now is: The PSD-per-unit-time converges to finite values at all frequencies except those where h(t) has a discrete sine-wave (or cosine-wave) component of finite amplitude. At those frequencies, it becomes a delta-function, ., a sharp spike | 500 Chapter 12. Fast Fourier Transform now is The PSD-per-unit-time converges to finite values at all frequencies except those where h t has a discrete sine-wave or cosine-wave component of finite amplitude. At those frequencies it becomes a delta-function . a sharp spike whose width gets narrower and narrower but whose area converges to be the mean square amplitude of the discrete sine or cosine component at that frequency. We have by now stated all of the analytical formalism that we will need in this chapter with one exception In computational work especially with experimental data we are almost never given a continuous function h t to work with but are given rather a list of measurements of h ti for a discrete set of tf s. The profound implications of this seemingly unimportant fact are the subject of the next section. CITED REFERENCES AND FURTHER READING Champeney . 1973 Fourier Transforms and Their Physical Applications New York Academic Press . Elliott . and Rao . 1982 Fast Transforms Algorithms Analyses Applications New York Academic Press . Fourier Transform of Discretely Sampled Data In the most common situations function h t is sampled . its value is recorded at evenly spaced intervals in time. Let A denote the time interval between consecutive samples so that the sequence of sampled values is hn h nA n . -3 -2 -1 0 1 2 3 . The reciprocal of the time interval A is called the sampling rate if A is measured in seconds for example then the sampling rate is the number of samples recorded per second. Sampling Theorem andAliasing For any sampling interval A there is also a special frequency fc called the Nyquist critical frequency given by fc -4 Jc 2A If a sine wave of the Nyquist critical frequency is sampled at its positive peak value then the next sample will be at its negative trough value the sample after that at the positive peak again and so on. Expressed otherwise Critical sampling of a sine wave is two sample points .
