## Geometric Signal Theory

This section follows Chapter 5 of the text.

### Vector Interpretation of Complex Numbers

Here's how Fig.5.1 may be generated in matlab:
>> x = [2 3];                  % coordinates of x
>> origin = [0 0];             % coordinates of the origin
>> xcoords = [origin(1) x(1)]; % plot() expects coordinates
>> ycoords = [origin(2) x(2)];
>> plot(xcoords,ycoords);      % Draw a line from origin to x


### Signal Metrics

The mean of a signal stored in a matlab row- or column-vector x can be computed in matlab as
mu = sum(x)/N

or by using the built-in function mean(). If x is a 2D matrix containing N elements, then we need mu = sum(sum(x))/N or mu = mean(mean(x)), since sum computes a sum along dimension 1'' (which is along columns for matrices), and mean is implemented in terms of sum. For 3D matrices, mu = mean(mean(mean(x))), etc. For a higher dimensional matrices x, flattening'' it into a long column-vector x(:) is the more concise form:
N = prod(size(x))
mu = sum(x(:))/N

or
mu = x(:).' * ones(N,1)/N

The above constructs work whether x is a row-vector, column-vector, or matrix, because x(:) returns a concatenation of all columns of x into one long column-vector. Note the use of .' to obtain non-conjugating vector transposition in the second form. N = prod(size(x)) is the number of elements of x. If x is a row- or column-vector, then length(x) gives the number of elements. For matrices, length() returns the greater of the number of rows or columns.I.1

#### Signal Energy and Power

In a similar way, we can compute the signal energy (sum of squared moduli) using any of the following constructs:
Ex = x(:)' * x(:)
Ex = sum(conj(x(:)) .* x(:))
Ex = sum(abs(x(:)).^2)

The average power (energy per sample) is similarly Px = Ex/N. The norm is similarly xL2 = sqrt(Ex) (same result as xL2 = norm(x)). The norm is given by xL1 = sum(abs(x)) or by xL1 = norm(x,1). The infinity-norm (Chebyshev norm) is computed as xLInf = max(abs(x)) or xLInf = norm(x,Inf). In general, norm is computed by norm(x,p), with p=2 being the default case.

### Inner Product

The inner product of two column-vectors x and y5.9) is conveniently computed in matlab as
xdoty = y' * x


### Vector Cosine

For real vectors x and y having the same length, we may compute the vector cosine by
cosxy = y' * x / ( norm(x) * norm(y) );

For complex vectors, a good measure of orthogonality is the modulus of the vector cosine:
collinearity = abs(y' * x) / ( norm(x) * norm(y) );

Thus, when collinearity is near 0, the vectors x and y are substantially orthogonal. When collinearity is close to 1, they are nearly collinear.

### Projection

As discussed in §5.9.9, the orthogonal projection of onto is defined by

In matlab, the projection of the length-N column-vector y onto the length-N column-vector x may therefore be computed as follows:
yx = (x' * y) * (x' * x)^(-1) * x

More generally, a length-N column-vector y can be projected onto the -dimensional subspace spanned by the columns of the N M matrix X:
yX = X * (X' * X)^(-1) * X' * y

Orthogonal projection, like any finite-dimensional linear operator, can be represented by a matrix. In this case, the matrix
PX = X * (X' * X)^(-1) * X'

is called the projection matrix.I.2Subspace projection is an example in which the power of matrix linear algebra notation is evident.

#### Projection Example 1

>> X = [[1;2;3],[1;0;1]]
X =

1   1
2   0
3   1

>> PX = X * (X' * X)^(-1) * X'
PX =

0.66667  -0.33333   0.33333
-0.33333   0.66667   0.33333
0.33333   0.33333   0.66667

>> y = [2;4;6]
y =

2
4
6

>> yX = PX * y
yX =

2.0000
4.0000
6.0000

Since y in this example already lies in the column-space of X, orthogonal projection onto that space has no effect.

