1
Blind SNR estimation of OFDM signals
Yunxin Li
NICTA, Locked Bag 9013, Alexandria, NSW 1435, Australia
Abstract—A low-complexity blind algorithm is presented for
estimating the Signal-to-Noise ratio (SNR) of an Orthogonal
Frequency Division Multiplexing (OFDM) signal transmitted
over a frequency-selective channel with colored or white noise.
The algorithm only uses the second moment statistics in the
frequency domain without requiring training symbols or pilot
subcarriers.
I. INTRODUCTION
O
FDM modulation scheme has been adopted in many
broadband systems. Compared to the conventional Single
Carrier (SC) modulation schemes, OFDM enables the efficient
use of the available channel bandwidth and the easy control of
the signal spectrum mask in the Transmitter (TX). At the
Receiver (RX) side, OFDM allows simple equalization and is
robust to constant timing offset due to the adoption of Cyclic
Prefix (CP). To optimize the system performance, the RX is
often required to estimate the SNR, e.g. for the purpose of
clear channel assessment, soft-decision channel decoding, TX
power control, adaptive coding and modulation, bit loading
and hand-off.
SNR estimation has been well investigated, especially in the
context of SC modulation[1-4]. Some of the SC SNR
estimation schemes can be directly adapted to OFDM
modulation[5]. OFDM SNR estimation methods can be
roughly classified into two categories: data-aided (DA), and
non-data-aided (NDA) also called blind schemes. DA schemes
require that some known digits are transmitted by the TX. For
example, the known digits can be put in some known pilot
subcarriers in the Payload field of Fig. 1, while the rest of the
subcarriers are used for data transmission. Alternatively, a
single or a sequence of training OFDM symbol(s) can be used
in the Preamble and the Channel Estimation (CE) fields in Fig.
1. OFDM SNR estimation schemes can further be divided into
time-domain (TD) and frequency-domain (FD) processing
algorithms. As a result, all OFDM SNR algorithms belong to
one or a hybrid of the following four types: DA-TD[6-8], DAFD[8-19], NDA-TD[20-22] and NDA-FD[23].
To the best of the author’s knowledge, [20-23] are the only
blind OFDM SNR estimation algorithms proposed so far, that
can be applied to the Payload field without the requirement of
known pilot subcarriers. The TD signal during the CP interval
and that towards the end of the OFDM symbol are highly
correlated. In [20,21], this correlation was used for SNR
estimation. The disadvantage of this approach is that the
number of useable samples in one OFDM symbol is very small
due to the fact that most of the CP interval is interfered by the
previous OFDM symbol. Therefore a large number of OFDM
symbols are needed to achieve accurate estimation. By
assuming that the signal and noise covariance matrices are
different and known, a Maximum-Likelihood (ML) method
with high computational complexity was proposed in [22].
These blind approaches are also limited to packet SNR
estimation and it is hard to adapt to subcarrier or sub-band
SNR estimation. The expectation maximization (EM)
algorithm was used in [23]. However, the algorithm assumed
the knowledge of the channel and is therefore dependent on
channel estimation accuracy. In [14,15], a special preamble
OFDM symbol with repeated signal segments in the TD (or
equivalently with periodic virtual subcarriers in the FD) is
used for SNR estimation. The limitation is that, for a
frequency-selective and/or fading channel, the SNR estimated
in the preamble may not accurately represent the SNR in the
CE and Payload fields. The contribution of this letter is
therefore to develop a generic low-complexity blind SNR
estimation algorithm that can be used for all OFDM signals.
The rest of the letter is organized as follows. In Section II,
we present the system model and other related work, which
serve as a short literature review and a motivation for this
letter. In Section III, we describe a low-complexity blind
OFDM SNR estimation algorithm and its performance is
analyzed. Practical implementation considerations are
presented in Section IV, and finally summary and conclusions
are drawn in Section V.
II. THE PROBLEM AND RELATED WORK
A. System Model
A generic packet format, consisting of the fields of
Preamble, Channel Estimation (CE) and Payload, is shown in
Fig. 1. The purpose of the preamble is to enable the RX to
reliably detect the arrival timing of a packet, estimate and
correct the carrier frequency offset. The CE field allows the
channel to be estimated so that frequency-domain equalization
can be easily performed to decode the Payload field. The
problem addressed in this paper is on how to estimate the SNR
in the Payload field. Although our algorithm does not require
the existence of the Preamble and CE fields, this generic
packet format reflects many practical OFDM implementations
2
and helps the understanding of other related work in OFDM
SNR estimation.
