5.1 Limitations of fourier analysis

Let's consider the Continuous-Time Fourier Transform (CTFT) pair: $$X(\Omega )=\int_{()} \,d t$$∞ ∞ x t Ω t $$x(t)=\frac{1}{2\pi }\int_{()} \,d \Omega$$∞ ∞ X Ω Ω t The Fourier transform pair supplies us with our notion of "frequency." In other words, all of our intuitions regarding therelationship 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 thetime-domain signal $x(t)$ can be expressed as a weighted summation of basis elements $\{b_{\Omega }(t)\colon ()\}()$∞ Ω ∞ b Ω t , where $b_{\Omega }(t)=e^{i\Omega t}$ is the basis element corresponding to frequency $\Omega$ . In other words, the basis elements are parameterized by the variable $\Omega$ that we call frequency . Finally, $X(\Omega )$ specifies the weighting coefficient for $b_{\Omega }(t)$ . In the case of the CTFT, the number of basis elements is uncountably infinite, and thus we need an integralto 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 isperiodic. Recall that if $x(t)$ is periodic with period $T$ , then it can be expressed as a weighted summation of basis elements $(k, )$∞ ∞ b k t , where $b_k(t)=e^{i\frac{2\pi }{T}tk}$ : $$x(t)=\sum_{k=()}$$∞ ∞ X k 2 T t k $$X(k)=\frac{1}{T}\int_{-\left(\frac{T}{2}\right)}^{\frac{T}{2}} x(t)e^{-(i\frac{2\pi }{T}tk)}\,d t$$ Here the basis elements comes from a countably-infinite set, parameterized by the frequency index $k\in \mathbb{Z}$ . 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 withtime. For example, consider a signal composed of a periodic component plus a sharp "glitch" at time $t_0$ , illustrated in time- and frequency-domains, .

Fourier analysis is successful in reducing the complicated-looking periodic component into a few simpleparameters: the frequencies $\{{\Omega }_1, {\Omega }_2\}()$ and their corresponding magnitudes/phases. The glitch component, described compactly in terms of the time-domainlocation $t_0$ and amplitude, however, is not described efficiently in the frequency domain since it produces a wide spread of frequencycomponents. Thus, neither time- nor frequency-domain representations alone give an efficient description of theglitched periodic signal: each representation distills only certain aspects of the signal.

As another example, consider the linear chirp $x(t)=\sin (\Omega t^2)$ illustrated in .

Though written using the $\sin ·$ function, the chirp is not described by a single Fourier frequency. We might try to be clever and write $$\sin (\Omega t^2)=\sin (\Omega t·t)=\sin (\Omega (t)·t)$$ where it now seems that signal has an instantaneous frequency $\Omega (t)=\Omega t$ which grows linearly in time. But here we must be cautious! Our newly-defined instantaneous frequency $\Omega (t)$ is not consistent with the Fourier notion of frequency. Recall that the CTFT says that asignal can be constructed as a superposition of fixed-frequency basis elements $e^{i\Omega t}$ 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 capableof uncovering this simple structure.

As a third example, consider a sinusoid of frequency ${\Omega }_0$ that is rectangularly windowed to extract only one period ( ).

Instantaneous-frequency arguments would claim that $$\forall , \Omega (t)=\begin{cases}{\Omega }_0 & \text{if $t\in \text{window}$}\\ 0 & \text{if $t\notin \text{window}$}\end{cases}\colon x(t)=\sin (\Omega (t)·t)$$ where $\Omega (t)$ takes on exactly two distinct "frequency" values. In contrast, Fourier theory says that rectangular windowing inducesa frequency-domain spreading by a $\frac{\sin \Omega }{\Omega }$ profile, resulting in a continuum of Fourier frequency components. Here again we see that Fourier analysisdoes not efficiently decompose signals whose "instantaneous frequency" varies with time.