VFDM for single user

i
TITLE: VANDERMONDE SUBSPACE FREQUENCY DIVISION
MULTIPLEXING FOR SINGLE USER (VFDM-SU)
ABSTRACT
This report consists of the summary of what I have been doing for my work. Its aim is to
introduce a technique for interference cancellation in overlay networks that comprises
a secondary network operating simultaneously with a primary network on the same
frequency band. In fact, this technique allows a primary and a secondary user wish to
communicate with their corresponding receivers simultaneously over frequency selective
fading channels. The techniques to be introduced is nothing but Vandermonde Subspace
Frequency Division Multiplexing(VFDM).Although VFDM can be used for both Single
User and Multi Users, my work is only concentrated on VFDM for Single user(VFDM-
SU). In this work a Vandermonde precoder cancels the interference from the secondary
user by exploiting the redundancy of a cyclic prefix. I t ( V F D M ) allows the
coexistence of a downlink orthogonal frequency division multiplexing(OFDM)
macro-cell and a cognitive small-cell in a time division duplex mode . In this work, it
was shown to be able to completely cancel the interference towards a primary
macro-cell system at the price of perfect channel state information (CSI) at the
opportunistic secondary small-cell. Briefly, i t uses a linear Vandermonde-
based precoder to project the signal to the secondary receiver on the null space of the
interfering channel from the secondary transmitter to the primary receiver.
The work to be done consists of two parts such as VFDM-SU under realistic assumptions
i.e. the secondary transmitter has no side information about the primary’s message and
each transmitter knows only its local channels. The second part consists of VFDM-SU
channel stats Awareness and Estimation. To simulate this work, I have been
concentrating on one part (the 1st part i.e. VFDM-SU under realistic assumptions).
The simulation of 2nd
part will be the rest of next works. In the simulated part of this
work, we have seen that numerical and simulation results were presented to prove
theoretical derivations. It has been shown that, achievable bit rates (ABR) of Secondary
User could be achieved with optimal power allocation scheme in Cognitive Radio
Network (CRN). In addition, we extended the assessment of VFDM by analyzing
the bit error rate and sum rate capacity of practical linear receiver structures for the
VFDM-based secondary system. Our numerical examples have show that VFDM, with
an appropriate design of the input covariance, enables the secondary user to achieve a
considerable rate while generating zero interference to the primary user. In order to assess
its performance, we have adopted a QPSK bit-mapping.

ii
Contents
ABSTRACT......................................................................................................................................i
I. INTRODUCTION................................................................................................................... 2
I.1.GENERAL INTRODUCTION.............................................................................................. 2
I.2.OBJECTIVE OF THE PROJECT.......................................................................................... 2
i. MAIN OBJECTIVE OF THIS PROJECT ........................................................................... 2
ii. SPECIFIC OBJECTIVES.................................................................................................... 3
II. SYSTEM MODEL.......................................................................................................... 4
II.1.EXPLAINATION OF A DIAGRAM................................................................................... 4
II. 2.SYSTEM OVERWIEW ...................................................................................................... 5
III. VFDM MODEL .................................................................................................................. 7
PRECODER DESIGN. ............................................................................................................... 9
IV. VFDM’S PERFORMANCE ............................................................................................. 13
1. OPTIMAL RECEIVER......................................................................................................... 13
V. SIMULATING RESULT AND NUMERICAL ANALYSIS ............................................. 16
1. OPTIMAL RECEIVER.................................................................................................... 17
2. SUB-OPTIMAL LINEAR RECEIVER............................................................................ 20
EQUALIZERS STRUCTURES FOR VFDM....................................................................... 20
PERFORMANCE ................................................................................................................. 22
VI. CONCLUSION ................................................................................................................. 29
VII. REFERENCE .................................................................................................................... 30

