SNR Estimation for Low Bit Rate OFDM Systems in AWGN Channel

Doukas Athanasios, Grigorios Kalivas

Department of Electrical and Computer Engineering, University of Patras
Campus of Rion, Achaia, 26500, Greece
{adoukas, kalivas}@ee.upatras.gr

Abstract applications such as digital video broadcasting (DVB)

[1] and broadband indoor wireless systems [2].
Orthogonal Frequency Division Multiplexing In order to exploit all these advantages and
(OFDM) is a promising approach to achieve higher maximize the performance of OFDM systems; using
data rates with sufficient performance. However a these high data rates, channel state information (CSI) is
measure of channel state is desired in order to decide necessary. Key role in the CSI has the Signal-to-Noise
upon transition to these rates. Signal-to-Noise Ratio Ratio (SNR). An on-line SNR estimator, operating in
(S1NR) is such a measure and an estimator operating low bit rates, would provide the knowledge to decide
in low transmission rates, such as BPSK or using the whether a transition to higher bit rates would be
preamble data would be proper. In this work two favorable or not.
online SNR estimators are developed for OFDM The SNR estimators to the best of the authors’
systems, operating on BPSK modulation or on the knowledge in most cases handle the issue considering
preamble data. It is shown that the convergence of the only single carrier schemes [3-5]. A Split Symbol
algorithms to the actual SNR value is achieved from Moments (SSM) SNR Estimator is presented in [3]. Its
about 0 dBs or even lower. The impact of the channel computational complexity (it requires the solution of a
estimation used on the SNR estimation accuracy is system of equations) limits its functionality in our
given and improvements, regarding the computational application. Another SNR of prohibiting computational
cost, on one of them are given. Finally a transmission cost is presented in [4]. The need for a continuous
procedure for an OFDM modem using these SNR calculation of an exponential function and its iterative
estimators is given. nature in order to get the SNR estimate, make it
inappropriate for our case. In [5], an SNR estimator for
QPSK modulated data is presented, which can not be
1. Introduction easily modified for BPSK modulation.
In the context of OFDM transmission, we modify
Fourth Generation wireless and mobile systems are and evaluate the performance of two SNR estimation
currently the focus of research and development. algorithms, one operating on BPSK modulation and
Broadband wireless systems based on orthogonal one on the preamble data, in Additive White Gaussian
frequency division multiplexing (OFDM) will allow Noise (AWGN) channel.
high-rate data communication. A major advantage of The first algorithm is the Squared Signal-to-Noise
OFDM systems is its ability to divide the input high- Variance [3], which operates on BPSK data. We
rate data stream into many low-rate streams [1] that are modify it properly for a multicarrier system such as
transmitted in parallel, thereby increasing the symbol OFDM and evaluate its performance. In addition the
duration and reducing the intersymbol interference connection among the SNR estimation accuracy and
over frequency-selective fading channels. This and the channel estimation method used is examined.
other features of equivalent importance have motivated In HiperLAN/2, like in the most of the transmission
the adoption of OFDM as a standard for several schemes, before the transmission of the useful data
packet there is the part of the preamble data which is
sent in order to achieve frequency and frame
1
synchronization, phase tracking, frequency offset
This work was co-funded by the European Union and the General
Secretariat of Research and Technology of Greece, under the
correction etc. An SNR estimator operating on
program EPAN PENED 03Eǻ829

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 by a data symbol Xi,n , where i represents the OFDM
symbol number and n, ranging from 0 to 63, represents
the subcarrier number. The modulated data Xi,n are
mapped on the subcarriers and passed through an
Inverse Fast Fourier Transform (IFFT). Through the
carrier mapping and the IFFT the data are assigned on
different frequencies. A Guard Interval (GI), which
consists of the last 16 parts of each OFDM symbol, is
inserted at the start of each OFDM symbol, forming
Figure 1. Baseband Model the OFDM Symbol. Scope of this insertion is to
preserve the orthogonality between the subcarriers and
cancel the intersymbol interference of the delayed
signals introduced in the channel beyond the GI.
Finally data Dži= [si,0,si,1,…, si,78, si,79] are serially
transmitted over the channel.
At the receiver side we get

