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Giacinto Gelli » 9.Introduction to digital communications


Physical layer
Analog vs. digital communication systems
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

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.

Analog communication system

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.

Digital communication system

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.

Source encoding/decoding

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).

Channel encoding/decoding

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.

Physical channel

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

Advantages of digital transmission

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).

Digital modulator

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

  • mi = (b1b2…bK) M=2K different symbols

Every symbol mi is mapped into a waveform si(t)

  • M = number of different waveforms (cardinality of modulation).

Each waveform si(t) carries K = log2M bits

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

Bit-rate, symbol-rate, signaling rate

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).

Example: binary PAM (M=2)

Example: quaternary PPM (M=4)

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).

Energy and power

The average energy of the transmitted waveforms is Es

  • evaluate the energy for each waveform and then average over all waveforms
  • the average energy per bit is Eb = Es /K.

Since a waveform is transmitted every Ts seconds, the average transmitted power is Pt = Es /Ts=Eb/Tb.

Digital demodulator

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 si(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):

\[\mbox{BER} = P_b = P\{ \hat{b}_k \ne b_k \}\]

  • 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):

\[\mbox{SER} = P_s = P\{ \hat{m}_i \ne m_i \}\]

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.

Signal-to-noise ratio (SNR)

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


\[\mbox{SNR} = \frac{\mbox{useful signal power}}{\mbox{noise power}} = \frac{P_r}{P_n} =\frac{P_r}{N_0 B}\]


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:

  • B = one-sided bandwidth of s(t);
  • Pr = power of sr(t);
  • N0/2 = PSD of white noise n(t) (measured in W/Hz=J).

SINR more common measure when also interference is present → \[\mbox{SINR} = \frac{P_r}{N_0 B + P_i}\]

where Pi = interference power.

Energy contrast per symbol/per bit

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

\[\gamma_s = \frac{E_s}{N_0}\qquad\mbox{(per symbol)}\qquad\qquad\gamma_b = \frac{E_b}{N_0}\qquad\mbox{(per bit)}\]


  • Es average energy of s(t) (energy per TX symbol)
  • Eb = Es/K=Es/(log2M) (energy per TX bit)

Relation with received SNR (neglecting path loss):

\[\mbox{SNR} = \frac{P_r}{N_0 B} = \frac{P_t}{N_0 B} = \frac{E_s/T_s}{N_0 B}=\frac{\gamma_s}{B T_s}\]

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 classes

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.

Baseband modulation

The waveforms si(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)

Passband modulation

The waveforms si(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 fc
  • suitable for long-range transmission over wired media or for wireless radio transmission

Types of passband modulations

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.

Geometrical representation of signals

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:

\[s_i(t) = \sum_{j=1}^{N} s_{ij} \, \phi_j(t)\]


\[s_{ij} = \int_{0}^{T_s} s_i(t) \, \phi_j(t) \, d t\]


\[\int_{0}^{T_s} \phi_i(t) \, \phi_j(t) \, d t = \delta_{ij} =\left\{\begin{array}{ll}1 & i = j \\0 & i \ne j\end{array}\right.\]

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.

Example: sine and cosine (N=2)

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

\phi_1(t) &=& (2/T_s)^{1/2} \cos( 2 \pi f_c t) \nonumber \\

\phi_2(t) &=& -(2/T_s)^{1/2} \sin( 2 \pi f_c t) \nonumber

It can be easily checked that for fc Ts >> 1 one has:

\[\int_{0}^{T_s} \phi_1(t)^2 \, dt = \int_{0}^{T_s} \phi_2(t)^2 \, dt = 1\]

\[\int_{0}^{T_s} \phi_1(t) \, \phi_2(t) \, d t = 0\]

Signal constellation

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:

  • || s1|| = vector norm
  • || s1s2 || = distance between two vectors
  • < s1 , s2 > = scalar product between two vectors
  • One-to-one correspondence with signal operations (e.g., the squared norm is the energy of the signal)

Example: QPSK

Quadrature Phase Shift Keying.

M=4 → each symbol carries two bits    \[s_i(t) = A \, (2/T_s)^{1/2} \cos( 2 \pi f_c t + \theta_i)\]

  • θi=2π(i-1)/4+Φ0, i=1,…,4.
  • Φ0 constellation displacement

\[s_i(t) = A \, (2/T_s)^{1/2} \cos( 2 \pi f_c t) \cos(\theta_i) -A \, (2/T_s)^{1/2} \sin( 2 \pi f_c t) \sin(\theta_i)\]

\phi_1(t) &=& (2/T_s)^{1/2} \cos( 2 \pi f_c t) \nonumber \\

\phi_2(t) &=& -(2/T_s)^{1/2} \sin( 2 \pi f_c t) \nonumber

\[\mathbf{s}_i = (s_{i1}, s_{i2} ) = [ A \, \cos(\theta_i), A \sin(\theta_i) ]

I materiali di supporto della lezione

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

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