#### Projection Example 2

Let X and PX be defined as Example 1, but now let
>> y = [1;-1;1]
y =

1
-1
1

>> yX = PX * y
yX =

1.33333
-0.66667
0.66667

>> yX' * (y-yX)
ans = -7.0316e-16

>> eps
ans =  2.2204e-16

In the last step above, we verified that the projection yX is orthogonal to the projection error'' y-yX, at least to machine precision. The eps variable holds machine epsilon'' which is the numerical distance between and the next representable number in double-precision floating point.

### Orthogonal Basis Computation

Matlab and Octave have a function orth() which will compute an orthonormal basis for a space given any set of vectors which span the space. In Matlab, e.g., we have the following help info:
>> help orth
ORTH  Orthogonalization.
Q = orth(A) is an orthonormal basis for the range of A.
Q'*Q = I, the columns of Q span the same space as the
columns of A and the number of columns of Q is the rank
of A.


Below is an example of using orth() to orthonormalize a linearly independent basis set for :
% Demonstration of the orth() function.
v1 = [1; 2; 3];  % our first basis vector (a column vector)
v2 = [1; -2; 3]; % a second, linearly independent vector
v1' * v2         % show that v1 is not orthogonal to v2

ans =
6

V = [v1,v2]      % Each column of V is one of our vectors

V =
1     1
2    -2
3     3

W = orth(V)  % Find an orthonormal basis for the same space

W =
0.2673    0.1690
0.5345   -0.8452
0.8018    0.5071

w1 = W(:,1)  % Break out the returned vectors

w1 =
0.2673
0.5345
0.8018

w2 = W(:,2)

w2 =
0.1690
-0.8452
0.5071

w1' * w2  % Check that w1 is orthogonal to w2

ans =
2.5723e-17

w1' * w1  % Also check that the new vectors are unit length

ans =
1

w2' * w2

ans =
1

W' * W   % faster way to do the above checks

ans =
1    0
0    1

% Construct some vector x in the space spanned by v1 and v2:
x = 2 * v1 - 3 * v2

x =
-1
10
-3

% Show that x is also some linear combination of w1 and w2:
c1 = x' * w1      % Coefficient of projection of x onto w1

c1 =
2.6726

c2 = x' * w2      % Coefficient of projection of x onto w2

c2 =
-10.1419

xw = c1 * w1 + c2 * w2  % Can we make x using w1 and w2?

xw =
-1
10
-3

error = x - xw

error = 1.0e-14 *

0.1332
0
0

norm(error)       % typical way to summarize a vector error

ans =
1.3323e-15

% It works! (to working precision, of course)

% Construct a vector x NOT in the space spanned by v1 and v2:
y = [1; 0; 0];     % Almost anything we guess in 3D will work

%  Try to express y as a linear combination of w1 and w2:
c1 = y' * w1;      % Coefficient of projection of y onto w1
c2 = y' * w2;      % Coefficient of projection of y onto w2
yw = c1 * w1 + c2 * w2  % Can we make y using w1 and w2?

yw =

0.1
0.0
0.3

yerror = y - yw

yerror =

0.9
0.0
-0.3

norm(yerror)

ans =
0.9487

While the error is not zero, it is the smallest possible error in the least squares sense. That is, yw is the optimal least-squares approximation to y in the space spanned by v1 and v2 (w1 and w2). In other words, norm(yerror) is less than or equal to norm(y-yw2) for any other vector yw2 made using a linear combination of v1 and v2. In yet other words, we obtain the optimal least squares approximation of y (which lives in 3D) in some subspace (a 2D subspace of 3D spanned by the columns of matrix W) by projecting y orthogonally onto the subspace to get yw as above. An important property of the optimal least-squares approximation is that the approximation error is orthogonal to the the subspace in which the approximation lies. Let's verify this:
W' * yerror   % must be zero to working precision

ans = 1.0e-16 *

-0.2574
-0.0119


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The DFT
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Factoring Polynomials in Matlab