Refreshing idea on Fourier analysis

Fumihiko Ishiyama
NTT Space Environment and Energy Labs.
Nippon Telegraph and Telephone Corp.
Tokyo, Japan
[email protected]

Abstract

The “theoretical limit of time-frequency resolution in Fourier analysis” is thought to originate in certain mathematical and/or physical limitations. This, however, is not true. The actual origin arises from the numerical (technical) method deployed to reduce computation time. In addition, there is a gap between the theoretical equation for Fourier analysis and its numerical implementation. Knowing the facts brings us practical benefits. In this case, these related to boundary conditions, and complex integrals. For example, replacing a Fourier integral with a complex integral brings a hybrid method for the Laplace and Fourier transforms, and reveals another perspective on time-frequency analysis. We present such a perspective here with a simple demonstrative analysis.

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1 Introduction

The “theoretical limit of time-frequency resolution of Fourier analysis” is thought to originate in certain mathematical and/or physical limitations, such as “it is from quantum mechanics $|\Delta f\Delta t|\geq h/2\pi$ ,” where $h$ is Planck constant.

This, however, is not true. The actual origin arises from the numerical method deployed to reduce computation time. In addition, there is a gap between the theoretical equation for Fourier analysis and its numerical implementation.

Let us remind ourselves of the equation for Fourier transform

S(f)=\int_{-\infty}^{\infty}s(t)e^{-2\pi ift}dt.

(1)

The equation requires an infinite continuous time series. However, the actual time series which in practice we have for analysis is always finite and discrete, and does not satisfy this requirement of the equation. Therefore, some modifications have to be made to the finite and discrete time series in order to satisfy the requirements of the equation, and it is these modifications that are the origin of the time-frequency limitation currently under discussion.

For this reason, we are using this article to present a refreshing idea. Breaking the time-frequency resolution limitation is a practically useful proposition.

First, we need an infinite time series. For this purpose, the “periodic boundary condition” which is shown in Fig. 1, is commonly introduced [1]. This condition is implicitly applied to all conventional linear methods, and replacing it with an alternative condition is the first step in the process of clearing the time-frequency resolution limitation. This means that all of the conventional linear methods share limitation in common that, as we will demonstrate below, can be overcome.

Refer to caption — Figure 1: Periodic boundary condition.

What the periodic boundary condition does is as follows. Assume that the finite time series, which is available for analysis, lines in the hatched area of the figure. From this position, we simply repeat the hatched time series infinitely, to fill the infinite time series.

Now, we have an infinite time series for the Fourier transform. However, the preparatory step shown in Fig. 1 throws up some awkward questions. It is undeniably the case that the possible frequencies are limited to the harmonics within the hatched area. Even though we have an infinite time series, the harmonics are limited, and herein lies the origin of the time-frequency resolution limitation shared by all the conventional linear methods. Indeed, this condition does boast the merit of reducing computation time [1, 2], and has been historically useful. However we would suggest that this merit is an anachronism in any case, on account of the improvements we have witnessed in computation power.

Therefore, we suggest replacing this condition with an alternative to fill out the infinite time series, and to clear the time-frequency resolution limitations that the conventionally applied condition introduces.

Our choice for an alternative is the linear extrapolation condition, shown in Fig. 2. However, this condition itself is unbound for $t\rightarrow\pm\infty$ , and some tricks are required [3].

In the following sections, we expand the Fourier analysis with some tricks, present a demonstrative analysis with the use of frequency modulated time series, and conclude this paper.

2 Expanding Fourier analysis

Our method looks very different from a conventional Fourier analysis, because of various unfamiliar concepts in the model equation. However, it will be found that the proposed method is a natural expansion of the familiar Fourier analysis.

Our method of analysis is not based on a Fourier integral, but on a mode decomposition with general complex functions, which organize nonlinear oscillators [3], from which we calculate the local linearized solution [4] of the decomposition.

More complete details of our method are set out in previously published papers [2, 3, 5, 6], and we ask the readers to please refer to them for further understanding. What we present here is just an outline.

2.1 Model equation

We expand the given time series $S(t)\in\mathbb{R}$ with general complex functions $H_{m}(t)\in\mathbb{C}$ as

S(t)=\sum_{m=1}^{M}e^{H_{m}(t)},

(2)

where $M$ is the number of complex functions.

The complex functions are expressed as

H_{m}(t)=\ln c_{m}(t_{0})+\int_{t_{0}}^{t}[2\pi if_{m}(\tau)+\lambda_{m}(\tau)]{\rm d}\tau,

(3)

where $f_{m}(t)~{}\in~{}\mathbb{R}$ represents the frequency modulation (FM) terms, which is known as instantaneous frequency [7] by van der Pol, $\lambda_{m}(t)~{}\in~{}\mathbb{R}$ represents the amplitude modulation (AM) terms, which is our original [3, 8], and $c_{m}(t_{0})\in\mathbb{C}$ represents the amplitudes of the oscillators at $t=t_{0}$ .

