Physical layer

Analog vs. digital communication systems

Modulation/demodulation

Modulation parameters:

- bit-rate and symbol rate
- BER and SER
- signal to noise ratio (SNR) and energy contrast

Modulation classes:

- baseband vs. passband modulation

Geometric representation of signals and constellations

The physical layer deals with “physical” aspects of data transmission (e.g., voltages, waveforms, noise, etc.).

Due to the hostile radio channel, the task of the physical channel in wireless communications is **more challenging** than in wired ones:

- advanced signal processing and coding techniques must be used to assure performance comparable to wired networks

First wireless systems (e.g., 1G cellular networks) were mainly based on **analog** communication techniques.

Starting from the 2G cellular networks, the shift is toward **digital** communication techniques.

The **source** is **analog** (e.g. voice, video, music)

The **modulator** adapts the signal emitted by the source to the characteristics of the **physical channel.**

**Example**: it performs frequency upconversion of the source signal when the channel operates at radio frequencies.

The **demodulator** reverses the operations made by the modulator in order to deliver at the destination a **close copy** of the source signal.

More complicated (but also more powerful) than analog communication systems.

Classical scheme thanks to **Claude Shannon**, the “inventor” of information theory.

The **source** is **digital** (data, email, text).

Analog sources can be converted in digital form (a stream of 0/1) by means of A/D conversion (sampling + quantizing + coding).

The reverse operation (D/A conversion) is performed at the **destination**.

The goal of **source encoding** is to eliminate/reduce the data redundancy of the source.

**Lossless encoding**: perfectly reversed by the source decoder (e.g. zip/rar file compression).

**Lossy encoding**: not perfectly reversed by the decoder, some information is “lost” in the process (e.g., MP3, JPG, MPEG2).

The goal of **channel encoding** is to introduce a **controlled** amount of redundancy to achieve reliable transmission over the channel (e.g., parity bits).

**Error detection**: redundancy is used only to **detect** channel errors and trigger retransmissions in ARQ (Automatic Repeat Request) schemes.

**Forward error correction (FEC)**: redundancy is used also to **correct** channel errors.

The **modulator** converts the input bits into an electrical/optical waveform suitable for transmission over the channel.

The **demodulator** performs the reverse operation at the receiver.

**Modem=modulator/demodulator.**

In digital modulation schemes the waveforms transmitted over the channel belong to a **finite** set.

It provides the electrical/optical/radio connection between the source and the destination.

It introduces several degradations in the transmitted signal:

- time/frequency dispersion due to multipath and mobility
- fading
- noise and interference
- other impairments

Cost-effectiveness due to advances in electronic technologies (VLSI, DSP, FPGA, etc.).

Noise immunity and robustness to channel impairments.

Multiplexing of heterogeneous services (voice, video and data).

Security and privacy (encryption).

Error correction techniques (channel coding).

Source compression techniques (source coding).

Spectral efficiency.

Design flexibility and reconfigurability (software radio).

The input binary stream is segmented in blocks of K bits (messages or **symbols**)

- m
_{i}= (b_{1}b_{2}…b_{K}) M=2^{K}different symbols

Every symbol m_{i} is mapped into a waveform s_{i}(t)

- M = number of different waveforms (
**cardinality**of modulation).

Each waveform s_{i}(t) carries K = log_{2}M bits

- binary modulation (M=2) is the simplest case.

The **bit-rate **R_{b}=1/T_{b} (bit/s) is the rate with which bits are presented at the modulator input.

The symbols m_{i} are generated each T_{s} = K *T_{b} seconds → the **symbol-rate** is R_{s} = 1/T_{s} (symbol/s or baud).

R_{s} represents also the rate with which the waveforms s_{i}(t) are generated and sent over the physical channel (**signaling rate**).

Strings of K=2 bits are mapped to M=4 delayed versions of the same pulse.

Adjacent pulses correspond to couples differing in one bit (**Gray mapping**) → minimize the bit error rate (BER).

The average energy of the transmitted waveforms is E_{s}

- evaluate the energy for each waveform and then average over all waveforms
- the average energy per bit is E
_{b}= E_{s}/K.

Since a waveform is transmitted every T_{s} seconds, the average **transmitted power** is P_{t} = E_{s} /T_{s}=E_{b}/T_{b}.

