OFDM Channel Estimation Using LS and Low
Complex MMSE Channel Estimators
Yogita Pradeep Lad, Vaishnavi Sunil Navale
[email protected], [email protected]
AISSMS College of Engineering, Pune
Abstract: Main aim of this paper is to implement low
complex MMSE channel estimation technique. In this
paper implementation of OFDM channel estimation
using pilot based block type channel estimation
techniques i.e. LS, MMSE, MMSE using SVD and
Proposed MMSE with low complexity discussed and
simulated. Implementation of these techniques in
MATLAB along with comparison of performance with
respect to MSE and BER is given. At the end, results of
different simulations are compared with each other
with the conclusion that, LS algorithm is less complex
but MMSE provides better results.
I.
INTRODUCTION
High data rate transmission capability and high
bandwidth efficiency are the main features, because
of
which
Orthogonal
Frequency
Division
Multiplexing is mostly used digital communication
technique in wireless applications. There are two
types of channel estimation techniques, first is the
pilot assisted estimation and second is the blind
channel estimation. The blind channel estimation
techniques are not using pilots. There are two main
difficulties in designing pilot assisted estimators, first
is arrangement of pilot information and second is
designing of an estimator with not only low
complexity, but also good tracking ability[1].
Depending upon the manner of pilot information
insertion, there are two types of channel estimation,
block type channel estimation and comb type channel
estimation. Least Square(LS), Minimum Mean
Square Error(MMSE) are the block type techniques
of pilot assisted estimations and LS estimator with
1D interpolation, ML estimator, Parametric Channel
Modeling Based(PCMB) are the comb type channel
estimators. In this paper different block type channel
estimators i. e. LS, MMSE, MMSE using SVD and
proposed method of MMSE with low complexity are
implemented. In Section II, the OFDM system based
on pilot channel estimation and channel estimation is
described. In Section III, proposed method of low
complex MMSE estimation is discussed. In Section
IV, comparative simulation results given and in last
Section V concluding remarks are mentioned.
II.
SYSTEM DESCRYPTION
The basic idea underlying OFDM systems is the
division of the available frequency spectrum into
several subcarriers. To obtain a high spectral
efficiency, the frequency responses of the subcarriers
are overlapping and orthogonal, hence the name
OFDM. This orthogonality can be completely
maintained with a small price in a loss in SNR, even
though the signal passes through a time dispersive
fading channel, by introducing a cyclic prefix (CP).
A block diagram of a baseband OFDM system is
shown in Figure 2.1
Figure 2.1: Implementation of a Baseband OFDM System.
First randomly generated binary data is coded and
modulated using signal mapper. Then guard band is
added, and data is converted in parallel form using
serial to parallel converter. Parallel stream of data is
converted in time domain using an N-point inverse
discrete-time Fourier transform (IDFTN) block. To
avoid intersymbol interference and intercarrier
interference, cyclic prefix of length Tg is added. It is
followed by converter, containing low-pass filters
with bandwidth 1/TS, where TS is the sampling
interval. Then analog signal is travelling through
channel, having impulse response g(t), affected by,
the complex additive white Gaussian noise (AWGN)
n(t).
At the receiving end, data is converted into digital
form, then cyclic prefix is removed and parallel data
obtained from serial to parallel converter is
transformed into frequency domain using DFT block.
Finally, using signal demapper, binary information
data is obtained [1].
III.
CHANNEL ESIMATION
Where,
m = number of taps
N = number of subcarriers
𝛾𝑚 = value of the tap
By eliminating inter symbol interference using the
cyclic prefix system modeling can be written as
yk = HkXk + wk, k = 0 … … N-1
(3)
where Hk is the frequency response of h , given by,
As discussed above, there are two types of channel
estimation i. e. block type and comb type. Block
diagram for block type channel estimation is as
shown below.
H = [H0 H1 … HN-1]
Similarly,
W = [w0 w1 … wN-1]
We can write equation (3) in matrix form as below,
y = XFh + w
(4)
where,
X = diag{xo x1 … xN-1}
Figure 3.1: Channel Estimation using LS/MMSE Estimators
y = [y0 y1 …… yN-1]T
In block-type pilot based channel estimation,
each subcarrier in an OFDM symbol is used in
such a way that all sub-carriers are used as
pilots. The estimation of the channel is then
done using Least Square Estimator and
Minimum Mean Square Error Estimator. [5],[6].
