On the Extended Relationships Among EVM, BER
and SNR as Performance Metrics
Rishad Ahmed Shafik
Islamic University of Technology
Gazipur 1704, Bangladesh
rishad09@gmail.com
Md. Shahriar Rahman
Islamic University of Technology
Gazipur 1704, Bangladesh
shahriar@iut-dhaka.edu
AHM Razibul Islam
Kyung-Hee University
South Korea
razib3002@gmail.com
Abstract—In this paper, we relate the error vector magnitude
(EVM) bit error rate (BER) and signal to noise ratio (SNR).
We also present the fact that with such relationship it would be
possible to predict or in cases substitute EVM in places of BER or
even SNR. In doing so, we first define EVM with normalization
so that the definition stands for multi-modulation systems, viz.
binary phas shift keying (BPSK), quadrature phase shift keying
(QPSK) etc. We also compare among the different performance
metrics and show that EVM can be equivalently useful as signal
to noise ratio and bit error rate. The relationships are based
on stream based communication systems. A few Monte Carlo
simulations are carried out to illustrate the performance of EVM
based on these relationships.
I. INTRODUCTION
Signal to noise ratio (SNR), error vector magnitude (EVM)
and bit error rate are common performance metrics for as-
sessing the quality of communication [7]. SNR is a direct
measure of the relative power of the noise compared to signal.
Since noise is source of all errors for simplistic Gaussian noise
channel model, SNR can be used to predict the performance
of the system in terms of correctness of the reception. The
later can be measured using the bit error rate (BER), which
gives a simple one-to-one binary decision as to whether a
bit is erroneous or not. Due to the simplicity of comparison,
BER has been a major choice to engineers, industries and
researchers[6], [5], [7]. BER performance against SNR is the
popular performance criterion that is used in today’s commu-
nications systems. On the other hand, EVM is a measure of
errors between the measured symbols and expected symbols.
The use of EVM as a performance metric is limited to radio
frequency engineering to infer reception the performance at
the receiver earlier than the BER. In this paper, we show the
relationship among BER, SNR and EVM extended from [7]
and [2]. However, the relationships are based on assumption
that the communication is stream based or the size of the
packet is such that the number of samples is much higher
than the number of distinct modulation symbols.
The relationships among these metrics are vital to allow
re-thinking on using these parameters towards making adap-
tive systems. As an example, for BER measurement, it is
incumbent that signal must be demodulated first at the receiver
side. For many of todays adaptive systems, viz. minimum bit
error rate (MBER) based adaptive modulation systems etc.,
this means that for every update in the adaptive algorithm,
it has to receive feedback from the receiver end[8]. Since
EVM can give the desired performance metric before the
demodulation can actually takes place, often BER calculations
can be avoided for large packets. This can be done without
any major change of algorithm in adaptive systems, since BER
is a direct consequence of EVM. But again, due to complex
mathematical operations that needs to be done in the digital
signal processor for calculation of EVM, for performance
measures for systems with small packet communication BER
versus SNR can be easier and more useful. In Section III
and IV, BER and EVM will be separately derived from SNR
and later EVM definition would be presented in terms of BER.
In VI, theoretical and empirical observations would be shown
through simulations showing the relation of EVM with BER.
Finally, in Section VII, we extend discussions on these results
and show future directions for such performance metric.
II. SIGNAL TO NOISE RATIO
Signal to noise ratio is a relative measure of the signal power
compared to the noise power. Assuming Gaussian noise model
for wireless channels and complex signals, SNR can be defined
as
SNR =
Signal P ower
Noise P ower
=
"
1
T
P
T
t=1
(I
t
)
2
+ (Q
t
)
2
1
T
P
T
t=1
|n
I,t
|
2
+ |n
Q,t
|
2
#
. (1)
Here I
t
and Q
t
are the in-phase and quadrature signal
amplitudes of the M-ary modulations, n
I,t
and n
Q,t
are the
in-phase and quadrature noise amplitudes of the complex noise
being considered. Equation 1 shows a direct measure of SNR,
which can be used in Monte Carlo simulation procedures for
large symbol streams, such that T >> N, where N is the
number of unique modulation symbols. Often these estimates
are simplified by considering the measure of the ratio of
variances of signal and the noise, when both are zero mean
processes. For systems, which are sampled at data rate,
E
s
N
0
gives the signal to noise ration directly, where E
s
is the symbol
energy and
N
0
2
gives the noise power spectral density. It is to
be noted that E
s
= log
2
MEb, for such systems.
4th International Conference on Electrical and Computer Engineering
ICECE 2006, 19-21 December 2006, Dhaka, Bangladesh
408
III. BIT ERROR RATE
Bit Error Rate (BER) is a commonly used performance
metric which describes the probability of error in terms of
number of erroneous bits per bit transmitted. BER is a direct
effect of channel noise for Gaussian noise channel models.
