[s] algorithms for estimating instantaneous frequency

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Signal Processing 84 (2004) 14231427

www.elsevier.com/locate/sigpro

Algorithms for estimating instantaneous frequency

Jaideva C. Goswami, Albert E. Hoefel

Schlumberger Technology Corporation, 110 Schlumberger Drive, Sugar Land, TX 77478, USA

Received 28 March 2003; received in revised form 22 April 2004

Abstract

Three algorithms to estimate instantaneous frequency of a frequency-modulated signal is discussed. These algorithms are

based 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 is

also computationally more ecient and can be implemented in real-time. Results for both synthetic and experimental data

are shown.

?2004 Elsevier B.V. All rights reserved.

Keywords: Frequency modulation; Phase-shift keying; Frequency-shift keying; Wavelet; Hilbert transform; Matrix pencil

1. Introduction

The problem of estimating instantaneous frequency

of a received signal is very important in many com-

munication areas. In this paper, we discuss this prob-

lem in the context of wireless data acquisition in

oileld industry. In some oil-eld exploration

applications, a small amount of measured data are

transferred between a measurement sensor and a

host unit where, both the sensor and host units are

located at a few kilometers underground. Binaryfrequency-shift keying (BFSK) scheme is often used

for data communication. The major problem for such

data communication comes because of highly lim-

ited availability of space and power. Demodulation

algorithms should be easily implementable in hard-

ware and rmware. With this application in mind,

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

transform [6], Haar wavelet [1], and generalized pen-

cil of function (GPOF) methods [2,3,5]. In this section

we briey review frequency modulation scheme. Sec-

tions24describe algorithms for estimating instanta-

neous frequency by Hilbert transform, wavelet trans-

form, and GPOF methods, respectively. And nally,

in Section5, we discuss numerical and experimental

results.

In frequency modulation (FM), the instantaneous

frequency !i of the carrier varies linearly with the

modulating signalm(t)[4]. Thus!i can be written as

!i=!c+kfm(t); (1)

where!c is the carrier frequency and kfis a constant.

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
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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1424 J.C. Goswami, A.E. Hoefel/ Signal Processing 84 (2004) 1423 1427

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 as

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)

varies linearly with the modulating signal, namely

(t) =!ct+kpm(t) (5)

and instantaneous frequency !i varies linearly with

the derivative of the modulating signal

!i=d

dt =!c+kpm(t): (6)

It is, therefore, clear that PM and FM are intimatelyrelated. One can be obtained from the other as shown

in Fig.1.The commonly used digital modulation tech-

niques, BPSK (binary phase-shift keying) and BFSK

are special cases of PM and FM, respectively, when

the modulating signal m(t) takes binary values. Al-

though all the examples discussed in this paper pertain

to FM, the algorithm, with trivial modication, can be

used for PM as well.

2. Hilbert transform algorithm

The Hilbert transform of a signal s(t) produces a

signalsh(t) that is orthogonal tos(t). The angle of the

complex signal s(t) + jsh(t) then gives the instanta-

neous phase, and its derivative, the instantaneous fre-

quency of the signal.

Let s(!) represent the Fourier transform of a

real-valued signals(t). A typical magnitude spectrum

ofs(t) is shown in Fig. 2.We can construct a signal

s+(t) that contains only positive frequencies of s(t)

cc

|s ()|^

Fig. 2. A typical magnitude spectrum of a signal.

by multiplying its spectrum s(!) with a unit step func-

tion as

s+

(!) = s(!)u(!); (7)

where u(!) is the unit step function, dened in the

usual way as

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

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()

t d; (11)

whereF1

represents inverse Fourier transform. It iseasy to verify thats(t); sh(t) = 0 sh(t) s(t).Once we have the orthogonal signal sh(t) of s(t),

the instantaneous phase and frequency ofs(t) can be

obtained by [6]

(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 84 (2004) 1423 1427 1425

0 10.5

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 transform

method, namely nding orthogonal signal of a given

frequency-modulated signal. From wavelet theory [1],

we can decompose a given function s(t) into orthog-

onal signals f(t) andg(t) such that

f(t) =

k

ck(at k); (14)

g(t) =

k

dk(at k); (15)

where parameter a depends on the sampling rate ofs(t). The bases (t) and(t) are called scaling func-

tion and wavelet, respectively. These two functions

are orthogonal to each other, i.e.(at); (at )=0; Z:= {: : : ;1; 0; 1; : : :}.