The system shown in Fig. 3.1 is modeled using
the following equation:
y = DFTN (IDFTN(X)ʘ
ℎ
√𝑁
+𝑤
̃)
w = [w0 w1 …… wN-1]T
h = [h0 h1 …… hN-1]T
𝑊𝑁00
F=[
⋮
(𝑁−1)0
𝑊𝑁
(1)
where,
𝑊𝑁𝑛𝑘 =
𝑤
̃ =[𝑤
̃0 𝑤
̃ 1 …… 𝑤
̃ N-1 ]T
h = [h0 h1 …… hN-1]T
The vector
is the observed channel impulse
√𝑁
response when the frequency of g(t) is sampled and is
given by,
√𝑁
∑𝑚 𝑒
]
𝑛𝑘
1
√𝑁
𝑒 −𝑗2𝜋 𝑁
̂ LS = X-1 y
𝐻
−1
̂
𝐻 MMSE = FRhy𝑅𝑦𝑦
y
ℎ
Hk =
⋮
(𝑁−1)(𝑁−1)
𝑊𝑁
If the channel vector h I Gaussian and is not
correlated with the noise of the channel, then the
frequency domain LS and MMSE estimation [7] is
given by,
y = [y0 y1 …… yN-1]T
𝜋
−𝑗 (𝑘+(𝑁−1)𝛾𝑚)
𝑁
0(𝑁−1)
𝑊𝑁
F is the matrix of DFT with corresponding weights
given by,
x = [ x0 x1 …… xN-1]T
1
⋯
⋱
⋯
Where,
Rhy = E (hyH) = RhyFHXH
sin(𝜋)𝛾𝑚)
𝜋
𝑁
sin( ( 𝛾𝑚−𝑘))
(2)
Ryy = E (yyH ) = XFRhhFHXH +𝜎𝑛2 I
(5)
(6)
Where,
Rhy is the cross correlation matrix between h and y,
Ryy is the autocorrelation matrix of y,
Eq.(@) becomes,
̂ = RH (RH + 𝜎𝑛2 [(PΛx)-1]H ּ (PΛx)-1)-1 𝐻
̂ LS
𝐻′
̂ LS
=RH𝑅𝑌−1 𝐻
(11)
Using SVD algorithm, RY can be described by,
Rhh is the autocorrelation matrix of h and
RY = RH + 𝜎𝑛2 [(PΛx)-1]H ּ (PΛx)-1
𝜎𝑛2 = is the noise variance[8]
IV.
= RH + 𝜎𝑛2 UUH
PROPOSED METHOD OF MMSE
ESTIMATION
It is well known that LS is the simple and easy
estimator and its implementation is also less
complex, comparing with other estimators; but its
drawback is having high MSE. Mean square error of
LS estimator is given by,
̂ LS = X-1 y
𝐻
(7)
̂ LS
= U (Λ - 𝜎𝑛2 I) UH (UΛUH)-1 𝐻
𝚲
̂ LS
) UH 𝐻
(13)
Where the diagonal matrix (
as diag (
Mean square error of MMSE Estimator in terms of
LS estimator is given by,
̂ MMSE = RH(RH + 𝜎𝑛2 (XXH) -1) -1 𝐻
̂ LS
𝐻
(8)
The main drawback of MMSE estimator is its high
computational
complexity,
which
increases
exponentially with the observation samples. As data
in X changes every time the matrix inversion is
required. Edfors and Sandell have proposed to use an
algorithm called singular value decomposition (SVD)
to further simplify the estimator by replacing the
term (XXH)-1 in Eq.(8) with its expectation (XXH)-1
and exclusion of base vectors corresponding to the
smallest singular values [7]. Although this low-rank
approximation can reduce the computational
complexity, it will inevitably cause attenuation of
performance. In order to reduce the complexity of
MMSE channel estimation with little or no
attenuation, we should find new method to replace
(XXH)-1. According to the theory of diagonal matrix
in [5]-[8], we have
X = PΛxP-1
(9)
where Λx is a diagonal matrix and P is a Hermitian
matrix.
Then (XXH)-1 can be expressed as
-1
̂ ' = (RY - 𝜎𝑛2 UUH 𝑅𝑌−1 𝐻
̂LS
𝐻
2I
𝚲−𝜎𝑛
y0
y= ⋮
yN-1
(12)
Where U = [(PΛx)-1]H is a unitary matrix and Λ=
diag(λ1, λ2, ….., λi, ……, λk ). Then eq.() can be
rewritten as,
= U(
Where, X = diag {x0, x1, ….., xN-1} and
H -1
= U Λ UH
−1 H −1
(XX ) = (PΛxP . (PΛxP ) )
= [(PΛx)−1]H . (PΛx)−1
(10)
2
λi−𝜎𝑛
λi
2I
𝚲−𝜎𝑛
𝚲
) can be rewritten
).