Figure 2.The common part of all PHY bursts Ri [ri ,0 , ri ,1 ,, ri ,78 , ri , 79 ] data. Under the
preamble data would be useful in order to estimate the assumptions that a) the guard interval duration is
value of SNR and decide upon the best constellation longer than the channel maximum excess delay b) the
scheme to use in the following transmission. channel is quasi stationary (i.e. the channel does not
Thus the Minimum Mean Square Error algorithm change within one OFDM symbol duration) and c) the
[5], an algorithm operating on preamble data, is synchronization is perfect the i–th received OFDM
modified properly for OFDM transmission and its symbol at the n-th subcarrier, Yi,n can be represented
performance is evaluated in AWGN channel. In by
addition we show that it can operate using only the real
or the imaginary part of the received signal and Yi,n X i, n ˜ H i, n  n i,n (1)
evaluate its performance. This expands its functionality
while it results in significantly lower complexity. where ni,n is a white complex Gaussian noise with
The rest of this paper is organized as follows. In variance ı2, Xi,n is the i-th transmitted symbol at the n-
Section 2 the system and channel models used are th subcarrier, Hi,n is the channel frequency response
illustrated, along with the channel estimators used to given by
improve signal’s reception. In Section 3, the SNR
estimators optimized for OFDM transmission are nWl
L  j2 S
presented along with the MMSE functional extension H i,n ¦ h l (iTs ) ˜ e NT (2)
using real or imaginary data only. Section 4 provides l 1
the performance analysis through computer
simulations of the proposed SNR estimators. In where hl(iTs) denotes the channel l-th path gain during
Section 5 a transmission procedure for an OFDM the i-th OFDM symbol, Ts=Tu+į is the duration of a
modem using these SNR estimators is presented. whole OFDM symbol including the guard interval,
Finally concluding remarks are given in Section 6. 1/Tu is the OFDM subcarrier spacing ,į is the guard
interval length ,IJl are the different path time delays, L
is the number of paths, N are the subcarriers used and
2. OFDM Physical Layer T the sampling interval.

2.1. HIPERLAN/2 Baseband and Channel 2.2. Channel Estimators

Model
In the HIPERLAN/2 transceiver platform we
The HIPERLAN/2 OFDM baseband model is perform channel estimation using the preamble
shown in Fig. 1. The OFDM modem simulated into transmitted in the beginning of each physical (PHY)
this work fully complies with the HIPERLAN/2 layer [6]. The last part of the preamble contains 2
specifications [6]. identical OFDM symbols of 64 samples protected by a
At the transmitter side each subcarrier is modulated CP of 32 samples, as can be seen in Fig. 2. The two
identical OFDM symbols of the preamble are averaged

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 to enhance the performance of the estimator. The result Now (3) becomes
is then divided by the known transmitted symbol to 2
§1 N  ·
implement the Least Square Channel Estimator (LS- ¨ ¦ SI X ( k ) ¸
CE). Further performance enhancement is achieved by
SNR ©N k 1 ¹
using the Linear Minimum Mean Square Error N  N  2
Channel Estimator (LMMSE-CE) [7]. Channel 1 §1 ·
estimation improves signal reception. Thus it is used in ¦ X S (k )  ¨© N ¦
Nk1 I
2

k 1
X SI (k ) ¸
¹
the baseband receiver to obtain the data. These data are
(6).
then used in the SNR estimation methods.
3.2 MMSE SNR Estimator (MMSE-SNR)
3. SNR Estimators
The Minimum Mean Square Error estimation
3.1 Squared Signal-to-Noise Variance SNR (MMSE) [5] method uses a training sequence
Estimator. a={a1,a2,…,aL} of length L and is expressed
mathematically by the orthogonality between the
The Squared Signal-to-Noise Variance SNR (SNV- estimation error and the estimate as follows
SNR) Estimator [3] uses a vector of N received BPSK-
modulated data. It is a data-aided (DA) estimator that  
( y  f ˜ a )(f ˜ a ) H 0 (7)
uses an estimate of the transmitted data sequence from
receiver decisions (RX) and is denoted as RXDA. The 
SNR is estimated by where f is the estimated attenuation factor, which is
assumed to be constant, y={y1,y2,…,yL} is the received
2 signal and H stands for the conjugate transpose.
§1 N ·
¨ ¦ rk ¸ The estimated SNR is given by the formula
©Nk 1 ¹ (3)
SNR 2
1 N 2 §1 N 2
· C
¦ rk  ¨ ¦ rk ¸ SNR (8)
Nk 1 ©Nk 1 ¹ 2 2
a E C
where rk and N represent the received signal and the
vector’s size respectively. The length N of the where C=yaH and E is the received signal energy.
measured symbols characterizes the accuracy of the We modify MMSE-SNR for a multicarrier system
estimator. such as OFDM. The training sequence will be the
In the OFDM receiver, previously added GI and result of carrier mapping and IFFT on the common
pilot signals are removed. FFT is then applied in order part of the preamble defined as a={ai,n : i=1,2 and
to have the suitable data for application of the n=1,…80} and L=160(adding pilot data is not referred
 because in preamble data we do not add pilot
algorithm. The estimated received symbol X i, n now
subcarriers).

represents estimated data. X i, n can be associated to In the case of AWGN, ƨi,n has a constant value and
 can be represented by ƨ. (7) becomes
the two transmitted symbols X S by
I
 
(Y  H˜a )(H ˜ a )H 0 (9)
   i, n i, n i, n
X SI ^
X i, n : X i, n S I ` (4)
where Yi,n is the received signal, propagated through
where Î is the actual in phase value of the received the AWGN channel, and ai,n is the transmitted
symbol. preamble sequence.
Selecting the appropriate boundaries for Î, we The estimated SNR will be given by
associate I, the theoretical value of the constellation
2
points, with a group of values of Î. Thus C
SNR (10)
2 2
a E C
°­S Î corresponds to  1 for S Î t 0
® (5)
°̄S Î corresponds to  1 for S Î  0 where C=Yi,n·aH and E is the received signal energy.

