Communication Theory - 5ETB0

5ETB0 Communication Theory

• Area: Communications and Networking
• Topics: Filters · Random Signals and Power Spectral Density · Noise · Modulation · Digital Signaling and Pulse Shaping · Detection and Matched Filters · Channel Capacity
• Related courses: Communication 1 - 5ETC0 · Math II - 5EMA0 · Signals II (DSP Fundamentals) - 5ESC0

requirements

formula sheet

Formula_Sheet.pdf

Course Introduction, Motivation, and Math Preamble

M1.1.pdf

M1.2.pdf

Analog vs Digital Communication

Math II - 5EMA0
Signals II (DSP Fundamentals) - 5ESC0
M2.1.pdf

Conditional PDF formula can be thought of of as Discrete Input, Continuous Output probability. .

M2.2.pdf

stationary autocorrelation function means
autocorrelation is correlation of a signal with a delayed copy of itself. similarity between observations of a random variable a function of the time lag between them.

higher bandwidth for an analog channel would let in higher frequency range which may let in more noise.

DSB-SC might not change SNR but as we see diagram we see increase in bandwidth…

discrete encoder, since the information is of the sampled signal it reduced the information, while error correction encoder adds redundant bits to later retrieve original signal in case noise is added: Like hannon encoding etc (info bits).

COS 126: Prefix Codes
ocw.tudelft.nl/wp-content/uploads/Algoritmiek_Huffman_Codes.pdf

Side Quest: Signals

Communication 1 - 5ETC0
Bandwidth of a signal is defined as the difference between the highest and the lowest frequency

Baseband vs Bandpass
bandpass basically produces another carrier signal

Bandpass signal is created when modulation or a constant carrier signal is added to the orignal message signal which is important for open air transmission in a noisy environment so you can occupy your specific carrier frequency spectrum and your bandwidth of it.

Sampling in Digital Communication - GeeksforGeeks

after sampling we can encode and digitize our signal to then transmit this information.

Modulation maps the digital signal onto a carrier wave, which is better suited for transmission over the chosen medium. It modifies the carrier wave’s properties (amplitude, frequency, or phase) to encode the information.

modulation can be on digital and continuous.

NOTE: PCM is technically on analog data since it converts analog signal to digital.
What is Modulation ? Why Modulation is Required ? Types of Modulation Explained. - YouTube

Decisions Rules for DIDO Channels

Topic: Detection and Matched Filters
M3.1.pdf

M3.2.pdf

MAP Interpretation Comparison

MAP is always better but when equally likely symbols, ML is the same.
We still care about ML, since with it you don’t need to inform the receiver what the a-priori probabilities are. Its more adaptable, overall the ML is simpler to implement.

Maximum Likelihood (ML) detection is generally simpler and computationally more efficient compared to Maximum A Posteriori (MAP) detection, especially when the messages are equally likely. In such scenarios, MAP simplifies to ML because the a-priori probabilities are uniform, making MAP and ML equivalent. The ML decision rule, defined as:

focuses only on maximizing the likelihood , bypassing the need to compute a-priori probabilities . This reduces complexity, especially in systems with unknown or uniform priors. While MAP minimizes error probability by incorporating prior probabilities, ML offers a more practical and scalable solution in many applications where computational simplicity is essential.

Decision Rules for DICO Channels

Topic: Detection and Matched Filters
M4.1.pdf

M4.2.pdf

M4.3.pdf

square of all elements in a vector:

Take the union, basically find a sum of points in that area we can confirm the points are in that region by seeing that the distance between the correct and the wrong signal is different as the one that is correct is always FURTHER away.
(minimum euclidean distance decision rule)

Important Definitions

MAP Decision Rule

Error Probability (Scalar System)

Decision Variable (Gaussian Noise)

Threshold for Decision Rule

(when equal probabilities)

Error Probability for AGN Channel

where is the Q-function.

Multi-Vector Channel Decision Variables

Minimum Euclidean Distance Decision Rule

Additive Gaussian Noise Channel (Vector Form)

, where

Optimum Receiver Decision Threshold

• For a scalar bi-AGN channel, the threshold for the optimum receiver is:
  
• When , the threshold simplifies to:
  

Error Probability (Scalar System)

• General formula:
  

Error Probability Components

• :
  

• :
  

Error Probability for AGN Channel

• Combining the above:
  

Error Probability Using Distance

• For vector channels:
  
  where is the half-distance between signal points.

Error Probability Upper Bound

• Upper bound using union bound:
  

Q-function and Properties

• Q-function definition:
  
• Symmetry property:
  

Probability of Error for a Hyperplane

• For the AGN vector channel:
  
  where is the perpendicular distance from the signal point to the hyperplane.

