ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Chapter 5
Chapter 5:
5.1
5.2
5.3
5.4
5.5
5.6
ECE 6640
Carrier and Symbol Synchronization
Signal Parameter Estimation
Carrier Phase Estimation
Symbol Timing Estimation
Joint Estimation of Carrier Phase and Symbol
Timing
Performance Characteristics of ML Estimators
Bibliographical Notes and References
Problems
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
290
290
295
315
321
323
326
327
2
Synchronization Issues
• There are times when the coherence assumption is invalid.
– The channel can introduces random changes to the signal as either
a random attenuation or a random phase shift. Chapter 10 deals
with equalization and Chapter 13 deals with fading channels.
– Alternately, imperfect knowledge of the signals at the receiver
arises when the transmitter and the receiver are not perfectly
synchronized. It can use only signals in the form of {sm(t − td )},
where td represents the time difference and is model as a random
variable or as a random received signal phase.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
3
Signal Parameter Estimation
• Due to the unknown “time-of-flight” of the RF signal.
• Due to the potential system digital sampling times
compared to the “optimal” sampling times.
– a non-integer number of clocks or samples per symbol can easily
exist.
– system time is based on “internal” clock oscillators that are never
“perfect”.
• We must be able to estimate and correct for carrier
frequency offsets and symbol timing offsets.
• A means to estimate these parameters is required for
optimal processing … coherent detection.
ECE 6640
4
Phase Parameter
•
The received signal is based on time, time offset and phase offset.
– To simplify the notation, we let θ denote the parameter vector
{φ, τ}, so that s(t; φ, τ) is simply denoted by s(t; θ).
•
Two criteria that are widely applied to signal parameter estimation: the
maximum-likelihood (ML) criterion and the maximum a posteriori
probability (MAP) criterion.
– In the MAP criterion, the signal parameter vector θ is modeled as random
and characterized by an a priori probability density unction p(θ).
– In the maximum-likelihood criterion, the signal parameter vector θ is
treated as deterministic but unknown.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
5
Phase Estimation
• The joint PDF of the random variables (r1 r2 · · · rN ) in the
expansion can be expressed as p(r|θ).
– Then, the ML estimate of θ is the value that maximizes p(r|θ).
– The MAP estimate is the value of θ that maximizes the a posteriori
probability density function
– if there is no prior knowledge of the parameter vector θ, we may
assume that p(θ) is uniform (constant) over the range of values of
the parameters. In such a case, the MAP and ML estimates are
identical.
– In the textbook treatment of parameter estimation given below, we
view the parameters φ and τ as unknown, but deterministic. Hence,
we adopt the ML criterion for estimating them.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
6
ML Estimation
• In the ML estimation of signal parameters, we require that
the receiver extract the estimate by observing the received
signal over a time interval T0 ≥ T , which is called the
observation interval.
– One shot estimation may be useful and sufficient.
– Typically, estimation is performed continuously.
• It is possible to derive the parameter estimates based on the
joint PDF of the R.V. but it is convenient to deal directly
with the signal waveforms
– develop a continuous-time equivalent
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
7
ML Estimation
• The pdf is
– where
– expressed in terms of the time signals
– maximization of p(r|θ) with respect to the signal parameters θ is
equivalent to the maximization of the likelihood function.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
8
Example of Receivers with
Parameter Estimation
•
•
•
•
BPSK: Figure 5.1-1
MPSK: Figure 5.1-2
M-ary PAM: Figure 5.1-3
QAM: Figure 5.1-4
• Notice the similarity of functional diagrams.
• Significant processing is implied in the receivers in
addition to symbol filtering/correlation and detection
ECE 6640
9
BPSK and MASK Receiver
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
10
MPSK and QAM Receiver
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
11
Carrier Phase Estimation
Two approaches:
• Embedded a carrier in the waveform for detection and
estimation
– time multiplexed with data symbols
– a pre-amble or post-amble to a burst
– use a phase-locked loop (PLL) to lock onto the carrier
• Derive the carrier phase estimate from the symbols
– does not require multiplexing or non-data signaling
– more complex, but usually the performed method
ECE 6640
12
Carrier Phase Errors
•
QAM and PSK Signal representation
– Mixing by a carrier with phase error
– Results in in-phase and quadrature symbols with phase offsets
•
The offset misadjust detected signal vectors. There is a reduction in the
desired signal component value and a potential for cross-talk between
the in-phase and quadrature-phase components.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
13
ML Carrier Phase Estimation
•
Assume that the delay τ is known and, in particular, we set τ = 0. The
function to be maximized is the likelihood function given in Equation
5.1–8.With φ substituted for θ, this function becomes
•
The first term of the exponential factor does not involve the signal
parameter φ. The third term, which contains the integral of s2(t; φ), is a
constant equal to the signal energy over the observation interval T0 for
any value of φ.
Only the second term, which involves the cross correlation of the
received signal r (t) with the signal s(t; φ), depends on the choice of φ.
•
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
14
ML Carrier Phase Estimation (cont)
•
Therefore, the likelihood function (φ) may be expressed as two
“constants” and a term of interest
– Note that the “relative gain scaling” is represented as a constant as well.
•
As the exponential function is monotonic, we can concentrate on
maximizing the argument
•
Moving forward an example helps to define a process …
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
15
ML Carrier Phase Est. Example
• Example 5.2-1: Carrier phase estimation for an
unmodulated carrier.
s t , LO   cos2  f c  t  LO 
r t   A  cos2  f c  t  err   nt 
2
 r t   cos2  f c  t  LO 
N 0 TC
2A
 L   
 A  cos2  f c  t  err   nt  cos2  f c  t  LO   dt
N 0 TC
 L   
 L   
2A
2A
  cos2  f c  t  err   cos2  f c  t  LO   dt 
 nt   cos2  f c  t  LO   dt
N 0 TC
N 0 TC
 L   
 L   
A
2A
  coserr  LO   cos4  f c  t  err  LO  dt 
 n2 t   dt
N 0 TC
N 0 TC
A
A
2A
  coserr  LO   dt 
  cos4  f c  t  err  LO   dt 
 n2 t   dt
N 0 TC
N 0 TC
N 0 TC
 L   
ˆML
ECE 6640
A
 coserr  LO   dt
N 0 TC
 A

