EC312 Lesson 13: Gaining the power
Objectives:
(a) Define gain and attenuation and describe their application to wireless communication. Gain at transmitter
and then attenuation at the receiver.
(b) Calculate power gains for single and multiple stage systems. Describe what a Watt is.
(c) Express power gain in decibels. Review logarithms and the log scale.
(d) Express power levels in dBW/dBm and other reference levels.
(e) Discuss noise as interference in a communication scheme. Calculate signal to noise ratio (SNR).
(f) Point out critical frequencies and power ratios in a graph of a bandpass filter.
I’m so excited for you. You are about to be privileged (Yes, I used a word from the first six-weeks, although
unrelated) with some of the most invigorating material you’ve ever been exposed to in your life.
1. Gain/Attenuation.
Let’s do a little role playing. Imagine it’s a Friday afternoon and you’re doing your favorite thing. What is it
you’re doing? Come on! Don’t kid yourself. You’re listening to JB on the radio! Nope, it’s no typo (JT).
You’re a die-hard Biebliever.
Now you happen to be listening to this classic of a song, Baby ft. Ludacris, on your car radio. You say to
yourself, “I’d just love to have the Beibster here with me, but isn’t it awesome how I can have his music playing
in my car even though the radio station is so many miles away. Golly Gee, I hope I can jam out to this music
when I’m out at sea on my first ship.”
You know how the power in a modulated signal is distributed amongst the sidebands and the carrier. You also
know that the highest baseband signal that you can modulate in AM Radio Band is 5kHz to stay within FCC
guidelines for a maximum Bandwidth of 10kHz for a station. So you know that you are listening to FM and
have been told that there are many more sidebands generated for frequency modulation and that is why FCC
gives stations in the FM Band a maximum bandwidth of 180kHz. Below is a picture with typical powers at the
radio transmitter and receiver ...
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What really intrigues that sharp midshipman mind of yours is all those large numbers (i.e. 15, 500 and
0.00000000000001) and what’s up with the W.
That W, my studious one, stands for Watt. Oh, this is a word with which you want to be familiar. The Watt
(W) is the electrical unit of measurement for power. Power is the driving force behind the communication
systems (the Bieberism) that rock your world. The more power, the further your communication signal can
travel. Conversely, the less power, the shorter distance that signal will travel.
So how are you going to get your signal to travel further? Turn up the power. That seems pretty straight
forward, right? Ever heard of amplifiers? To amplify means to make more powerful.
The term gain usually refers to amplification. For an amplifier, the gain (A) is the ratio of the output to the
input. Most amplifiers are power amplifiers, so to calculate power gain (AP) where Pin is the power input and
Pout is the power output:
Pout
Pin
Power gain (Ap) = Pout / Pin
Using gain, much like at a rock concert, is how you can take a communications signal and increase its power
large enough to transmit many miles, much like the above example of a tower transmitting at 15,500W.
Similarly, elements of communication systems can also reduce the power of a signal. Reduction or signal loss
is termed attenuation.
Attenuation (A) = output/input = Pout/Pin
Circuits that introduce attenuation have a gain that is less than 1.
Putting together what we’ve learned, we have our signal feeding an amplifier that that increases the power of
the signal. The signal that is broadcast out of the transmitter (tower) where the signal is attenuated as it travels
through the air. Finally the significantly reduced signal is picked up by the receiver, and the receiver recovers
the original signal.
Amplifier
(Gain)
Amplifier
Amplifier
(Gain)
(Gain)
Free Space
(Attenuation)
Example
The power output of an amplifier is 6 watts (W). The power gain is 80. What is the input power?
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Example
The power output is 0.00000000000001W. The input power is 15,500W. Is this system associated with
amplification or attenuation? What is the gain of this system?
Oh my gosh! After that last problem does anybody else have a headache this big?
2. Decibels.
As engineers, (even though some of you may think I’m lying) we just want our lives to be as easy as possible.