For fading channels, BER performance of any communication
system is worse and can be directly related to that of the
Gaussian noise channel performance[4]. Considering M-ary
modulation with coherent detection in Gaussian noise channel
and perfect recovery of the carrier frequency and phase, it can
be shown that[4]
P
b
=
2
1 −
1
L
log
2
L
Q
"
s
3log
2
L
L
2
− 1
2E
b
N
0
#
, (2)
where L is the number of levels in each dimension of the
M-ary modulation system, E
b
is the energy per bit and N
0
/2 is
the noise power spectral density. Q[.] is the Gaussian co-error
function and is given by[3]
Q(x) =
Z
∞
x
1
√
2π
e
−y
2
2
dy . (3)
Assuming raised cosine pulses with sampling at data rate,
Equation 2 also gives the bit error rate in terms of signal to
noise ratio as
P
b
=
2
1 −
1
L
log
2
L
Q
"
s
3log
2
L
L
2
− 1
2E
s
N
0
log
2
M
#
, (4)
where
E
s
N
0
is the signal to noise ratio for the M -ary modula-
tion system (as shown before) and raised cosine pulse shaping
at data rate. Equation 4 defines the BER performance in terms
on SNR.
IV. ERROR VECTOR MAGNITUDE
Error Vector Magnitude (EVM) measurements are often
performed on vector signal analyzers (VSAs), real-time an-
alyzers or other instruments that capture a time record and
internally perform a Fast Fourier Transform (FFT) to enable
frequency domain analysis. Signals are down converted before
EVM calculations are made[2]. With the insurgence of high
speed communication requirements, highly efficient multiplex-
ing systems like orthogonal frequency division multiplexing
(OFDM) as specified in [6] is becoming the hinge of fu-
ture communication systems. OFDM is now being used in
different carrier standards in Wireless Local Area Networks
(WLANs) and also being considered as a potential fourth
generation communication system. The IEEE802.11a-1999
specification describes a set of different schemes that are
used in adaptive fashion: binary phase shift keying (BPSK),
4 quadrature amplitude modulation (4-QAM), 16 quadrature
amplitude modulation (16-QAM), 64 quadrature amplitude
modulation (64-QAM) etc. The fact that pilots and training bits
are always BPSK modulated, it is also possible to have more
than one modulation scheme within a burst[7]. This requires
that these modulation schemes are normalized to facilitate the
calculation of EVM.
The normalization is derived such that the mean square
amplitude of all possible symbols in the constellation of any
modulation scheme is one. Thus, EVM is defined as the
root-mean-square (RMS) value of the difference between a
collection of measured symbols and ideal symbols. These
differences are averaged over a given, typically large number
of symbols and are often shown as a percent of the average
power per symbols of the constellation. As such EVM can be
mathematically given as[1]
EV M
RMS
=
1
N
P
N
n=1
|S
n
− S
0,n
|
2
1
N
P
N
n=1
|S
0,n
|
2
, (5)
where S
n
is the normalized nth symbol in the stream of
measured symbols, S
0,n
is the ideal normalized constellation
point of the nth symbol and N is the number of unique
symbols in the constellation. The expression in Equation 5
cannot be replaced by their unnormalized value since the
normalization constant for the measured constellation and the
ideal constellation are not the same. The normalization scaling
factor for ideal symbols is given by[7]
|A| =
s
1
P
v
T
=
r
T
P
v
, (6)
where P
v
is the total power of the measured constellation of
T symbols. For RMS voltage levels of in-phase and quadrature
components, V
I
and V
Q
and for T >> N, it can be shown
that P
v
can be expressed as
P
v
=
T
X
t=1
(V
I,t
)
2
+ (V
Q,t
)
2
(W ) . (7)
The normalization factor for ideal case can be directly
measured from N unique ideal constellation points and is
given by
|A
0
| =
s
N
P
N
n=1
(V
I
0,n
)
2
+ (V
Q
0,n
)
2
. (8)
Hence Equation 5 can be further extended using normaliza-
tion factors in Equations 6 and 8 as
EV M
RMS
=
"
1
T
P
T
t=1
|I
t
− I
0,t
|
2
+ |Q
t
− Q
0,t
|
2
1
N
P
N
n=1
[(I
0,n
)
2
+ (Q
0,n
)
2
]
#
1
2
, (9)
where I
t
= (V
I
t
)|A| is the normalized in-phase voltage for
measured symbols and I
0,t
= (V
I
0,t
)|A
0
| is the normalized
in-phase voltage for ideal symbols in the constellation, Q
t
=
(V
Q
t
) |A| is the normalized quadrature voltage for measured
symbols and Q
0,t
= (V
Q
0,t
) |A
0
| is the normalized quadrature
voltage for ideal symbols in the constellation. This is the
definition which is now being used as the standard definition
of the EVM in IEEE 802.11a − 1999
T M
[6], [5].
