g _« }y _{re and}

JK. = foist VK+N* ^{■■ ■} » VK+W-DN] ^Γ Let = V-'9 \y y$ _ _: ;T

where

Therefore, from rearranging equation (17) as Di =

inserting it into (19):

where q _K includes the first column of the circulant matrix

1

(circ{g _K }) scaled by vTv ^{7 '} . Due to the fact that the

coefficients of the prototype filter are known, the vectors q _K 's

can be calculated offline. Additionally, since the prototype filter coefficients are real, q _K 's are also real. From equation (20), one may realize that calculation of the vector y ^~ needs N number of M-point circular convolutions. After acquiring y ^~ , the ZF estimates of the transmitted symbols can be obtained as

(21)

As can be inferred from equation (21 ), finding d ZF from y ^~ requires M number of N-point inverse DFT (IDFT) operations.

2 MMSE receiver

To implement the MMSE receiver of the present invention, put equation (15) into equation (1 0) and

where ^ = ^ + ^MN = d g{T ⁾ _u . . . . , l ^j v-i } and

Recalling the circulant property of Di from equation (17), it can be understood that is also circulant and can be expanded as

where ** =

Let

^■ y = [ f , ^■ y5- i] ^T ₌ χτ ¹ ^ A ^H y one can write

= p«@y«, (23)

where p _K includes the first column of the circulant matrix

Since, in an MMSE receiver, the matrix ^ depends on ^v and the receiver cannot be simplified as in equation (20) for the ZF receiver, the circular convolution of equation (23) needs to be calculated in the frequency domain, known as fast convolution, in order to have the lowest complexity. After obtaining y~, the MMSE estimates of the transmitted symbols can be found as dMMSE = ^ff- (24)

3. MF Receiver

To implement the MF receiver of the present invention, due to the fact that ^ u = IMN , the equation d _M F = A ^H y can be written as d.MF = _b A ^H y

= ^ry, where Γ is a sparse matrix with only NM ² non-zero elements, and multiplication of to the resulting vector T can be implemented by using M number of N-point inverse DFT operations. Thus, it will be appreciated that the same receiver structure as ZF and MMSE receivers can also be used for the matched filter receiver.

It should be appreciated that matrix A could equally be implemented with a different order of the columns of the matrix, such as for example as described in the publication by N. Michailow, M. Matthe, I. Gaspar, A. Caldevilla, L. Mendes, A. Festag, and G. Fettweis, "Generalized Frequency Division Multiplexing for 5th Generation Cellular Networks (Invited Paper)," Communications, IEEE Transactions on, vol. PP, no. 99, pp. 1 -1 , 2014. However, it should be noted that the same block circulant properties still exist for this alternative matrix implementation, and hence, following the same line of derivations as described above, a similar receiver structure can be designed.

The GFDM receivers of the present invention can be implemented by cascading Figures 3 (a) and (b). It should be noted that the commutator on the right hand side of Figure 3 (a) will turn after collecting M samples from the i ^th branch, i.e., Mx 1 vector y ^" 7y~i, in the counter clockwise direction. In the ZF receiver of the invention, the vectors are replaced by qi's, while in MMSE receiver of the invention, they will be replaced by pi's. Due to the fact that in ZF receiver, the vector qi is fixed and only depends on the prototype filter coefficients, it can be calculated offline and hence there is no need for its real-time calculation. However, in MMSE receivers, the vectors pi depend on the signal to noise ratio. Therefore, these vectors they should be calculated in real-time. As mentioned previously, the circular convolutions in the MMSE receiver need to be performed by taking advantage of fast convolution to keep the complexity low.

Figure 4 details the computational complexity of a GFDM transmitter implemented with direct matrix multiplication versus the GFDM transmitter of the present invention based on the number of complex multiplications.

As discussed previously, the GFDM transmitter of the present invention involves two steps. The first step includes M number of N-point FFT operations that requires 2 ^Z CMs.

The second step needs N number of M-point circular convolutions. Recalling equation (13), since g _K 's are real-valued vectors, one may realize that each M- point circular convolution demands ² number of CMs. If M is a power of two, the complexity can be further reduced by performing the circular convolutions in frequency domain. This is due to the fact that circular convolution in time is multiplication in frequency domain. Thus, to perform each circular convolution, what is required is a pair of M-point FFT and I FFT blocks together with M complex multiplications to the filter coefficients in frequency domain.

Similar to Figure 4, Figure 5 summarizes the computational complexity of a direct ZF and MMSE GFDM receiver implementation versus the ZF and MMSE GFDM receivers of the present invention based on the number of complex multiplications. The parameter I is the number of iterations in the algorithm with interference cancellation.

