Cynosure对《Fast Fourier Transform and Its Applications》的笔记(1)

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If the waveform s(t) is not periodic, then the Fourier transform will be a continuous function of frequency, 
The Fourier transform is then a frequency-domain representation of a function. 
Fourier analysis allows one to examine a function from another point of view, the transform domain. 
FT --(applied to computer)--> numerical integration --(some problems do not have a closed-form FT solution)--> resort to DFT --(improve performance)--> FFT
We use this terminology throughout the book: 
- t is time and f is frequency. 
- a lowercase symbol represents a function of time, e.g. h(t); 
- the Fourier transform of this time function(e.g. h(t)) is represented by the same uppercase symbol as a function of frequency, e.g. H(f). 
H(f): the Fourier transform of h(t)
It is important to develop the Fourier transform of impulse functions because their use greatly simplifies the derivation of many transform pairs. 
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Appendix: THE IMPULSE FUNCTION: A DISTRIBUTION 
Even though the concept of the impulse function is correctly applied in the solution 
of many problems, the basis or definition of the impulse is normally  
mathematically meaningless. To ensure that the impulse function is well-defined, 
we must interpret the impulse not as a normal function but as a concept in 
the theory of distributions. 
The various properties of impulse functions can be determined directly 
from these definitions((A.l) (A.2) ). However, in a strict mathematical sense, these  
definitions are meaningless if we view b(t) as an ordinary function. If the impulse 
function is introduced as a generalized function or distribution, then these 
mathematical problems are eliminated. 
DISTRIBUTION CONCEPTS (p388) 通俗地解释了impulse function的意义