A Comparison Between Instantaneous Frequency Estimation Methods of Frequency Modulated Signals

Covered with Gaussian Noise

Reiz Romulus Electronics and Telecommunications Department

University of Oradea Oradea, Romania

Abstract—This paper presents a comparison between several instantaneous frequency estimation methods. Some practical instantaneous frequency estimation examples are presented, using as test signals nonstationary frequency modulated signals covered with zero mean gaussian noise.

Keywords: estimation; instantaneous frequency; time-frequency representations

I. INTRODUCTION One of the most important parameters that describe a

sinusoidal signal is its frequency. Most real-world signals have at least one parameter that is time dependant. These signals are often called nonstationary signals. In the case of these signals, when the signal is frequency modulated, the notion of frequency loses its meaning, since the signal’s frequency is a function of time. For nonstationary frequency modulated signals, the frequency at a certain moment of time is being properly described by the notion of “instantaneous frequency.” As these nonstationary signals are the most common in real-world environments, the estimation of the instantaneous frequency is very important. [1].

A definition of the instantaneous frequency for a real-time signal was given by Ville as [2]:

( ) ( )dttdtFI φ

π21= (1)

where ( )tφ is the phase of the analytic signal associated with the real-time signal.

There have been developed many methods to estimate the instantaneous frequency of such signals. The most important ones are presented in [3]. A few of these methods called auto-regressive (AR) represent the signal using a model based on a rational transfer function. Other estimation methods use the definition of the instantaneous frequency given by Ville and calculate a phase difference in discrete time. This way a forward and backward finite difference (FFD and BFD) can be calculated and used as an estimator.

The instantaneous frequency may also be estimated using the central finite difference with the advantage that this is an unbiased estimator.

There are also adaptive instantaneous frequency estimation methods that are based on algorithms such as RLS and LMS.

Another family of instantaneous frequency estimation techniques is based upon the use of time-frequency representations. One can use the first moment of such representations as estimator. For instance, the Wigner-Ville Distribution can give the instantaneous frequency through its first moment [4], and also other time frequency representations like the Short-Time Fourier Transform can produce an approximation of the instantaneous frequency using their first moment. The use of time-frequency representations to obtain the instantaneous frequency present also the advantage of spreading the noise in the time frequency plane, thus yielding better estimations at higher noise levels. Also time-frequency representations were developed in the first place to process nonstationary signals.

These Time-frequency representations as the Short-time Fourier, Gabor, Wigner-Ville, Choi-Williams and “wavelet” transforms of a signal contain very important information concerning the regions from the time-frequency plane where the signal's energy is higher. These maximums points are localized around the instantaneous frequency (IF) of the signal, which means that the detection of the ridges offers also the possibility to estimate the IF and to reconstruct the original signal [1] [3].

Stationary signals can be analyzed using classical Fourier transform methods in which the signal can be expressed as a sum of sine and cosine waves. Even so, as it was shown, most real-world signals are nonstationary and have highly complex time-frequency components.

Nonstationary signals can be processed with similar Fourier based methods if these signals are analyzed over short periods of time, where the signal can be considered as stationary This way a Fourier transform can be calculated for each of these short periods of time. In this case the Fourier transform analyses the signal through a window that is localized in time.

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The Short-time Fourier transform defines a two-dimensional time-frequency representation [3]:

( ) ( ) ( )∫∞

∞−

−−= τττω ωτdetwxtTF jSTFTx , (2)

where ( )w τ represents the windowing function. Usually it is considered that the windowing function is a unit energy signal.

The peaks of this representation can be used to extract the instantaneous frequency of the signal; however the method works only with simple signals, having only one dominant frequency at one moment of time. One of the main advantages of the linear type time-frequency representations such as the Short-time Fourier is that they do not produce any interference terms. Even so the localization of the signal in the time-frequency plane is poorer than the one obtained with bilinear representations such as the Wigner-Ville representation.

Most real-time signals are covered with noise, so it is useful to test these methods of instantaneous frequency estimation with signals covered with noises, having different SNRs. A common noise model is the Gaussian noise which provides a good model of noise in many signal processing systems. Its probability density function (pdf) for a zero mean noise is [6]:

( ) 2

2

222

1 σ

πσ

n

enx−

= (3)

where parameter 2σ is the variance of the noise.