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I. INTRODUCTION
I.1.GENERAL INTRODUCTION
To start developing this work, it is better to have over view of introduction of this work
for challenges of current Telecommunication technology .In the recent century;
regulatory communication agencies face a new challenge: how to increase the radio
capacity[1] using a limited radio spectrum. The fact that the allocated radio spectrum
is underutilized has pushed these regulatory communications agencies towards
adopting a flexible spectrum management model, different from their approach. As I
have mentioned in abstract, our system consists of two networks such as Primary
network and second one. The Primary network is licensed while the second one is
opportunistic .These networks are arranged in overlay manner supporting reutilizing
resource .Cognitive radios [2,3,4] are envisioned to adopt spectrum sharing techniques to
offer a solution to the spectrum shortage problem. To do this a technique which is
called Vandermode Subspace Frequency Division Multiplexing (VFDM) has
introduced. Normally,VFDM is nothing but an overlay spectrum sharing technique for
cognitive radio. It makes use of a precoder based on a Vandermonde[5,6] structure to
transmit information over a secondary system, while keeping an orthogonal frequency
division multiplexing (OFDM) based primary system interference-free. I addition ,it
is based on a linear precoder that allows a secondary transmitter to precode its signal
on the null-space of the interference channel, therefore incurring zero interference at
the primary receiver .It exploits the unused resources created by frequency selective
and the use of guard symbols in block transmission systems at the primary system.
Herein, a global view of VFDM has been presented, including also practical aspects
such as linear receivers and the impact of channel estimation (CSI). We have also
presented some key design parameters for its future implementation and a feasible
channel estimation protocol. Finally we will show that, even when some of the
theoretical assumptions are relaxed, VFDM provides non-negligible rates while
protecting the primary system.
I.2.OBJECTIVE OF THE PROJECT
i. MAIN OBJECTIVE OF THIS PROJECT
The main objective of this research is nothing but
To design a Precoder that will be able to cancel interference at primary receiver.

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Maximizing achievable rate at secondary system.
ii. SPECIFIC OBJECTIVES
Understand new technology of wireless communication such as:
 Cognitive radio
 Dynamic spectrum access(spectrum sharing)

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II. SYSTEM MODEL
Consider a system with a single Primary and Secondary Network as show by Figure
1.Each Network comprises of a single receiver and a single transmitter. In this system
model, secondary system wishes to communicate over the same frequency band as the
primary system while generating no interference. But we keep in our mind that the
primary system has no information about the presence of Secondary network within the
system. The Scenario is show by figure bellow.
Figure 1 Diagram of two networks sharing same band.[6]
II.1.EXPLAINATION OF A DIAGRAM
The figure above, the two networks (Primary and Secondary) wish to communicate with
their corresponding receivers simultaneously. Let Channel (1,1) be noted as h(1,1)
,channel
(1,2) as h(1,2)
,channel(2,1) as h(2,1)
while channel (2,2) is h(2,2)
.In the above figure ,all
transmitters and receivers have a single antenna. The cognitive interference antenna is
characterized as a primary system (the pair TX1-RX1), that communicates a message S1
over licensed band ,while a secondary system (the pair TX2-TR2),exploits the band
opportunistically to communicate its own message S2,while avoiding harmful
interference to primary receiver. The primary system, being the legal license of the band,
does not need to avoid interference to the secondary system.
There is also a case where by both secondary and primary systems can fully cooperate
by sharing information trough an unlimited backhaul,i.e they belong to the same
operator ,then primary and secondary systems can be considered as part of a network
multiple input, multiple output(MIMO) [7]system . Therefore, if all messages (S1 and
S2) are known prior to transmission at all the transmitters (TX1 and TX2), then the
cognitive interference channel can be generated to a 2X2 MIMO broadcast channel[7].
In this case, interference is suppressed on both cross links (h (12)
and h (21)
). The main
difference with what we have seen previously with what concerns the second receivers is
that, this time, each one will need to cancel its own interference to all the primary users.

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The secondary receivers this time will not only suffer interference from the primary
system, but also from other secondary transmitters. It is why the secondary transmitters
can coordinate to form a network to deal with multiuser interference[7,8]. As I have
mentioned in abstract ,I’m not interested in that later case .I’m only concentrating on
the first case i.e case under realistic assumption.
II. 2.SYSTEM OVERWIEW
As we have already mentioned, a primary system is the legacy RAT (Radio Access
Technology) which is already in operation, the licensee of the spectrum. This primary
system can be a PCS system, such as a global system for mobile communications (GSM)
cellular system, or a broadcast network, such as analog television or radio. These primary
systems are also usually seen as immutable, normally due to high costs involved in
changing their underlying technologies. A secondary system, on the other hand, is the
cognitive opportunistic system, aiming at finding free resources from the primary system
to transmit its own data.
The primary network is composed of macrocells while Secondary one composed of
small cells. The system looks like the following
Figure 2. The legacy cellular system (Primary Network)[9,10]
Figure 3.Small Cells (Secondary Network)[9,10]

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Figure 4.. Coexistence with legacy[9,10]
N.B: Although there is coexistence with legacy, the Small-cells must not interfere with it
(legacy).

