Anyone who has spent more than ten minutes researching digital communications has run across
the cryptic notation Eb/No.
Usually this shows up when discussing bit error rates or modulation
methods. You may have a vague feeling that it represents something important about a digital
communication system, but can't really put a finger on what or why. So let's take a look at just
what this Eb/No thing is and
why it's important.
First of all, how do you pronounce Eb/No?
Most engineers that I know say "E bee over en zero," though some of the more fastidious
ones say "E sub bee over en sub zero". At any rate, even though "No" is
usually written with an "Oh" instead of a zero, it is not pronounced as the
word "no".
Eb/No is classically defined as
the ratio of Energy per Bit (Eb) to the Spectral Noise Density (No). If this definition leaves
you with a empty, glassy-eyed feeling, you're not alone. The definition does not give you any
insight into how to measure Eb/No
or what it's used for.
Eb/No is the measure of signal
to noise ratio for a digital communication system. It is measured
at the input to the receiver and is used as the basic measure of how strong the signal is.
Different forms of modulation -- BPSK, QPSK, QAM, etc. -- have different curves of theoretical bit
error rates versus Eb/No as shown
in Figure 1. These curves show the communications engineer
the best performance that can be achieved across a digital link with a given amount of RF power.
Figure 1. BER vs Eb/No
(Thanks, Intersil for this figure)
In this respect, it is the fundamental prediction tool for determining a digital link's
performance. Another, more easily measured predictor of performance is the carrier-to-noise
or C/N ratio.
So let's pretend that we are designing a digital link, and see how to use Eb/No and C/N to
find out how much transmitter power we will need. Our example will use differential quadrature
phase shift keying (DQPSK) and transmit 2 Mbps with a carrier frequency of 2450 MHz. It will
have a 30 dB fade margin and operate within a reasonable bit error rate (BER) at an outdoor distance
of 100 meters. Hold on to your hat here! Remember that when we play with dB or any log-type
operation, multiplication is replaced by adding the dBs, and division is replaced by subtracting
the dBs.
Our strategy for determining the transmit power is to:
- Determine Eb/No for our desired BER;
- Convert Eb/No to C/N at the receiver using the bit rate; and
- Add the path loss and fading margins.
We first decide what is the maximum BER that we can tolerate. For our example, we
choose 10-6 figuring that we can retransmit the few packets that will have errors at
this BER.
Looking at Figure 1, we find that for DQPSK modulation, a BER of 10-6 requires
an Eb/No of 11.1 dB.
OK, great. Now we convert Eb/No
to the carrier to noise ratio (C/N) using the equation:
Where:
fb is the bit rate, and
Bw is the receiver noise bandwidth. [EDITOR'S NOTE: See Phil Karn's comment below concerning
this equation.]
So for our example, C/N = 11.1 dB + 10log(2x106 / 1x106) = 11.1 dB + 3dB = 14.1dB.
Since we now have the carrier-to-noise ratio, we can determine the necessary received carrier
power after we calculate the receiver noise power.
Noise power is computed using Boltzmann's equation:
N = kTB
Where:
k is Boltzmann's constant = 1.380650x10-23 J/K;
T is the effective temperature in Kelvin, and
B is the receiver bandwidth.
Therefore, N1 = (1.380650x10-23 J/K) * (290K) *(1MHz) =
4x10-15W = 4x10-12mW = -114dBm
Our receiver has some inherent noise in the amplification and processing of the signal.
This is referred to as the receiver noise figure. For this example, our receiver has a 7 dB
noise figure, so the receiver noise level will be:
N = -107 dBm.
We can now find the carrier power as C = C/N * N, or in dB C = C/N + N.
C = 14.1 dB + -107dBm = -92.9 dBm
This is how much power the receiver must have at its input. To determine the transmitter
power, we must account for the path loss and any fading margin that we are building in to the
system.
