Communication Systems Chapter 8: Fourier Analysis

Communication Systems and Computer Networks (1404703) - Part 8

• The physical layer studies mechanical, physical, electrical, and timing properties of data transmission.
• It analyzes different transmission media: guided, wireless, and satellite.
• Data communication relies on functions (voltage or current) that can be mathematically analyzed.
• Fourier analysis is a crucial technique for analyzing these functions.
• Periodic functions can be represented by an infinite combination of sine and cosine waves.
• Knowing the harmonics allows reconstructing the function.
• Signal analysis helps determine proper sampling.

Fourier Series

• A Fourier series represents a function as a sum of harmonically-related sinusoids with varying amplitudes and phases.
• The amplitudes and phases of these sinusoids are combined into Fourier coefficients (complex numbers).
• Fourier series are only applicable to periodic functions.
• For functions defined over a finite interval, the values outside that interval are irrelevant.
• The Fourier series formula is given as a0/2 + Σ [an cos(nx) + bn sin(nx)] where the summation is from n = 1 to infinity

Bandwidth-Limited Signals

• Key factors include RMS (Root Mean Square) amplitude, bandwidth (related to construction, medium thickness and length), and cutoff frequency (fc).
• Bandwidth-limited signals are crucial for data transmission.
• Successive approximations to the signal using different harmonics provide a clearer illustration.
• Increasing the number of harmonics closer approximates the original signal.

Bandwidth-Limited Signals (2)

• A binary signal and its Fourier amplitudes showcase how successive approximation improves the signal reconstruction.

Bandwidth-Limited Signals (3)

• A table illustrates the relationship between data rate (bps), time (msec), first harmonic (Hz), and the number of harmonics sent for different data rates.
• This demonstrates how increasing data rates increase the number of harmonics required.

Shannon's Maximum Data Rate of a Channel

• Shannon's theorem states the maximum achievable data rate (bits per second) of a noisy channel.
• This depends on the bandwidth (H, in Hz) of the channel and the signal-to-noise ratio (S/N).
• The maximum data rate is given by the formula Max # of bits/sec = H log2(1 + S/N).

Guided Transmission Media

• Selection of transmission media depends on bandwidth, delay, cost, and ease of installation.
• Magnetic media (skipped) are less relevant
• Common types of guided media include twisted pair, coaxial cable, and fiber optics.

Twisted Pair

• Twisted pair consists of two insulated copper wires (1mm thick).
• Twisting eliminates the antenna effect of parallel lines.
• Twisted pairs are commonly used in telephones.
• They can transmit data over several kilometers before requiring repeaters for extended distances.
• Different categories (e.g., Category 3, 5 UTP) define the number of twists per centimeter affecting characteristics like crosstalk and bandwidth.

Coaxial Cable

• Coaxial cables offer better shielding compared to twisted pairs.
• They are used for longer distances and faster data rates and also have higher bandwidth (sometimes up to 1 GHz).

Fiber Optics

• Fiber optics are ultra-thin glass fibers that transmit light signals converted to electrical signals.
• Theoretical data rates are exceptionally high (50,000 Gbps) but the actual ability is only around 10 Gbps due to losses.
• Light signals travel through the fiber in a process called total internal reflection.
• Fiber optics are characterized by high bandwidth and low error rates, but are more expensive.

Fiber Cables (2)

• A comparison table for LEDs and semiconductor lasers as light sources in fiber optic systems.
• The table highlights factors such as data rate, fiber type, distance range, lifetime, temperature sensitivity and cost.