Thus the calculation of finding the inverse matrix
̂ LS can be replaced in the
RH(RH + 𝜎𝑛2 (XXH) -1) -1 𝐻
form of SVD, through inversing the diagonal matrix
𝚲−𝜎 2 I
( 𝑛 )
to reach the purpose of reducing
𝚲
computational burden. But unlike [4], eq.(13) do not
use E (XXH)-1 approximation and low rank
approximation, therefore, it shows little attenuation
of performance, which is consistent with the
simulation results.
V.
SIMULATIONS AND RESULTS
In this section simulation results are discussed.
For simulation different
shown in below table.
parameters
used
are
Table 1: SIMULATION PARAMETERS
Parameters
Modulation
Demodulation
Channel Model
Noise Model
Sub-carrier Number
FFT & IFFT Point (N)
Symbol Length
&
Specification
16- QAM
AWGN and Rayleigh
independent
and
identically distributed
AWGN
64
64
64Microseconds
Figure 4.1 shows comparison of BER of LS, MMSE,
MMSE using SVD and Proposed method for AWG
channel and Fig 4.2 shows comparison of BER of LS
MMSE, MMSE using SVD and Proposed method for
Rayleigh channel.
Figure 4.3: Graph of MSE Vs SNR for AWG Channel
Figure: 4.1 Graph of BER Vs SNR for AWG Channel
Observing graph of BER Vs SNR, one come to know
that, Proposed method of MMSE estimation is giving
better result comparing with other methods, for
MMSE result is good but there is problem of
comutational complexity.
Figure 4.4: Graph of MSE Vs SNR for Rayleigh Channel
Observing graph of MSE Vs SNR one can conclude
that, MSE is more for LS and less for MMSE
estimation but complications are more. In Proposed
method, MSE is less than LS but, more than MMSE
with reduction in computational complexity.
VI.
Figure: 4.2 Graph of BER Vs SNR for Rayleigh Channel
Figure 4.3 shows comparison of MSE of LS, MMSE,
MMSE using SVD and Proposed method for AWG
channel and Fig 4.4 shows comparison of MSE of LS
MMSE, MMSE using SVD and Proposed method for
Rayleigh channel.
CONCLUSION
This paper is an implementation of low complex
MMSE channel estimator, its performance is
numerically confirmed for the OFDM system
proposed in the IEEE 802.16 standard. From the
simulation results one can conclude that, as
compared with conventional MMSE, proposed
scheme can reduce the computational complexity
but suffers with little attenuation of
performances. So, it is useful for practical
application in OFDM-based communication
systems.
REFERENCES:
[1] Tian-Ming Ma, Yu-Song Shi, and Ying-Guan Wang , “A Low Complexity MMSE
for OFDM Systems over Frequency-Selective Fading Channels ,” IEEE
COMMUNICATIONS LETTERS, VOL. 16, NO. 3, MARCH 2012
[2] Sajjad Ahmed Ghauri, et al , “Implementation OF OFDM and Channel Estimation
Using LS and MMSE Estimators”, International Journal of Computer and
Electronics Research VOL 2, Issue 1, February 2013.
[3] R. Prasad, OFDM for Wireless Communications Systems. Artech House, 2004.
[4] M. Morelli and U. Mengali, “A comparison of pilot-aided channel estimation
methods for OFDM systems,” IEEE Trans. Signal Process., vol. 49, no. 12, pp.
3065–3073, Dec. 2001.
[5] K. Todros and J. Tabrikian, “Fast approximate joint diagonalization of positive
definite Hermitian matrices,” Acoustics, Speech and Signal Process., vol. 3, pp.
1373–1376, Apr. 2007.
[6] S. Baig and Fazal-ur-Rehman, “Frequency domain channel equalization using
circulant channel matrix diagonalization,” in Proc. 2005 Interna- tional Multitopic
Conference, pp. 1–5. December 2005.
[7] O. Edfors, M. Sandell, and J.-J. van de Beek, “OFDM channel estimation by singual
value decomposition,” IEEE Trans. Commun., vol. 46, no. 7, pp. 931–939, July
1998.
[8] S. Baig and Fazal-ur-Rehman, “Frequency domain channel equalization using
circulant channel matrix diagonalization,” in Proc. 2005 Interna- tional Multitopic
Conference, pp. 1–5. December 2005.