MIT1: Physical Layer

• Radio waves
  \[s(t)=A\cdot\sin(2\cdot\pi ft+\varphi)\]
  \(A:\) Amplitude
  \(f:\) frequency
  \(t:\) time
  \(\varphi:\) phase
  • wave length \(\lambda=c/f,c=\) speed of light
• Frequency allocation
  • ISM band (Industrial, scientific, medical)

Modulation

• Analog modulation: set parameters to an arbitary value
• Digital modulation (Digital keying): set parameters from a finite set of values

Amplitude Shift Keying (ASK)

• Encode data with different amplitude
• needs not much bandwidth
• susceptible against distortion

Fourier Transform
Each periodic siganls can be represented by a composition of harmonic signals
\[g(t)=\frac{1}{2}c+\sum_{n=1}^\infty a_n\sin (2\pi n f t)+\sum_{n=1}^\infty b_n\cos (2\pi n f t)\]
Digital modulation \(\Rightarrow\) Analog signal\(=\) periodic signal
Analog signal has infinite bandwidth
bandwidth: width of frequency band needed to represent a signal
wireless communication: restricted bandwidth

• Relationship between bandwidth and signal rate
  \[S_{max}[\text{Hz,baud}]=2\cdot B[\text{Hz}]\]
  \(S:\) signal rate
  \(B:\) bandwidth
• Nyquist Theorem relationship between signal rate and data rate
  \[R_{max}[\text{bit/s}]=2\cdot B[\text{Hz}]\cdot Id(n)[\text{bit}]\]
  \(R:\) data rate
  \(Id(n):\) \(n\) parameters to represent \(m=Id(n)\) bits
  \[\text{Data rate}\Leftrightarrow \text{Bandwidth}\]

1. Digital modulation

• analog baseband signal
• low-pass filtering

2. Analog modulation

• Shift the baseband signal to a given frequency band with carrier frequency \(f_c\)

Frequency Shift Keying (FSK)

• encode data with different frequency
• needed higher bandwidth than ASK
• bandwidth depends on \(f_1,f_2\)

Advanced FSK- MSK (Minimum Shift Keying)

• encode bit transitions instead of bits

• make the transition more smooth

  ||
  |–-|–-|–-|–-|–-|–-|
  |even|0|1|0|1|
  |odd|0|0|1|1|
  |siganl|h|l|l|h|
  |value|-|-|+|+|
  h: high frequency
  l: low frequency
  -: inverted signal
  +: original signal

Advanced FSK-GMSK

• Gaussian MSK
• cut away high frequency side-band

Phase Shift Keying (PSK)

• more robust against distortion than ASK

Advanced PSK- BPSK (Binary)

• bit 0: sine wave
  bit 1: inverted sine wave
• robust
• low spectral efficiency
  • spectral efficiency: relationship between data rate and bandwidth

Advanced PSK-QPSK (Quaternary)

• Two bits coded per signal
• higher spectral efficiency than BPSK
• susceptible to distortion than BPSK
• DQPSK: Different QPSK

Quadrature Amplitude Modulation (QAM)

• Amplitude and phase modulation
• code \(m\) bits per siganl, \(m=2:\) QPSK
• \(m\) increases \(\Rightarrow\) bit error rate increase
• error rate less than PSK
• Example: 16-QAM (4 bits = 1 symbol)

Signal-to-Noise ratio

• \(SNR=S/N\)
• \[\text{dezibel[dB]:}SNR_{max}=10\cdot\log_{10}(S/N)\]

Shannon Theorem

• Noise level has influence on achievable data rate
  \[R_{max}[\text{bit/s}]=B[\text{Hz}]\cdot Id(1+S/N)\text{[bit]}\]

Minimum of Shannon theorem and Nyquist theorem gives maximum achievable data rate

Hierarchical modulation to deal with noise

• DVB-T high priority stream in low priority stream
• poor reception: high priority stream (10)
  good reception: low priority stream (1001)

