Removal of noise by auto-correlation/cross-correlation
Aim: To verify theremoval of noise by auto correlation/cross correlation.
PC with windows (95/98/XP/NT/2000).
Detection of a periodic signal masked by random noise is of great importance .The noise signal encountered in practice is a signal with random amplitude variations. A signal is uncorrelated with any periodic signal. If s(t) is a periodic signal and n(t) is a noise signal then
Qsn(T)= cross correlation function of s(t) and n(t) Then Qsn(T)=0
Detection of noise by Auto-Correlation:
S(t)=Periodic Signal (Transmitted) , mixed with a noise signal n(t).
Then f(t) is received signal is [s(t ) + n(t) ]
Let Qff(T) =Auto Correlation Function of f(t)
Qss(t) = Auto Correlation Function of S(t)
Qnn(T) = Auto Correlation Function of n(t)
The periodic signal s(t) and noise signal n(t) are uncorrelated
The Auto correlation function of a periodic signal is periodic of the same frequency and the Auto correlation function of a non periodic signal is tends to zero for large value of T since s(t) is a periodic signal and n(t) is non periodic signal so Qss(T) is a periodic where as aQnn(T) becomes small for large values of T Therefore for sufficiently large values of T Qff(T) is equal to Qss(T).
Detection by Cross Correlation:
c(t)=Locally generated signal with same frequency as that of S(t)
Qfc (t) = Lim 1/T ? [s(t)+n(t)] [ c(t-T)] dt
C(t) is periodic function and uncorrelated with the random noise signal n(t). Hence Qnc(T0=0) Therefore Qfc(T)=Qsc(T).
Applications:detection of radar and sonal signals ,detection of cyclical component in brain analysis meterology etc.