The advantage of using a time-frequency representation is that generally these representations produce a diffusion of the noise covering the signal in the time-frequency plane. Also a good concentration of the signal’s energy around the instantaneous frequency can be achieved.

An example of bilinear time-frequency representation is the Wigner-Ville representation. Using the kernel [5]:

( ) ⎟⎠⎞

⎜⎝⎛ −⋅⎟

⎠⎞

⎜⎝⎛ +=

22, * τττ txtxtKwv (4)

the Wigner-Ville representation is defined by:

( ) ∫∞

∞−

−⋅⎟⎠⎞

⎜⎝⎛ −⋅⎟

⎠⎞

⎜⎝⎛ += τττω ωτ detxtxtTF jWV

x 22, * (5)

This representation has some useful properties such as a better localization of the signal’s energy in the time frequency plane. However, one of the main disadvantages of this representation is the presence of interference terms, which can seriously affect the results obtained in estimating the instantaneous frequency of a signal.

A time-frequency representation that combines the useful properties of the Short-time Fourier and Wigner-Ville representations would be very useful in estimating instantaneous frequency of a signal. In this paper we use one such representation that is obtained by computing a multiplication between the modulus of the STFT

xTF transform

and the WVxTF transform. This might produce a transform with

better localization of the signal in the time-frequency plane (due to the use of the Wigner-Ville representation) and reduced interference terms (because the Short time Fourier transform “windows” the noise). Using an algorithm that detects the maximum values of this representation, the instantaneous frequency of the signal can be estimated.

II. ANALYSIS METHOD

In order to compare the performances of the instantaneous frequency estimation methods some test signals were generated. These test signals were single component nonstationary signals. The results presented in this paper are those obtained using linear and sinusoidal frequency modulated signals. The frequency of these signals changes over time following a linear or sinusoidal law from 1 kHz to 4 kHz, the sampling frequency being 10 kHz. All the programs used were based upon the “Time-Frequency toolbox “for Matlab, available online [7]. The generated signals have 2048 samples. These signals then were covered by zero mean Gaussian noise, obtaining test signals having several different signal to noise ratios (SNRs). These signals were then used to test the estimation methods. The following estimation methods were used: instantaneous frequency estimation using phase differentiation, estimation using the first order moment of the Wigner-Ville representation and using an auto-regressive model. Also the peaks of time-frequency representation were calculated and used to obtain an estimation of the IF. The Short-time Fourier, Wigner-Ville and Choi Williams time-frequency representations were used to achieve this. Also, a more complex time-frequency representation was calculated by

multiplying the modulus of the STFTxTF transform and

theWVxTF transform. The result this multiplication has the

effect of obtaining a better localization of the signal in the time-frequency plane (due to the use of the Wigner-Ville representation) and diminished interference terms (because those are filtered by the short time Fourier transform). Using an algorithm that detects the maximum values of these time-frequency representations, the instantaneous frequency of the signal can be estimated. A root mean squared error (rmse) was calculated to indicate the difference between the ideal law of the IF and the estimated one, at different signal to noise levels, using the ideal and estimated curve of the instantaneous frequency, for both the linear and the sinusoidal frequency modulated test signals. In figure 1 is presented the Short-time Fourier transform of the linear fm signal, at 0 dB noise.

g

Timp [s]

Fre

cven

ta [

Hz]

200 400 600 800 1000 1200 1400 1600 1800 20000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 1. Short-time Fourier representation of the linear frequency

modulated signal covered with noise

To estimate the instantaneous frequency the peak values from the time-frequency plane were detected, and used to approximate the IF law. Then, as it was mentioned before, an error was calculated by comparing the estimated and ideal instantaneous frequency laws. Several time-frequency representations were tested. For example in figure 2 is presented the Wigner-Ville representation of the same signal. As it can be observed, this representation provides a much better localization of the signal in the time-frequency plane. Also in the case of the linear FM signal, the interference terms aren’t important. In the case of more complex signals, however, these interference terms can seriously affect the accuracy of the IF estimation, when using the Wigner-Ville representation. A time-frequency representation that has a good localization of the signal in the time-frequency plane and has reduced interference terms would be very useful for IF estimation. Such a representation can be obtained by multiplying the Short time Fourier transform and the Wigner –Ville transform. To demonstrate de reduction of the noise and interference terms, in figure 3 a cross section of the Wigner-Ville representation from figure 2 is presented. The presence of noise in this figure is obvious.