Demodulation

• Receiver decode signal to \(0/1\)
• Problem caused by channel imperfections
  • Carrier synchronization
  • Bit (signal) synchronization
  • Frame synchronization
  • (most) interference

Antannas

• electronic siganl \(\Leftrightarrow\) electromagnetic waves
• Ideal isotropic antanna: radiation pattern = circle/sphere
• Real antanna: size \(\propto\) wavelength
  • best reception with wavelength \(\lambda\):
    • dipole of size \(\lambda/2\)

Antanna gain

• Maximum power in direction of main lobe (葉) compare to isotropic radiator
  \[g[\text{dB}]=10\cdot\log_{10} (P_{antanna}/P_{isotropic})\]
• Whole input power is radiated
  • the smaller radiation area, the higher the signal power
• Directed antanna
• Sectorized antanna

Antanna Diversity

• two or more antannas
• to get receiver antanna gain
• Diversity combining:
  1. increase strength
  2. construct directed antannas
  • but interference

Beamforming

• connecting antannas with cables of different length
• to avoid interference
  \(\Rightarrow\) Transmit signals with different delays causing different phases. (Through a transmitter)

Signal Propagation

Receiver Power

• in vacuum \[P_r=P_t\cdot(\frac{\lambda}{4\pi d})^2\cdot G_r\cdot G_t\]
  \(P_i:\) power
  \(G_i:\) antanna gain
  \(d:\) distance
  \(r:\) receiver
  \(t:\) transmitter

• In reality \[P_r\propto (1/d)^\alpha\]
  \(\alpha\) depends on the material

• Received power influenced by

  • attenuation (衰減)
  • shadowing
  • reflection (反射)
  • refraction (折射)
  • scattering
  • disfraction (散射) at edges

Multipath propagation

• Signals may be directed on several paths to receiver
• Effects:
  • receiver has different signal strength
  • Time dispersion: signal is dispersed (分散) over time
    • constructive or destructive interference
    • inner-system interference (ISI)

Mobility

• Quick change \(\Rightarrow\) short term fading
  slow change \(\Rightarrow\) long term fading

• Conclusion

  reason	effect	Cause
  Multipath Propagation	fast fading	fast fluctuation position
  Multipath Propagation	Time dispersion Delay spread	Signal consists of several components \(\Rightarrow\) ISI
  Movement	Doppler Effect Frequency Dispersion	distance change
  Attenuation	Path loss	especially rain and fog
  Shading	slow fading	slow fluctuation

Solution to interference, ISI, movement effects

• Antanna diversity
• Signal spreading: increase bandwidth of a signal
• Orthogonal Frequency Division Multiplexing (OFDM): reduce the signal rate but send on several channels simultaneously
• Synchronization sequence: allow receiver to calculate signal distortion

Multiplexing

• Subdivide given spectrum for simultaneous use

Frequency Multiplex

• Seperation of spectrum into smaller frequency bands
• Advantages:
  1. No dynamic coordination
  2. Works also for analog signals
• Disadvantages:
  1. Waste of bandwidth
  2. Inflexible
  3. Guard spaces
• Frequency divistion duplex
  • Used for duplex communication: uplink / downlink

Time Multiplex

• Advantages:
  1. Only one carrier in a medium at any time
  2. Throughput even high for many users
• Disadvantages:
  1. Precise synchronization
  2. Guard time (waste transmission capacity)

  RTT: round-trip time

• Time division duplex
  • one frequency carrier
  • uplink / downlink at different time
  • Flexible: adjust time slots

Time and frequency multiplex

• Advantages:
  • protect against tapping
  • protect againt frequency selective interference
  • data rate > code multiplex
• Disadvantages:
  • precise coordination

Code division multiplex (CDM)

• each channel (transmitter) has a unique code
• Advantages:
  • protect against tapping and interference
  • bandwidth effeciently used
• Disadvantages:
  • complex signal regeneration, coordination, synchronization
• to avoid noise: orthogonal codes
  \[S\cdot T=0\\ S\cdot\bar{T}=0\\ S\cdot S=1\]

