K. C. Chen, for NTUEE Course “Mobile Communications” Only
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2 Frequency Estimation
Once the receiver lowers the frequency into baseband range, the immediate challenge
is to achieve synchronization so that other signal processing and detection/decoding
can proceed. As the received waveform has ambiguity in time,
frequency/phase/carrier, we have to decide what to do first to effectively extract
useful signal. This is a critical and fundamental step to design receiver architecture
and algorithms.
In case the sampling period is Ts and we suppose the overall phase θ T (t ) = Ωt + θ ,
the maximal frequency ambiguity can be represented as Ω max . Frequency
offset/ambiguity can come from oscillator error or instability between transmitter and
receiver, LNB, Doppler spread due to fading and mobility, etc. We may come out a
rule of thumb: in case | ΩT |<< 1 , we usually first recover the timing and perform
frequency estimation in a time-directed (TD) mode. Otherwise, we estimate frequency
first to recover carrier, then recover timing, which is known as non-time-directed
(NTD) mode. Along with well known data-aided (DA) (i.e. with pilot tones/signals or
preamble as reference signal) and non-data-aided (NDA), we can proceed frequency
estimation in different combination of ways.
2.1 NDA-NTD Frequency Control
The most fundamental frequency estimation might be non-data-aided and
non-time-directed (NDA-NTD), which has been used in a wide range of
communications, broadcasting, and other application scenarios. In this section, we are
introducing a few technological milestones in automatic frequency control (or
tracking) in digital communications.
2.1.1 Costas Loop
The most well known frequency/carrier recovery scheme might be Costas loop.
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Costas Loop
LPF
cos
VCO
Loop
Filter
Data
Decision
90o phase
shift
sin
Mobile Communications
LPF
KC Chen, NTU EE
1
Figure 2-1: Costas Loop
The major application constraint for Costas loop is small loop bandwidth like the
same reason for typical phase-locked loop (PLL), which suggests small pull-in and
locking frequency range. In case the maximum possible frequency deviation is only a
small fraction of signal bandwidth, Costas loop is simple and effective.
2.1.2 Automatic Frequency Control (AFC) Algorithms
Generally speaking, automatic frequency control (AFC) or frequency estimation or
tracking can be considered as the general structure of the following figure.
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3
I
X
H(f)
D(f)
X
AFC Discriminator
(Frequency Detector)
& Filtering
r(t)
X
H(f)
D(f)
+
+
-
e(t)
X
Q
VCO
Loop
Filter
Figure 2-2 General AFC Loop Configuration
The key of design is how to realize the AFC discriminator. Traditionally, there are a
few common approaches:
n
n
n
Delay-and-multiply
Cross-product
Discrete Fourier transform
In many cases, synchronizers adopt non-linearity to remove modulated signal
information and to generate desirable spectrum features. Delay-and-multiply is a well
known structure for this purpose.
The S-curve of the differentiate-and-multiply AFC (AFC-Di) has pull-in region
around 0.75 BLP and linear contraction region larger than 0.5 BLP , where BLP is the
bandwidth of low pass filter. In digital realization, we are likely to use interpolation of
samples to animate differentiation, which would result in some reduction of pull-in
region and lock-in region.
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I
X
LPF
d/dt
X
+
+
-
r(t)
X
LPF
d/dt
e(t)
X
Q
VCO
Loop
Filter
Figure 2-3 Differentiating AFC
A more practical digital realization is to replace differentiators by delays. Since the
frequency is the differentiation of phase, it is also very reasonable for such a
substitution. The LPFs of AFC-Di are implemented by integrate-and-dump circuits
that can be considered as a sort of LPF, and we call this scheme as cross-product AFC
