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# Limitations of Fourier Analysis

Module by: Phil Schniter. E-mail the author

Let's consider the Continuous-Time Fourier Transform (CTFT) pair: XΩ=xte(jΩt)d t X Ω t x t Ω t xt=12πXΩejΩtd Ω x t 1 2 Ω X Ω Ω t The Fourier transform pair supplies us with our notion of "frequency." In other words, all of our intuitions regarding the relationship between the time domain and the frequency domain can be traced to this particular transform pair.

It will be useful to view the CTFT in terms of basis elements. The inverse CTFT equation above says that the time-domain signal xt x t can be expressed as a weighted summation of basis elements bΩt bΩt <Ω< bΩ t Ω bΩ t , where bΩt=ejΩt bΩ t Ω t is the basis element corresponding to frequency ΩΩ. In other words, the basis elements are parameterized by the variable ΩΩ that we call frequency. Finally, XΩ X Ω specifies the weighting coefficient for bΩt bΩ t . In the case of the CTFT, the number of basis elements is uncountably infinite, and thus we need an integral to express the summation.

The Fourier Series (FS) can be considered as a special sub-case of the CTFT that applies when the time-domain signal is periodic. Recall that if xt x t is periodic with period TT, then it can be expressed as a weighted summation of basis elements bk t|k= k bk t , where bk t=ej2πTtk bk t 2 T t k : xt=k=Xkej2πTtk x t k X k 2 T t k Xk=1TT2T2xte(j2πTtk)dt X k 1 T t T 2 T 2 x t 2 T t k Here the basis elements comes from a countably-infinite set, parameterized by the frequency index kZ k . The coefficients Xk|k= k X k specify the strength of the corresponding basis elements within signal xt x t .

Though quite popular, Fourier analysis is not always the best tool to analyze a signal whose characteristics vary with time. For example, consider a signal composed of a periodic component plus a sharp "glitch" at time t0 t0, illustrated in time- and frequency-domains, Figure 1.

Fourier analysis is successful in reducing the complicated-looking periodic component into a few simple parameters: the frequencies Ω1 Ω2 Ω1 Ω2 and their corresponding magnitudes/phases. The glitch component, described compactly in terms of the time-domain location t0t0 and amplitude, however, is not described efficiently in the frequency domain since it produces a wide spread of frequency components. Thus, neither time- nor frequency-domain representations alone give an efficient description of the glitched periodic signal: each representation distills only certain aspects of the signal.

As another example, consider the linear chirp xt=sinΩt2 x t Ω t 2 illustrated in Figure 2.

Though written using the sin· · function, the chirp is not described by a single Fourier frequency. We might try to be clever and write sinΩt2=sinΩt·t=sinΩt·t Ω t 2 Ω t · t Ω t · t where it now seems that signal has an instantaneous frequency Ωt=Ωt Ω t Ω t which grows linearly in time. But here we must be cautious! Our newly-defined instantaneous frequency Ωt Ω t is not consistent with the Fourier notion of frequency. Recall that the CTFT says that a signal can be constructed as a superposition of fixed-frequency basis elements ejΩt Ω t with time support from to + + ; these elements are evenly spread out over all time, and so there is noting instantaneous about Fourier frequency! So, while instantaneous frequency gives a compact description of the linear chirp, Fourier analysis is not capable of uncovering this simple structure.

As a third example, consider a sinusoid of frequency Ω0Ω0 that is rectangularly windowed to extract only one period (Figure 3).

Instantaneous-frequency arguments would claim that xt=sinΩt·t  ,   Ωt={Ω0  if  twindow0  if  twindow    Ω t Ω0 t window 0 t window x t Ω t · t where Ωt Ω t takes on exactly two distinct "frequency" values. In contrast, Fourier theory says that rectangular windowing induces a frequency-domain spreading by a sinΩΩ Ω Ω profile, resulting in a continuum of Fourier frequency components. Here again we see that Fourier analysis does not efficiently decompose signals whose "instantaneous frequency" varies with time.

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