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 3.3 Improved MMSE-SNR estimation scheme 4. Simulation Results
In what follows we show that (10) can be used with We use LS-CE and LMMSE-CE to improve signal
the real/imaginary part of the received signal only. reception. The resulting received samples are then used
Using only the real part, denoted by superscript R, the to obtain SNV-SNR and MMSE-SNR estimates.
estimation error and the estimate are mutually Since that the use of LMMSE-CE needs a relatively
orthogonal for MMSE so accurate SNR estimate to operate, it is not feasible to
use it while we try to estimate SNR. However we can
  H  H
(Yi,nR  H ˜ a R )(H ˜ a R )H 0 Ÿ Yi,nR ˜ a R H ˜ aR ˜ aR make this assumption temporarily in order to assess the
different SNR estimators.
(11)
4.1 SNV-SNR Estimator.
Taking into account that ĮR is real we have

H T
SNV-SNR estimator operates on BPSK
aR aR (12) modulation. Simulations were performed for Modes 1
and 2 of HiperLAN/2, which use BPSK modulation
where ȉ represents the transpose operation. Equation and differ only in the bit rate.
(11) gives then Satisfactory SNR estimation can be achieved using
1000 OFDM samples. Simulation results for SNR
T estimation are presented in Figs. 3 to 5, where
 Yi,nR ˜ a R C1
H (13) LMMSE-CE, LS-CE and no CE are used respectively.
T 2
aR ˜ aR a In the presented results the LMMSE-CE uses the
actual SNR, not the estimated SNR because channel
T estimation method precedes the SNR estimation
where C1 Yi,nR ˜ a R . method.
SNR will be given by From these figures we can see that the algorithm
converges to the actual SNR from 3 dB, in all cases. At
 2 values over 7 dB, the estimated SNR is smaller that the
H ˜aR
SNR (14). actual SNR. This should happen in small SNR values
 2
too, but it does not because of the bias of the
Yi,nR  H ˜ a R
algorithm.
For the same reason, there is no obvious advantage
The denominator is calculated as follows of using LMMSE-CE with respect to the LS-CE or
without CE. Taking under consideration the
  
R
Yi,n  H ˜ aR
2 R 2
Yi,n
2
 H ˜ a R  2 Re Yi,n
R
˜ (H ˜ a R)H ^ ` complexity of each CE and the SNR estimation
accuracy in every case, we can say that the best
2 2
C1 C1 combination of CE and SNR estimation is that of using
E 2
2 2
(15)
a a 1000 S A M P LE S , LM M S E E S TIM A TOR , A W GN CHA NNE L
20

R 2
where E Yi,n . 15

Taking into account (13) and (15), the SNR

S NR E stim ation (dB )

10
estimation from (14) becomes
5
2
C1 A ctual S NR

SNR (16). E stim ated S NR Mode 1

E stim ated S NR Mode 2
2 2
a E  C1 0

-5
The use of only the real or the imaginary part of the -5 0 5
S NR (dB )
10 15 20

received signal results in a reduction of the necessary

additions and multiplications by a factor of 5 and 6 Figure 3.Estimated SNR (SNV-SNR) vs. actual
respectively. SNR for LMMSE-CE

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 1000 S A M P LE S , LS E S TIM A TOR, A W GN CHA NNE L MMSE SNR estimator for preamble
20 20

15

15
10

5
S NR E stim ation (dB )

SNR estimation (dB)

10
0

-5
5
A ctual S NR -10
E stim ated S NR Mode 1
E stim ated S NR Mode 2
0
-15
Actual SNR
MMSE estimation for preamble
-20 MMSE estimation for preamble
with real/imaginary part only
-5 -25
-5 0 5 10 15 20 -15 -10 -5 0 5 10 15
S NR (dB ) SNR (dB)

Figure 4.Estimated SNR (SNV-SNR) vs. actual Figure 6.Estimated SNR (MMSE-SNR) vs.
SNR for LS-CE actual SNR for preamble

20
1000 SAMPLES, CE is not used, AWGN CHANNEL Table 2. Variance of MMSE Estimator using all
preamble data
15
SNR Variance SNR Variance
(db) (db) (db) (db)
SNR Estimation (dB)