The path loss in dB for an open air site is:
PL = 22 dB + 20log(d/λ)
Where:
PL is the path loss in dB;
d is the distance between the transmitter and receiver; and
λ is the wavelength of the RF carrier (= c/frequency)
This assumes antennas with no gain are being used. For our example,
PL = 22 dB + 20log(100/.122) = 22 + 20*2.91 =
22 + 58.27 = 80.27 dB
Finally, adding our 30 dB fading margin will give the required transmitter power:
P = -92.9 + 80.27 + 30 = 17.37 dBm = 55 mW
Our result, 55 mW, is well within a reasonable power level for spread spectrum links in the
2.4 GHz band. So we see that, in this example, our 100 meter range is a very reasonable
expectation.
So, what is all this Eb/No stuff?
Simply put, it's one of the "secrets" used by top RF
design engineers to evaluate options for digital RF links, and is a crucial step in the design of
systems that will meet performance expectations.
Comments from Phil Karn
From: Phil Karn
To: Jim Pearce
Sent: Monday, April 23, 2007 3:47 AM
Subject: Eb/No Explained
[Editor's Note: Phil is a Qualcomm engineer who is very well known in the radio community.
His website is at www.ka9q.net, and has a number of articles of interest to
electronics/wireless aficionados/practitioners.]
Hi Jim,
I found your article "What's All This Eb/No Stuff, Anyway?" while
looking for references that would help me better explain this stuff.
It's a good paper, but I have a tiny little nit. Your first equation says:
C/N = Eb/No * fb/Bw, where
fb is the bit rate, and
Bw is the receiver noise bandwidth
Usually I see this stated as
C/N = Eb/No * (R/B), where
R = bit rate
B = channel bandwidth
I.e., "channel bandwidth" instead of "receiver noise bandwidth".
I see two problems with using receiver noise bandwidth in this equation.
First, Eb/No is supposed to be a universal figure of merit for any kind
of receiver, so it's measured at the receiver input terminals and is
independent of anything inside that receiver. Different receiver designs
for the same signal might use multiple filters with different shapes and
bandwidths, but that would not affect the Eb/No of the signal at their
inputs.
The other problem is that there's more than one definition of bandwidth.
Noise bandwidth is just one of many. In fact, that's precisely why Eb/No
is such a useful figure in the first place: it completely avoids
arguments over the exact system bandwidth and/or which definition of
bandwidth to use to measure it. No is the noise power spectral density
in units of watts/Hz (or milliwatts/Hz), so the only filter bandwidth
that's relevant is that of the spectrum analyzer being used to measure it.
The procedure I like for measuring Eb/No on the bench is to use an
analyzer to measure the signal power with a resolution bandwidth wide
enough to capture all of the signal. Then I turn off the signal source,
turn on the noise generator, and measure the noise power on the analyzer
with the resolution bandwidth set to the user data rate. (Naturally I
have to ensure that both signal and noise swamp the analyzer's own noise).
Then I calculate Eb/No by simply subtracting the noise power measurement
from the signal power measurement. Setting the analyzer RBW to the user
data rate simplifies the calculation by causing the data rate and noise
bandwidth terms to cancel and fall out of the equation.
I found your article while trying to explain to another person that
his Eb/No measurement methods are wrong. This fellow claims to have invented a
family of "ultra narrow band" modulation methods that are in fact ultra
wide band (UWB) plus a very strong carrier that wastes most of the
signal power. Among many other mistakes, he has fallen into the trap of
confusing noise bandwidth with other, more relevant definitions of
bandwidth, and his receivers have filters with noise bandwidths that are
much smaller than the Nyquist rate. This is how he has fooled himself
into thinking that his signals are narrow band.
Anyway, thanks again for the article you published way back in 2000.
Regards, Phil
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Related Links
Intersil Tutorial on Basic Link Budget Analysis, by Jim Zyren and Al Petrick
Adobe Acrobat format -- 80K
Link Analysis with the Iridium System, MLDesign Technologies
Crosslink Channel Analysis (with the Iridium satellite), MLDesign Technologies
An Interesting Link Budget Analysis for the Mars Pathfinder
Williamson Labs Link Budgets using Satmaster Pro MK 4.0c
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