Physical Layer:
Channel models and digital modulation
techniques
Maria Papadopouli
Department 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 timethe receiver is unable to distinguish them
Channel models
• Large-scale propagation models
Predict 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
n
 d 
 in dB
PL( d )  PL( d 0 ) or
 d0 
• d: transmitter-receiver distance
• d0: reference distance
• n: path loss exponent
d 
PL(dB)  PL(d 0 )  10n log  
 d0 
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)
d 
PL(d )  PL(d )  X   PL(d 0 )  10n log    X 
 d0 
Xσ : zero mean Gaussian distributed random variable (in dB) with standard
deviation σ (also in dB)
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 dB
correspond 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:
N 1
hb (t )   ai exp( j i ) (t  ti )
i 0
• αi & θi : the amplitude & phase of the ith multipath copy
• ti corresponds to the time of arrival of the ith copy
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:
 ai  i
2

i
a
i
i
 P( )

 P( )
i
i
i
2
i
i
P(τi) is the received power at time τi (arrival of the i-th multipath copy).
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:

    
2
a 

a
2
where
2
i
2
i
i
2
i
i
2
 P( )

 P( )
i
2
i
i
i
i
• Maximum excess delay (X dB): The time delay during which the
multipath energy falls to X dB bellow the maximum.
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
Frequency
signal strength as grey-scale intensity
Time
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:
BC 
1
50 t
• If the coherence bandwidth is defined as the zone over which the amplitude
correlation> 0.5 :
BC 
1
5 t
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:
fd 

cos( )

• v: velocity of the receiver
• θ: direction of arrival of the received signal
• λ: wavelength
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:
Tc 
K
fd
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
Range of frequencies
of the transmitted signal
Coherence bandwidth
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
 r

r2
 2 exp  

2 2

p(r )  

0,


σ2
:

,

0r 
r0
time averaged power of the received signal before envelope
detection
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
r
p(r )  2 e

 ( A2  r 2 )
2 2
 r 
0  2 
 
• A: peak amplitude of the dominant signal
• I0(): modified Bessel function of the first kind & zero order
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 KN)
– 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
 2E

cos( 2Fi t ), 0  t  T
Si (t )   T

0
elseware

E: ενέργεια
amplitude
Example BFSK
n 1
F0 
2T
F1 
• Bit 0 corresponds to:
2E
cos( 2F0t )
T
• Bit 1 corresponds to:
2E
cos( 2F1t )
T
n2
2T
FSK demodulation
• Consider a vector space with base vectors:
bi 
2
cos( 2Fi t ), i  1,2,..., M

• 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
BFSK demodulation
Assumptions:
When the received signal is
• bellow the dashed line, bit 0 is
transmitted
• Otherwise, bit 1 is transmitted
PSK modulation
Assume an alphabet of M = 2K different symbols
• To transmit the ith symbol, we send the following signal:
 2E

cos( 2Fct  i ), 0  t  T
Si (t )   T

0
elseware

• Signals Si(t) are linearly dependent & can be represented by
linear combination of the vectors:
2
b1 
cos(2Fct )

b2 
2
sin( 2Fct )

Example BPSK
   / 2
1   / 2
• Bit 0 corresponds to :
2E
cos( 2Fct   / 2)
T
• Bit 1 corresponds to:
2E
cos( 2Fct   / 2)
T
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:
b1 
2
cos( 2Fct )

b2 
2
sin( 2Fct )

• Difference: not only the phase but also the amplitude of the
carrier signal may vary
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
•
The resulted signal is
– These points can be represented as complex numbers which are then fed
to the module which performs FFT-1
– 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 followed
1. The signal is brought down to baseband
2. It is converted from analog to digital
3. FFT is performed, producing the estimations of transmitted symbols
4. 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.