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Breaking the Limits -- Redefining the Instantaneous Frequency

The Carson and Fry (1937) introduced the concept of variable frequency, required to the theory of frequency modulation (FM), as a generalization of the definition of constant frequency. Moreover, the nonstationary nature of the signals and nonlinear systems require the idea of instantaneous frequency (IF). It is also the basis of the time-frequency-energy (TFE) analysis of a signal. The IF is the time derivative of the instantaneous phase and it is well-defined only when this derivative is positive. If this derivative is negative, the IF creates problem because it does not provide any physical significance. This study proposes a mathematical solution and eliminate this problem by redefining the IF such that it is valid for all monocomponent and multicomponent signals which can be nonlinear and nonstationary in nature. This is achieved by using the property of the multivalued inverse tangent function. The efforts and understating of all the methods based on the IF would improve significantly by using this proposed definition of the IF. We also demonstrate that the decomposition of a signal, using zero-phase filtering based on the well established Fourier and filter theory, into a set of desired frequency bands with proposed IF produces accurate TFE distribution that reveals true nature of signal and, therefore, real physics of underlying system. Simulations and numerical results demonstrate the efficacy and superiority of the proposed IF that makes zero-phase filter based decomposition most powerful, for the TFE analysis of a signal, as compared to other existing methods in the literature.


INTRODUCTION
The time-domain representation and the frequency-domain representation are two classical representations of a signal. In both domains, the time (t) and frequency (f ) variables are mutually exclusive [2].
The Time-Frequency Distributions (TFD) representation on the other hand, provides localized signal information in time and frequency domain. The TFD provides insight into the complex structure of a signal consisting of several components. There exist many types of time-frequency analysis methods such as shorttime Fourier transform (STFT), Gabor transform, Wavelet transforms, Wigner-Ville distribution.
The instantaneous frequency (IF) is the basis of the TFD or time-frequency-energy (TFE) representation and analysis of a signal. The IF is a practically important parameter of a signal which can reveal the underlying process and provides explanations for physical phenomenon in many applications such as vibration, acoustic, speech signal analysis [13], meteorological and atmospheric applications [1], seismic [13], radar, sonar, solar physics, structural engineering, communications, health monitoring, biomedical and medical applications [16], cosmological gravity wave and financial market data analysis.
The IF is the time derivative of the instantaneous phase (IP) and it is well-defined only when this derivative is positive. If this derivative is negative, it creates problem because it does not provide any physical significance. In order to avoid this problem, recently many nonlinear and nonstationary signal representation, decomposition and analysis methods, e.g. empirical mode decomposition (EMD) algorithms [1,[3][4][5][6][7][8], synchrosqueezed wavelet transforms (SSWT) [9], variational mode decomposition (VMD) [10], eigenvalue decomposition (EVD) [11] and Fourier decomposition methods (FDM) [12-15, 17, 18, 20] based on the Fourier theory, are proposed. The Fourier theory is the only tool for spectrum analysis of a signal and the FDM has established that it is a superior tool for nonlinear and nonstationary time series analysis. The main objective of all these methods is to obtain the signal representation such that the IF of a signal understudy is always positive.
Unlike these decomposition methods, the IF proposed in this study does not necessitate to decompose a signal into a set of narrow band components, which satisfy certain properties, to generate the TFE distribution of a signal. That is, without any decomposition the TFE distribution of a signal can be obtained. It also provide freedom and potential to decompose a signal into a set of desired (preferably orthogonal or LINOEP) frequency bands by a zero-phase filtering approach to obtain TFE distribution of a signal. All these features are obtained by redefining the IF when it is negative and thus defining the IF for both monocomponent and multicomponent signals. In order to redefine the IF, we use the fact that inverse tangent is a multivalued (i.e. one-to-many mapping) function. So defined IF of a signal is always positive and valid for any signal.
This study is organized as follows: The proposed methodology is presented in Section 2. Simulation results and discussion are presented in Section 3. Section 4 presents conclusion of the work.

METHODS
The concept of variable frequency with application to the theory of frequency modulation (FM) is proposed in [21], and it is postulated that the notion of IF is a generalization of the definition of constant frequency. A definition of the IF by analyzing an expression for simple harmonic motion (SHM) is considered in [22] as where the argument of the cosine function is the phase ψ(t) = [ t 0 2πf (t) dt + θ]. This leads to the definition of instantaneous frequency [22] f (t) = 1 2π The concept of instantaneous frequency was enhanced in [23] where a method for generating a unique complex signal z(t) from a real signal x(t) and its Hilbert transformx(t) was proposed. This method obtains where and z(t) is the Gabor's complex signal (well-known as the analytic signal) andx(t) is the Hilbert Transform (HT) of x(t), defined aŝ where p.v. denotes the Cauchy principal value of the integral [24] and * denotes convolution operation. The work done in [21] and [23] was unified in [25] to define the IF of a signal x(t) = a(t) cos(φ(t)) as where z(t) is the analytic signal given by (3) and arg[z(t)] = φ(t) as defined in (4). The IF, as defined in (6), provides physical meaning only when it is positive and it becomes meaningless when it is negative [1, 3-5, 7, 9-13, 26, 27]. This is where we improve and provide the solution to make the IF positive for all time t. From (4) and (6), one can obtain the TFE distribution by 3-D plot of {t, f (t), a 2 (t)}. However, it is to be noted that the frequencies, of signal x(t) = a(t) cos(φ(t)), obtained by the Fourier transform and Hilbert spectrum (6) are same only if a(t) is a constant, otherwise Hilbert spectrum gives frequencies of signal cos(φ(t)) and not of signal x(t). Before evaluating the time derivative of phase in (6), phase unwrapping is necessary to ensure that all appropriate multiples of 2π have been included in phase φ(t). Phase unwrap operation corrects the radian phase angles by adding multiples of ±2π when absolute jumps between consecutive elements of a phase vector are greater than or equal to the default jump tolerance of π radians [28]. This is being done for phase delay and IF determination by all the methods available in literature.
As is well-known that the tangent is a surjective (many-to-one mapping) function. The domain, range and period of tan(x) are {x|x = π 2 +nπ, ∀n ∈ Z}, all real numbers R = (−∞, ∞) and π, respectively. The inverse tangent is the multivalued function. The domain of tan −1 (x) is all real numbers, R, and range is (− π 2 , π 2 ). If z = x + jy, then the range of tan −1 (y/x) is (−π, π], sign of x and y is used to determine the specific quadrant. In order to obtain the IF positive for all the time, we consider the discrete signal processing which is the only practical way to process data by a processor. The discrete time version of the equations (3), (4), (5) and (6) and respectively. By considering the phase unwrapping fact and multivalued nature of the inverse tangent function, we redefine the IF ω[n], defined in (7e) as a discrete version of (6), as A dual to IF (8), the group delay (GD) is defined as the negative frequency derivative of the phase in the Fourier domain. It measures the relative delay of different frequencies from the input to the output in a system. Thus, similar to (8), we can modify the GD definition to make it always positive and valid for all signals.
A MATLAB implementation code of the proposed method is outlined in Algorithm 1. In order to avoid unnecessary variations in the IF, before evaluating the IF, the mean and dominating low frequency component like trend present in the signal can be removed. These can be easily removed by any zero-phase low pass filtering operation [13,18]. Moreover, depending upon the requirements, signal can be decomposed into a set of desired frequency bands.
In the examples presented in the following section, to decompose a signal x[n] into a set of desired frequency bands, we have used the signal model where c 0 is the mean-value of signal are the M orthogonal and LINOEP components, respectively. A simple block diagram of the zero-phase filter-bank (i.e. Figure 1. The frequency response of i-th band of the discrete Fourier transform (DFT) based zero-phase filterbank can be defined as where i = 1, 2, · · · , M ; K 0 = 0 and K M = (N − 1).
can be computed as where In this work, we use non-causal finite impulse response (FIR) and infinite impulse response (IIR) filter to decompose a signal into a set of LINOEP vectors by the filter mode decomposition (FMD) algorithm proposed in [18], as summarized in Algorithm 2. In this algorithm, for each iteration, F ILT ER i is zero-phase high pass filter (e.g. filtfilt function of MATLAB) with desired cutoff frequency f ci , and value of α i is obtained such that c i ⊥c i+1 . It is to be noted that, in general, filter is not ideal (or non brick wall frequency response) and therefore c i ⊥ c l for i, l = 1, 2, . . . , M − 1 and We advocate to use zero-phase filtering because it preserves salient features such as minima and maxima in the filtered waveform exactly at the position where those features occur in the unfiltered waveform. It is pertinent to note that the conventional (i.e. non zero-phase) filtering shifts these features in the filtered waveform and therefore cannot be used to obtain a meaningful TFE distribution, which is clearly demonstrated in simulation results. The zero-phase filtering of a signal can be obtained by the non-causal FIR and IIR filters, or via other decomposition methods like FDM, VMD, EVD and EMD algorithms.
. . .   z=hilbert(x);amp=abs(z);phi=unwrap(angle(z)); Algorithm 2: An FMD algorithm to obtain LINOEP vectors c i from decomposition of a signal x such that in all these signals. Figure 5 shows the TFE estimates of nonstationary signal, case (c), by this proposed IF: (top figure) with zero-phase FIR filter based decomposition into 100 bands; and (bottom figure) with conventional (non zero-phase) FIR filter based decomposition into 100 bands of equal frequencies, which is not able to detect and track true frequencies present in the signal.  Figure 7 shows the TFE distribution of a nonstationary signal (sum of five linear chirp signals) (1) by this proposed method with zero-phase FIR filter based decomposition into 10 bands (upper figure), and with conventional (non zero-phase) FIR filter based decomposition into 10 bands of equal frequencies (middle figure); and (2) by the EMD algorithm (lower figure). Clearly, both the conventional and EMD algorithm not able to detect true frequencies present in the signal. clearly not able to reveal true frequencies present in the signal.
Discussion: From Examples (1, 2 and 3), it is clear that the zero-phase DFT or zero-phase FIR filter based decomposition is able to track TFE distribution present in a signal. However, conventional filtering cannot be used to obtain meaningful TFE distribution of a signal, which is clearly demonstrated in bottom figures of Figure 5 π −π exp(jω(n − n 0 )) dω. This representation demonstrate that it is a superposition of equal amplitude sinusoidal functions of all frequencies [0π). The Nyquist frequency (F s /2) is the highest frequency that can be present at a given sampling rate, F s , in a discrete-time signal. The analytic representation of this signal is given by [13]  , and φ[n] = π 2 (n − n 0 ). Thus, frequency ω[n] = φ[n + 1] − φ[n] = π 2 , which corresponds to half of the Nyquist frequency, i.e. F s /4. Figure 10 shows the TFE estimates of this signal (with n 0 = 1999, F s = 1000 Hz and length N = 4000) using the EMD (top figure) and proposed method without decomposition (bottom figure). We again observe that the frequency present in this TFE plot is a average frequency (because delta function contains equal amplitude sinusoids of all frequencies 0 to F s /2). This example also demonstrate that the TFE plot, obtained by the Hilbert spectrum, is not limited by uncertainty principle and signal can be highly concentrated in time and frequency plane. However, it is to be noted that this TFE plot is not providing the true frequencies of delta function.
Discussion: In order to explain a average frequency effect in Figure 2, Figure 6, Figure 8 and Figure 10, let us consider a sum of sinusoids of equal amplitudes x(t) = N k=1 A cos(ω 0 kt). Its analytic representation is given by z(t) = then resultant IF would not be a constant (average) frequency but it would be a variable one.
Example 5: Figure 11 shows the TFE analysis of a white Gaussian noise (with zero mean, unit variance, 10240 samples and sampling frequency F s = 100 Hz) obtained from the EMD (top one) and this proposed method without decomposition (bottom one), which shows that the energy is randomly distributed across all the frequency range.
Example 6: An Earthquake time series signal is a nonlinear and nonstationary data. The Elcentro Earthquake data (sampled at F s = 50Hz) has been taken from [29] and is shown in Figure 12 (top one). The critical frequency range that matter in the structural design is less than 10Hz, and the Fourier based power spectral density (PSD), Figure 12  maximum energy concentration is around 1.7Hz and 2 second. The TFE plot provide details of how the different waves arrive from the epical center to the recording station, e.g. the compression waves of small amplitude but higher frequency range of 10 to 20Hz, the shear and surface waves of strongest amplitude and lower frequency range of below 5Hz which does most of the damage, and other body shear waves which are present over the full duration of the data span.
Discussion: It is clear from the above examples that different methods are producing the different TFE distributions of a signal. So, before concluding anything one needs to be careful while doing the analysis. For example, frequencies present in the Fourier spectrum is telling that these frequencies are present all the time in a signal under analysis, which may not be true. Consider another example, Figure 10 (bottom), of unit sample sequence which is telling that this signal is concentrated in time (which is true) and also concentrated in frequency (which is not true). If this were true, then we could transmit delta function in almost zero time with zero bandwidth through any system or channel. Thus, we conclude that the TFE representation depends on the number of bands in which data has been divided. In order to further illustrate this point we divided the same data, which is used for Figure 10, by DFT based zero-phase filtering into five and ten bands to obtain top and bottom figures in Figure 15, respectively, which reveal many those frequencies of data which are not present in Figure 10. From the Figures (2, 6, 8, 10 and 15), it is clear that the proposed method using the Hilbert spectrum produces average frequencies and good time resolution if envelope of signal is smooth. However, if envelope of a signal is fluctuating randomly or rapidly, e.g. consider the case of Gaussian white noise and Earthquake time series, the TFE plot has good time and frequency resolution as shown in Figure 11 and Figure 14 (top one). It is also clear, form Figures (4, 6, 8, 10 and 15), that as the signal is decomposed to more number of narrow bands, true frequencies present in the signal under analysis are revealed, frequency resolution is also increasing while the time resolution is reducing marginally.

CONCLUSION
The instantaneous frequency (IF) is an important parameter for the analysis of nonstationary signals and nonlinear systems. It is the basis of the time-  frequency-energy (TFE) analysis of a signal. The IF is the time derivative of the instantaneous phase (IP) and, originally, it is well-defined only when this derivative is positive. That is, the IF is valid only for monocomponent signals. If time derivative of IP is negative, i.e. the IF is negative, then it does not provide any physical significance. This study proposed a mathematical solution and eliminate this problem by modifying the present definition of IF. This is achieved by using the property of the multivalued inverse tangent function.
There are two fundamental and important conceptual innovation of this work. First, the extension of the traditional definition of IF by redefining it such that it is always positive. This proposed IF is valid for all types of signals such as monocomponent and multicomponent, narrow-band and wideband, stationary and nonstationary, linear and nonlinear signals. The understating of the time-frequency-energy representation, by all the methods which are using the IF, would improve significantly by using this definition. Second, we have also demonstrated that the zero-phase filtering based decomposition of a signal into a set of desired frequency bands with proposed IF accurately reveals the TFE distribution. Whereas, conventional (non zero-phase) filtering based decomposition cannot be used to obtain correct and meaningful TFE distribution. The Fourier and filter theory are well established, fully matured and developed, thus the zero-phase filtering based decomposition of a signal is most powerful which   Figure 15: The TFE analysis of unit sample sequence δ[n − n 0 ] (with, n 0 = 1999, sampling frequency Fs = 1000 Hz, length N = 4000) by proposed method with DFT based decomposition into five (top) and ten (bottom) bands of equal frequencies.