• CH3. data transmission
  • 1. Data transmission media
    • 1-1 Types of transmission media
    • 1-2 Connection structure
    • 1-3 Transmission direction
    • 1-4 4 factors to consider when designing/selecting a medium
  • 2. Signal representation
    • 2-1 Analog vs Digital
    • Sine wave
  • 3. Frequency Domain
    • 3-1 Fourier analysis
    • 3-2 Time domain vs frequency domain
    • 3-3 Spectrum and Bandwidth
    • 3-4 Why these concepts matter
  • 4. Data Rate ↔ Bandwidth
    • Why go digital
  • 5. Transmission Impairments
    • 5-1 Attenuation
    • 5-3 Delay Distortion
    • 5-4 Noise
  • 6. Channel Capacity
    • 6-1 What is Channel Capacity?
    • 6-2 Two formulas
    • 6-3 Bandwidth vs SNR

CH3. data transmission (how to turn data into signals and send them)

1. Data transmission media

Transmitter ↔ Transmission medium ↔ Receiver

1-1 Types of transmission media

Classified by whether the signal travels only inside a physical medium (cable)

1-2 Connection structure

How the devices connected to the medium are arranged

• Direct link: no devices between the transmitter and receiver other than amplifiers or repeaters
  • Amplifiers and repeaters are equipment that restores signal strength, so they are not counted as "intermediate devices"
• Point-to-Point: a direct link, with 2 devices sharing the medium
• Multi-point: more than two devices share the same medium

1-3 Transmission direction

Which direction does the signal flow? Can it flow both ways at once?

1-4 4 factors to consider when designing/selecting a medium

Frequency-domain graph

Amplitude ↑
     │     ┌───────────┐
     │     │           │
     │     │           │
     └─────┴───────────┴──────→ Frequency
          300Hz       3400Hz

          ←─── bandwidth ───→
              = 3100Hz

• attenuation: the signal weakens in proportion to distance

2. Signal representation

2-1 Analog vs Digital

1. Analog signal

• A signal that changes continuously (can take infinitely many values)
• Examples: natural sounds, the human voice
• Directly affected by noise
• Hard to recover (once it's damaged, that's it)

2. Digital signal

• A signal that changes in discrete steps, like a staircase
• Usually represented with two values (0 and 1, +v and -v)
• Range of values: discrete (usually 2)
• Indirectly affected by noise (bit errors)
• Easy to recover (you only need to decide whether it's 0 or 1)

Sine wave

• Why sine waves matter: because every complex signal can be decomposed into a sum of sine waves
• Every signal we process, whether a human voice, digital bits or music, is ultimately a combination of sine waves
• The 3 parameters that define a sine wave
  1. Amplitude (A)
     • The maximum magnitude of the signal
     • The height along the y-axis
     • Unit: volts (V)
     • "How big the swing is"
  2. Frequency (f)
     • Number of oscillations per second
     • Unit: Hz (Hertz)
     • The time for one oscillation = Period (T)
     • Key formula: T = 1/f
  3. Phase (φ)
     • Where within one period the signal starts

Amplitude ↑
        │      ╱╲
      A ┤    ╱    ╲
        │  ╱        ╲
        │╱            ╲
       0├──────────────────────→ Time
        │              ╲      ╱
     -A ┤                ╲  ╱
        │                  ╲╱
        ←──── T (period) ──→

• Wavelength (λ)
  • How many meters does it travel in one cycle?
  • If the signal's propagation speed is v:
    • λ = v × T

One cycle
      ←──── λ (wavelength) ────→
   ┌─────────────────────────────┐
   │      ╱╲              ╱╲     │
   │    ╱    ╲          ╱    ╲   │
   │  ╱        ╲      ╱        ╲ │
   │╱            ╲  ╱            │
   │               ╲╱            │
   └─────────────────────────────┘
   x-axis: distance (space)

3. Frequency Domain (frequency-domain analysis)

3-1 Fourier analysis

• A mathematical operation that converts a time-domain signal into the frequency domain Every signal is a sum of sine waves of various frequencies
• The viewpoint of looking at a signal on the frequency axis instead of the time axis (the bandwidth graph)

Complex signal         Sum of sine waves

                   = 1Hz sine wave (large amplitude)
   ╱╲╱╲╱╲╱╲         + 3Hz sine wave (medium amplitude)
  ╱        ╲        + 5Hz sine wave (small amplitude)
 ╱          ╲       + 7Hz sine wave (smaller amplitude)
                    + ...

