g*********u
发帖数: 21 | 1 假设r(tau) is the autocorrelation function of a complex random process.
by the definition of autocorrelation, r() is conjugate symmetric:
r(tau) = r^*(-tau)
what are the other conditions needed to garantee that r(tau) is an
autocorrelation function?
for example, a rectangular function f(tau) = 1 for abs(tau) <0.5 and 0
elsewhere. can not be an autocorrelation function.
because its fourier transform is sinc(f) and is negative at some frequencies
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r******u
发帖数: 50 | 2 R_X(\tau) has to be continuous over R
frequencies
【在 g*********u 的大作中提到】
: 假设r(tau) is the autocorrelation function of a complex random process.
: by the definition of autocorrelation, r() is conjugate symmetric:
: r(tau) = r^*(-tau)
: what are the other conditions needed to garantee that r(tau) is an
: autocorrelation function?
: for example, a rectangular function f(tau) = 1 for abs(tau) <0.5 and 0
: elsewhere. can not be an autocorrelation function.
: because its fourier transform is sinc(f) and is negative at some frequencies
: .
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g*********u
发帖数: 21 | 3 when r(\tau) = delta(\tau), which is still a valid autocorrelation.
but it is not continuous at 0.
【在 r******u 的大作中提到】
: R_X(\tau) has to be continuous over R
:
: frequencies
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r******u
发帖数: 50 | 4 Actually these three conditions are equivalent:
A WSS random process X is m.s. continuous
R_X(\tau) is continuous around 0
R_X(\tau) is continuous over R
I think you are talking abour Gaussian white noise, which is not m.s. contin
uous.
【在 g*********u 的大作中提到】
: when r(\tau) = delta(\tau), which is still a valid autocorrelation.
: but it is not continuous at 0.
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g*********u
发帖数: 21 | 5 thanks for your reply. suppose R_X(\tau) is defined to have the same shape
as the frequency response of a raised cosine filter
http://en.wikipedia.org/wiki/Raised-cosine_filter
then the Fourier transform of R_X(\tau) is S(w) will have negative value at
some frequency points. therefore, R_X(\tau) is not a valid autocorrelation
function.
however, R_X(\tau) is continuous over R.
which is inconsisitant with the conditions you mentioned.
contin
【在 r******u 的大作中提到】
: Actually these three conditions are equivalent:
: A WSS random process X is m.s. continuous
: R_X(\tau) is continuous around 0
: R_X(\tau) is continuous over R
: I think you are talking abour Gaussian white noise, which is not m.s. contin
: uous.
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