Physical Layer:Channel models and digital modulation

techniques

Maria PapadopouliDepartment of Computer Science

University of Crete

Thanks to G. Fortetsanakis for generating most of these slides

Multi-path • A signal transmitted from a transmitter may have multiple

copies traversing different paths to reach a receiver• The received signal: sum of all these multi-path signals

– The paths traversed by these signals are different (of various length)– The one at the direction of light of signal (LOS) should be the shortest

• Channel fading: interaction of these signals with each other:– If signals are in phase, they would intensify the resultant signal– Otherwise, the resultant signal is weakened due to out of phase– This phenomenon is called channel fading.

• Criteria to measure channel fading:– Doppler spread– delay spread

Delay spread

• The signals on shorter paths reach the receiver earlier than those on longer paths

• The direct effect of these unsimultaneous arrivals of signal causes the spread of the original signal in time domain

– This spread is called delay spread – It constrains the maximum transmission capacity on the wireless channel

• If the period of baseband data pulse is larger than the delay spread, inter-symbol interference (ISI) will be generated at the receiver– The data signals on two neighbouring pulse periods are received

at the same timethe receiver is unable to distinguish them

Channel models

• Large-scale propagation modelsPredict the mean signal strength at large transmitter-receiver distances

• Small-scale or fading models Characterize the rapid fluctuations of the received signal strength over

very short distances or time durations

Large-scale models

• Log distance path-loss– The average large scale path loss is a function of distance

or in dB

• d: transmitter-receiver distance • d0: reference distance

• n: path loss exponent

n

d

ddPLdPL

00 )()(

00 log10)()(

d

dndPLdBPL

Path loss exponents for different environments

Log-normal shadowing• Surrounding environment varies at different locations

Measured signals are different from the average value predicted by the log-distance model

• Path loss is random, distributed log-normally (mean distance-dependent value)

Xσ : zero mean Gaussian distributed random variable (in dB) with standard deviation σ (also in dB)

Xd

dndPLXdPLdPL

00 log10)()()(

Parameter estimation

• Parameters n & σ are estimated using linear regression

The difference between the measured and estimated path losses is minimized in a mean square error sense

Path loss model for various cities in Germany

Parameters • n=2.7 • σ= 11.8 dBcorrespond to all cities

Fading models

• Fading describes the rapid fluctuations of amplitudes, phases or multipath delays of a radio signal

Fading is caused by interference between two or more versions of the transmitted signal which arrive at the receiver at different times

Channel impulsive response

• By sending a pulse of very small duration to the channel, the impulsive response can be estimated

At the output, the duration of the pulse is extended due to multipath

Analytical expression

• If the channel is stationary over a small time interval, the channel impulsive response (h) may be written as:

• αi & θi : the amplitude & phase of the ith multipath copy

• ti corresponds to the time of arrival of the ith copy

1

0

)()exp()(N

iiiib ttjath

Time dispersion parameters• Power delay profile: Received power as a function of the excess

(additional) delay

• Mean excess delay: The first moment of the power delay profile. It can be expressed as:

P(τi) is the received power at time τi (arrival of the i-th multipath copy).

ii

iii

ii

iii

P

P

a

a

)(

)(

2

2

Time dispersion parameters (con’td)

• rms delay spread: Square root of the second central moment of the power delay profile. It can be defined as:

where

• Maximum excess delay (X dB): The time delay during which the multipath energy falls to X dB bellow the maximum.

22

ii

iii

ii

iii

P

P

a

a

)(

)( 2

2

22

2

Example of Time Dispersion

Flat Channel

Definition• When the channel gain remains the same across all the frequencies

of the transmitted signal• The channel impulsive response is approximately the same in all the

frequencies of the transmitted signal• The “manner” that the transmitted signal fades at each frequency is

the same

Frequency-selective time-varying fading

Frequency-selective time-varying fading causes a cloudy pattern to appear on a spectrogram

signal strength as grey-scale intensity

Time

Frequency

Coherence Bandwidth • The range of frequencies over which the channel is considered flat

– The channel passes all spectral components with approximately equal gain

– Frequency components have strong potential for amplitude correlation

• If the coherence bandwidth is defined as the zone over which the amplitude correlation> 0.9:

• If the coherence bandwidth is defined as the zone over which the amplitude correlation> 0.5 :

tCB 50

1

tCB 5

1

Flat vs. frequency selective fading• Flat fading: Bandwidth of transmitted signal < coherence bandwidth (Bs < Bc)

The channel gain is equal for all frequency components of the transmitted signal

• Frequency selective fading: Bandwidth of transmitted signal > coherence bandwidth (Bs > Bc)

The channel gain may vary for different frequencies of the transmitted signal

Doppler spread

When a sinusoidal pulse of frequency fc is transmitted over a multi-path channel, the received spectrum will have components in the range fc- fd to fc- fd

fd: Doppler spread:

• v: velocity of the receiver• θ: direction of arrival of the received signal• λ: wavelength

)cos(

df

Coherence time

Definition:• The time interval over which the channel impulsive response is considered

stationary

Typical assumption:

• Doppler spread and coherence time are related by the formula:

dc f

KT

K: constant in the range of 0.25 to 0.5

Fast vs. slow fading

• Fast fading: The symbol duration > the coherence time (Ts> Tc)

The channel impulsive response varies during the symbol duration

• Slow Fading: The symbol duration is much less than the coherence time (Ts<<Tc)The channel impulsive response does not change for many symbol

intervals

Fading phenomena

Coherence bandwidth

Range of frequencies of the transmitted signal

Observations• Strong destructive interference is frequently referred to as a deep

fade and may result in temporary failure of communication due to a severe drop in the channel signal-to-noise ratio

