# 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</br>Propagation|fast fading|fast fluctuation </br>position|
|Multipath</br>Propagation|Time dispersion</br>Delay spread|Signal consists of several components </br>$\Rightarrow$ ISI|
|Movement|Doppler Effect </br> 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)