Calculation of the Fourier Transform and Fourier Series using the FFT

Most theoretical and mathematical analysis of signals and systems use the Fourier series, Fourier transform, Laplace transform, discrete-time Fourier transform (DTFT), or the z-transform, however, when we want to actually evaluate transforms, we calculate values at sample frequencies. In other words, we use the discrete Fourier transform (DFT) and, for efficiency, usually evaluate it with the FFT algorithm. An important question is how can we calculate or approximately calculate these symbolic formula-based transforms with our practical finite numerical tool. It would certainly seem that if we wanted the Fourier transform of a signal or function, we could sample the function, take its DFT with the FFT, and have some approximation to samples of the desired Fourier transform. We saw in the previous section that it is, in fact, possible provided some care is taken.

Summary

For the signal that is a function of a continuous variable we have

FT: f ( t ) f ( t ) → → F ( ω ) F ( ω ) DTFT: f ( T n ) f ( T n ) → → 1 T F p ( ω ) = 1 T ∑ ℓ F ( ω + 2 π ℓ / T ) 1 T F p ( ω ) = 1 T ∑ ℓ F ( ω + 2 π ℓ / T ) DFT: f p ( T n ) f p ( T n ) → → 1 T F p ( Δ k ) 1 T F p ( Δ k ) for Δ T N = 2 π Δ T N = 2 π

For the signal that is a function of a discrete variable we have

DTFT: h ( n ) h ( n ) → → H ( ω ) H ( ω ) DFT: h p ( n ) h p ( n ) → → H ( Δ k ) H ( Δ k ) for Δ N = 2 π Δ N = 2 π

For the periodic signal of a continuous variable we have

FS: g ˜ ( t ) g ˜ ( t ) → → C ( k ) C ( k ) DFT: g ˜ ( T n ) g ˜ ( T n ) → → N C p ( k ) N C p ( k ) for T N = P T N = P

For the sampled bandlimited signal we have

Sinc: f ( t ) f ( t ) → → f ( T n ) f ( T n ) f ( t ) = ∑ n f ( T n ) sinc ( 2 π t / T − π n ) f ( t ) = ∑ n f ( T n ) sinc ( 2 π t / T − π n ) if F ( ω ) = 0 F ( ω ) = 0 for | ω | > 2 π / T | ω | > 2 π / T

These formulas summarize much of the relations of the Fourier transforms of sampled signals and how they might be approximately calculate with the FFT. We next turn to the use of distributions and strings of delta functions as tool to study sampling.