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{\bf R}^N$) or complex ( $ x\in{\bf 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})$.

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.

An Example Vector View: $ N=2$

Consider the example two-sample signal $ x=(2, 3)$ graphed in Fig.5.1.

Figure 5.1: A length 2 signal $ x=(2, 3)$ plotted as a vector in 2D space.
\includegraphics[scale=0.7]{eps/vec2d}

Under the geometric interpretation of a length $ N$ signal, each sample is a coordinate in the $ N$ dimensional space. Signals which are only two samples long are not terribly interesting to hear,5.2 but they are easy to plot geometrically.


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