lecture02: Continuous Fourier Transforms

The lecture concerns the definition and properties of the continuous Fourier transform. The course itself will be primarily occupied by the discrete Fourier transform, but the two are directly analogous, and many of the properties are more straight-forward to prove in the continuous case.

Topics covered

• Jean Baptiste Joseph Fourier
• Fourier series
• Fourier series as a representation
• Complex Fourier series
• Fourier series for other periods
• Example Fourier Series
• Integral transforms
• Fourier Transform
• Example: FT of a delta function
• Gaussian
• Example: FT of a Gaussian
• FT of some simple functions
• Deriving a Fourier Transform
• FTs of sin and cos
• The Fourier Transform: definitions
• Wave terminology
• RMS power of a sin wave
• Measuring power
• Decibels and sounds
• Power Spectra
• Phase
• Properties of the Fourier transform
• Properties: Linearity
• Properties: time shift
• Properties: Time scaling
• Properties: Duality
• Properties: Frequency shift
• Properties: Modulation
• Convolutions
• Properties: Convolution
• Convolution example
• Limiting convolutions
• Convolution example: interpolation
• Properties: Diff. $\frac{d^n{dt^n} f(t) \rightarrow (i 2 \pi s)^n F(s) $}
• Properties: Differentiation II %
• Properties: Integration
• Example: FT of a Gaussian
• Some useful rules for FTs
• Properties: Existence
• Properties: Invertible
• Trigonometric basis
• Measurement of Spectra