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III. VFDM MODEL
The figure below(figure 5) shows the simplification of cognitive interference channel.
Figure 5. Cognitive interference channel[9,10]
In figure1,we let h(ij)
denotes the L+1 tap channel impulse response vector between
transmitter i and receiver j. For Simplicity purposes, the channel’s entries are made to be
unit norm, independent and identically distributed (i.i.d), complex circularly symmetric
and Gaussian CN (0,IL+1 /(L+1)) .The channels are i.i.d over any pair i,j.In order to avoid
block- interference, we apply orthogonal frequency division multiplexing(OFDM) with N
subcarriers with cyclic prefix of size L.
The received signals at both the primary and Secondary receivers are given by:
y1=F(Ƭ(h(11)
X1+ Ƭ(h(21)
)X2+n1) .…………………………..(1)
y2=F(Ƭ(h(22)
X2+ Ƭ(h(12)
)X1+n2) …………………………(2)

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Schematically, the system will look like
s1
s2
y2 = (F Ƭ (h(22)
)X2) + Ƭ h(12)
X1 + n2)
y1= (F Ƭ (h(11)
)X1) +Ƭ (h(21)
)X2) +n1)
Figure 6: VFDM model of figure (1)[9,10]
Where ∈ C Nx(N+L)
is matrix with a Toeplitz structure constructed from the
channel’s coefficients given by
…………………………(3)
F is a NXN Discrete Fourier Transform (DFT) matrix with
……………………. (4)
For k,l=0,..,N-1,and xi denotes the transmit vector of user i of size N+L subjected to the
individual power constraint given by
tr(E [xixiH
]) ≤ (N+L)Pi …………………………………………….(5)
and ∼ C N (0,σn
2
IN) is an N-sized additive white Gaussian noise (AWGN) noise

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vector. The transmit power per symbol is Pi=1
For the Primary user, the considered modulated OFDM modulated Symbol
X1= AF
−1
s1 ………………….(6)
where A is a (N +L)×N a cyclic prefix precoding matrix that appends the
last L entries of F
−1
s
1
and s1 is a symbol vector of size N and unitary norm.
At the other hand regarding the secondary user, the transmit vector is given by
X2= Es2, ………………………..(7)
where E ∈C
(N+L)xL
is a linear precoder and s2 is a unitary norm symbol vector.
PRECODER DESIGN.
As previously stated, the secondary system tries to cancel its interference to the primary
one, while the primary system remains oblivious to the presence of the secondary one.
For doing this, is nothing but to cancel the term Ƭ (h(21)
)X 2 in expression of y1.
As it shown in figure below the term Ƭ(h(21)
)X2 in the expression of primary receiver
have to be cancelled.
As it is shown in [1,9,10,11,12,]the orthogonal condition Ƭ(h(21)
)X2=0 ………(8)
must be satisfied such that E belongs to the null space of Ƭ(h(21)
) .
s1 X2 = Es2
X1 = AFH
s1
s2
y2 = F Ƭ (h(22)
)X2+ Ƭ (h(12)
)X1 + n2
y1= F Ƭ (h(21)
)x2 + Ƭ (h(11)
)x1 + n1
Figure7: Cognition to create E[9,10]

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To evaluate interference cancelation in primary receiver, let us simplify equation (1) and
(2)
y1=FƬ(h(11)
x1+ FƬ(h(21)
)x2+F(n1) ……. …………………………..(9)
y2=FƬ(h(22)
x2+ FƬ(h(12
))x1+F(n2)
By substituting (6) and (7) in (8), we get
y1=FƬ(h(11)
AF−1
s1+ FƬ(h(21)
)Es2+F(n1) ……. …………………………..(10)
y2=FƬ(h(22)
Es2+ FƬ(h(12)
)AF−1
s1+F(n2) ………………………. ………(11)
When we make interference part in (10) equal to zero ,the signal received at the primary
system becomes
y1= H11s1+ ν1, …………………………………………………………….(12)
Where H11= FƬ(h(11)
)AF−1
is an N ×N diagonal overall channel matrix for the primary
system and ν1 the Fourier transform of the noise n1, has the same statistics as n1.
As we have previously said at the beginning of this project in our system i.e single user
VFDM, the primary system does not cooperate with the secondary one, which performs
single user decoding at the secondary receiver.
Therefore the received signal at secondary system becomes
y2= H22s2+ H11s1+ ν2, ………………………………………..(13)
where H22 = FƬ(h(22)
)E is an N ×L overall channel matrix for the secondary system,
H11= FƬ(h(12)
)AF−1
is an N×N diagonal overall channel matrix for the primary
system(interference w.r.t. the secondary receiver) and ν2, the Fourier transform of the
noise n2, has the same statistics as n2. The primary system does not cooperate with the
secondary one, which performs single user decoding at the secondary receiver.
Let H11s1+ ν2=η …………………………………..(14)
as the interference plus noise component, obtained when we substituting in (12)
By substituting (14) in (13) the received signal in second system becomes
y2=H22s2+ η …………………………………………(15)