So rather than work with these terribly tedious numbers, we often use something called decibels. The decibel is
a logarithmic measure that provides more convenient gain and attenuation figures. Now, I know you remember
logarithms.
The logarithm of a number is the exponent to which
another fixed value, the base, must be raised to produce
that number. For example, the logarithm of 1000 to
base 10 is 3, because 1000 is 10 to the power 3: 1000 =
10 × 10 × 10 = 103. More generally, if x = by, then y is
the logarithm of x to base b, and is written y = logb(x),
so log10(1000) = 3.1
It this class, we will be using base 10 for our
calculations.
The great attribute of the log scale to the right is that
you can graph values over a very large range. In this
case, the range is from 1 to 1000.
Wikipedia Logarithmic Scale
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Wikipedia Logarithm, http://en.wikipedia.org/wiki/Logarithm
The formula for computing decibels (dB) is:
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Power Gain or Attenuation: dB = 10 log Pout / Pin
Gains greater than 1 (amplification) will result in a positive (+) dB value.
Gains less than 1 (attenuation) will result in a negative (-) dB value.
Now that we have expressed
So then for the above example that gave us a headache, we see:
dB = 10log(0.00000000000001/15500) = -181.9 dB
Yes, please.
Example
Convert the two gains to decibels (dB).
A couple of very common power gains are:
A power amplification by a factor of two will result in a power gain of +3 dB.
dB  10log10 (2)  3 dB
An attenuation by a factor of one-half will result in a power gain of -3 dB.
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dB  10 log10 ( )  3 dB
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I mean so extremely common, you really better know these.
Now, I hear your next question, and I’m ready to answer it. You want to know how to find power gain as a
ratio if given power gain in dB? No problem! Just rearrange the dB equation from earlier and you get:
dB
Pout
AP 
 10 10
Pin
Example
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Convert the following from decibels to ratios.
25 dB: AP =
-6 dB: AP =
Sometimes, you may be asked to use decibels to indicate the ratio of a power level with respect to a fixed
reference level. Here are two common times. In later lectures you will be given others.
dBm: The reference power level is 1 mW and dBm is expressed mathematically as
P(dBm)  10 log10
P
0.001 W
dBW: The reference power level is 1 W and dBW is expressed mathematically as
P(dBm)  10 log10
P
1W
Example
Express Pin = 2 watts in decibels (dBm) and (dBW).
Now, you may be saying to yourself, “Wow, I’m so glad I learned about dB.” But wait… There’s more.
The really beauty of decibels is about to be revealed.
When looking at electronic systems, they either add amplification or attenuation. So guess what. It is very
possible and most probable that you will see gains cascaded together.
So, if we leave the gains in ratio form, then the total gain of the system will be the product of all the gains
multiplied together.
AT = A1 x A2 x A3
These, numbers can be quite tedious to calculate. As alluded to earlier, engineers are lazy-ish. We have
something I know you’ll love.
Another wonderful thing about decibels is that you don’t have to multiply them. You just add them. In fact,
DON’T EVER MULTIPLY DECIBELS. This is a huge common mistake. So common I’ll say it again.
DON’T EVER MULTIPLY DECIBELS.
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With decibels you find:
So, if you are supplied with dBs, which most practically you mostly likely will be, you just have to sum them.
Example
The circuit below represents the first three stages of a typical AM or FM receiver Find the following quantities.
a. AP1(dB), AP2(dB), and AP3(dB)
b. APT(dB)
c. P1, P2, and Pout.
d. Pin (dBm), P1 (dBm), P2 (dBm), and Pout (dBm).
3. Noise and the Signal-to-Noise Ratio.
Let’s take a moment to think about your original question. Will you be able to listen to JB on the radio while
out at sea on your ship. We know that the power of the signal plays into answering that question, but what else.
Electrical noise is one of the principle limiting factors in the performance of communication systems.