• Since different frequency components of the signal are affected independently, it is highly unlikely that all parts of the signal will be simultaneously affected by a deep fade

Rayleigh fading

In flat-fading channels, the envelope of the received signal follows a Rayleigh distribution

σ2 : time averaged power of the received signal before envelope detection

0,0

0,2

exp)(

2

2

2

r

rrr

rp

Ricean fading

• When there is a dominant signal component present, such as a line of sight propagation path, the received signal envelope follows the Ricean distribution

• A: peak amplitude of the dominant signal• I0(): modified Bessel function of the first kind & zero order

202

)(

2

2

22

)(

re

rrp

rA

Digital Modulation• Frequency Shift Keying (FSK)

– Use of different carrier frequencies to encode the various symbols

• Phase Shift Keying (PSK)– Use of a single carrier frequency– The various symbols are encoded by the phase

• Quadrature Amplitude Modulation (QAM)– Both phase & amplitude are used for encoding the various

symbols

FSK modulation

• An alphabet of M symbols is used (M = 2K for some KN)– Each symbols corresponds to a combination of K bits

• ith symbol is mapped to carrier frequency Fi = (n+i)/2T– T: symbol duration, n: an arbitrary integer

• To transmit the ith symbol, the following signal is used

elseware

TttFT

EtS i

i

0

0),2cos(2

)( E: ενέργεια amplitude

Example BFSK

• Bit 0 corresponds to:

• Bit 1 corresponds to:

)2cos(2

0tFT

E

)2cos(2

1tFT

E

T

nF

2

10

T

nF

2

21

FSK demodulation• Consider a vector space with base vectors:

• The transmitted & received signal correspond to different points on this vector space– This is due to noise & channel gain

• The largest coordinate of the received signal corresponds to the transmitted signal with high probability

MitFb ii ,...,2,1),2cos(2

BFSK demodulation

When the received signal is • bellow the dashed line, bit 0 is

transmitted

• Otherwise, bit 1 is transmitted

Assumptions:

PSK modulationAssume an alphabet of M = 2K different symbols

• To transmit the ith symbol, we send the following signal:

• Signals Si(t) are linearly dependent & can be represented by linear combination of the vectors:

elseware

TttFT

EtS ic

i

0

0),2cos(2

)(

)2cos(2

1 tFb c

)2sin(2

2 tFb c

Example BPSK

• Bit 0 corresponds to :

• Bit 1 corresponds to:

2/

2/1

)2/2cos(2 tFT

Ec

)2/2cos(2 tFT

Ec

QPSK

Assumptions:If the received signal lies in the:• 1st quadrant, 00 is transmitted

• 2nd quadrant, 01 is transmitted etc…

8PSK

• If the received signal lies in the first area it is assumed that the combination 000 is transmitted.

• If it lies in the second area it is assumed that 001 is transmitted etc.

QAM modulation

• This modulation scheme is an expansion of PSK– Again a single carrier frequency is used (Fc)

– Transmitted & received signal are represented as linear combinations of:

• Difference: not only the phase but also the amplitude of the carrier signal may vary

)2cos(2

1 tFb c

)2sin(2

2 tFb c

Example: 16QAM

• The constellation point that is closer to the received signal is assumed to correspond to the transmitted bit combination

The matched filter

• Suppose that r(t) is the received signal which corresponds at a particular symbol interval.

• The estimation of the transmitted symbol is done using the matched filter.– r(t) is multiplied with each signal that belongs to the basis of the

signal space. – The result of each multiplication is then integrated into the

whole symbol interval. – That way the coordinates of the received signal can be

estimated.

The matched filter for QPSK

PDF of the received signal• The probability that the received

signal would lie at a particular point is given by a 2-D Gaussian distribution

• The probability space of the PDF is the vector space of the signals

• The peak of the distribution corresponds to the transmitted signal

BER calculation

• To calculate BER: compute the integral of signal PDF in red zone• For 8PSK: the red zone is larger & yields higher BER• Also the additional red zones in 8PSK have large probability mass which

means that the BER is significantly higher than in QPSK

OFDM Demodulation

• At the receiver the inverse procedure is followed.• First the signal is brought down to baseband and is converted

from analog to digital. Then FFT is performed which produces the estimations of the transmitted symbols.

• Finally parallel to serial conversion is performed resulting in an estimation of the transmitted bit stream.

Orthogonal frequency division multiplexing (OFDM)

• When the bandwidth of a communication system is larger than the coherence bandwidth of the channel, ISI is introduced (Frequency Selective Fading)

• To reduce the effect of frequency selective fading, we use OFDM– The total available bandwidth is divided into N frequency bins– N is selected such that the channel frequency response is almost

constant at each bin (Flat fading)

OFDM modulation

• Bit stream is divided into N parallel sub-flows• Bits of each sub-flow are modulated using MPSK or MQAM • After modulation, symbols are mapped to points on a signal constellation

– These points can be represented as complex numbers which are then fed to the module which performs FFT-1

• The resulted signal is – converted from analog to digital– brought to the RF frequencies – fed to the antenna of the transmitter

OFDM Demodulation

At the receiver the inverse procedure is followed1. The signal is brought down to baseband 2. It is converted from analog to digital3. FFT is performed, producing the estimations of transmitted symbols4. Parallel to serial conversion is performed resulting in an estimation of the

transmitted bit stream

References

• Theodore S. Rappaport “Wireless communications principles and practice”, Chapters 4,5.

• Simon Haykin “Communication systems”, chapter 10.

• John G. Proakis Masoud Salehi “Telecommunication systems”, Chapter 8.