Spread spectrum

• Application of CMD priciples for robust transmission
  \[\text{narrowband signal}\xRightarrow[]{CMD}\text{broadband signal}\]

1. Original signal of sender
2. spreading to braodband \(\Rightarrow\) signal power is distributed
3. \(+\) broadband interference
   \(+\) narrowband interference
4. Receiver reconstruct the signal \(\Rightarrow\) narrowband interference is spread
5. Use band pass filter \(\Rightarrow\) remove interference

Directed Sequence Spread Spectrum (DSSS)

• \[\text{User data }\oplus \text{ Chipping sequence (random) = resulting signal}\]
• Advantages:
  • Reduces frequency selective fading
  • Base station can use the same frequency band \(\Rightarrow\) soft handover
• Disadvantages:
  • precise power control

  Near-far problem
  signal near base band can wipe out signal far from base band
  \(\Rightarrow\) base band has to measure and adjust transmit power

Frequency Hopping Spread Spectrum (FHSS)

• Discrete changes of carrier frequency
• Fast hopping: channels/bit
  Slow hopping: bits/channel
• Advantages:
  • frequency selective fading limited to short period
  • Simple implementation
  • Use only small portion of spectrum at any time
• Disadvantages:
  • robust < DSSS
  • simpler to tap

Orthogonal Frequency Division Multiplexing (OFDM)

• Varient of FDM: sub-channels are orthogonal
  \(f_n\) maximum power = \(f_{n-1},f_{n+1}\) minimum power
• still only one channel (merged before)
• Advantages:
  • Robust against narrowband co-channel interference
  • Robust against ISI
  • High spectral efficiency > FDM
  • Use Fast Fourier Transform (FFT)
  • Low sensitivity to time synchronization errors
  • Sub-channel can be adapted to dynamic channel conditions

Space Multiplexing

• Limited by transmit power

• Advantage:

  • No coordination

• Disadvantages:

  • No mobility
  • Only useful with other multiplexing

• Cell structure

  • Advantages:
    • Frequency re-use
    • Less transmission power
    • Robust, decentralize
  • Disadvantages:
    • Handover
    • Interference between cells

• Clusters

  • maximum distance \(D\) between 2 cells
  • \[D=R\cdot\sqrt{3k}\]
    \(R:\) cell radius
    \(k:\) cells per cluster

Coding

• BER: Bit Error Rate
• Packet error probability
  \[P_p=1-(1-\text{BER})^n\]
  \(n:\) bit length
• How to deal with high BER?
  • Error detecting code
    • CRC
  • Error correction code
    • Block error correction:
      • Hamming code
      • BCH code
      • Reed-solomon code
    • Convolutional code
      • Trellis code
      • Turbo code

Block code

• process blocks of data independently
• \((n,k)\) protect \(k\) bits data with \((n-k)\) FEC-bits
  • ex. \((255,247)\)
• Disadvantage:
  • large \(k\) causes expensive calculations periodically
  • convolutional code: generate redundant bits continuously

Convolutional code

• Dispere (分散) each bit over time
• \((n,k,K)\)
  \(K:\) constraint factor ("memory")
  \(n=f(k,K)\)
  \(n\) bits output depends on \(k\) and \(K-1\) input bits
• Example
  • \(V_{n1}=u_{n-2}\oplus u_{n-1}\oplus u_{n}\)
    \(V_{n2}=u_{n-2}\oplus u_{n}\)
    \(V_i:\) output bits
    \(u_i:\) input bits (state)

Trellis Diagram

• Hamming distance
  \[d\geq2\cdot t+1\]
  code can correct \(t\) errors
• Search trellis for possible paths

Viterbi Algorithm

• Check if received string gives a valid path through trellis
• Calculate hamming distance, choose the shortest path
• High complexity

Turbo codes

• Faster decoding > Viterbi Algorithm
• RSC codes: recursively systematic convolutional codes
  • systematic: input bit taken over as output bit
  • take interleaved bit sequence
  • guess missing check bit (or set to 0)