(AFC-CP). AFC-CP is very suitable for digital implementation and suitable for
tracking CW or FSK signals.
However, the effective loop bandwidth to pull-in and to lock-in signal is less than that
of AFC-Di. Typical discriminator curve can effectively pull-in and lock-in around
0.2 / δ .
To more fit QPSK signals, we may introduce an angle doubling mechanism by
squaring operation and other techniques discussed in later section.
DFT approach requires timing information to execute DFT (or FFT in practical
implementation) and we will discuss in NDA-TD section.
K. C. Chen, for NTUEE Course “Mobile Communications” Only
I
kδ
2BLP∫
X
(k−1)δ
Delay
δ
dt
5
X
+
+
-
r(t)
X
Q
kδ
2BLP∫
(k−1)δ
Delay
δ
dt
X
N
∑
Loop
Filter
VCO
e(t)
1
Figure 2-4 AFC-CP
2.1.3 Quadri-correlator Frequency Control
In [1], two frequency detector structures (quadriccorrelator and rotational frequency
detector) are described, and later modified into widely applied AFC for possible large
frequency deviation, which will be discussed hereafter.
FDD
I
X
H(f)
D(f)
X
+
+
-
r(t)
X
H(f)
D(f)
Q
VCO
Figure 2-5
Loop
Filter
X
e(t)
K. C. Chen, for NTUEE Course “Mobile Communications” Only
6
Quadricorrelator AFC has a balanced structure and is very similar to AFC-CP, except
different filter design. Such an AFC family is effective and robust in tracking
frequency. Compared with Costas loop, such a mechanism has wider frequency
pull-in and locking range, at half of the signal bandwidth (or symbol rate more
precisely).
2.1.4 Maximum Likelihood Frequency Estimation
In certain wireless communication systems (most likely in satellite communications),
frequency deviation can be significant and even larger than signal bandwidth.
Effective frequency estimation is greatly needed to ensure coherent communications
usually with better performance.
In the presence of frequency offset Ω , phase process could be modeled as
θ T (t ) = Ωt + θ . It is required that the frequency range
| ω |≤ 2πB + | Ω max |
(2.10)
where B is 1-side signal bandwidth and ± Ω max is the maximum frequency uncertainty.
The sampling rate must satisfy
|Ω |
1
> B + max
2Ts
2π
(2.11)
s f (t , Ω) = s (t ) ⊗ c (t ) ⊗ f (t )e jΩt is the frequency-translated signal s f (t )e jΩt , and
N −1
s f (t , Ω) = ∑ an g (t − nT − εT )e j (Ωt +θ )
(2.11)
n =0
The matched filter output
z n (ε , Ω) =
∞
∑r
k = −∞
f
(kTs )e − jΩkTs g MF (nT + εT − kTs )
(2.12)
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Modulator
s(t )e jθT ( t )
C ( f ) or c(t )
Channel
Pre-filtering
F(f) or f(t)
ˆ
εˆ
gMF (kTs )
X
kT s
Detection
e − jθ
Interpolator
Decimator
X
ˆ
e − jΩkTs
Figure 2-6 Frequency Estimation
For NTD case, we start from low SNR approximation of likelihood function.
L(ε , Ω) =
L
∑ | z (lT + εT , Ω) |
l =− L
2
= c0 + 2 Re{c1e j 2πε } + random disturbanc e
(2.13)
The value Ω̂ which maximizes the coefficient c0 (Ω) is an unbiased estimate, as
the following proposition.
Proposition 2.1: Under the conditions,
(i)
Sampling rate fulfills 1 / Ts > 2(1 + α ) / T (i.e. twice the rate required for data
(ii)
(iii)
path)
T / Ts = M s is an integer
Data {an } i.i.d.
Then,
ˆ = arg max
Ω
Ω
LM s −1
∑ | z (lT , Ω) |
l = − LM s
defines an unbiased estimate of frequency.
s
2
(2.14)
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Consequently, our exploration of maximum likelihood (ML) frequency estimator is
similar to the ML timing estimator.
We usually do NOT implement the ML frequency estimator. Instead, we adopt the
following PLL-type structure to (sub-)optimally realize ML frequency estimator.
Signal
Matched
Filter
r(t)
FDD
Error
Generator
X
e(k)
Frequency
Matched
Filter
fˆk ,θˆk
VCO
Loop
Filter
Figure 2-7
Above structure is based on the maximum-likelihood principle. The key challenge is
how to implement the filters. In [1], rotational frequency detector (FD) was suggested
to realize, which is more suitable for digital realizations.
Rotational FD has no filtering functions, and decides by observing transitions of
