Short-Term Autocorrelation and Covariance
Recall our definition of the autocorrelation function:
	
Note:
·	this can be regarded as a dot product of  and .
·	let's not forget preemphasis, windowing, and centering the window w.r.t. the frame, and that scaling is optional.
What would C++ code look like:
We note that we can save some multiplications by reusing products:

This is known as the factored autocorrelation computation.
It saves about 25% CPU, replacing multiplications with additions and more complicated indexing.
Similarly, recall our definition of the covariance function:
	
Note:
·	we use N-p points
·	symmetric so that only the  terms need to be computed
This can be simplified using the recursion:

Autocorrelation functions for the word "three" comparing the consonant portion of the waveform to the vowel (256-point Hamming window).

Note:
·	shape for the low order lags - what does this correspond to?
·	regularity of peaks for the vowel - why?
·	exponentially-decaying shape - which harmonic?
·	what does a negative correlation value mean?
array-style:
for(k=0; k<M; k++) {
r[k] = 0.0;
for(n=0; n<N-k; n++) {
r[k] += s[n] * s[n+k]
}
}
pointer-style:
for(k=0; k<M; k++) {
*r = 0.0; s = sig; sk = sig + k;
for(n=0; n<N-i; n++) {
*r += (s++) * (sk++);
}
}
Example of the Autocorrelation Function
Other Short-Term Spectral Measures

Power Density Spectrum:

Cross-Correlation:

Cross-Power Density Spectrum:

Example: Lag Estimation
The cross-correlation can be used to find the location of the best match, in a least-square error sense, between  and  (for example, in multipulse linear predictive coding).
Other Interesting Short-Term Time-Domain Measures
Average Magnitude Difference Function (AMDF):

Why? What does this look like:
Zero-Crossing Measure:

Gives a crude estimate of the fundamental frequency.
An Application of Short-Term Power: Speech SNR Measurement
Problem: Can we estimate the SNR from a speech file?
Energy Histogram:
Cumulative Distribution:
The SNR can defined as:


What percentiles to use?
Typically, 80%/20%, 85%/15%, or 95%/15% are used.