2. The Discrete Fourier Transform
Society for Industrial and Applied Mathematics eBooks · 1995
2.1. Introduction We must first agree on what the words discrete Fourier transform mean. What, exactly, is the DFT? The simple response is to give a formula, such as Fk = 1 N ∑ n=− N 2 +1 N 2 ƒn e−i2πnk/N , 2.1 and state that this holds for k equal to any N consecutive integers. Equation (2.1) is, in fact, a definition that we shall use, but such a response sheds no light on the original question. What, then, is the DFT? Is it a Fourier transform, as its name might imply? If it is not a Fourier transform, does it approximate one? The adjective discrete suggests that it may be more closely related to the Fourier series than to the continuous Fourier transform. Is this the case? There are no simple answers to these questions. Viewed from certain perspectives, the DFT is each of these things. Yet from other vantage points it presents different faces altogether. Our intent is to arrive at an answer, but we shall not do so in the span of two or three pages. In fact, the remainder of this book will be devoted to formulating an answer.