Time and Frequency
Bohr maintains that there are two “complementary aspects of reality.” These are like two distinct planes which we cannot exactly focus simultaneously.— Louis de Broglie (1892–1987)
One standard way of generating a time-frequency response builds on the Fourier transform (FT). We are going to develop concepts related to the continuous wavelet transform (CWT) and related somewhat to the short-time Fourier transform (STFT).
Begin with the idea of an experiment that yields a time series. It could be a seismic experiment in the field or in a laboratory, an audio recording, an electrocardiogram of the electrical activity of the heart, or any other measurement that has time-variable output. The data can have various properties. A time function can be periodic, such as a single sine or cosine wave, a voice perfectly holding one musical note, or the regular resting heartbeat. Ideally, the first two contain only one frequency, whereas the last has many frequencies but is still periodic. For such cases, the FT is ideal. If a heartbeat is recorded for, say, 1000 s and we do a Fourier transform, the amplitude spectrum will precisely identify the frequencies contained in the signal. That is possible because the frequency content is unchanging; the heart beats in a regular and repeatable pattern for the entire measurement period. The signal is stationary and unchanging, and the FT is the ideal tool for analysis.
However, we also have to deal with nonperiodic signals. One example would be a heartbeat under stress such as while taking a university examination