Signals

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.

Signal Properties

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) = a² * exp(jw(q-p))
• R = dd^H 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)]

The Matrix Reference Manual is written by Mike Brookes, Imperial College, London, UK. Please send any comments or suggestions to mike.brookes@ic.ac.uk