From Figure 2, it can be understood that the proposed receivers of the present invention, namely the ZF, MMSE and MF receivers, involve N and M numbers of M-point circular convolutions and N-point IDFT operations, respectively. IDFT operations can be efficiently implemented using an N-point

FFT algorithm which requires T ^ CMs. As mentioned earlier, in the ZF receiver, the vectors have fixed values and hence can be calculated and stored offline. Furthermore, ^s are real-valued vectors. Thus, the number of complex multiplications needed for N number of M-point

NM ²

circular convolutions is ²

In contrast to the ZF receiver, in the MMSE receiver of the present invention, the vectors ^s are not fixed and depend on the signal-to-noise ratio (SNR). Hence, they need to be calculated in real-time. To this end, as highlighted previously, those operations can be performed by using M-point DFT and IDFT operations. Due to the fact that + ^σ Λ« ) j _S a real-valued diagonal matrix, its inversion and multiplication to ^K only needs 2 CMs. The resulting diagonal matrix multiplied into an M χ 1 vector which needs M CMs. Since M is not necessarily a power of 2, complexity of M-point DFT and IDFT operations in the implementation of the circular convolutions is considered as M ² . Obviously, if M is a power of 2, a further complexity reduction is possible by taking advantage of FFT and I FFT algorithms. Therefore, the complexity of the MMSE receiver of the present invention only differs from the complexity of the ZF receiver of the present invention in the implementation of the circular convolution operations.

Figure 5 also presents the complexity of the direct ZF and MMSE detection techniques, i.e., direct solutions to the equations (9) and (10), respectively. Those solutions involve direct inversion of an MN ^χ MN matrix which has the complexity of ^σ ( ³ Α* ³ ) _anc j two vector by matrix multiplications with the computational burden of 2(MN) ² CMs. The present invention provides a transceiver for GFDM systems which provides a number of advantages when compared to existing GFDM transceiver techniques. The transceiver of the present invention reduces the computational cost without incurring any performance loss penalty. Due to the fact that the transceiver structure involves block diagonalization and diagonalization of matrices, the memory requirements of the system can be substantially reduced. Furthermore, as the processing steps of the present invention are direct, the receivers do not incur any performance loss compared with the optimal ZF, MMSE and MF receivers. Another advantage of the receiver structure with respect to interference cancellation receivers is that it is not iterative. Hence, the computations can run in parallel, which can in turn reduce the overall processing delay of the system.

It will be appreciated GFDM is a non-orthogonal waveform that is circularly pulse-shaped. It is also more bandwidth efficient than OFDM as only one CP is used for each data packet. Through filtering each subcarrier band with a well-designed prototype filter, GFDM gains resiliency against CFO as only its adjacent subcarriers overlap in frequency domain and hence the leakage among subcarriers is reduced. The price to pay for gaining all the aforementioned advantages is some bit error rate (BER) performance loss compared with OFDM when linear detectors like zero forcing (ZF) or minimum mean square error (MMSE) are used. This performance loss is due to the fact that GFDM is a non- orthogonal waveform.

To tackle the performance degradation of GFDM, a multicarrier technique inspired by GFDM can be implemented that is able to preserve the orthogonality among the subcarriers. In these systems, hereinafter referred to as circular filter bank multicarrier (C-FBMC), similar to filter bank multicarrier systems with linear pulse shaping, a carrier phase of ττ/2 is applied to the adjacent subcarriers containing real data symbols to satisfy the orthogonality in real domain. Hence, a simple matched filter (MF) detector leads to the same BER performance as that of OFDM. C-FBMC may be thought of as a modified version of GFDM.

This modification can be easily applied to the proposed transceiver structure with MF receiver. The only difference is that rather than QAM symbols, real data symbols are fed into the proposed structure doubling the number of subcarriers in order to be able to transmit the same number of bits. Applying the aforementioned carrier phase of ττ/2 to the adjacent subcarriers, will result in real orthogonality and hence, even values of M preferably powers of two can be chosen to get the same BER performance as OFDM and circumvent the BER loss of GFDM due to its non-orthogonality. It will also be appreciated that even values of the parameter M which are powers of two allows utilization of FFT and IFFT algorithms to implement circular convolution operations involved in the proposed transceiver structure. The embodiments in the invention described with reference to the drawings comprise a computer apparatus and/or processes performed in a computer apparatus. However, the invention also extends to computer programs, particularly computer programs stored on or in a carrier adapted to bring the invention into practice. The program may be in the form of source 5 code, object code, or a code intermediate source and object code, such as in partially compiled form or in any other form suitable for use in the implementation of the method according to the invention. The carrier may comprise a storage medium such as ROM, e.g. CD ROM, or magnetic recording medium, e.g. a floppy disk or hard disk. The carrier may be an electrical or optical signal which may be transmitted via an electrical or an optical cable or by radio or other means. In the specification the terms "comprise, comprises, comprised and comprising" or any variation thereof and the terms include, includes, included and including" or any variation thereof are considered to be totally interchangeable and they should all be afforded the widest possible interpretation and vice versa.

The invention is not limited to the embodiments hereinbefore described but maybe varied in both construction and detail.