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From those last equation of received signals i.e. equation (12) and equation (15) we
remark that VFDM converts the frequency-selective interference channel (1) and (2) (or
X interference channel)into a one-side vector Z interference channel where the primary
receiver sees interference-free N parallel channels and the secondary receiver sees the
interference from the primary transmitter . Hence, employing an equalizer able to deal
with the interference without the knowledge of the transmitted primary symbols is of
interest for VFDM .Due to particular structure of FƬ(h(21)
)E=0,it is not difficult to show
that a matrix E ,capable of yielding FƬ(h(21)
)E=0,has to evaluate the polynomial
of ……..(16)
where {a1,...,aL} are the roots
with L+1 coefficients of the channel h(21)
. We have, therefore, called this technique
Vandermonde-subspace Frequency Division Multiplexing (VFDM).Vandermonde matrix
is interesting due to its property to evaluate a polynomial of a certain values.Thus it is
straightforward to see that
.......................(17)
V=
defines the null space of Ƭ(h(22)
) and without the loose of generality, we can further
choose the columns of E as any liner combination based on the columns of v ,such that
……..(18)
Where is the the (kth
, lth) element of ΓLxL
,a coefficient matrix.Therefore from
(17) we can finally define E as
E VΓ……………………….(19)
In practice, we select Γ to obtain orthonormal columns for E.
This can be accomplished numerically by:
1. QR decomposition[13] of V;
2. Gram-Schmidt process[13] on the columns of V;

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3. Singular value decomposition (SVD)[14]of Ƭ (h(21)
).
For QR decomposition, we apply its value of V=EΓ-1
where Γ-1
is an upper-triangular
matrix and E is orthonormal .For the case of Gram-Schmidt process on the columns of
V,it can be numerically obtained by performing on the column of V .By using SVD of
Ƭ (h(21)), E can be constructed. where if Ƭ(h(21)) = U Ƭ Λ Ƭ VƬ
H
, then
E =[VƬN| ···| VƬ(N+L)−1| VƬN+L] and V Ƭ has the form [VƬ1| VƬ2|···|V Ƭ N+L]
Find a new precoder, that possess better conditioning, but conserving the same properties
of V.
Note that roots of a Gaussian polynomial tend to fall on the unit circle and some roots fall
outside and deteriorate the conditioning of V. Remember that the single antenna
interference is channel is the target of our work. If multiple antennas were to be adopted
at the secondary transmitter, then the spatial dimension could be used to manage
interference to the primary receiver, by means of techniques such as Zero Forcing (ZF)
precoding.

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IV. VFDM’S PERFORMANCE
To achieve its best performance, the secondary system must be designed with two goals
in mind such:
1. Maximize the achievable rate at the secondary system
2. Enforce the interference protection at the primary system
This is equivalent by finding S2 and E that solve by the following optimal problem
)……...(20)
where S2is the covariance matrix of s2, Sη= H12S1H12
H
+ σn
2
IN is the covariance matrix of
η, the first constraint comes from (5) and the second constraint comes from (8).We
approximate η to a zero-mean Gaussian random vector. Note that, the objective function
in (20) does not take into consideration the rate at the primary system. This is the case
since, by guaranteeing zero interference from the secondary system, the primary system
can achieve maximum capacity through the optimization of its own input power
allocation [15].
1. OPTIMAL RECEIVER
To determine achievable rates at secondary system, we simplify (20) by choosing a Γ
matrix that provides a well-behaved E with orthogonal columns, and optimize only S2.
The optimization problem now becomes
)…(21)
……(22)
N.B: We have dropped the second restriction since it becomes implicit from precoder
design.