This expansion corresponds to a mode decomposition with general complex functions, noting that

H_{m}^{\prime}(t)=2\pi if_{m}(t)+\lambda_{m}(t).

(4)

Additionally, note that the case

H_{m}^{\prime}(t)=2\pi i\frac{m}{M\Delta T},

(5)

becomes a Fourier series expansion

S(t)=\sum_{m=1}^{M}e^{H_{m}(t)}=\sum_{m=1}^{M}c_{m}(t_{0})e^{2\pi i\frac{m}{M\Delta T}(t-t_{0})},

(6)

itself. That is, our model equation contains Fourier analysis as a special case, and is a natural expansion of the analysis.

2.2 Locally linearized solution

As it is known that our model equation does not have a unique solution, due to the non-linearity [9, 10], we need a special idea to make our model equation uniquely solvable [2, 4]. This is a part of our tricks, which corresponds to the linear extrapolation condition shown in Fig. 2.

We expand our model equation Eq. (2) as

S(t)|_{t\sim t_{k}}\simeq\sum_{m=1}^{M}e^{H_{m}(t_{k})+H_{m}^{\prime}(t_{k})(t-t_{k})+O\left((t-t_{k}\right)^{2})},

(7)

around $t\sim t_{k}=t_{0}+k\Delta T$ , consider a short enough time width, and ignore the higher order terms $O((t-t_{k})^{2})$ .

Then, the equation becomes a simple linear equation

	$\displaystyle S(t)\|_{t\sim t_{k}}$	$\displaystyle\simeq$	$\displaystyle\sum_{m=1}^{M}e^{H_{m}(t_{k})+H_{m}^{\prime}(t_{k})(t-t_{k})}$		(8)
		$\displaystyle=$	$\displaystyle\sum_{m=1}^{M}c_{m}(t_{k})e^{[2\pi if_{m}(t_{k})+\lambda_{m}(t_{k})](t-t_{k})},$		(9)

and we can obtain unique $H_{m}^{\prime}(t_{k})$ easily by applying the numerical method of linear predictive coding (LPC) with $N$ samples, noting that we must use a non-standard numerical method [2, 11] to hold the condition shown in Fig. 2.

The standard method of LPC is not adequate, because it contains the unfavorable condition [1] shown in Fig. 1, and an approximation [12] to reduce computation cost.

Subsequently, we calculate the complex amplitudes $c_{m}(t_{k})$ of the oscillators $c_{m}(t_{k})e^{H_{m}^{\prime}(t_{k})(t-t_{k})}$ as

\mathop{\arg\min}\limits_{c_{m}(t_{k})}\sum_{n=0}^{N-1}\left(S(t_{k}+n\Delta T)-\sum_{m=1}^{M}c_{m}(t_{k})e^{nH_{m}^{\prime}(t_{k})\Delta T}\right)^{2}.

(10)

2.3 Instantaneous spectrum

The equation for instantaneous spectrum is given as follows.

We transform the locally linearized time series $S(t)|_{t\sim t_{k}}$ around $t\sim t_{k}$ (Eq. (9)) with a complex integral as

$\displaystyle F(f,t_{k})$	$\displaystyle=$	$\displaystyle\int_{C}S(t)\|_{t\sim t_{k}}e^{-2\pi ift}dt$	(11)
	$\displaystyle=$	$\displaystyle\sum_{m}\int_{C}c_{m}(t_{k})e^{[\lambda_{m}(t_{k})+2\pi if_{m}(t_{k})]t-2\pi ift}dt$
	$\displaystyle=$	$\displaystyle\sum_{m}c_{m}(t_{k})\int_{C}e^{[\lambda_{m}(t_{k})+2\pi i(f_{m}(t_{k})-f)]t}dt$
	$\displaystyle=$	$\displaystyle\sum_{m}\frac{c_{m}(t_{k})}{\lambda_{m}(t_{k})+2\pi i(f_{m}(t_{k})-f)}.$

Note that the Laplace transform is written as

L(s)=\int_{0}^{\infty}s(t)e^{-st}dt=i\int_{0}^{-i\infty}s(i\tau)e^{-is\tau}d\tau,

(12)

and is understood as a Fourier integral on the imaginary axis. Therefore, the Laplace transform of $S(t)|_{t\sim t_{k}}$ around $t\sim t_{k}$ becomes

\Lambda(\lambda,t_{k})=\sum_{m}\frac{c_{m}(t_{k})}{(\lambda_{m}(t_{k})-\lambda)+2\pi if_{m}(t_{k})}.

(13)

As Eq. (11) is for a continuous time series, some modifications for its application to a discrete time series are required. That is, specifically, a modification from a Fourier transform to a Fourier series expansion.

For example, when $\lambda_{m}(t_{k})=0$ , the equation is unbound at $f=f_{m}(t_{k})$ , and is not practical. The practical value is the maximum value $|c_{m}(t_{k})|$ for discrete system.

Therefore, we take the absolute value of each term, and adjust the maximum values at $f=f_{m}(t_{k})$ so as to be $|c_{m}(t_{k})|$ [2].