relative phases. If the received waveform carrier has frequency f c and estimated
frequency fˆc . When fˆc = f c , the transitions of f c maintain a fixed relationship to
fˆ
those of fˆc . Otherwise, the angle of rotation is shown to be 2π ( c − 1) , which is
fc
counterclockwise if fˆc > f c and clockwise if fˆc < f c . By “measuring” phase change,
we can track the frequency.
K. C. Chen, for NTUEE Course “Mobile Communications” Only
The useful range of the FD is | ∆ω |< πf c =
9
ωc
and the largest FD (linearity) output is
2
ωc / 4 .
2.2 Digital Signal Processing Carrier Frequency Recovery
Schemes that we discussed in earlier section are basically with the closed loop
structure to recover carrier frequency. With advance of microprocessors and digital
signal processors, a new thinking based on processing digital samples of the received
waveform has been introduced into digital communication transceiver design, and has
been widely applied in many modern digital communication systems. As packet
communications becomes a main stream in packet switching networks (such as
Internet), bursty digital transmission based on packets or frames suggests importance
of open loop carrier frequency estimation.
2.2.1 Nonlinear Estimation of Modulated Carrier
Viterbi and Viterbi [8] developed a non-linear estimation of PSK modulated carrier
phase for such a purpose.
I
( n +1 / 2 ) T
xsn
( n −1 / 2 )T
Optimal
Estimate
Function
For Certain
y sn Modulation
∫
X
r(t)
X
Q
VCO
Figure 2-8
∫
( n +1 / 2 ) T
( n −1 / 2 )T
xn
N
X
1
∑
2 N + 1 n=− N
1
Y 
tan −1  
m
X 
yn
N
1
Y
∑
2 N + 1 n=− N
1
Y 
θˆ(m) = tan −1   for m − PSK
m
X
θˆ
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In above figure, xn = x sn and y n = y sn represents m = 1 and ∆f = 0 for un-modulated
1
Y 
carrier. This estimator θˆ(m) = tan −1   for m − PSK
m
X
variance of the phase estimator is
var(θˆ) =
is unbiased and the
1
[1 + O(2 N + 1) −1 ]
(2 N + 1)(2 ES / N 0 )
while the variance corresponds to the Cramer-Rao lower bound. This linear case is not
useful for m-PSK modulated waveform, and we shall have a nonlinear function as
typical synchronization solutions, which can be modeled as
xn + jy n = F ( ρ n )e jmϕ n
where ρ n2 = xn2 + y n2 and ϕ n = tan −1 ( y n / x n ) . Multiplying the phase by m, along with
the final operation of dividing tan −1 by m, gives m-fold phase ambiguity of carrier
phase estimator. A practical way to adjust this phase ambiguity is through the (error
correcting) coding, rather than data itself. Typical selections are F ( ρ ) = 1, ρ 2 , ρ 4 .
Finally, we may note the similarity between the selection of F(.) and square-loop etc.
2.2.2 Frequency Offset Estimation
Let us consider the ML estimation of frequency ∆f of a complex-valued oscillation
by observing the sampled signal [9]
rk = e j ( 2π∆fkTS +θ ) + υ k , 1 ≤ k ≤ N
where TS ≤ 1 /( 2∆f ) is the sampling interval, and θ is the unknown random phase
uniformly distributed in (0, 2π ) , each component of complex υ k is independent
Gaussian with autocorrelation Rυ (k ) = σ 2δ k . This is a well-known problem to detect
frequency in modern spectral estimation, and it leads to the maximization of
equivalent likelihood function
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N
N N
~
~
~
Λ(∆f ) =| ∑ ri e − j 2π∆fTS |2 = ∑∑ rk rl* e − j 2π∆fTS ( k −l )
i =1
(2.20)
k =1 l =1
~
where ∆f is a tentative value for ∆f . Above maximization leads to ML
estimate ∆f ML
that is consistent and asymptotically efficient. For
N →∞
and ρ = 1 /(2σ ) as carrier-to-noise ratio, the Cramer-Rao lower bound is
2
2
σ CR
=
3
1
2 2
2π TS ρN ( N 2 − 1)
Unfortunately, (2.20) does not have a closed form solution. Considering only
~
necessary condition and taking derivative of (2.20) with respect to ∆f , we can get
N
N
∑∑ (k − l )r r e
k =1 l =1
~
* − j 2π∆f TS ( k −l )
k l
=0
By defining R(k ) as the estimated autocorrelation of rk , that is,
R(k ) =
N
1
ri ri*−k 0 ≤ k ≤ N − 1
∑
N − k i =k +1
The necessary condition is therefore
~
 N −1