3-2 Time domain vs frequency domain

• Time graph

Amplitude ↑
     │  ╱╲    ╱╲╱╲
     │ ╱  ╲  ╱    ╲
     │╱    ╲╱
     ├──────────────→ Time (s)
     │
     y-axis: magnitude of the signal (amplitude)
     x-axis: time

• Frequency graph

Amplitude ↑
     │
   3 ┤ █
     │ █
   2 ┤ █     █
     │ █     █
   1 ┤ █     █     █
     │ █     █     █
     ├─────────────────→ Frequency (Hz)
       1Hz   3Hz   5Hz

     y-axis: magnitude of that frequency component
     x-axis: frequency

3-3 Spectrum and Bandwidth

• Spectrum: the range of frequencies a signal contains

• The example signal above consists of 1Hz, 3Hz and 5Hz, so spectrum = 1Hz ~ 5Hz

• Absolute Bandwidth = (highest frequency) - (lowest frequency) = 5Hz - 1Hz = 4Hz

• Effective Bandwidth: the narrow frequency range where the energy is concentrated

• In general, plain "bandwidth" means effective bandwidth

Amplitude ↑
     │
     │  ████████
     │  ████████
     │  ████████
     │  ████████ █
     │  ████████ █  ▪
     ├─────────────────→ Frequency
        ←──────→
         effective
         bandwidth
        (most of the energy gathers here)

• DC (direct current) component: the 0 Hz component, a signal that does not change over time

3-4 Why these concepts matter

• Bandwidth isn't just a frequency range; it directly determines how much data can be sent over that medium
• Complex signal = needs many frequency components = needs wide bandwidth

Simple signal (slow changes):
   ╱╲      ╱╲
  ╱  ╲    ╱  ╲     → can be expressed with a few low frequencies → narrow bandwidth
 ╱    ╲  ╱    ╲

Complex signal (fast changes):
   ┌┐ ┌┐ ┌┐
   ││ ││ ││         → needs everything up to high frequencies → wide bandwidth
 ──┘└─┘└─┘└──

• Why does a square wave (digital signal) have infinite bandwidth?
  • A digital signal is a square wave where 0 and 1 switch abruptly at right angles
  • To build this right-angled shape from a sum of sine waves, you theoretically need infinitely many frequencies
  • But in practice most of the energy is in the first few components, so it can be approximated with a finite bandwidth.
  • However, the narrower the bandwidth, the more rounded the shape becomes, making 0 and 1 harder to tell apart

4. Data Rate ↔ Bandwidth

The larger the bandwidth, the higher the data rate

[1] Every transmission system has only a limited frequency band
                    ↓
[2] This limit restricts the data rate that can be sent over the medium
                    ↓
[3] A square wave (digital signal) has infinitely many frequency components
   (= needs infinite bandwidth)
                    ↓
[4] But most of the energy is concentrated in the first few components
                    ↓
[5] Limiting the bandwidth distorts the signal

By frequency

Original (pulses before transmission):
  ┌─┐ ┌──┐ ┌─┐
  │ │ │  │ │ │
──┘ └─┘  └─┘ └──    perfect square wave


Bandwidth limited to 500Hz:
   ╱╲    ╱╲
  ╱  ╲__╱  ╲___       almost a sine wave, hard to tell 0/1 apart


Bandwidth 900Hz:
   ╱╲   ╱╲╱╲
  ╱  ╲_╱     ╲       slightly angular shape


Bandwidth 1700Hz:
   ┌╲   ┌─╲
  ╱  ╲_╱   ╲___      similar to a square wave


Bandwidth 4000Hz:
   ┌─┐  ┌──┐ ┌─┐
  ─┘ └──┘  └─┘ └──   almost the original

Bandwidth ∝ Data Rate (directly proportional)

Intuitive analogy Bandwidth = number of lanes on a road Data Rate = number of cars passing per hour

Why go digital

5. Transmission Impairments

Analog signal → degraded signal quality (a crackling radio) Digital signal → bit errors (0 misread as 1, 1 misread as 0)

Transmission Impairments
│
├─ 1. Attenuation
│
├─ 2. Delay distortion
│
└─ 3. Noise
   ├─ Thermal noise
   ├─ Intermodulation noise
   ├─ Crosstalk
   └─ Impulse noise

5-1 Attenuation

• The weakening of signal strength with distance.

1. Weakening with distance

Start:  ████████████  (strong)
Middle: ████████      (weaker)
End:    ████          (even weaker)

2. Happens differently depending on frequency (higher frequencies weaken faster)

After traveling the same distance:

Low-frequency component:  ████████  (weakened less)
Mid-frequency component:  █████     (medium)
High-frequency component: ██        (weakened a lot)

A digital signal is a sum of many frequencies, so if only the high frequencies weaken, the shape of the signal is ruined. This is called attenuation distortion.

So the receiver has 2 conditions to satisfy

1. Strong enough to be detected
2. Higher than noise

Solutions

5-3 Delay Distortion

• Different frequency components arrive at different speeds
• Condition: occurs only in guided media (twisted pair, coaxial cable, fiber)
  • Does not occur in wireless (in wireless (through the air), all frequencies travel at the same speed, the speed of light)

Departure (simultaneous):
   Low freq  ████
   Mid freq  ████
   High freq ████  → transmit

In transit (speed difference):
   Low freq  ━→
   Mid freq  ━━→ (a bit faster)
   High freq ━━━→ (faster)

Arrival (time difference):
   Low freq  ████      ← arrives late
   Mid freq   ████
   High freq   ████    ← arrives early

   → combine into a shape different from the original

A phase difference arises between the frequencies

The phase difference can cause Intersymbol Interference (ISI).

