Analog vs. digital communication systems
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:
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.
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:
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).
Design flexibility and reconfigurability (software radio).
The input binary stream is segmented in blocks of K bits (messages or symbols)
Every symbol mi is mapped into a waveform si(t)
Each waveform si(t) carries K = log2M bits
The bit-rate Rb=1/Tb (bit/s) is the rate with which bits are presented at the modulator input.
The symbols mi are generated each Ts = K *Tb seconds → the symbol-rate is Rs = 1/Ts (symbol/s or baud).
Rs represents also the rate with which the waveforms si(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 Es
Since a waveform is transmitted every Ts seconds, the average transmitted power is Pt = Es /Ts=Eb/Tb.
Due to channel and noise effects, the RX waveform is different from the TX one:
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):
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/log2M.
The signal to noise-ratio at the RX is defined as
Pt and Pr are related through path loss: Pr = Pt/Pl.
In the following we assume for simplicity that Pl = 1 (0 dB) so that Pr = Pt:
SINR more common measure when also interference is present →
where Pi = interference power.
Instead of SNR/SINR, the energy contrast is often utilized:
Relation with received SNR (neglecting path loss):
Since BTs ≈ 1 for many modulations → SNR ≈ γs.
For many modulation schemes and channel models the SER can be expressed as a function of γs and the BER as a function of γb.
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 si(t) is a linear function of the symbol mi.
Nonlinear modulation: the waveform si(t) is a nonlinear function of the symbol mi.
The waveforms si(t) have a significant spectral content around f = 0:
The waveforms si(t) have a significant spectral content around f = ± fc ≠ 0:
A sinusoidal carrier can be modulated by changing its three basic features: see figure on the top.
Some combinations possible:
QAM = Quadrature Amplitude Modulation.
Modulation schemes can be conveniently analyzed by resorting to geometrical interpretation.
Every set of M signals s1(t), …sM(t) can be represented as the linear combination of N ≤ M orthonormal functions:
N = dimensionality of the signal set.
N ≈ 2BTs for signals of duration Ts and bandwidth B (dimensionality theorem).
φ1(t), …φN(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 fc Ts >> 1 one has:
Vector si = (si1,…,siN) associated to signal si(t) can be represented in an N-dimensional Euclidean space → signal space.
Signal constellation: the set of vectors s1, s2 , …,sM representing the waveforms si(t), i =1,2,…, M.
Linear algebra concepts can be utilized:
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