Y ( j, k ) .
Hˆ ( j , k ) =
X ( j, k )
From (3) and (4), it is apparent that simple LS channel
estimate cannot be used for SNR estimation as it leads to
ψˆ = ∞ . In [9], the channel is assumed to be independent of
Fig. 1. The packet format
the
In FD, the received OFDM signal Y ( j , k ) at time j and
subcarrier k can be represented by
Y ( j, k ) = X ( j, k ) H ( j, k ) + N ( j, k ) ,
(1)
X ( j, k ) is the transmitted value, H ( j , k ) is the
channel and N ( j , k ) is the noise. The instantaneous SNR
ψ within time interval J and the set of subcarriers K is
where
defined as the random variable
∑ ∑ | X ( j, k ) H ( j, k ) |
2
j∈ J k ∈ K
∑ ∑ | N ( j, k ) |
2
.
j∈ J k ∈ K
∑ ∑ | X ( j, k ) H ( j, k ) |
=
subcarriers
and
a
single
channel
estimate
Hˆ ( j , k ) = Hˆ ( j ) is used in (3) for SNR estimation. The
B. SNR definition
ψ =
(4)
(2)
2
j∈ J k ∈ K
∑ ∑ | Y ( j, k ) − X ( j, k ) H ( j, k ) |
2
j∈ J k ∈ K
In this letter, the focus is on the estimation of the true ‘local’
SNR as defined above and ψ is thus a random variable, while
some of the literature has been on the ‘global’ expected value
of ψ , which is a constant in a static environment. For a time
and frequency selective channel, the ‘local’ SNR is a more
useful parameter.
C. Related Work
X ( j, k ) and the channel
If both the transmitted value
approach is thus not applicable to frequency selective
channels. An alternative is to use the linear minimum mean
square error (LMMSE) channel estimate for SNR estimation.
Interestingly, to estimate the channel by the LMMSE method,
the knowledge of SNR is required. To avoid this deadlock, a
nominal value of SNR is used in the channel estimation and
then the estimated channel is used for SNR estimation[10].
The SNR is highly correlated between adjacent subcarriers and
adjacent OFDM symbols. Therefore, the SNR estimates can be
further improved by 2D filtering across time and
frequency[11]. Although many of the previously proposed
algorithms can be used in estimating the SNR in the Payload
field, they suffer from one or more of the following
disadvantages[8].
1. The non-blind SNR estimation accuracy depends on the
accuracy of channel estimate and/or on the probability
of error in data symbol estimate. Usually this
dependence leads to under estimation of the noise and
therefore over estimation of the SNR. Sometimes this
dependence prevents the use of the simple LS channel
estimate.
2. There are no blind approaches that can be easily
adapted to estimate the SNR in different time and
frequency scopes (e.g. SNR per packet, per OFDM
symbol, per sub-band or per subcarrier).
3. Some blind algorithms need a long observation time
interval to converge and others need a high
computational power.
H ( j , k ) were known by the RX, the SNR can be estimated
In the following section, we describe a blind algorithm that
overcomes the above disadvantages.
by its definition (2). During the Preamble and the CE interval,
X ( j, k ) is known. Therefore, the Preamble and/or the CE
III. A BLIND SNR ESTIMATION ALGORITHM
Almost all practical OFDM systems define a set of
subcarriers Q , on which no signal is transmitted. Q is
fields can be used for SNR estimation as follows.
∑ ∑ | X ( j , k ) Hˆ ( j , k ) |
ψˆ =
2
,
j∈ J k ∈ K
∑ ∑ | Y ( j , k ) − X ( j , k ) Hˆ ( j , k ) |
(3)
2
j∈ J k ∈ K
where Hˆ ( j , k ) is the estimated channel and ψˆ is the
estimated SNR. The accuracy of SNR estimation is therefore
dependent on the accuracy of channel estimation. The simple
least square (LS) estimate of the channel can be represented by
normally located at the two ends of the channel bandwidth.
The existence of Q allows a practical filter to be used to
create a guard band between adjacent channels to avoid
adjacent channel interference. At the RX, the FD samples for
the subcarriers Q are primarily due to the noise N ( j , q ) , i.e.
Y ( j, q ) = N ( j, q )
q∈Q .
