D02 - BIEN 130 Bioinstrumentation D02 May 6 2009 Hyle Park...

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Unformatted text preview: BIEN 130: Bioinstrumentation D02 May 6, 2009 Hyle Park Fourier transforms • Complex numbers • Definition • The point of Fourier transforms is to identify what frequency components make up any particular function cos sin jx e x i x = + ( 29 ( 29 ( 29 1 2 j t v t v t e dt V ϖ π ϖ- ℑ = = ∫ ( 29 ( 29 ( 29 1 1 2 j t V V e d v t ϖ π ϖ ϖ ϖ- ℑ = = ∫ Fourier transform pairs • Common examples • identity • Complex exponential • Cosine • Sine • Gaussian ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 1 2 2 2 1 2 2 cos sin t j t j t v t V v t e dt e t t j e e σ ϖ σ ϖ π ϖ π π ϖ πδ ϖ πδ ϖ ϖ ϖ δ ϖ ϖ δ ϖ ϖ ϖ δ ϖ ϖ δ ϖ ϖ σ---- =- + +- +-- ∫ Fourier transform properties • Useful rules • Reversibility • Linearity • Scaling • Shifting • Frequency shifting ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 1 1 2 2 1 2 1 j t a a j t j t v t V v t e dt...
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D02 - BIEN 130 Bioinstrumentation D02 May 6 2009 Hyle Park...

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