Fourier transforms & the convolution theorem - UGAstro

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Fourier Transforms ∞ ∫ −∞ H( f )= h(t)e −2πift dt h(t)= ∞ ∫ −∞ H ( f )e 2πift df • If t is measured in seconds, then f is in cycles per second or Hz • Other units – E.g, if h=h(x) and x is in meters, then H is a function of spatial frequency measured in cycles per meter
Fourier Transforms • The Fourier transform is a linear operator – The transform of the sum of two functions is the sum of the transforms h 12 = h 1 + h 2 H 12 ( f )= ∞ ∫ −∞ ∞ h 12 e −2πift dt = ∫ ( h 1 + h 2 )e −2πift dt = ∫ h 1 e −2πift dt + h 2 e −2πift dt −∞ = H 1 + H 2 ∞ −∞ ∞ ∫ −∞
Page 1 and 2: Convolution, Correlation, & Fourier
Page 3: Time & Frequency Domains • A phys
Page 7 and 8: FT Symmetries If… h(t) real h(t)
Page 9 and 10: Convolution • With two functions
Page 11 and 12: Correlation • The correlation of
Page 13 and 14: Autocorrelation • The correlation
Page 15 and 16: The Response Function • The effec
Page 17 and 18: Fourier Transforms & FFT • Fourie
Page 19 and 20: Discrete Convolution • In the dis
Page 21 and 22: Discrete Convolution s j r k (s*r)
Page 23: Suppose that f and g are functions

Fourier transforms & the convolution theorem - UGAstro

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