September 2014
Intermediate to advanced
776 pages
24h 3m
English
So far, we have studied Fourier representations for finite sequences (the discrete Fourier transform), infinite sequences (the discrete-time Fourier transform), and functions on a finite real interval (the Fourier series).
| Transform | Time domain | Frequency domain |
| Discrete Fourier transform | Discrete, bounded | Discrete, bounded |
| n ∈ {0, 1, …, N − 1} | m ∈ {0, 1, …, N − 1} | |
| Fourier series | Continuous, bounded | Discrete, unbounded |
| x ∈ [0, L] | ||
| Discrete-time Fourier transform | Discrete, unbounded | Continuous, bounded |
| θ ∈ [ − π, π) | ||
| Fourier transform | Continuous, unbounded | Continuous, unbounded |
Matching these up, there is a pattern: if one domain is bounded, the other domain is discrete, and if one domain is unbounded, the other domain is continuous. The one transform we have yet to explore should map a continuous, unbounded time domain to a continuous, unbounded frequency domain. That fourth member of the Fourier family is the (continuous-time) Fourier transform:
This chapter begins with a heuristic development of the Fourier transform ...