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In order to adapt the precoder to the channel fluctuations, perfect knowledge of the
h(21)
channel state information (CSI) is required. In addition to that, perfect knowledge of
the interference plus noise covariance Sη of the secondary receiver is also required at the
secondary transmitter.
Remember that the primary network’s capacity is maximized by the classical water filling
(WF) algorithm, due to the OFDM modulation and N parallel channels; and we consider
that the interfering signals from the primary network are seen as noise on the secondary
network.
Finally, we let S2= VGD2VG
H
with D2∈ L×L
being diag[d1,··· ,dL], and the new
definitions of (21) and (22) becomes
)…(23)
……(24)
Where G= and GNxL
to be an equivalent channel.The SVD of equivalent channel
where UG ∈ N×N
and VG ∈ L×L
are unitary matrices and
ΛG ∈ N×L
contains a top diagonal with the r ≤ L eigenvalues of GH
G, with
[ΛG]i,i≥ 0.
(23) and (24) can be further rewritten in scalar for as
……….(25)
.
The optimization problem in its new form (25) can be solved using the Karush-Kuhn-
Tucker (KKT) conditions which lead to the classical water-filling solution .
The solution to (25) is given by S2= VGD2VG
H
, where the ith
component of the matrix D
is the weighted water-filling solution given by
……………………….(26)
where µ, known as the “water level”, is determined to fulfill the total power constraint
(N+L)P2. Since we have chosen Γ such that E is orthonormal, it follows that ∀i,[Q]i,i= 1,

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and therefore, the maximum achievable spectrum efficiency for the secondary system is
finally given by
……………(27)
PERFORMANCE
In this section , we have evaluated representative numerical examples to illustrate the
performance of Vandermonde precoder with the proposed power allocation schemes, as
well as the achievable rate of Secondary user with the premise of guaranteeing zero-
interference to Primary user. The evaluation of the result is based on the performance of
VFDM’s optimal receiver .Therefore, numerical result were produced through Monte
Carlo based simulations. In this case, transmit powers are considered to be unitary for
both primary and secondary system, and the signal to noise ratio (SNR) is controlled by
varying the noise variance σ2
n.As we have mentioned in this the previous sections , E is
generated by a Gram-Schmidt process on the columns of V .For some of the presented
results, in order to control the secondary system’s performance with respect to the
interference coming from the primary system, an interference weighting factor α has been
added to (14) such that
………………………(29)
Equation of the signal received at the second system becomes
y2=F (Ƭ(h(22)
x2+ αƬ(h(12))x1+n2) ………………………….(30)
With α( interference factor) α∈ [0, 1] to scale the interference coming from the primary
system .
Note that:
1. Perfect interference cancelation is obtained only for perfect h(21)
knowledge;
2. VFDM’s performance depends on the number of available dimensions;

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V. SIMULATING RESULT AND NUMERICAL ANALYSIS
A summary of the simulation parameters is presented in table 1.
Parameter Value
Number of carriers(N) 64
Cyclic prefix size (L) {8,16,32}
Channels (CN (0,IL+1/(L + 1))
Additive noise CN (0,σ2
nIN)
E construction Gram-Schimidt process
Interference factor (α) α{0,0.1,0.5,1}
Table1. Simulation parameters
In figures 9,10,11 VFDM’s average achievable spectrum efficiency using an optimal
receiver is given for N = 64 and three sizes of channel L ∈ {8,16,32} taps with cross
interference from primary to Secondary equal to zero(i.e α=0 ) for the purpose of
isolating the performance of the secondary system. The spectrum efficiency is seen to
suffer a higher penalty for smaller values of L, since this directly translates into a smaller
number of available precoding dimensions. On other hand, the figure12 shows VFDM’s
average achievable spectrum efficiency for N = 64 and size of channel L=16 with
different values of cross interference from primary system to the secondary one i.e α=[0
0.1 0.5 1].This figure12 also compares each the rate of primary systems with the rates of
secondary systems for each interfering factor.

