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Vector Spaces

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Review :: Operations :: Geometric Representation

Geometric Representation of Signals

All of the above ideas carry over to vector spaces of signals. We can define the length of a signal and the angle and distance between two signals. First we need to define a basis for our vector space of signals $ V$. Let $ x_1(t), x_2(t), \cdots, x_N(t)$ be a basis for our space. This means that they span the space and are linearly independent. So, any other signal $ y(t)$ in the space can be represented as a linear combination

$\displaystyle y(t) = \sum_{i=1}^N a_i x_i(t)$    

where the coordinates $ a_i$ are unique.

If we say that the basis is orthonormal then we know that

$\displaystyle \langle x_i(t), x_j(t) \rangle = \left\{ \begin{array}{ll} 1 & \text{if $i=j$,} \\ 0 & \text{otherwise} \end{array} \right.$    

but how do we calculate the inner product $ \langle x_i(t), x_j(t) \rangle$? There are many ways that this could be done. We will abide by the following definitions.

Definition: Inner product and norm of signals

$\displaystyle \langle x_i(t), x_j(t) \rangle = \int_0^T x_i(t) x_j(t) dt$    
$\displaystyle \Vert x_i(t) \Vert = \sqrt{ \langle x_i(t), x_i(t) \rangle } = \sqrt{ \int_0^T x_i(t) x_j(t) dt}$    

Note that the norm or length of a signal is just the square root of the energy of the signal.

$\displaystyle \Vert x_i(t) \Vert = \sqrt{ E_{x_i} }$    

Hence, the basis $ \{ x_i(t) \}$ is orthonormal if $ x_i(t)$ has unit energy and is orthogonal to $ x_j(t)$ for all $ j \not = i$.

Let $ y(t)$ be an arbitrary signal and let $ V$ be an $ N$-dimensional signal space with $ \{ x_i(t) \}$ an orthonormal basis. Then $ y(t)$ can be decomposed into two parts

$\displaystyle y(t) = y_x(t) + y_n(t)$    

where $ y_x(t)$ is the part of $ y(t)$ ``living'' in $ V$ and $ y_n(t)$ is orthogonal to $ V$. We have the following relations.

$\displaystyle y_x(t) = \sum_{i=1}^N \langle y(t), x_i(t) \rangle \; x_i(t) = \sum_{i=1}^N a_i x_i(t)$    
$\displaystyle y_n(t) = y(t) - y_x(t)$    
$\displaystyle \langle y_x(t) , y_n(t) \rangle = 0$    

In digital communications, this idea will be used in the following way. The signal space will be spanned by a basis of orthonormal pulses. The transmitted signal will be a linear combination of the basis pulses. The received signal is equal to the transmitted signal plus noise. The noise signal has a component that lives in the signal space and an orthogonal component. The orthogonal component is removed by the receiver. Hence, the only part of the noise that may cause errors at the receiver is the noise component in the signal space.

Definition: Unit energy signal

$\displaystyle u(t) = \frac{y(t)}{\Vert y(t) \Vert} = \frac{y(t)}{\sqrt{E_y}} \qquad \qquad \Vert u(t) \Vert = \sqrt{E_u} = 1$    

Once an orthonormal basis for the $ N$-dimensional signal space is specified, any vector in the space is represented uniquely by the numbers

$\displaystyle a_i = \langle y(t) , x_i(t) \rangle \qquad \qquad y(t) = \sum_{i=1} a_i x_i(t)$    

Because of the unique relationship

$\displaystyle y(t) \leftrightarrow \mathbf{a}= \begin{bmatrix}a_1 \\ a_2 \\ \vdots \\ a_N \end{bmatrix}$    

we can work with the vector $ \mathbf{a}$ rather than the signal $ y(t)$. This is useful because we can do all our analysis and design without having to worry about the waveforms. It will also make calculations easier and lead to some intuition that we would not get otherwise. The main requirement to utilize the relationship between the signal $ y(t)$ and its vector representation $ \mathbf{a}$ in Euclidean space is that the the elements of the vector $ \mathbf{a}$ must be the coordinates of $ y(t)$ with respect to an orthonormal basis for the signal space. That is why our text book goes through the Gram-Schmidt orthonormalization process.

Let $ \mathbf{a}$ be the vector representation of the signal $ y(t)$ and let $ \mathbf{b}$ be the same for the signal $ z(t)$. The following relationships hold.

$\displaystyle E_y = \int_0^T y^2(t) dt = \Vert y(t) \Vert^2 = \Vert \mathbf{a}\Vert^2 = \sum_{i=1}^N a_i^2$    
$\displaystyle \langle y(t) , z(t) \rangle = \int_0^T y(t) z(t) dt = \sum_{i=1}^N a_i b_i = \langle \mathbf{a}, \mathbf{b}\rangle$    
$\displaystyle d(y(t), z(t)) = \Vert y(t) - z(t) \Vert = \Vert \mathbf{a}- \mathbf{b}\Vert = d(\mathbf{a}, \mathbf{b})$    
Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, June 29). Vector Spaces. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Communication_Systems_I_1/vectors_spaces_3.htm. This work is licensed under a Creative Commons License. Creative Commons License
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