• Intersymbol Interference (ISI): after one bit ends, the tail of its signal spills into the next bit and interferes with reading the next bit

Bit string to send: 1 0 1 1
            ┌┐ ┌┐┌┐
            ││ ││││
            ┘└─┘└┘└

After delay distortion:
            ┌┐_┌┐┌┐
             ⌒⌒⌒⌒⌒⌒  ← the end of one bit leaks into the next bit

                → bit boundaries blur → hard to tell 0/1

5-4 Noise

• Unwanted signals that intrude between the transmitter and the receiver.

• Noise is the main limiting factor on communication system performance.

• Attenuation and delay distortion can be compensated for, but noise fundamentally cannot be removed. So it is the ultimate limit of communication. The Shannon formula in step 6 defines exactly this limit mathematically.

• 4 kinds of noise

1. Thermal Noise

• Caused by the thermal motion of electrons
• Distributed evenly across all frequencies (hence also called white noise)
• Cannot be removed

2. Intermodulation Noise

• Caused by nonlinearity in the transmitter/receiver/medium
• An ideal system produces output proportional to input (linear), but in reality there is some nonlinearity, so when two frequencies go in, new frequencies are formed

Input signal:   f₁ = 100Hz, f₂ = 150Hz
                  ↓ (passes through a nonlinear system)
Output signal:  f₁ = 100Hz (normal)
                f₂ = 150Hz (normal)
                + 250Hz (= f₁ + f₂) ← newly created noise
                + 50Hz  (= f₂ - f₁) ← newly created noise
                + ...

3. Crosstalk

• Caused by electrical coupling between adjacent lines
• The signal flowing in the neighboring wire leaks into my wire

Wire A: ━━━ signal A ━━━━━━━━━━
              ↕  (part of the signal leaks over through electrical coupling)
Wire B: ━━━ signal B + part of signal A ━━

• Solution: shielding. This is why UTP is more expensive than STP but has less crosstalk.

4. Impulse Noise

• Caused by external electromagnetic interference, lightning, etc.

• Characteristics

  • Irregular short spikes
  • Short duration + high amplitude
  • Non-continuous

• To the human ear it's just heard for a moment, so why is it fatal for digital signals?

  • Digital sends many bits in a short time, so even a single short spike corrupts several bits at once.

• If even one bit is different, things like images get completely garbled, yet in reality bit errors inevitably occur. How do we manage them?

6. Channel Capacity

6-1 What is Channel Capacity?

• The maximum data rate that can be transmitted over a communication channel under given conditions

Channel Capacity (C)
= "the maximum number of bits per second this channel can carry"
= unit: bps (bits per second)

• The 4 factors that determine channel capacity

1. Data Rate
   • Unit: bps
   • Meaning: the speed you want to send at
2. Bandwidth
   • Unit: Hz
   • Meaning: the width of frequencies the channel passes
3. Noise
   • Unit: dB
   • Meaning: the noise level of the channel
4. Error rate
   • Unit: ratio
   • Meaning: how often errors occur

• Bandwidth ↑ → Data rate ↑ (learned in step 4)
• Bandwidth ↑ → cost ↑
• Noise ↑ → Error rate ↑ → effective Data rate ↓
• Noise is the most fundamental constraint (learned in step 5)

6-2 Two formulas for finding channel capacity

1. Nyquist Bandwidth

   • How high can it go if there is no noise?
   • The limit of an ideal noise-free channel
   • Formula: C = 2B log₂ M
     • C = channel capacity, B = bandwidth, M = number of signal levels (number of distinct signal elements)
   • If M is increased to infinity, does the data rate become infinite too?
     • In theory yes, but the spacing between signal levels narrows and it becomes vulnerable to noise
     • So a real limit that accounts for noise is needed

2. Shannon Capacity

   • How high can it go in the real world, with noise?
   • The theoretical upper bound of a real channel
   • Data rate ↑ → time per bit ↓ → even short noise corrupts many bits → error rate ↑

   First, quantify the noise level SNR (Signal-to-Noise Ratio)

           signal power
   SNR = ─────────
           noise power
   

   Used as is, the numbers get too large or too small, so it is converted to dB (decibels)

   SNR_dB = 10 × log₁₀ (signal/noise)
   

   • Shannon formula
     • C = B × log₂ (1 + SNR)
   • SNR: signal-to-noise ratio (a ratio, not dB!)
   • Large SNR (little noise) → log₂(1+SNR) is large → C is large (can send a lot)
   • Small SNR (lots of noise) → log₂(1+SNR) is small → C is small (can send only a little)
   • Shannon capacity is an unreachable theoretical upper bound

6-3 Which is more efficient to increase, Bandwidth or SNR?

• Capacity is linear in bandwidth and logarithmic in SNR, so in theory increasing bandwidth is more efficient.
• But in practice, increasing SNR is easier
• So in reality, SNR is squeezed out to the point of diminishing returns, and then B is increased.