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1. OPTIMAL RECEIVER
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3
VFDM performance N = 64 L = 8
1/n
2
=SNR [dB]
Ropt[bps/Hz]=Rate[bps]
R1(Rates of Primary sys.)
R2(Rates of Secondary sys.),= 0
Figures 9 .VFDM’s average achievable spectrum efficiency for N = 64, L=8 and α=0

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0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3
1/n
2
=SNR [dB]
Figures 10 .VFDM’s average achievable spectrum efficiency for N = 64, L=16 and α=0

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0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3
1/n
2
=SNR [dB]
Figures 11 .VFDM’s average achievable spectrum efficiency for N = 64 ,L=32 and α=0

20
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3
1/n
2
=SNR [dB]
R2(Rates of Secondary sys.),= 0.1
R2(Rates of Secondary sys.),= 0.5
Figures 12 .VFDM’s average achievable spectrum efficiency for N = 64, L=16 and α=[0
0.1 0.5 1]
In figure 12, the effect of interference coming from the primary system on the spectral
efficiency performance of VFDM is shown. As expected, the secondary system quickly
becomes interference limited the higher the. Nevertheless, it is interesting to see that even
in the worst case scenario; VFDM is still able to offer non-negligible rates.
2. SUB-OPTIMAL LINEAR RECEIVER
EQUALIZERS STRUCTURES FOR VFDM
In this section, we shift our focus to linear equalizers, moving one step closer to the
implementation of a practical VFDM system. We analyze the choice of three classical
linear equalizers: minimum mean square error (MMSE), ZF and matched filter (MF) in
the performance of VFDM.
As a starting point to construct the equalizers for VFDM let us consider the estimated
symbols at the secondary receiver as

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……………………………………(31)
where C is the linear equalizer .In the following we derive the expressions for G for each
of the studied receivers.
a. Minimum Mean Square Error(MMSE)
The MMSE is a well known receiver for its good performance in the presence of
interference, maximizing the SINR . From the MMSE definition in [13,16,17,18] we
have that
CMMSE= SyySys,……………………………….(32)
where Syy is the covariance of the received signal with itself and Sys is the covariance of
the received signal with the transmitted signal. By further developing (32) with the
elements of (13) we get
CMMSE= H22
H
(Sη + H22 H22
H
)-1
…………………(33)
where Sη is the covariance of the interference plus noise. By looking closely into (13) we
can compute RηI as
Sη = H11H11
H
+ σ2
IN.
In order to properly compute CMMSE, we are considering that the secondary receiver
knows perfectly each of the overall channels H22 and H11. The channels can be obtained
by pilot estimation.
We can compute the effective SINR as
…..(34)
Where γk is the SINR contribution of the Kth received symbol,ck is the kth column of C
and h(.) is the kth column of H(.).
In the specific case for the MMSE, this expression can be further simplified to
γMMSEk= h2k
H
(Sη + U2UH
2)−1
h2k,………………(35)
where U2 is an N×(L−1) matrix representing H2excluding of the kth
column.
b. Zero Forcing

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The ZF equalizer achieves a zero inter-symbol interference by assuming a peak distortion
factor of zero.
CZF is the ZF equalization filter and according to [13,16,17,18], CZF is given by
CZF= H22
−1
.
In the case of the rectangular overall channel matrix H22,the inversion is accomplished by
the pseudo-inverse operation(defined by A*
= (A2
H
A2)−1
A2
H
), and thus
GZF= H22
*
…………………………………..(36)
Here we consider that the secondary receiver possesses only an estimate for the overall
channel H22, obtained as described in the MMSE case. The ZF does not take into account
the outside interference as for the case of the MMSE, and is thus simpler. Similar to the
MMSE case, the ZF equalizer’s SINR is given by taking ck as the kth
column of CZF in
(34).
c. Matched Filter
Unlike the previous two equalizers, the MF correlates the received symbols with a filter
that matches the channel, hence its name. This is accomplished by convolving the
received signal with a time reversed version of the overall channel matrix, and therefore,
CMF is the MF equalization filter given by
CMF= H22
H
……………… (37) [13,16,17,18]
PERFORMANCE
In order to better understand the performance of VFDM based linear receivers, Monte
Carlo simulations were executed, following the same configuration parameters as in the
previous section. In figure 11 the spectral efficiency is given for all three equalizers as
well as the optimal receiver, for reference. Therefore the spectral efficiency for the linear
receivers Rlin is obtained by substituting k into the general capacity expression.
……………………………(38)
Again, we isolate the performance of the secondary system by setting α to zero. The
optimal receiver clearly outperforms the three other equalizers with an almost constant
gap of about 2 dB with respect to MMSE and around 4 dB with respect to the ZF. Even
though there is no interference coming from the primary system, the MMSE takes into

23
consideration the characteristics of the noise, which explains its best performance among
the three linear receivers. The MF presents the worst performance, with capacity
saturating at around 16 dB.
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1/n
2
[dB]
Ropt
,Rlin
[bps/Hz]
MF(=0)
ZF(=0)
MMSE(=0)
Optimal Receiver(=0)
Figure 13. VFDM’s Ropt and Rlin for the MMSE, ZF and MF equalizers for N = 64, L =
16 and α = 0.