Im∑ k ( N − k ) R(k )e − j 2π∆fkTS  = 0
 k =1

(2.24)
Please note that the possibility of “false maxima” can be avoided by appropriate
restricting the operating range of the estimator. We may further note that (2.24) can be
considered as the DFT of R(k ) , weighted by a parabolic windowing
function w(k ) = k ( N − k ) . Via replacing a sub-optimal window with uniform weights,
we can have
~
M

Im∑ R(k )e − j 2π∆fkTS  = 0
 k =1

(2.25)
~
For an ideal noiseless channel, R(k ) = e − j 2π∆fkTS and ∆f = ∆f is still a trivial solution
K. C. Chen, for NTUEE Course “Mobile Communications” Only
12
of (2.25). When the noise is present, with proper choice of M, the mean squared
distance is expected to be negligible as CNR increases. For high CNR and low
frequency deviation ( M∆fT << 1 ), replacing the exponential in (2.25) by its Taylor
series expansion truncated to the linear term, we can reach
M
∆fˆ ≅
1
2πTS
∑ Im{R(k )}
k =1
M
∑ k Re{R(k )}
(2.26)
k =1
If we consider assumptions and use Taylor expansion (to linear term with small noise)
of R(k ) , we can argue
M

R
k
≅
Im
{
(
)
}
arg
∑ R(k )
∑
k =1
 k =1

M
M ( M + 1)
k Re{R(k )} ≅
∑
2
k =1
1
M
M
(2.27)
It leads to a simple estimate
∆fˆ ≅
1
M

arg∑ R(k )
πTS ( M + 1)  k =1

(2.28)
It is true only when the first equation of (2.27) not exceeding π . Consequently, the
operating range of this frequency estimation is limited as
| ∆f |<
1
TS ( M + 1)
(2.29)
References:
[1] D.G. Messerschmidt, “Frequency Detectors for PLL Acquisition in Timing and
Carrier Recovery”, IEEE Tr. On Communications, Sep. 1979, pp.1288-1295.
[2] F.D. Natali. “AFC Tracking Algorithms”, IEEE Tr. On Communications, Aug.
1984, pp.935-947.
[3] T. Alberty, V. Hespelt, “A New Pattern Jitter Free Frequency Error Detector”,
IEEE Tr. On Communications, Feb. 1989, pp.159-163.
K. C. Chen, for NTUEE Course “Mobile Communications” Only
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[4] A.N. D’Andrea, U. Mengali, “Performance of a Quadricorrelator Driven by
Modulated Signals”, IEEE Tr. On Communications, Nov. 1990, pp.1952-1957.
[5] ------, “Design of a Quadricorrelator for Automatic Frequency Control Systems”,
IEEE Tr. On Communications, June 1993, pp.988-997.
[6] ------, “Noise Performance of two Frequency-Error Detectors Derived from
Maximum Likelihood Estimation Methods”, IEEE Tr. On Communications,
Feb/Mar/Apr 1994, pp.793-802.
[7] G. Karam, I. Jeanclaude, H. Sari, “A reduced-Complexity Frequency Detector
Derived from the Maximum-Likelihood Principle”, IEEE Tr. On
Communications, Oct. 1995, pp.2641-2650.
[8] A. Viterbi, A. Viterbi, “Nonlinear Estimation of PSK-Modulated Carrier Phase
with Application to Burst Digital Transmission”, IEEE Trans. On Information
Theory, no. 4, vol.29, July 1983.
[9] M. Luise, R. Reggianni, “Carrier Frequency Recovery in All-Digital Modems for
Burst-Mode Transmissions”, IEEE Trans. On Communications, no. 2/3/4, vol. 43,
Feb/Mar/Apr 1995.
[10] H. Meyr, Digital Communication Receiver, 1999.