# Signals

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## Signal Properties

In this section, *n* denotes the time index of a signal, *m* is
the length of the observation vector and **R** is the *m*#*m*
autocorrelation matrix.

### Observation Vector

If *s(n)* is a sampled signal, the observation vector of order *m*
is **x***(;n)* = [*s(n) s(n-1) ... s(n-m+1)*]^{T}.
Thus **x***(i;n)* = *s(n-i+1)*

### Correlation Matrix

The *m'*th order correlation matrix of a stationary stochastic process
is E(**xx**^{H}) where **x***(;n)* is the
corresponding observation vector

## Special Signals

### Complex Sinewave

If *s(n) = a**exp*(jwn)*.

- The correlation matrix is
**R** where **R***(p,q) =
a*^{2} *** exp*(jw(q-p))*
**R = dd**^{H} where **d***(p) = a **
exp*(-jwp)*
- The only non-zero eigenvalue of
**R** has multiplicity 1 and is equal to
*a*^{2}.

The corresponding eigenvector is conj(**d**), where **d** is as defined
above.

### Sinewave

If *s(n)=a**sin*(wn)*,

- the correlation matrix is
**R** where
**R***(p,q)=a*^{2}*/*2*** cos*(w(q-p))*
**R = DD'** where **D** has dimension *m*#2 with
**D**(*p*,:) = *a/*sqrt(2) *** [cos*(w(p-*1*))*
sin(*w(p-*1*)*)]
- The two non-zero eigenvalues of
**R** have multiplicity 1 and are
*a*^{2}/4 * [*m*+sin(*wm*)/sin(*w*)
*m*-sin(*wm*)/sin(*w*)].

Writing *k=m*-1, the eigenvectors are the columns of
[sin((0:*k*)**w*) sin((*k*:-1:0)**w*)]*[1 1 ; 1 -1] =
2[sin(*kw*/2)*cos((-*k*/2:*k*/2)**w*)
cos(*kw*/2)*sin((-*k*/2:*k*/2)**w*)]

This page is part of The Matrix Reference
Manual. Copyright © 1998-2022 Mike Brookes, Imperial
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instructions. Please send any comments or suggestions to "mike.brookes" at
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Updated: $Id: signal.html 11291 2021-01-05 18:26:10Z dmb $