Waveform Channels

Topic: Random Signals and Power Spectral Density
M5.1.pdf

M5.2.pdf

Energy of a Waveform

Orthogonality of Waveforms

Synthesis of Waveforms

Recovery of Signal Vector

Additive Noise in Signal Vector

Vector Representation of Noise

Relevant Noise Covariance

Signal Space Projection

Gram-Schmidt Process (See Question 5.1)

Step 1: Define the First Basis Function

Step 2: Define the Auxiliary Signal for Subsequent Functions

where

Step 3: Compute Energy of Auxiliary Signal

Step 4: Define the New Orthogonal Basis Function

Step 5: Express Each Signal in Terms of Basis Functions

see 5.1.c for a different distance based approach

see 5.4 and 5.7 for an intuition on gram-schmidt process

Receiver Implementation, Matched Filters

Topics: Filters · Detection and Matched Filters
M6.1.pdf

16 messages → 16 filters needed (needed same amount as messages)

M6.2.pdf

Signal Energy Considerations, Orthogonal Signals

Topic: Noise
M7.1.pdf

M7.2.pdf

Translation and rotation of signal structures maintain the signal energy .

Average energy:

Binary Orthogonal vs. Binary Antipodal Signals: - For binary orthogonal signaling, the error probability under AWGN is - For binary antipodal signaling, the error probability is To achieve the same , orthogonal signaling requires twice the energy of antipodal signaling (a 3 dB disadvantage).

Antipodal signaling is more energy-efficient than orthogonal signaling in the binary case.

Message Sequences, Bandwidth

Topic: Digital Signaling and Pulse Shaping
M8.1.pdf

M8.2.pdf

Capacity of the Baseband and Wideband Channels

Topic: Channel Capacity
M9.1.pdf

M9.2.pdf

Pulse Transmission

Topic: Digital Signaling and Pulse Shaping
M10.1.pdf

M10.2.pdf

Pass-Band Channels

Topic: Modulation
M11.1.pdf

M11.2.pdf

Q1: .
Q2: Set frequency (2nd addition of sine is now 0) and the number of cosine building block waveforms .
Q3: 1

Inside Wireless: QAM modulation (Quadrature Amplitude Modulation) - YouTube

Quadrature Amplitude Modlation (QAM): Explained - YouTube

Quadrature amplitude modulation - Wikipedia

What You Need to Know about Wideband Signal Analysis

Random Carrier-Phase

Topic: Modulation
coherent vs incoherent reception: no knowledge of means there is only cosine but it can be decomposed to have the sine component.

M12.1.pdf

M12.2.pdf

optimum incoherent reception (and when equal energy)

envelope simplified representation

the green box the linear filter is a basically a mix of baseband and passband(carrier) signals.

i. we have a binary message, ii. vector representation of the received signal in the time domain, iii. linear filter: invert + delay, iv: what you get out of the linear filter, v: then you get the envelopes, and now you can optimally detect messages (receive something close to 1 then message 2 and if close to 0 then message 1).

If message 1 if just inverted of message 1: would be identical to , so cant detect message differences: antipodal signaling BUT in coherent reception we have the phase so we can differentiate between them. For incoherent we dont have the phase so we cant differentiate. ANTIPODAL SIGNALING DOES NOT WORK IN INCOHERENT RECEPTION.

only orthogonal signaling possible…

Nice to knows

signal like this can be split in the following way:

where:

• Message component: encodes the actual data.
• Building block: structures the signal in time, shapes it in frequency and minimizes the inter-symbol interference.
• Carrier component: modulates the signal to the desired transmission frequency while preserving orthogonality between in-phase and quadrature components.

Invited Lecture

5ETB0_Invited_Lecture_Optical_Satellite_Communications.pdf

• Satellites operate on RF band → could be a problem?

• Infrared c-band spectrum → allows a much larger bandwidth that classic RF like FM or 5G MMWAVE BAND.

  • Sat RTT: 46ms and
  • Great circle fibre: 55ms → takes longer over long distance than satellite.
  • free space optical communication is better over long distance than RF.

FSO vs RF
Receive power
proportional to:
Example: Ka-band vs FA
with RF Ka-band (30GHz) vs FSO C-band (193 THz).
This gives about 75 dB gain in receive power for FSO over RF.
Assuring antenna and transmit power distance all the same.

objective of communication networks:

• throughput
• reliability
• high-availability
• low-latency

minimizing:

• cost per bit
• bandwidth / wavelength
• power/safety
• form factor (size/weight)
• complexity of the system

UP link: diversity gain: temporal diversity, spatial diversitty, site diversity, frequency diversity.