 N   sin err  LO   dt 
0 TC
  tan 1  sin err  LO    err  LO
 tan 1 
 A

 coserr  LO  
 N   coserr  LO   dt 
 0 TC

Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
16
Estimators:
PLL or One Shot Direct Computations
• Based on the error oscillator model, the following
estimators can be used.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
17
Phase Lock Loop (PLL) Operation
•
•
PLLs are discussed in the time domain and then, with assumptions that
the loop is in or near lock, analyzed in the Laplace domain.
Time domain first …
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
18
PLL Laplace Domain
• When the loop is locked, it can be analyzed in the phase or
phase error domain as a linear system.
• In integrator in the feedback loop allows errors to be
zeroed in the steady state response of the circuit.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
19
PLL Loop Filter Based Responses
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
20
Noise in PLL
•
The PLL approach functions in a sufficiently high SNR environment.
In this region, there is a derived noise term to the estimate that is
presented and discussed.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
21
Decision-Directed Loops
•
When the signal carries the information sequence, the estimation
process is more complex.
– We must either use the probabilistic properties of the information. or
– We have to assume symbols are known or correctly detected
(thereby the reference to decision directed).
•
If we know the information,
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
22
Decision-Directed Loops
•
This approach has isolated the phase term so that
•
It is easily shown (Problem 5.10) that the mean value of φML is φ, so
that the estimate is unbiased. Furthermore, the PDF of φML can be
obtained (Problem 5.11) by using the procedure described in Section
4.3–2.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
23
PAM Decision-Directed Loop
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
24
QAM Decision-Directed Loop
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
25
MPSK Decision Directed PLL
This M-phase tracking loop has a phase
ambiguity of 360/M, necessitating the need to
differentially encode the information sequence
prior to transmission and differentially decode
the received sequence after demodulation to
recover the information.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
26
Matlab Simulation
•
MATLAB comm.CarrierSynchronizer System object
•
The CarrierSynchronizer System object™ compensates for carrier
frequency and phase offsets for single carrier modulation schemes.
– [1] Rice, Michael. Digital Communications: A Discrete-Time Approach.
Upper Saddle River, NJ: Prentice Hall, 2009, pp. 359–393.
– [2] Huang, Zhijie, Zhiqiang Yi, Ming Zhang, and Kuang Wang. "8PSK
Demodulation for New Generation DVB-S2." International Conference on
Communications, Circuits and Systems, 2004. ICCCAS 2004. Vol. 2,
2004, pp. 1447–1450.
ECE 6640
MATLAB Material, see
http://www.mathworks.com/help/comm/ref/comm.carriersynchronizer-class.html
27
Non-Decision Directed Loops
•
•
The first approach described assumes symbol synchronization …
The squaring loop (or nth power loop) functions by taking the input to
the nth power where any symbol phase ambiguity would be removed.
An n x fc frequency reference is formed and divided by n before being
used for coherent signal demodulations.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
28
Costas Loop
•
Another method for generating a properly phased carrier for a doubles
ideband suppressed carrier signal is illustrated by the block diagram
shown in Figure 5.2–15. This scheme was developed by Costas (1956)
and is called the Costas loop.
– Highly used and useful in analog communications for carrier phase
recovery. This “structure” has already been seen in the other approaches!
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
29
Symbol Timing Estimation
• In a digital communication system, the output of the
demodulator must be sampled periodically at the symbol
rate, at the precise sampling time instants tm = mT+τ ,
where T is the symbol interval and τ is a nominal time
delay that accounts for the propagation time of the signal
from the transmitter to the receiver.
• To perform this periodic sampling, we require a clock
signal at the receiver. The process of extracting such a
clock signal at the receiver is usually called symbol
synchronization or timing recovery.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
30
Maximum-Likelihood Timing
Estimation
For PAM, QAM
and MPSK
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
31
Early-Late Gate Synchronization
•
Another non-decision-directed timing estimator exploits the symmetry
properties of the signal at the output of the matched filter or correlator.
– The output of the filter matched to s(t) attains its maximum value at time t
= T.
– Sample early, at t = T−δ and late at t = T + δ. The absolute values of the
early samples |y[m(T − δ)]| and the late samples |y[m(T + δ)]| will be
smaller than the peak. Use them to align the time delay.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
32
Early-Late Gate Diagram
• The early–late gate synchronizer described above is a nondecision-directed estimator of symbol timing that
approximates the maximum-likelihood estimator.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
33
Joint Estimation of Carrier Phase and
Symbol Timing
•
Based on the similarity of structures the processes can (should) be
merged.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
34
Performance
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
35