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Signals as Vectors

For the DFT, all signals and spectra are length $ N$ . A length $ N$ sequence $ x$ can be denoted by $ x(n)$ , $ n=0,1,2,\ldots N-1$ , where $ x(n)$ may be real ( $ x\in\mathbb{R}^N$ ) or complex ( $ x\in\mathbb{C}^N$ ). We now wish to regard $ x$ as a vector5.1 $ \underline{x}$ in an $ N$ dimensional vector space. That is, each sample $ x(n)$ is regarded as a coordinate in that space. A vector $ \underline{x}$ is mathematically a single point in $ N$ -space represented by a list of coordinates $ (x_0,x_1,x_2,\ldots,x_{N-1})$ called an $ N$ -tuple. (The notation $ x_n$ means the same thing as $ x(n)$ .) It can be interpreted geometrically as an arrow in $ N$ -space from the origin $ \underline{0}
\isdef (0,0,\ldots,0)$ to the point $ \underline{x}\isdef
(x_0,x_1,x_2,\ldots,x_{N-1})$ .

<11088>> Another notation commonly used for vectors is matrix notation which is covered in any course on linear algebra [49]. A point $ x$ in $ N$ -space is normally expressed as a column vector

$\displaystyle \underline{x}= \left[\begin{array}{c} x_0 \\ [2pt] \vdots \\ [2pt] x_{N-1}\end{array}\right]

as opposed to a row vector $ \underline{x}= (x_0, \cdots, x_{N-1})$ . However, when working with Matlab, using row vectors by default saves screen space when typing them out. For that reason, we will adopt the row-vector convention. In state space analysis of dynamic systems, the column-vector convention is always used.

We define the following as equivalent:

$\displaystyle x \isdef \underline{x}\isdef x(\cdot)
\isdef (x_0,x_1,\ldots,x_{N-1})
\isdef [x_0,x_1,\ldots,x_{N-1}]
\isdef [x_0\; x_1\; \cdots\; x_{N-1}]

where $ x_n \isdef x(n)$ is the $ n$ th sample of the signal (vector) $ x$ . From now on, unless specifically mentioned otherwise, all signals are length $ N$ .

The reader comfortable with vectors, vector addition, and vector subtraction may skip to §5.6.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2019-01-09 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University