(5)
Please note that carrier frequency offset and other
nonlinearity in the analogue front end of the transceiver may
3
cause inter-carrier interference and spectrum leakage that can
potentially invalidate (5). However, it is expected that the
concerned impairments have been compensated adequately in
the stage of payload processing so that (5) is approximately
correct. It is worth noting that the reliable detection of the
payload data also depends on the condition (5). Therefore (5)
is not an extra condition in payload processing. By assuming
| N ( j , q ) | 2 > 0 , we define the noise-to-noise ratio
∑∑
j ∈ J q ∈Q
(NNR) η as follows.
∑ ∑ | N ( j, q) |
η=
2
.
j ∈ J q ∈Q
∑ ∑ | N ( j, k ) |
(6)
2
From (8) and (10), it is apparent that the mean E {δ } = 0 and
the mean E {ψˆ } = ψ . Therefore the SNR estimator (9) is
unbiased.
The estimation error δ in (10) is a linear combination of the
unknown noise samples. Therefore, if the noise is Gaussian
distributed, δ will also be Gaussian distributed. In fact,
irrespective of the noise distribution, the estimation error δ is
a Gaussian distributed random variable as long as || J |||| K || is
reasonably large. This is the direct implication of the central
limit theory. We will show that || J |||| K || needs to be
reasonably large to achieve the required accuracy. From (10),
the mean square error (MSE) of the SNR estimate, denoted by
mse (ψˆ ) , can be easily derived:
j∈ J k ∈ K
If the noise is white, e.g. the classical example of additive
white Gaussian noise (AWGN), the expected value of NNR
becomes a known constant:
η =|| Q || / || K || ,
(7)
where || o || represents the number of subcarriers in the
respective set. From (2) and (6), we can easily derive the
following SNR estimate ψˆ .
mse (ψˆ ) = E (δ 2 ) =
2ψ
.
|| J |||| K ||
(13)
The MSE of SNR estimate is proportional to the SNR
ψ itself and inversely proportional to the total number of
samples in the concerned space, i.e. || J |||| K || . As long as the
SNR ψ is finite, the MSE of the estimate can be made as small
as desired by increasing || J |||| K || , i.e.
lim || J |||| K || → ∞ mse (ψˆ ) = 0 .
ψˆ = ψ + δ .
(8)
∑ ∑ | Y ( j, k ) |
ψˆ =
∑ ∑ | Y ( j, q) |
From an engineering point of view, it is more convenient to
define the normalized MSE (NMSE) in decibels, i.e.
2
j∈ J k ∈ K
2
η −1 .
(9)
nmse (ψˆ ) dB = 10 log 10
j ∈ J q ∈Q


2 Re  ∑ ∑ X ( j , k ) H ( j , k ) N * ( j , k ) 
 j∈ J k ∈ K
.
δ =
∑ ∑ | N ( j, k ) |2
(10)
The superscript (o ) * in (10) represents complex conjugate.
Please note that the estimator (9) is independent of the
characteristics of the signal X ( j , k ) and the channel H ( j , k ) .
Therefore the proposed SNR estimator can be equally applied
to any signal constellation. In contrast most of the previously
proposed SNR estimators assumed a constant-amplitude signal
constellation. The proposed estimator is also equally
applicable to frequency selective and fading channels. Assume
that the mean of the concerned noise samples are zero and that
at least one of the concerned noise samples is non-zero, i.e.
E {N ( j , k )} = 0 .
∑ ∑ | N ( j, k ) |
2
(11)
> 0.
mse (ψˆ )
ψ
2
.
(15)
= 10 log 10 2 − 10 log 10 (|| J |||| K ||) − ψ dB
j∈ J k ∈ K
j∈ J k ∈ K
(14)
(12)
ψ dB = 10 log 10 ψ .
(16)
As shown in Fig. 2, we can use (15) to accurately predict the
SNR estimate error for a target SNR once the parameter
|| J |||| K || has been chosen. We can also use (15) to
determine the required value of || J |||| K || to achieve the
target error at the target SNR. The latter application is shown
in Fig. 3. Please note that the MSE of the estimate is proportional to
the true SNR ψ , but the NMSE is inversely proportional to ψ .
If our objective is to minimize the MSE of the estimate, the
proposed algorithm works more favorably in the lower SNR
range. On the other hand, the same algorithm works more
favorably in the higher SNR range if the objective is to
minimize the NMSE. This relationship has not been well
recognized in the literature and according caution has to be
taken to determine the estimation accuracy and its relationship
with the true SNR.
