The Discrete Fourier Transform

Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you can find more thorough discussions in Oppenheim's book. Please let me know if you find any typos. In this lecture, we discuss the Discrete Fourier Transform (DFT), which is a fourier representation for finite length signals. The main practical importance of this new representation is that (unlike the DTFT), it can be computed very efficiently, for arbitrary inputs. This makes it the primary tool for performing frequency-domain analysis of signals on a computer. The discrete-time Fourier transform is an extremely useful tool for analyzing signal processing systems. However, for practical calculation, it has the disadvantage that X(e jω) is defined on ω ∈ R, and hence the full transform cannot be directly calculated. It is also worth noting that the discrete-time Fourier transform is defined via an infinite summation X(e jω) = ∞ n=−∞ x[n] exp(−jωn), (1.1) while, in practice we always work with signals x[n] of finite length. The discrete Fourier transform (DFT) is a Fourier representation for finite-length signals We say that such an x has length N .