Due to channel and noise effects, the RX waveform is different from the TX one:

- the task of the demodulator is to determine which one of the M waveforms s
_{i}(t) was actually transmitted - classical problem in
**detection theory**

The performance of the demodulator is measured by its **bit/symbol error rate** (expressed as a probability).

The most common performance measure is the **bit error rate **(BER):

- it depends not only on the modulation scheme and of the channel, but also on mapping between bits and symbols/waveforms
- target BER values are application-dependent

Simpler to evaluate is the **symbol error rate** (SER):

It represents the probability that the demodulator decides on a different waveform than the one actually transmitted.

When M=2 → BER = SER.

When M>2 with Gray mapping → BER ≈ SER/log_{2}M.

The** signal to noise-ratio** at the RX is defined as

.

.

P_{t} and P_{r} are related through path loss: P_{r} = P_{t}/P_{l}.

In the following we assume for simplicity that P_{l} = 1 (0 dB) so that P_{r} = P_{t:}

- B = one-sided bandwidth of s(t);
- P
_{r}= power of s_{r}(t); - N
_{0}/2 = PSD of white noise n(t) (measured in W/Hz=J).

SINR more common measure when also interference is present →

where P_{i} = interference power.

Instead of SNR/SINR, the energy contrast is often utilized:

where:

- E
_{s}average energy of s(t) (energy per TX symbol) - E
_{b}= E_{s}/K=E_{s}/(log_{2}M) (energy per TX bit)

Relation with received SNR (neglecting path loss):

Since BT_{s} ≈ 1 for many modulations → SNR ≈ γ_{s}.

For many modulation schemes and channel models the SER can be expressed as a function of γ_{s} and the BER as a function of γ_{b}.

**Modulation without memory**: the waveform emitted in the k-th signaling interval depends only on the symbol emitted in the same interval.

** Modulation with memory**: the waveform emitted in the k-th signaling interval depends also on symbols emitted in previous intervals.

** Linear modulation**: the waveform s_{i}(t) is a linear function of the symbol m_{i}.

** Nonlinear modulation**: the waveform s_{i}(t) is a nonlinear function of the symbol m_{i}.

The waveforms s_{i}(t) have a significant spectral content around f = 0:

- suitable for short-range transmission over wired media
- used by some IR schemes and UWB transmission over wireless media
- linear modulations (PAM=Pulse Amplitude Modulation) versus nonlinear modulation (PPM=Pulse Position Modulation, PDM/PWM=Pulse Duration/Width Modulation)

The waveforms s_{i}(t) have a significant spectral content around f = ± fc ≠ 0:

- typically obtained via multiplication of a baseband signal with a sinusoidal signal with
**carrier frequency**f_{c} - suitable for long-range transmission over wired media or for wireless radio transmission

A sinusoidal carrier can be modulated by changing its three basic features: **see figure on the top**.

Some combinations possible:

- Example: QAM = ASK + PSK.

QAM = Quadrature Amplitude Modulation.

Modulation schemes can be conveniently analyzed by resorting to geometrical interpretation.

Every set of M signals s_{1}(t), …s_{M}(t) can be represented as the linear combination of N ≤ M orthonormal functions:

.

.

N = **dimensionality** of the signal set.

N ≈ 2BT_{s} for signals of duration T_{s} and bandwidth B (dimensionality theorem).

φ_{1}(t), …φ_{N}(t) can be found with the **Gram-Schmidt** procedure.

Many bandpass modulations schemes can be described using only N=2 orthonormal functions:

It can be easily checked that for f_{c} T_{s} >> 1 one has:

Vector **s**_{i} = (s_{i1},…,s_{iN}) associated to signal s_{i}(t) can be represented in an N-dimensional Euclidean space → **signal space**.

** **

**Signal constellation**: the set of vectors **s _{1}**,

Linear algebra concepts can be utilized:

- ||
**s**|| = vector norm_{1} - ||
**s**–_{1}**s**|| = distance between two vectors_{2} - <
**s**,_{1}**s**> = scalar product between two vectors_{2} - One-to-one correspondence with signal operations (e.g., the squared norm is the energy of the signal)

*3*. Current and emerging wireless systems

*5*. Shadowing

A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (selected parts of chaps. 5 and 6)

C.E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal, vol. 27, pp. 379-423, 623-656, July, October 1948

Supplementary material eventually available on the website

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion