Time-frequency equivalence using chirp signals for frequency 1 response analysis

7 Frequency response analysis (FRA) of systems is a well-researched area. For years, FRA has been 8 performed using input signals, which are a series of sinusoids or a sum of sinusoids. This results in 9 large experimentation time, particularly when the system has to be probed at lower frequencies. In this 10 work, we describe a previously unknown time-frequency duality for linear systems when probed through 11 chirp signals. We show that the entire frequency response can be extracted with a single chirp signal by 12 extending the notion of instantaneous frequency to both the input and output signals. It is surprising 13 that this powerful result had not been uncovered given that FRA has been used in multiple disciplines 14 for more than hundred years. This result has the possibility of completely revolutionizing methods used 15 for frequency response analysis. Simulation studies that support the main result are described. While 16 this result is of relevance in multiple areas, we demonstrate the potential impact of this result in electro- 17 chemical impedance spectroscopy. 18


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A system can be characterized by how it responds to sinusoidal input perturbations, also known as the 22 frequency response analysis (FRA). The frequency response at a particular frequency can be specified as a 23 ratio of the output to input, represented as a complex number. Since frequency response is computed from 24 time series data, an equivalence between time and frequency needs to be established. This is directly realized 25 through the well-known Fourier Transform, which allows any time domain signal to be decomposed into its 26 constituent frequency components. A standard approach for FRA of a system is to perturb the system with 27 an input, which is usually a series of sine signals or a sum of sine (multi-sine) signal [29,25] and identify 28 the frequency response from the output data. A key observation here is that to generate one point in the 29 frequency domain, all the time domain data needs to be processed. This is referred to as the localization 30 problem. A direct consequence of this problem is that large experimentation times are needed for generating 31 the complete frequency response of the system and there are also other issues related to deconvolution of 32 the various frequency components from the time domain signal. 33 There have been several attempts that have been made over the years to address the localization problem 34 [17,11]. The ideal case would be for a single time point to be localized to a single frequency, which 35 is theoretically not possible. Short term Fourier transforms (STFT) [1] and Wavelet transforms (WT) 36 [11, 17] are some of the time frequency localization approaches that have been attempted. Hilbert-Huang-37 Transforms (HHT) is another approach that is focused on addressing this problem [16]. In HHT, from in chirp signals is due to the fact that it is possible to define a "so called" instantaneous frequency, which is a 44 differential of the phase function of a sinusoid. As a result, a notional frequency can be assigned to every time 45 point in the input signal. Although this notion of one-to-one mapping between time and frequency could 46 be carried over to the output response for linear systems, work in extant literature is focused exclusively on using chirp signals for data generation to be processed by other techniques such as STFT [10,30,26] 48 or WT [3] and less on exploring the implications of the interesting time-frequency localization that chirp 49 signals afford. This might also be because instantaneous frequency as a concept itself is not well accepted 50 and/or understood [4,21]. There has been interest in interpreting instantaneous frequency and exploring 51 connections between standard techniques such as FFT and chirp, but still only in terms of information 52 content in the signal and not from viewing two time series (input and output) as having the same frequency 53 variation across time [21,20]. Our prior work [6, 28] comes closest to exploring the time-frequency equivalence 54 proposed here; however, we just proposed an algorithm for FRA of electrochemical systems. We claimed 55 that our algorithm was an approximate method for FRA; the impact of time-frequency equivalence was 56 neither clearly understood nor carefully explored at that time. Summarizing, a fundamental question that 57 is of interest is the following: is there a direct one-to-one equivalence between the time domain behavior and 58 the frequency domain behavior that can be established by assigning a single frequency to every time point in

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A chirp signal is a signal with time-varying frequency. The generic form of a chirp signal is u(t) = A sin(φ(t)) where φ(t) is the instantaneous phase. The instantaneous angular frequency of the signal at any instant t is defined as the differential of the instantaneous phase of the sinusoid at time t ω(t) = 2πf (t) = dφ(t) /dt . One can see from the definition of chirp signal that the phase function φ(t) is not assumed to take any particular form. Linear chirp defined below has been a popular choice.
where f (t) = f 0 + h 1 t is the linear instantaneous frequency and φ 0 is the initial phase. One could generalize 70 this linear chirp to n th order polynomial chirp, whose phase function φ(t) = P n+1 (t), is an (n + 1) th order 71 polynomial.

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We start with a very well-known result in the area of system identification.

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Lemma 1. When a stable, strictly causal linear system G(s) is perturbed with an input sine signal (u s (t) = A in sin ωt; ω = 2πf ), as time t tends to infinity, output of the system x(t) is also a sine signal with the same frequency as the input but with an amplitude ratio and phase lag.
where AR(ω) and φ L (ω) are the amplitude ratio and phase lag at angular frequency ω. Also, E(t) t→∞ = 0 and thus, This result has been used for decades now and is the foundation on which FRA has progressed. Using 75 this result, the frequency response of the system as a complex number can be identified at each frequency by 76 perturbing the system at every frequency of interest. However, a major disadvantage of this result is that, to 77 derive the complete frequency response, the system has to be perturbed at several frequencies individually.

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This is sometimes simplified using a sum of sines input and deconvolution of the output using fast Fourier 79 transform (FFT) [7]. Notice that this is an asymptotic result and hence one would have to wait for a certain 80 amount of time for the transients to dissipate before the frequency response is identified. We now present 81 the main result derived in this paper and contrast that with Lemma 1.

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Main Result. When a stable, strictly causal linear system G(s) is perturbed with a chirp signal (u(t) = A in sin φ(t)), as time t tends to infinity, the output of the system is also a chirp signal such that the instantaneous amplitude ratio (AR ch ) and phase lag (φ ch L ) of the chirp signal are same as the true amplitude ratio and phase lag of the system corresponding to the instantaneous frequency.
Angular frequency, ω = ψ(t) = dφ(t) dt , is a known quantity from the one-to-one mapping between time and 83 frequency of the input chirp signal. Second-order critically damped system

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Second-order overdamped system with a zero in left half plane 4 System with real repeated poles 5 2 (s+400) 2 (s 2 +200s+10000) 4 th -order system with real repeated poles and complex conjugate poles 6 20 (s+100+400i) 2 (s+100−400i) 2 4 th -order system with repeated complex-conjugate pairs as poles In summary, the asymptotic output response of the system to an input chirp signal can be written as: We will now validate the claims proposed in this paper through simulation studies. While we have  Table 1. To validate the claim, we compare the true chirp response (x(t)) of these systems to unit 88 amplitude chirp input and the asymptotic output behavior x(t) t→∞ as predicted by (8)  The first thing to notice about the main result is that this is also an asymptotic result (much like Lemma  practical value much like the result described in Lemma 1, which has been used for decades now.

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The most important implication of this result is that the time required for identifying the FR of the 126 system can be brought down dramatically. This is illustrated in Figure 1, where one sees that a single point 127 in the Nyquist plot corresponds to a signal in FR analysis. In the series of sines approach, these signals are 128 combined serially and this increases the testing times significantly. In the sum of the sines approach, these 129 signals are overlaid; however, to generate a point in the Nyquist plot, the output has to be deconvolved as   No. of data points in the plot Same as no. of signals Same as total samples in the signal (b) Example with fourth order chirp input that sweeps through the frequency range 0.001Hz to 10000Hz at a sampling rate, r = 10, 000 samples/sec ScienDirect and Scopus has 38,727 and 59,856 articles with the same keyword for the same duration. 153 We are now in a position to describe the impact of the main result reported in this paper on EIS. If the 154 chirp analysis procedure is followed instead of a series of sinusoidal signals for EIS, then the significance will 155 become apparent. In summary, a novel result of this work is that it is possible to extract the entire frequency response from iterative algorithms for processing the chirp response data can be developed that can minimize, even more, 192 the effect of error terms in the initial segment of data and the corresponding frequency response identification.

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Other than the literature associated with system identification, the approach described in this paper 194 has a role to play in all fields where impedance is used. Impedance being a fundamental characteristic of