algorithms for inst freq estimation
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Algorithms for estimating instantaneous frequency
Jaideva C. Goswami, Albert E. Hoefel3
Schlumberger Technology Corporation, 110 Schlumberger Drive, Sugar Land, Texas 77478, USA
Received 28 March 2003; received in revised form 22 April 20045
Abstract
Three algorithms to estimate instantaneous frequency of a frequency-modulated signal is discussed. These algorithms are7based on Hilbert transform, Haar wavelet, and generalized pencil of function (GPOF) methods. While GPOF-based frequency
detection method appears to be least sensitive to noise, wavelet-based method is easiest to implement. The latter method is9also computationally more ecient and can be implemented in real-time. Results for both synthetic and experimental data
are shown.11? 2004 Elsevier B.V. All rights reserved.
Keywords: Frequency modulation; Phase-shift keying; Frequency-shift keying; Wavelet; Hilbert transform; Matrix pencil13
1. Introduction
The problem of estimating instantaneous frequency15
of a received signal is very important in many com-
munication areas. In this paper, we discuss this prob-17
lem in the context of wireless data acquisition in
oileld industry. In some oil-eld exploration19
applications, a small amount of measured data are
transferred between a measurement sensor and a21
host unit where, both the sensor and host units are
located at a few kilometers underground. Binary23frequency-shift keying (BFSK) scheme is often used
for data communication. The major problem for such25
data communication comes because of highly lim-
ited availability of space and power. Demodulation27
algorithms should be easily implementable in hard-
ware and rmware. With this application in mind,29
Corresponding author.
E-mail addresses: [email protected] (J.C. Goswami),
[email protected] (A.E. Hoefel).
we have studied three dierent methods for estimating
instantaneous frequencies. These are based on Hilbert 31
transform [6], Haar wavelet [1], and generalized pen-
cil of function (GPOF) methods [2,3,5]. In this section 33
we briey review frequency modulation scheme. Sec-
tions24describe algorithms for estimating instanta- 35
neous frequency by Hilbert transform, wavelet trans-
form, and GPOF methods, respectively. And nally, 37
in Section5, we discuss numerical and experimental
results. 39
In frequency modulation (FM), the instantaneousfrequency !i of the carrier varies linearly with the 41
modulating signalm(t)[4]. Thus!i can be written as
!i=!c+kfm(t); (1)
where!c is the carrier frequency and kf is a constant. 43
The phase,(t), is given by
(t) =
t
[!c+kfm()] d (2)
=!ct+kf
t
m() d: (3)
0165-1684/$ - see front matter? 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.sigpro.2004.05.016
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ModulatingSignal
PhaseModulator
FM
Signal
ModulatingSignal Frequency
Modulator
PM
SignalDifferentiator
Integrator
Fig. 1. Relationship between phase and frequency modulation.
An FM signal, therefore, can be represented as1
s(t) =A cos!ct+kf t
m() d : (4)It is worth mentioning here that in a phase-modulated
(PM) signal, instead of frequency, the phase (t)3
varies linearly with the modulating signal, namely
(t) =!ct+kpm(t) (5)
and instantaneous frequency !i varies linearly with5
the derivative of the modulating signal
!i=d
dt =!c+kpm(t): (6)
It is, therefore, clear that PM and FM are intimately7related. One can be obtained from the other as shown
in Fig.1.The commonly used digital modulation tech-9
niques, BPSK (binary phase-shift keying) and BFSK
are special cases of PM and FM, respectively, when11
the modulating signal m(t) takes binary values. Al-
though all the examples discussed in this paper pertain13
to FM, the algorithm, with trivial modication, can be
used for PM as well.15
2. Hilbert transform algorithm
The Hilbert transform of a signal s(t) produces a17
signalsh(t) that is orthogonal to s(t). The angle of the
complex signal s(t) +jsh(t) then gives the instanta-19
neous phase, and its derivative, the instantaneous fre-
quency of the signal.21
Let s(!) represent the Fourier transform of a
real-valued signals(t). A typical magnitude spectrum23
ofs(t) is shown in Fig. 2.We can construct a signal
s+(t) that contains only positive frequencies of s(t)25
cc
|s ()|^
Fig. 2. A typical magnitude spectrum of a signal.
by multiplying its spectrum s(!) with a unit step func-
tion as 27
s+(!) = s(!)u(!); (7)where u(!) is the unit step function, dened in the
usual way as 29
u(!) =
1 ! 0
0 otherwise:(8)
From (7)we have
2 s+(!) = s(!)[1 + sgn(!)]
