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 vector^5.1 $\underline{x}$ 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:

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

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.