By Jyrki Kauppinen

This contemporary method of the topic is obviously and logically established, and provides readers an figuring out of the essence of Fourier transforms and their functions. All vital elements are integrated with appreciate to their use with optical spectroscopic facts. in line with well known lectures, the authors give you the mathematical basics and numerical purposes that are crucial in useful use. the most a part of the booklet is devoted to functions of toes in sign processing and spectroscopy, with IR and NIR, NMR and mass spectrometry handled either from a theoretical and sensible standpoint. a few points, linear prediction for instance, are defined the following completely for the 1st time.

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10 we see that the inverse Fourier transform of h(t) = cos(2π f 0 t) is H ( f ) = 12 δ( f − f 0 ) + 12 δ( f + f 0 ). 3. 12, by setting h(t) = 1, and using the fact that −1 {1} = δ( f ). Generally, the convolution of δ( f ± f 0 ) with another function is ∞ δ( f ± f 0 ) ∗ G( f ) = δ(u ± f 0 )G( f − u) du = G( f ± f 0 ). 11) 38 3 Discrete Fourier transform Thus, the spectrum of a truncated cosine wave is HT ( f ) = 2T sinc(2π f T ) ∗ H ( f ) = T sinc [2π( f − f 0 )T ] + T sinc [2π( f + f 0 )T ] . 4.

It is recommended that you solve Problem 4 ﬁrst. ) 11. Applying the derivative theorem, compute −1 t exp −π t 2 . 34 2 General properties of Fourier transforms 12. A signal consists of two cosine waves, which both have an amplitude A. The frequencies of the waves are f 1 and f 2 . Derive the inverse Fourier transform of a differentiated signal, (a) by ﬁrst differentiating the signal and then making the transform −1 , (b) by ﬁrst transforming the original signal and then applying the derivative theorem.

1, 0, 1, . . , N − 1. 1) Generally, the recorded signal is a real function. 2, also the spectrum H ( f ) is real and symmetric, and H (− f ) = H ( f ). 2. 3) j=−N where T = N t. It is clear that a signal which consists of data at a ﬁnite number of discrete points cannot contain the same amount of information as an inﬁnitely long, continuous signal. This, inevitably, leads to some distortions, compared to the true case. In the following, we shall examine these distortions. 3 covers only a ﬁnite segment (−T, T ) of the signal h(t), unlike the Fourier integral, which extends from −∞ to +∞.