Sometimes only the estimation of the noise variance (rather
than the SNR) is required. The noise variance can be easily
4
obtained through (5), (6) and (7).
Fig. 2. Determine the NMSE based on SNR and ||J|||K|| parameters
Fig. 3. Determine ||J|||K|| based on the required NMSE and SNR
IV. IMPLEMENTATION CONSIDERATIONS
In the development of the algorithm, we have mainly
focused on wideband noise, e.g. the thermal noise. The
algorithm requires the knowledge of the NNR defined in (6).
Knowing the noise power spectrum density (PSD) of the noise
is sufficient but not necessary for the accurate determination of
the expected NNR value. For white noise (not necessarily
Gaussian-distributed), the expected NNR has been shown in
(7). Another likely scenario is that the colored noise is the
result of white noise filtered by the RX filter whose frequency
response at subcarrier k is Fk . In this case, (7) can be
modified as
η = ( ∑ | Fq | 2 ) /( ∑ | Fk | 2 ) .
q ∈Q
(17)
k∈K
In dealing with colored noise with unknown PSD in some
scenarios, the NNR may have to be estimated by its definition
(6). This is best done by designing a protocol that switches off
the TX during the time interval in which the RX is estimating
the NNR. For example, the inter-frame space between packets
can be used. Another strategy is to switch off the TX at the end
of the preamble for a short time to allow the RX to estimate
the NNR and adjust its other RX parameters such AGC gains
and carrier frequencies.
The impact of wideband interference can be treated the
same as noise. However, the algorithm is more applicable to
scenarios that the interference has wider bandwidth than the
target signal so that the interference occurs at both data
subcarriers and the guard-band subcarriers. For example, a
Wi-Fi receiver can use this algorithm to measure the
interference from ultra-wideband radios.
The proposed algorithm does require the existence of virtual
subcarriers on which no signal is transmitted. Almost all
practical OFDM systems have guard-band virtual subcarriers
towards the ends of the channel bandwidth. If all subcarriers
have been used for data and pilot signals, the RX can adopt an
oversampling strategy to make use of the proposed algorithm.
After over-sampling, it is expected that the signal is
concentrated in a fraction of the spectrum while the noise falls
into the whole spectrum.
Since the algorithm is blind, it is a trivial matter to apply to
multiple-input multiple-output (MIMO) OFDM systems. The
algorithm can be used directly before or after the MIMO
equalizer. When used before the MIMO equalizer the SNR for
a particular receiver can be measured. The MIMO equalizer
resolves the MIMO channel into multiple independent
channels, and the SNR on each resolved channel can thus be
measured after equalization.
The algorithm can be used in SC systems. The received
signal is first over-sampled and then a sequence of samples is
converted to the FD by fast Fourier transform (FFT). In the
FD, it is expected that the signal is concentrated in a fraction
of the spectrum while the noise falls into the whole spectrum
as shown in Fig. 4. It is worth noting that no training signal is
required. In contrast, the schemes in [3,4] requires training
symbols for channel estimation in the TD and then the
estimated channel impulse response is converted to FD for
SNR estimation.
Signal PSD
Noise PSD
Frequency
Fig. 4. An example of signal and noise PSD after over-sampling
Finally the proposed algorithm mainly depends on the fact
that the noise bandwidth is wider than the signal bandwidth in
over-sampled received data. In some situations, a TD
equivalent might exist, i.e. the received data include periods
for which only noise is present and other periods that contain
combined signal and noise. In these favorable scenarios, the
proposed algorithm can be directly applied in the TD for SNR
estimation, avoiding the complexity of FFT.
V. CONCLUSIONS
A low-complexity blind algorithm has been presented for
estimating the SNR of an OFDM signal transmitted over a
5
frequency-selective channel with wide-band colored/white
noises/interferences. The algorithm takes advantage of the fact
that the noise bandwidth is wider than the signal bandwidth,
without requiring training symbols or pilot subcarriers. The
proposed algorithm is low in complexity as only the second
moment statistics in the frequency domain are used. We have
demonstrated that the algorithm is unbiased and developed the
relationship between the NMSE, the target SNR and the size of
the required sample space. The application of the proposed
algorithm to SC and MIMO systems has been illustrated.
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