= s(!) +j[ j sgn(!) s(!)]
sh (!); (9)
where sgn(!) is the signum function dened as 31
sgn(!) =
1 ! 0
1 ! 0:(10)
In (9),sh(t) is the Hilbert transform ofs(t), dened as
sh(t) =F1{j sgn(!) s(!)}
=1
s()
td; (11)
whereF1
represents inverse Fourier transform. It is 33easy to verify thats(t); sh(t) = 0 sh(t) s(t).
Once we have the orthogonal signal sh(t) of s(t), 35
the instantaneous phase and frequency ofs(t) can be
obtained by [6] 37
(t) = arctan
sh(t)
s(t)
(12)
and
!i(t) =d
dt =!c+kfm(t): (13)
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J.C. Goswami, A.E. Hoefel / Signal Processing ( ) 3
0 1
0.5
0 1
(b)(a)
Fig. 3. Haar (a) scaling function, and (b) wavelet.
3. Wavelet algorithm
The principle behind wavelet-based instantaneousfrequency estimation is the same as Hilbert transform3
method, namely nding orthogonal signal of a given
frequency-modulated signal. From wavelet theory [1],5
we can decompose a given function s(t) into orthog-
onal signals f(t) andg(t) such that7
f(t) =
k
ck(atk); (14)
g(t) =
kdk(atk); (15)
where parameter a depends on the sampling rate of
s(t). The bases(t) and(t) are called scaling func-9
tion and wavelet, respectively. These two functions
are orthogonal to each other, i.e.(at); (at )=110; Z:= {: : : ;1; 0; 1; : : :}.
We will use the simplest scaling function and13
wavelet due to Haar, as shown in Fig. 3.For this case
the coecients{ck}and{dk}witha = 2 are given as15
2ck=s(k) +s(k+ 1); (16)
2dk= s(k) s(k+ 1): (17)
As with Hilbert transform case, once we have orthog-
onal signals, we can obtain the instantaneous phase17
k :=(tk); tk= kt
k= Aarctan
dk
ck
; (18)
whereA is a constant depending upon the sampling19
rate. Finite dierence of{k}then gives the instanta-neous frequency.
4. GPOF algorithm 21
Given a set of discrete data {fi : i = 0; 1; : : : ; N }
of a complex-valued functionf(t), generalized pencil 23of function method nds a set of complex coecients
{ck; k :k= 1; 2; : : : ; M }such that 25
fi :=f(ti) =
Mk=1
ckexp(kti); M N; (19)
whereti = itand tis the discretization step. The
GPOF method has better numerical performance than 27
the conventional Pronys method [7] which involves
solving an ill-conditioned matrix equation, and nd- 29
ing roots of a polynomial. The GPOF method also
has better noise sensitivity [3]. A detailed discussion 31on the subject may be found in [3]. The algorithm is
briey described below. 33
Consider vectors of discrete data{fi},
Fi := [fi fi+1 : : : fi+NL1]T; 06 i6L; (20)
where T stands for transpose, and form matrices of 35
size (N L) L,
A1= [F0 F1 : : : FL1] (21)
and 37
A2= [F1 F2 : : : FL]: (22)
Then, withzk := exp(kt), it can be shown that{zk}are the generalized eigen values of matrix pencil A2 39zA1. Once we have all the exponents, the coecients
{ck}can be easily computed from (19)since in{ck}, 41it is a linear equation.