24
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1/n
2
[dB]
Ropt
,Rlin
[bps/Hz] MF(=0.5)
ZF(=0.5)
MMSE(=0.5)
Optimal Receiver(=0.5)
Figure 14. VFDM’s Ropt and Rlin for the MMSE, ZF and MF equalizers for N = 64,L =
16 with α=0.5

25
0 2 4 6 8 10 12 14 16 18 20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1/n
2
[dB]
Ropt
,Rlin
[bps/Hz]
MF(=1)
ZF(=1)
MMSE(=1)
Optimal Receiver(=1)
Figure 15:VFDM’s Ropt and Rlin for the MMSE, ZF and MF equalizers for N = 64,L =
16 with α=1
For all the SINR expressions given previously, all symbols have the same statistical
behavior, and thus, we can consider that the average bit error probability for QPSK to be
given by
………………………………(39)
And Q (.) is the Q-factor.Once again, α is set to zero, to isolate the their performance.
MMSE outperforms the two other equalizers with a constant gap of about 4 dB compared
to the ZF .And the figure 16 shows the details of this Probability error.
In figures 17&18, we concentrate only on the best performing linear receivers (ZF and
MMSE) to minimize the clutter. This time, Pe curves are presented for increasing
interference (α  {0,0.5,1}) factors. As expected, the presence of interference severely
degrades the performance of both equalizers.

26
0 2 4 6 8 10 12 14 16 18 20
10
-3
10
-2
10
-1
10
0
1/n
2
[dB]
Pe
MF=0
ZF=0
MMSE=0
Figure 16: VFDM’s Pe for the MMSE, ZF and MF equalizers for N = 64, L = 16 and α=0.

27
0 2 4 6 8 10 12 14 16 18 20
10
-3
10
-2
10
-1
10
0
1/n
2
[dB]
Pe
MF=0.5
ZF=0.5
MMSE=0.5
Figure 17.VFDM’s Pe for the MMSE, ZF and MF equalizers for N = 64, L = 16 and α =
0.5

28
0 2 4 6 8 10 12 14 16 18 20
10
-3
10
-2
10
-1
10
0
1/n
2
[dB]
Pe
MF=1
ZF=1
MMSE=1
Figure 18.VFDM’s Pe for the MMSE, ZF and MF equalizers for N = 64, L = 16 and α =
1

29
VI. CONCLUSION
This work aimed at introducing a novel technology able to allow two radio access
technologies (RATs) to operate side-by-side in a cognitive radio (CR) setting, i.e. while
sharing the same band and protecting the legacy system from interference. As we have
seen the name of this introduced technique is called VFDM.But the work has concerned
only on single user i.e VFDM-SU. It was shown that VFDM can effectively share a band
with a primary system on a cognitive interference channel. It behaves as an opportunistic
radio system that makes use of the available free dimensions to transmit its own data at
the cost of channel knowledge and extra processing power. We showed that there exists a
set of pre-coders which lie in the nullspace of the secondary-to-primary interfering
channel, which are able to achieve our interference cancellation goals. As we have seen
in analysis result and simulation , due to the particular structure of the frequency selective
channel model adopted based on the Toeplitz matrix this nullspace can be easily found
bus constructing a specially built Vandermonde matrix based on the roots of a
polynomial created from the interfering channel coefficients. We named this the
Vandemonde-subspace. We have shown that, since VFDM is dependent and limited to
the available dimensions, restricting its performance. Furthermore, we have seen that
VFDM is still susceptible to interference coming from the primary transmitter, which
further limits its performance. In this contribution, we have rather studied VFDM’s
performance using practical linear equalizers, namely the MMSE, ZF and MF using
QPSK symbols.We showed that the best receiver, the minimum mean square error
(MMSE) lags behind by about 2 dB with respect to the optimal receiver, followed by the
zero forcing (ZF) with a gap of about 4 dB and the matched filter (MF) which could not
keep up with the others,saturating at 16 dB of signal to noise ratio (SNR).

