Signals as Vectors
For the DFT, all signals and spectra are length . A length
sequence
can be denoted by
,
, where
may be
real (
) or complex (
). We now wish to regard
as a
vector5.1
in an
dimensional vector space. That is,
each sample
is regarded as a coordinate in that space.
A vector
is mathematically a single point in
-space represented by a list of coordinates
called an
-tuple. (The
notation
means the same thing as
.) It can be interpreted
geometrically as an arrow in
-space from the origin
to the point
.
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}]
$](http://www.dsprelated.com/josimages_new/mdft/img687.png)




The reader comfortable with vectors, vector addition, and vector subtraction may skip to §5.6.
An Example Vector View:
Consider the example two-sample signal
graphed in
Fig.5.1.
Under the geometric interpretation of a length signal, each sample is a
coordinate in the
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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Vector Addition
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The DFT