Limitations of Fourier Analysis 2.0 2003/01/13 2003/01/13 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Fourier transform frequency Let's consider the Continuous-Time Fourier Transform (CTFT) pair: X Ω t x t Ω t 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 x t can be expressed as a weighted summation of basis elements bΩ t Ω bΩ t , where bΩ t Ω t is the basis element corresponding to frequency Ω. In other words, the basis elements are parameterized by the variable Ω that we call frequency. Finally, X Ω specifies the weighting coefficient for 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 x t is periodic with period T, then it can be expressed as a weighted summation of basis elements k bk t , where bk t 2 T t k : x t k X k 2 T t k 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 k . The coefficients k X k specify the strength of the corresponding basis elements within signal 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 , illustrated in time- and frequency-domains, .

Fourier analysis is successful in reducing the complicated-looking periodic component into a few simple parameters: the frequencies Ω1 Ω2 and their corresponding magnitudes/phases. The glitch component, described compactly in terms of the time-domain location t0 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 x t Ω t 2 illustrated in .

Though written using the · function, the chirp is not described by a single Fourier frequency. We might try to be clever and write Ω t 2 Ω t · t Ω t · t where it now seems that signal has an instantaneous frequency Ω t Ω t which grows linearly in time. But here we must be cautious! Our newly-defined instantaneous frequency 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 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 Ω0 that is rectangularly windowed to extract only one period ().

Instantaneous-frequency arguments would claim that Ω t Ω0 t window 0 t window x t Ω t · t where Ω t takes on exactly two distinct "frequency" values. In contrast, Fourier theory says that rectangular windowing induces a frequency-domain spreading by a 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.