409
V. RELATIONSHIP AMONG EVM, BER AND SNR
From Equation 9, it is evident that EVM is essentially the
normalized error magnitude between the measured constella-
tion and the ideal constellation. For Gaussian noise model,
Equation 9 can be simplified in terms of noise in-phase
component, n
I,t
and quadrature component, n
Q,t
as
EV M
RMS
=
1
T
P
T
t=1
h
|n
I,t
|
2
+ |n
Q,t
|
2
i
P
0
1
2
, (10)
where P
0
is the power of the normalized ideal constellation
or the transmitted constellation. The numerator of Equation 10
sets up the normalized noise power. However, for T >> N,
the ratio of normalized noise power to the normalized power
of ideal constellation can be replaced by their unnormalized
quantities, i.e. the Equation 10 rewritten as
EV M
RMS
≈
1
SNR
1
2
=
N
0
E
s
1
2
. (11)
In order to establish relationship between BER and EVM, SNR
in Equation 11 can be expressed in terms of EVM as
SNR ≈
1
EV M
2
. (12)
Combining Equations 12 and 4, we can now relate the bit error
rate directly with the error vector magnitude as follows
P
b
≈
2
1 −
1
L
log
2
L
Q
"
s
3log
2
L
L
2
− 1
2
EV M
2
RMS
log
2
M
#
, (13)
VI. SIMULATIONS
Few simulations have been carried out to illustrate the
relation established in Equation 13. The bits are gray coded
and then M -ary modulated, where M = 2 for BPSK, M = 4
for 4-QAM, M = 16 for 16-QAM and M = 64 for 64-QAM.
Monte Carlo Simulation techniques are carried out using 10
6
packets, each with the size of 1024 bits. The channel model
used is Gaussian noise model. The normalization factors will
be equal as shown in Equation IV since 1024 >> M for any
M-ary modulation scheme. The simulation results for BPSK,
4-QAM, 16-QAM and 64-QAM are presented in figures 1, 2
and 3, respectively.
Figure 1 shows the BER versus SNR performance of
different modulation systems. Note that due to the relationship
set up between BER and EVM in Equation 13, the BER versus
EVM curve shown in Figure. 2 shows the inverse relationship
that exists between BER and EVM (with power term in log
scale shown).
In order to also establish the fact that the normalized EVM
is same for all modulation schemes, we have also carried out
another simulation as shown in Figure 3. Due to normalization,
the power levels for all different M-ary modulations are same
and hence shows a one-to-one relationship between them.
In Figure 2, we note that there is a constant 2.89dB
difference between BPSK and 4-QAM, whereas there is a
6.85dB and 6.5dB difference between 4-QAM and 16-QAM,
and 16-QAM and 64-QAM.
0 5 10 15 20 25 30
10
−4
10
−3
10
−2
10
−1
10
0
SNR in dB
Bit Error Rate
Bit Error Rate Vs Signal to Noise Ratio
BPSK
4−QAM
16−QAM
64−QAM
BPSK(Theoritical)
4−QAM(Theoritical)
16−QAM(Theoritical)
64−QAM(Theoritical)
Fig. 1. Stylized BER versus SNR Performance Curves
−30 −25 −20 −15 −10 −5 0
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Bit Error Rate Vs Error Vector Magnitude(rms)
Error Vector Magnitude(rms) in dB
Bit Error rate
BPSK
4−QAM
16−QAM
64−QAM
BPSK(Theoritical)
4−QAM(Theoritical)
16−QAM(Theoritical)
64−QAM(Theoritical)
Fig. 2. Stylized BER versus EVM Performance Curves
VII. CONCLUSIONS
Extended relationships among the bit error rate, signal to
noise ration and error vector magnitude are shown in this
paper. Due to normalization, the EVM is the same for a given
SNR, and they maintain an inverse relationships between them.
Since error vector magnitude can be directly measured from
the down converted signals using vector signal analyzers, it
can save the extra calculations that may be required to find out
the bit error rates, which is more of a end to end comparison.
In many adaptive systems, this can also simplify the cost
function calculation greatly. However, for large streams or
packets the EVM calculation may be expensive. Effect of
410
0 5 10 15 20 25 30 35 40
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
5
SNR in dB
Error Vector Magnitude(rms) in dB
Error Vector Magnitude(rms) Vs Signal to Noise Ratio
BPSK
4−QAM
16−QAM
64−QAM
Fig. 3. Stylized EVM versus BER Performance Curves
different fading environments, effect of using EVM-adaptive
M-ary modulation systems instead of BER-adaptive systems
are now being considered as an extension of the work.
ACKNOWLEDGMENT
The authors would like to thank Mr. Nabil Shovon Ashraf
for his generous assistance throughout the work.
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