F_{\rm disc}(f,t_{k})=\sum_{m}\left|\frac{c_{m}(t_{k})\lambda_{m}(t_{k})}{\lambda_{m}(t_{k})+2\pi i(f_{m}(t_{k})-f)}\right|

(14)

This equation has several merits [2, 3]. For example, the instantaneous spectrum of each term is available, and this feature is valuable for signal separation, as we show below. In addition, $F_{\rm disc}(f,t_{k})^{2}$ becomes power spectrum, corresponding to the conventional Fourier power spectrum.

2.4 Revised time-frequency resolution

We briefly demonstrate how our method works [11]. The source code and its execution output of the followings are shown in the reference.

The time series for analysis shown in Fig. 3 is

S(t)=0.01+\sin 2\pi t,

(15)

and we take twelve samples with a sampling frequency of $10$ Hz, as shown in the figure. The samples correspond to $1.2$ cycles of the oscillation, and this small sample set and short time series are sufficient for our method [10], in contrast to conventional methods.

Following this, we apply our method to the twelve samples, with the parameters $M=7,~{}N=12$ , and plot each obtained term in Eq. (14) in Fig. 4.

In addition, we plot the conventional Fourier spectrum from the same twelve samples in the figure, which corresponds to a bin of short time Fourier transform (STFT).

Four spectra (no. 1 to 4), are shown in the figure. Each spectrum corresponds to (no. 1) a constant term with amplitude $0.01$ , (no. 2 and 3, the same spectra) a sinusoidal time series with a frequency of $1$ Hz, and (no. 4) computational error, which corresponds to white noise on the time series.

Note that white noise on the given time series (no. 4) is obtained as a mode with a flat spectrum, and we can remove this unnecessary term from Eq. (14) for plotting spectrum.

Obtained numerical results for no. 1 to 3 are shown in Table 1.

Table 1: numerical resolutions

term frequency (Hz) amplitude no. 1 0.0 0.009999999999492205 no. 2 & 3 0.9999999999999438 1.0000000000004858

Conventional Fourier analysis requires the time width $T$ s to obtain the frequency resolution $f_{\rm res}=1/T$ Hz, and it holds that $T\times f_{\rm res}=1$ . This is the limit of the time-frequency resolution for a conventional Fourier analysis.

In contrast, the frequency resolution $f_{\rm res}$ of no. 2&3 in Table 1 is $1-0.9999999999999438=5.62\cdot 10^{-14}$ Hz, and is obtained using the time width $T=1.2$ s. Therefore, the time-frequency resolution becomes $T\times f_{\rm res}=6.74~{}\cdot~{}10^{-14}$ , and this resolution corresponds to the computational resolution of the numerical data. That is, the time-frequency resolution of our method is bounded by the resolution of the given numerical data, and not by the method itself.

3 Demonstrative analysis

Now we show a sample of a spectrogram using Eq. (14). For the purpose, we employ a simple time series $S(t)$ with frequency modulation

S(t)=\sin 2\pi\left(ft+a\cos wt\right).

(16)

Assuming the parameters in Eq. (16) as $f=100$ Hz, $a=0.5$ , and $w=10$ , the time series of modulated frequency $f(t)$ becomes

$\displaystyle f(t)$	$\displaystyle=$	$\displaystyle\left(ft+a\cos wt\right)^{\prime}$	(17)
	$\displaystyle=$	$\displaystyle f-aw\sin wt$
	$\displaystyle=$	$\displaystyle 100-5\sin 2\pi\frac{10}{2\pi}t$
	$\displaystyle\simeq$	$\displaystyle 100-5\sin 2\pi\frac{1}{0.63}t.$

Note that the waveform $-\sin wt$ appears in the time series.

We show the time series for analysis Eq. (16) and its corresponding spectrogram Eq. (14), which has the time series of modulated frequency in Eq. (17), in Fig. 5.

The sampling frequency $f_{s}$ was set to $10$ kHz, and the other parameters for the analysis were set to $M=10$ , and $N=20$ .

Therefore, the range of the spectrogram is $5$ kHz, and the enlarged view (20 Hz width) is shown in the figure. In addition, the time width for a single analysis becomes $Nf/f_{s}=0.2$ cycles of the base frequency $f=100$ Hz. This time width is sufficient for our method of analysis, as is shown in Fig. 5.

The upshot of all this is that a “less-than-a-cycle” time-frequency analysis is available with our method of analysis, with the resolution shown in Fig. 5 and also in Table 1.

4 Conclusion

We presented a refreshing idea on Fourier analysis. The idea makes less-than-a-cycle time-frequency analysis a reality, offers signal to noise manipulation, and brings mathematical understanding on nonlinear systems, such as the origin of the modulated frequencies.

We think that our method will not be the final nor ultimate way of time-frequency analysis, but we believe it to be sufficiently effective in its current form.

The know-how required to apply our method, such as the setting of parameters, is still under investigation, and requires further work.

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