# Vector Spaces

## 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 . Let be a basis for our space. This means that they span the space and are linearly independent. So, any other signal in the space can be represented as a linear combination

where the coordinates are unique.

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

but how do we calculate the inner product ? There are many ways that this could be done. We will abide by the following definitions.

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Definition:
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Inner product
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and
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norm
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of signals

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

Hence, the basis is orthonormal if has unit energy and is orthogonal to for all .

Let be an arbitrary signal and let be an -dimensional signal space with an orthonormal basis. Then can be decomposed into two parts

where is the part of ``living'' in and is orthogonal to . We have the following relations.

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.

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Definition:
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Unit energy signal

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

Because of the unique relationship

we can work with the vector rather than the signal . 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 and its vector representation in Euclidean space is that the the elements of the vector must be the coordinates of with respect to an orthonormal basis for the signal space. That is why our text book goes through the Gram-Schmidt orthonormalization process.

Let be the vector representation of the signal and let be the same for the signal . The following relationships hold.