# 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(xxH) 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) = a2 * exp(jw(q-p))
• R = ddH where d(p) = a * exp(-jwp)
• R is of rank 1
• The only non-zero eigenvalue of R has multiplicity 1 and is equal to a2.
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)=a2/2* 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))]
• R is of rank 2
• The two non-zero eigenvalues of R have multiplicity 1 and are a2/4 * [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 College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "imperial.ac.uk".
Updated: \$Id: signal.html 11291 2021-01-05 18:26:10Z dmb \$