Noise is divided into two broad categories:

External noise is electrical noise introduced into the transmission channel. Examples are:

o Industrial noise caused by man-made electrical sources (motors, generators, ignition)
o Atmospheric noise due to naturally occurring disturbances in earth’s atmosphere (e.g., lightening)
o Extraterrestrial noise due to solar and cosmic activity.
Free Space
(Attenuation)
Internal noise is noise introduced inside the receiver itself.
o Thermal noise
o Semiconductor noise
o Intermodulation distortion
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The thing about noise is that, if significant enough, it can mask the original signal such that the signal becomes
unrecoverable. This is not much different than if an enemy were to flood the air with an erroneous signal at the
same frequency on which you were transmitting. If that erroneous signal was stronger at the receiver than your
signal, your signal would become unrecoverable.
How do we know the effect of noise on the signal? You always impress me with your spot on questions.
The signal-to-noise (S/N) ratio or SNR is used to quantify the effect of noise on a signal.
A strong signal and weak noise results in a high SNR.
A weak signal and strong noise results in a low SNR.
Signal without noise.
20 dB SNR
10 dB SNR
The signal-to-noise ratio indicates the relative strengths of the signal and the noise in a communication system.
The stronger the signal and the weaker the noise, the higher the SNR.
 Ps  signal power (W)
S
P
 s where 
N P Pn
 Pn  noise power (W)
(Ps)
(Pn)
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Signal to noise ratio is usually expressed in decibels
SNR ratio (dB) using power: dB  10log
Ps
Pn
Example
The signal power at the input to a receiver is 6.2 nW and the noise power at the input to a receiver is 1.8 nW.
Express the SNR as a ratio and as a value in decibels.
4. Tuned Filters.
Your company mates told you about a Whitesnake Youtube link. You watched the video and LOVE
Whitesnake! You’re not a Biebliever anymore. No more listening to 88.1 FM. You are going to tune into 90.9
FM radio station to listen to the BEST ROCK BAND in the world. What allows your radio to isolate one
station from all of the adjacent stations?
Filters are designed to pass some frequencies and reject others. A filter is a frequency-selective circuit.
There are four basic kinds of filters:
Low-pass filter. Passes frequencies below a critical frequency, called the cutoff frequency, and attenuates
those above.
High-pass filter. Passes frequencies above the critical frequency but rejects those below.
Bandpass filter. Passes only frequencies in a narrow range between upper and low cutoff frequencies.
Band-reject filter. Rejects or stops frequencies in a narrow range but passes others.
We focus on the Bandpass filter in this class. Believe when I say, the other filters are no less important in real
life practical application and communication systems you undoubted will use throughout your career in the
Navy or Marine Corp.
A bandpass filter (aka. passband) passes frequencies over a narrow range between lower and upper cutoff
frequencies.
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𝑃𝑜𝑢𝑡
𝑃𝑖𝑛
Look at the important attributes. Notice that ideally, the signal would be at full strength until the upper and
lower cutoff frequencies and then turn off. In reality, there is a gradual reduction in signal strength until the
upper and lower frequency at which point any more reduction in signal indicates the signal should not pass.
Any idea at what signal strength the signal is no longer acceptable to be passed? Oh, you’re good. I’m always
so impressed when you look at the material we just went over and realize the answer is there.
That’s right. The frequencies at which signal strength is at -3dB (1/2 power) from the signal strength at the
center frequency are the upper and lower cutoff frequencies.
So there you have it. By using a bandpass filter in your radio, you can tune into a radio station of your choice,
that choice being to listen to Whitesnake, and cutoff all the other frequencies to radio station s you don’t want to
hear. (i.e. Any that are playing JB.)
Remember my young Padawan, like Uncle Ben always said, “With great power comes great responsibility.”
Today you have learned much about power, so you are now responsible for the material.
LCDR Jesse Atwood, CDR Hewitt Hymas
Help us improve these notes! Send comments, corrections and clarifications to atwood@usna.edu
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