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Auto correlation along one axis.

Given a 1-D wide sense stationary (WSS) sequence X, the auto correlation RXX may be defined as (with E expectation and Conj complex conjugate)

RXX[m] := E{ W[m] Conj(W[0]) } = E{ W[0] Conj(W[-m]) },
W[n]   := (X[n] - MU) / S,
MU     := E{ X[0] },
S**2   := E{ (X[0] - MU) Conj(X[0] - MU) }.

This function takes the viewpoint that x is (along one axis) a finite sub-sequence of a realization of (WSS) X, and then uses x to produce an estimate of RXX[m] as follows:

After extending x from length L to inf by zero padding, the auto correlation estimate rxx[m] is computed for m = 0, 1, ..., max_lags as

rxx[m] := (L - m)**-1 sum_n w[n + m] Conj(w[n]),
w[n]   := (x[n] - mu) / s,
mu     := L**-1 sum_n x[n],
s**2   := L**-1 sum_n (x[n] - mu) Conj(x[n] - mu)

The error in this estimate is proportional to 1 / sqrt(len(x) - m), so users often set max_lags small enough so that the entire output is meaningful.

Note that since mu is an imperfect estimate of E{ X[0] }, and we divide by len(x) - m rather than len(x) - m - 1, our estimate of auto correlation contains a slight bias, which goes to zero as len(x) - m --> infinity.

x float32 or complex64 Tensor.
axis Python int. The axis number along which to compute correlation. Other dimensions index different batch members.
max_lags Positive int tensor. The maximum value of m to consider (in equation above). If max_lags >= x.shape[axis], we effectively re-set max_lags to x.shape[axis] - 1.
center Python bool. If False, do not subtract the mean estimate mu from x[n] when forming w[n].
normalize Python bool. If False, do not divide by the variance estimate s**2 when forming w[n].
name String name to prepend to created ops.

rxx: Tensor of same dtype as x. rxx.shape[i] = x.shape[i] for i != axis, and rxx.shape[axis] = max_lags + 1.

TypeError If x is not a supported type.