Communication Systems · 96 h · Topic 1 of 11

Introduction to communication systems

The same low-noise amplifier improves a receiver's sensitivity by 5 dB if it is placed at the antenna rather than after the cable. In a receiver, the order of the stages matters.

Decibel and dBm Thermal noise Noise figure Friis formula Sensitivity

01Carrying information from one place to another

Every communication system has the same blocks: a source that produces the message, a transmitter that adapts it to the medium (encodes, modulates, amplifies), a channel that carries it and degrades it (attenuates it, distorts it, adds noise and interference to it), and a receiver that tries to recover the original message. The channel can be a cable, a fiber, free space or a disk; the transmission media course studied them. This subject studies what is done with the signal at the ends. The introduction to modulation systems is in the technician program, under modulation systems.

There are three scarce resources to be shared: the transmitter power, the channel bandwidth and time. Each modulation and coding technique is a different way of trading one for another, and comparing systems means comparing how they use those resources.

02Decibels and dBm

Powers in communications range from kilowatts in a transmitter to femtowatts at a receiver antenna: twenty orders of magnitude. That is why we work in decibels, where gains and losses add:

\[ G_{dB} = 10 \log\frac{P_2}{P_1} \] \[ P_{dBm} = 10 \log\frac{P}{1\ \text{mW}} \] 0 dBm is 1 mW; 30 dBm, 1 W; −100 dBm, 0.1 pW. A ratio of voltages across the same resistance gives 20 log instead of 10 log.

03Thermal noise

Any resistor at temperature \( T \) generates noise through the thermal agitation of its electrons, with an available power that does not depend on its value, only on the bandwidth \( B \):

\[ N = k\, T\, B \] With \( k = 1.38 \times 10^{-23} \) J/K. At 290 K this gives −174 dBm per hertz: in 200 kHz, the channel of an FM radio, −121 dBm. It is the noise floor from which every receiver starts.

Since noise grows with bandwidth, a receiver should have exactly the bandwidth of the signal: a wider one lets in noise that contributes nothing.

04Noise figure and the Friis formula

Each stage of a receiver adds its own noise. Its noise factor \( F \) is how much it worsens the signal-to-noise ratio as the signal passes through it, and in decibels it is called the noise figure. For several stages in cascade:

\[ F = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots \] \[ S_{min} = -174 + 10\log B + NF + SNR_{min} \] The Friis formula for noise: the contribution of each stage is divided by the gain of all the preceding ones. The first stage decides almost everything, and a lossy cable or filter at the front has a noise figure equal to its attenuation. The second expression gives the sensitivity in dBm.

That is why satellite receivers, radio telescopes and base stations put a low-noise amplifier (LNA) right at the antenna: its gain makes everything that comes after it negligible, the cable included.

Lab · the noise figure of a receiver

A receive chain: an LNA with 20 dB of gain and a 1 dB noise figure, a lossy cable, a −7 dB mixer with a 7 dB noise figure, and an IF amplifier with 30 dB and 4 dB. The bars are the contribution of each stage to the total noise factor; the sensitivity assumes the demodulator needs 10 dB of SNR.

Cable loss =

05A limit for everything

Claude Shannon proved in 1948 that a channel of bandwidth \( B \) with signal-to-noise ratio \( S/N \) can carry information without errors up to a maximum rate, the capacity:

\[ C = B \log_2\left(1 + \frac{S}{N}\right) \] In bits per second. A 3.1 kHz telephone channel with 35 dB of SNR supports about 36 kbit/s. No system exceeds it; the best present-day codes come within 1 dB of it.

Capacity summarizes the trade-off among the resources: rate can be gained with more bandwidth or with more power, but power pays off only logarithmically. It is studied in depth in the unit on information theory.

06In the lab

Activity 1 · The noise floor

With a spectrum analyzer, measure the noise floor with the input terminated in 50 Ω for several resolution bandwidths and verify that it rises 10 dB per decade.

Activity 2 · The LNA

Measure the sensitivity of a radio receiver with and without an antenna preamplifier, using a long cable.

Activity 3 · Noise figure with a noise source

Measure the noise figure of an amplifier by the Y-factor method with a calibrated noise source.

07Common mistakes

Putting the amplifier after the cable. The cable has already degraded the signal-to-noise ratio.
Adding power decibels to voltage decibels. They are 10 log and 20 log.
Using a channel filter wider than the signal. Noise gets in without adding any signal.
Confusing dB with dBm. The first is a ratio; the second, an absolute power.

08Self-assessment

How many dBm is 50 W?

\( 10\log 50\,000 \approx 47 \) dBm.

How much thermal noise power is there in 1 MHz at room temperature?

\( -174 + 60 = -114 \) dBm.

A 15 dB LNA with NF 2 dB precedes a stage with NF 10 dB. What is the total noise figure?

\( F = 1.585 + 9/31.6 = 1.87 \): 2.7 dB.

What capacity does a 1 MHz channel with 20 dB of SNR have?

\( 10^6 \log_2 101 \approx 6.66 \) Mbit/s.

09Further reading

Simon Haykin and Michael Moher. Communication Systems. 5th ed., Wiley, 2009. The course textbook on analog and digital communication systems.
B. P. Lathi and Zhi Ding. Modern Digital and Analog Communication Systems. 5th ed., Oxford University Press, 2019. Very clear on signals, modulation and noise, with many examples.
Claude E. Shannon. A Mathematical Theory of Communication. Bell System Technical Journal, 27, 1948. The paper that founded information theory, surprisingly readable.

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