WV, lin. scale, imagesc, Threshold=5%

Timp [s]

Fre

cven

ta [

Hz]

200 400 600 800 1000 1200 1400 1600 1800 20000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 2. Wigner-Ville representation of the linear frequency modulated

signal covered with noise

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

200

400

600

800

1000

1200

1400

1600

1800

Figure 3. Cross-section of the Wigner-Ville representation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2000

4000

6000

8000

10000

12000

14000

Figure 4. Cross-section of the time frequency representation obtained by

computing a multiplication between the modulus of the STFTxTF transform

and theWVxTF transform

By multiplication with the modulus of the Short-time Fourier transform however, a reduction of this noise can be observed. This is evident in the cross section presented in figure 4. The peaks of all four (STFT, WV, CW and STFT*WV) time-frequency representations were used to estimate the instantaneous frequency of the test signals.

As it was mentioned before three other IF estimation methods were also used. In figure 5 is presented the result of the IF estimation using a phase difference estimator.

To compare the results of these methods, in figure 6 is presented a graph that shows the root mean squared error of the estimation in function of the SNR, for the linear FM signal. Figure 6 contains the errors for all estimation methods that were tested. Each estimation at every noise level were repeated for 16 different signals, the results presented in figure 6 being an average of the errors obtained. As it can be noticed, the methods based on the peaks of time frequency representations are much better. The performances of these methods are good even at higher noise levels.

0 500 1000 1500 2000 2500-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5Estimated IF

Ideal IF

Figure 5. Result of the IF estimation using a phase difference estimator

However, at the highest noise level (-15dB), the errors level becomes high for this type of methods too. This may be because the time frequency representations spread the noise in the time frequency plane, making the detection of the useful signal easier.

The same instantaneous frequency estimation methods were applied to the sinusoidal frequency modulated signal. The results in this case are synthesized in figure 7.

As is may be observed, the results for this type of signal differ significantly from the previous estimation results only for the error rates obtained using the peaks of the Wigner-Ville representation. This is caused by the auto-interference terms produced by this representation when the analysed signal is nonlinear (e.g. sinusoidal FM law).

15 10 5 2 0 -2 -5 -10 -15 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SNR(dB)

rmse

(Hz)

STFTWV

CW

STFT*WV

DPh

AR2Momtfr

Figure 6. IF estimation errors at different noise levels for the linear fm signal covered with noise

15 10 5 2 0 -2 -5 -10 -15 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SNR(dB)

rmse

(Hz)

STFTWV

CW

STFT*WV

DPh

AR2Momtfr

Figure 7. IF estimation errors at different noise levels for the sinusoidal fm signal covered with noise

III. CONCLUSIONS In this paper a comparison I made between different

instantaneous frequency estimation methods for nonstationary frequency signals. The methods based upon the use of the peak values of time-frequency representations, namely the Short-time Fourier transform, the Wigner-Ville, Choi-Williams and a proposed time frequency representation that is obtained by multiplication proved to be superior. The relation that exist between IF and time-frequency distributions is underlined. At smaller noise level all representations performed well, with the Wigner-Ville transform having the best results. These results are specific to these types of signals. Different signals may produce different estimation errors. Further work can be done using different combinations of more advanced time-frequency representations, such as smoothed and reassigned versions of time frequency distributions.

REFERENCES [1] Boashash B. 1992: “Estimating and Interpreting the Instantaneous

Frequency of a Signal-Part 1: Fundamentals “, Proceedings of the IEEE, vol 80, no.4;

[2] J. Ville, “Theorie et Application de la Notion de Signal Analytic,” Cables et Transmissions, vol. 2A, pp. 61-74, 1948.

[3] Boashash B. 1992: “Estimating and Interpreting the Instantaneous Frequency of a Signal-Part 2: Algorithms and Applications”, Proceedings of the IEEE, vol 80, no.4;

[4] T. A. C. M. Classen and W. F. G. Mecklenbrauker, “The Wigner Distribution-Part 11,” Philips Journal of Research, vol. 35, pp.276-300, 1980.

[5] A. Isar, I. Naforniţă, “Reprezentari timp-frecvenţă”, Ed. Politehnica Timisoara, 1998

[6] D. C. Montgomery, G. C. Runger, “Applied Statistics and Probability for Engineers”, Third Edition, John Wiley & Sons, Inc., 2003

[7] TFTB-Time-Frequency toolbox, ttp://tftb.nongnu.org/