34 IEEE Instrumentation & Measurement Magazine December 2010 1094-6969/10/$25.00©2010IEEE Shlomo Engelberg Measuring the Spectral Content of a Signal: An Introduction Shlomo Engelberg and Edmond Chalom F ourier analysis is often thought of as a technique for measuring the spectral content of a signal over the pe- riod in which the signal was measured. In this column, we discuss techniques that allow you to use Fourier analysis to provide accurate measurements of the instantaneous frequency of a signal. We discuss the short-time Fourier trans- form and its main problem; we explain why it gives a very coarse estimate of the frequency. We show how it is possible to use the phase information provided by the dis- crete Fourier transform (DFT) to produce a much sharper estimate of the frequency. Finally, we show how the improved es- timate can be used to build a virtual FM demodulator. What is the spectral content of a signal? There are several ways of looking at the spectral content, the “frequency content,” of a signal. Consider y(t), the signal of Fig. 1. You might say that the signal is zero from 0 to 10 ms, from 20 to 30 ms, from 40 to 50 ms, and then from 10 to 20 ms the frequency of the signal is 605 Hz, and from 30 to 40 ms the frequency is 205 Hz. If you said this, you would be right in one sense. You would be saying that what interests you is the instantaneous frequency. Another way an engineer in good standing might choose to look at the spectral content of the igure is to say that its frequency content is deined by the signal’s Fourier transform, (1) Assuming that the parts of the signal that are not shown are all zero, this is not a hard integral to calculate. The magnitude of the Fourier transform of the signal is given in Fig. 2. Using this deinition of spectral content we ind that while the peaks of the Fourier transform appear about where we would like them to, there is a lot of energy at frequencies that are unrelated to the frequencies we expect to see. In applications in which the instantaneous frequency is what you need to measure, such as in an FM demodulator, the irst perspective is the appropriate one to take. In other cases, such as when you need to design a linear time invariant (LTI) ilter that will pass a signal while blocking non-signal components, the Fourier transform perspective is the correct one to take. Measuring the spectral content of a signal One of the many ways of actually measuring the spectral content of a signal, y(t), is to take a series of bandpass ilters and to see how each of the ilters responds to the signal – which ilters’ outputs contain most of the energy in the input signal. By controlling the bandwidth of the ilters, one controls the resolution with which the bandpass ilters’ outputs determine the signal’s frequency content. One of the most popular methods of measuring frequency is to use the discrete Fourier transform (DFT) and, speciically, its eficient implementation, the fast Fourier transform (FFT). Given a sequence of N measurements of y(t) taken T s seconds apart, (2) instrumentation notes Fig. 1. A signal, y(t), composed of sinewaves of different frequencies and periods when the signal is zero.