Linear Predictive Coding
• The problem of linear prediction is the estimation of the set of coefficients ak from the input signal x(n). The standard solution minimizes the mean square error: p 1 1 e2(n)= [x(n) a x(n k)]2 N N k n n X X kX=1 p @E n = 2 [(x(n)x(n i) a x(n k)x(n i)] = 0 @a k i n X kX=1 p a (n) ( i k )= (i) s.t. (k)= x(n)x(n + k) k | | n kX=1 X Linear Predictive Coding
Can be written in matrix form as: a = where: = ( i k ) ik | | a = ak = (k) which can be solved using the Levinson-Durbin recursion Linear Predictive Coding
H(z) Remember Cepstrum?
• Treat the log magnitude spectrum as if it were a signal -> take its (I)DFT
• Measures rate of change across frequency bands (Bogert et al., 1963)
• For a real-valued signal it’s defined as:
c (l)=real(IFFT(log( FFT(x) ))) x | |
• Followed by low-pass “liftering” in the cepstral domain Cepstrum
• The real cepstrum can be weighted using a low-pass window of the form:
1ifl =0,L1 w (l)= 2if1 l