In the present problem of instantaneous frequency 43
estimation, we choose a window size that covers at
least one cycle of the highest carrier frequency and 45
then extract two exponents. The imaginary parts of
these exponents k give instantaneous frequency. By 47sliding the window, we can compute the instantaneous
frequency for the entire signal. 49
5. Results and discussions
As a rst step, we test and compare all three al- 51
gorithms discussed in this paper by applying them
to synthetic frequency-modulated data. The compari- 53
son is based on relative signal reconstruction error, ,
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Table 1
Error () in frequency estimation by three algorithmsHilbert
transform, Haar wavelet, and generalized pencil of function
(GPOF)for dierence noise level (as a percentage of signal
strength)
Noise Error
Hilbert transform Haar wavelet GPOF
10 0.00004 0.00024 0.00004
20 0.00019 0.00109 0.00005
30 0.00042 0.00273 0.00010
40 0.00076 0.00523 0.00016
50 0.00123 0.00850 0.00022
60 0.02897 0.02143 0.00030
70 0.03840 0.03500 0.00042
80 0.04707 0.06756 0.00062
90 0.05133 0.10211 0.00099100 0.05656 0.11747 0.01252
0 0.5 1 1.5 2 2.53
2
1
0
1
2
3
Time (seconds)
Frequency modulated signal
Fig. 4. A BFSK signal with additive Gaussian noise
(SNR = 3:0 dB).
dened as1
=
i |Si; nSi; 0|
2i |Si; 0|
2 ; (23)
where summation is over all samples, andSi; n, andSi; 0representith sample of reconstructed signal with and3
without noise in the modulated data, respectively. Ta-
ble1,shows the results for error with dierent levels5
of additive Gaussian noise as a percentage of signal
strength. To visually see the eect of noise on fre-7
quency demodulation, we have plotted the signal in
0 0.5 1 1.5 2 2.50
10
20
30
40
Frequency estimation ( Exact Computed)
Time (seconds)
Frequenc
y(Hz)
0 0.5 1 1.5 2 2.50
10
20
30
40
Time (seconds)
Frequency(Hz)
0 0.5 1 1.5 2 2.50
10
20
30
40
Time (seconds)
Frequency(Hz)
Hilbert transform
Haar wavelet
GPOF
Fig. 5. Frequency demodulation of a BFSK signal shown in Fig.
4 by using three algorithms based on Hilbert transform, Haar
wavelet, and GPOF.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (milli-seconds)
Frequency detection using Hilbert transform
Experimental modulating dataComputed normalized frequency
Fig. 6. Frequency demodulation of experimental BFSK data using
Hilbert transform.
Fig.4and the frequency demodulated signals in Fig.5 9
for all three algorithms. Finally, we apply these algo-
rithms to experimental data obtained by using a BFSK 11
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (milli-seconds)
Frequency detection using Haar wavelet
Experimental modulating dataComputed normalized frequency
Fig. 7. Frequency demodulation of experimental BFSK data using Haar wavelet.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (milli-seconds)
Frequency detection using GPOF
Experimental modulating dataComputed normalized frequency
Fig. 8. Frequency demodulation of experimental BFSK data using
GPOF.
transmitter and a pick-up coil. The results are shown1
in Figs.68.For experimental data, all three methodsperform well because of low noise environment.3
The results indicate that GPOF-based method is
least sensitive to noise. This is primarily due to the5
fact that in GPOF, the frequency is determined by
a least squared algorithm which essentially mini-7
mizes the error. Wavelet-based method is the sim-
plest to implement and fastest to compute. Further-9
more, wavelet-based method can be implemented
11
for real-time application, since it requires signal infor-
mation only in the vicinity of the current time loca- 13
tion. In Hilbert transform method, on the other hand,
we need the entire signal before estimating the in- 15
stantaneous frequency because of the global nature of
Fourier bases. The GPOF method can be implemented 17
in real-time, however, matrix operations are dicult
to be realized in hardware. 19
References
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[4] B.P. Lathi, Modern Digital and Analog Communication 29Systems, Holt, Rinehart & Wilson, New York, 1983
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435441. 35[6] J.G. Proakis, Digital Communication, McGraw-Hill, New
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