10
-5 0.212 3 0.213
-3 0.213 5 0.212
5
Actual SNR -1 0.21 7 0.211
Estimated SNR Mode 1
Estimated SNR Mode 2 1 0.212 9 0.21
0

-5
are presented in Fig. 6 and Table 2 respectively.
-5 0 5
SNR (dB)
10 15 20
Analogous results are taken in the cases of using LS-
CE or LMMSE-CE for both SNR estimation and
Figure 5.Estimated SNR (SNV-SNR) vs. actual
variance of the SNR estimate.
SNR without CE
From Fig. 6 we can see that convergence of both
types of SNR estimation, using the entire preamble or
Table 1. Variance of SNV-SNR Estimator using
the real/imaginary part, happens at the same point, at
LS-CE
about –6 dB. From the same figure is obvious that in
the case of using complex qualities convergence to
SNR Variance SNR Variance
(db) (db) (db) (db) actual SNR is better (by 1.5dB at low SNRs) than
when using only real/imaginary data. Furthermore,
0 0.231 8 0.221
from Table 2 we note that the variance of the estimate
2 0.225 10 0.223
is low and from the convergence point of -5 dB, it
4 0.223 12 0.219 stabilizes its value.
6 0.223 14 0.219
5. Transmission Procedure
the LS-CE.
The variance of the SNR estimation using the LS- The OFDM modem at first transmits the preamble
CE is shown in Table 1. The variance of the rest of the data. The MMSE estimator will give a first SNR
cases is analogous. We can see that the variance of the estimate. This estimate will guide the transmitter to
SNR estimation is low in every SNR and lowers as decide for the proper modulation scheme that fits the
SNR estimation converges to the actual SNR. current transmission characteristics, under the rule that
the better the SNR the higher the modulation scheme
4.3 MMSE-SNR applied to the transmission [8].
At the next step the transmitter will switch to the
The results of the simulation without using any proper modulation scheme. There are two cases: to
kind of CE, along with the variance of the estimation,

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 switch to the proper constellation scheme than the
current.

6. Conclusions

We have evaluated the performance of two SNR

estimators in a HiperLAN/2 digital modem.
SNV-SNR estimator uses BPSK data and
converges to the actual SNR at about 3dBs. We must
also notice that channel estimation, usually an integral
part of OFDM modems, affects the SNR estimation at
low SNR values. In addition we must say that SNR
estimation is more robust when LMMSE-CE is used in
the receiver.
MMSE-SNR estimator is used on the preamble
data and converges at SNR equal to -6 dB. It can also
operate with only the real or imaginary part of the
received signal, with small performance degradation.
Yet the decrease in the calculations needed makes it a
good candidate for a real system implementation.
Finally an algorithm for an OFDM modem using
the SNR estimators evaluated is presented. The
algorithm uses the SNR estimate to decide upon the
proper constellation scheme, the transmitter has to use.
The criterion to decide upon the proper constellation
used is the IER efficiency of the system in every case.

Figure 7. Transmission procedure algorithm 7. References

based on SNR estimation.
[1] R. Nee and R. Prasad, OFDM for Wireless Multimedia
switch to a BPSK modulation or to switch to a higher Communication, Artech House, London, 2000.
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In the first case of the BPSK modulation, the Digital Communications: Theory And Applications Of
receiver can at any time estimate the SNR. In this way OFDM, Springer Science, New York, 2004.
the receiver is again capable of making a decision on [3] D.R.Pauluzzi, N.C.Beaulieu, “A comparison of SNR
whether can switch to a higher modulation or remain estimation techniques for the AWGN channel” IEEE Trans.
in the same modulation. on. Comm., vol.48, October 2000, pp. 1681-1691.
[4] Li Bin et al., “A Low Bias Algorithm to Estimate
On the other case, of switching to a higher
Negative SNRs in an AWGN Channel”, IEEE Comm.
modulation, the receiver does not have the ability to Letters, vol. 6, no.11, November. 2002, pp. 469 – 471.
decide upon the SNR estimate but can use the [5] Dong-Joon Shin, Wonjin Sung, In-Kung Kim, “Simple
Instantaneous Error Rate (IER), using the pilot data in SNR Estimation Methods for QPSK Modulated Short
the OFDM symbol, in order to decide whether to Bursts,” IEEE GlobeCom 2001, vol. 6, pp. 3644-3647,
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increases it can switch to BPSK modulation, estimate [7] J.-J van de Beek et al., “On channel estimation in OFDM
systems” IEEE 45th Vehicular Technology Conference
SNR and based on that to decide whether the SNR
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favors a higher or a lower modulation scheme than the [8] A. Doufexi, S. Armour, A. Nix, and D. Bull, “A
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secondly will give the ability to the transmitter to

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