30
VII. REFERENCE
[1] Leonardo S. Cardoso, Francisco R. P. Cavalcanti, Mari Kobayashi and Mérouane
Debbah “Vandermonde-Subspace Frequency Division Multiplexing Receiver Analysis”,
in Alcatel-Lucent Chair - SUPÉLEC, Gif-sur-Yvette, France;Telecommunications Dept. -
SUPÉLEC, Gif-sur-Yvette, France;GTEL-DETI-UFC, Fortaleza, Brazil
[2] J. Mitola, “Cognitive radio an integrated agent architecture for software defined
radio,” Ph.D. dissertation, Royal Institute of Technology (KTH),May 2000.
[3].Natasha Detroyes,Patrick Mitran,and Vahid Tarokh, “Limitation on Communications
a cognitive Radio Channel ”;Harvard University,IEEE Communication magazine .
[4] Marco MASO” Flexible Cognitive Small-cells for Next Generation Two-tiered
Networks (Réseaux Small-cell Cognitifs pour la Prochaine Génération de Réseaux de
Transmissions Sans Fils)”, École Doctorale<<Sciences et Technologies de l’Information
des Télécommunications et des Systèmes>>, N◦d’ordre : 2013-07-TH,
SUPÉLEC,France.
[5]S.K.PADMANABHN,DR.T.JAYACHANDRA PRASAD”DESIGN AND
PERFORMANCE ANALYSIS OF PRECODED OFDM TRANSCEIVERS FOR
COGNITIVE RADIO”., Research scholar,Sathyabama
university,Channai,india.;Principal,RajeevGandhi Memorial college of Engineering and
Technology,Andra Pradesh,India.
[6] XU XiaoRong, ZHANG JianWu& ZHENG BaoYu” Achievable bit rates of cognitive
user with
Vandermonde precoder in cognitive radio network”, College of Telecommunication
Engineering, Hangzhou Dianzi University,Hangzhou 310018, China;Institute of Signal
Processing and Transmission, Nanjing University of Posts and
Telecommunications,Nanjing 210003, China, Received May 21, 2010; accepted July 1,
2010.
[7] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the gaussian
multiple input multiple-output broadcast channel,” Information Theory, IEEE
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[8] Marco Maso, Student Member, IEEE, Leonardo S. Cardoso, Member, IEEE,
Mérouane Debbah, Senior Member, IEEE, and Lorenzo Vangelista, Senior Member,
IEEE” Cognitive Orthogonal Precoder for Two-tiered Networks Deployment”

31
[9] Leonardo S. Cardoso” Orthogonal Precoder for Dynamic Spectrum Access in
Wireless Networks”,Doctoral Thesis Presentation Alcatel-Lucent Chair in Flexible Radio
Supélec — UFC-DETI
[10] Leonardo S. Cardoso” Vandermonde Frequency Division Multiplexing for Cognitive
Radio Networks”, Alcatel-Lucent - Supélec Chair in Flexibe Radio
[11] Leonardo S. Cardoso” Orthogonal Precoder for the Coexistence of Small Cell and
Cellular Network “,Alcatel-Lucent Chair on Flexible Radio Supélec”
[12] Leonardo S. CARDOSO” Précodeur Orthogonal pour l’Accès Dynamique au
Spectre dans les Réseaux Sans Fils (Orthogonal Precoder for Dynamic Spectrum Access
in Wireless Networks)”,PhD Thesis.
[13] C. Meyer, Matrix analysis and applied linear algebra.Society for Industrial
Mathematics, 2000.
[14] M. Kobayashi, M. Debbah, and S. Shamai, “Secured communication over frequency
selective fading channels: A practical vandermonde precoding,” EURASIP Journal on
Wireless Communications and Networking, vol. 2009, p. 2, 2009.
[15] W. Rhee and J. Cioffi, “Increase in capacity of multiuser ofdm system using
dynamic subchannel allocation,” in Vehicular Technology Conference Proceedings,
2000. VTC 2000-Spring Tokyo. 2000 IEEE 51st, vol. 2, 2000, pp. 1085 –1089 vol.2
[16]D. Tse and P. Viswanath, Fundamentals of wireless communication. Cambridge
University Press, 2005.
[17]. L. Rabiner and B. Gold, Theory and application of digital signal processing.
Prentice Hall, 1975.
[18]. S. Verdu, Multiuser detection.Cambridge University Press, 1998.

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