We will use the simplest scaling function and

wavelet due to Haar, as shown in Fig. 3.For this case

the coecients{ck}and{dk}witha = 2 are given as

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 phase

k :=(tk); tk= kt

k= Aarctan

dk

ck

; (18)

whereA is a constant depending upon the sampling

rate. Finite dierence of{k}then gives the instanta-neous frequency.

4. GPOF algorithm

Given a set of discrete data {fi : i= 0; 1; : : : ; N }

of a complex-valued functionf(t), generalized pencilof function method nds a set of complex coecients

{ck; k :k= 1; 2; : : : ; M }such that

fi :=f(ti) =

Mk=1

ckexp(kti); M N; (19)

whereti = itand tis the discretization step. The

GPOF method has better numerical performance than

the conventional Pronys method [7]which involves

solving an ill-conditioned matrix equation, and nd-

ing roots of a polynomial. The GPOF method also

has better noise sensitivity [3]. A detailed discussionon the subject may be found in [3]. The algorithm is

briey described below.

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

size (NL) L,

A1= [F0 F1 : : : FL1] (21)

and

A2= [F1 F2 : : : FL]: (22)

Then, withzk:= exp(kt), it can be shown that{zk}are the generalized eigen values of matrix pencil A2 zA1. Once we have all the exponents, the coecients

{ck}can be easily computed from (19)since in{ck},it is a linear equation.

In the present problem of instantaneous frequency

estimation, we choose a window size that covers at

least one cycle of the highest carrier frequency and

then extract two exponents. The imaginary parts of

these exponents k give instantaneous frequency. Bysliding the window, we can compute the instantaneous

frequency for the entire signal.

5. Results and discussions

As a rst step, we test and compare all three

algorithms discussed in this paper by applying them

to synthetic frequency-modulated data. The compari-

son is based on relative signal reconstruction error, ,


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1426 J.C. Goswami, A.E. Hoefel/ Signal Processing 84 (2004) 1423 1427

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 0.00099

100 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 as

=

i |Si; n Si; 0|

2i |Si; 0|

2 ; (23)

where summation is over all samples, andSi; n, andSi; 0representi th sample of reconstructed signal with and

without noise in the modulated data, respectively. Ta-

ble1,shows the results for error with dierent levels

of additive Gaussian noise as a percentage of signal

strength. To visually see the eect of noise on fre-

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)

Frequency(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

for all three algorithms. Finally, we apply these algo-

rithms to experimental data obtained by using a BFSK


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J.C. Goswami, A.E. Hoefel / Signal Processing 84 (2004) 1423 1427 1427

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


Frequency detection using Haar wavelet


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


Frequency detection using GPOF


Fig. 8. Frequency demodulation of experimental BFSK data using

GPOF.

transmitter and a pick-up coil. The results are shown

in Figs.68.For experimental data, all three methodsperform well because of low noise environment.

The results indicate that GPOF-based method is

least sensitive to noise. This is primarily due to the

fact that in GPOF, the frequency is determined by

a least squared algorithm which essentially mini-

mizes the error. Wavelet-based method is the sim-

plest to implement and fastest to compute. Further-

more, wavelet-based method can be implemented

for real-time application, since it requires signal infor-

mation only in the vicinity of the current time loca-

tion. In Hilbert transform method, on the other hand,

we need the entire signal before estimating the in-

stantaneous frequency because of the global nature of

Fourier bases. The GPOF method can be implemented

in real-time, however, matrix operations are dicult

to be realized in hardware.

References

[1] J.C. Goswami, A.K. Chan, Fundamentals of Wavelets: Theory,

Algorithms, and Applications, Wiley, New York, 1999.

[2] J.C. Goswami, R. Mittra, On the solution of a class of

large-body scattering problems via the extrapolation of FDTD

solutions, J. Electromagn. Waves Appl. 12 (1998) 229244.

[3] Y. Hua, T.K. Sarkar, Generalized pencil-of-function method

for extracting poles of an EM system for its transient response,

IEEE Trans. Antennas Propagat. 37 (1989) 229234.

[4] B.P. Lathi, Modern Digital and Analog Communication

Systems, Holt, Rinehart & Wilson, New York, 1983

(Chapter 4).[5] A.J. Mackay, A. McCowen, An improved pencil-of-function

method and comparisons with traditional methods of pole

extraction, IEEE Trans. Antennas Propagat. 35 (1987)

435441.

[6] J.G. Proakis, Digital Communication, McGraw-Hill, New

York, 1995, pp. 152157.

[7] M.L. Van Blaricum, R. Mittra, A technique for extracting

the poles and residues of a system directly from its transient

response, IEEE Trans. Antennas Propagat. 23 (1975) 777781.

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