MATLAB Function Reference

delay

Estimates the delay of y with respect to x using the information theoretic delay criterion

Syntax

```[lag,criterion] = delay(x,y)
[lag,criterion] = delay(x,y,shift)
[lag,criterion] = delay(x,y,shift,order)
[lag,criterion] = delay(x,y,shift,order,base)
```

Description

Some primitive delay estimation algorithms are implemented in time-domain; for example the cross-correlation method. Most advanced time delay estimation algorithms are implemented in frequency-domain; for example the generalized cross-correlation method. The spectral estimation in case of short signal segments is the most important weakness of the frequency-domain implementations. Amongst other reasons R. Moddemeijer searched in his thesis for a time-domain implementation of an advanced delay estimation algorithm and found the "information theoretic delay criterion".

After publication of the thesis there has not been any further research in this direction. On the other hand the proposed method deserves further investigations. The main advantage of the method is the proven unambiguity of the criterion for a large enough order. Therefore the method is a suitable delay-estimation method in case of for quasi-periodic processes.

In theory the criterion as proposed is only valid for an x and y signal which have a common multidimensional normal distribution. Like the cross-correlation function the information theoretical delay criterion can be applied as a general purpose method in a much wider context. For `order=1` the criterion reduces to ```-1/2 log(1-rho^2)``` where `rho` is the correlation coefficient.

It can be proven that the criterion is a time-domain implementation of the maximum likelihood delay estimation algorithm as publiced by Knapp and Carter.

The `shift` is in absolute value the maximum relative shift of the two signals. As a consequence the estimated delay `lag` is bounded `-shift <= lag <= shift`. The resulting `criterion` is computed from a lag of `-shift` at index `1` to a lag of `shift` at index `2*shift+1`.

The `order` determines the computational effort, the effort is appproximately quadratic in the `order` and linear in `length(x)`. For `order=1` the method is equivalent with the cross-correlation method. For larger orders the method is better capable of including the dependancies of subsequent samples into the estimate. For large enough order the criterion has in theory no local minima; as a consequence the criterion has only one maximum. The order should be choosen large enough; the criterion may not show any local minima. The information criterions measures the mutual informations of two signal segments of x resp. y of size `order`.

The `base` of the logarithm determines the unit of measurement of the information criterion. Default base e (nats) is used, alternative choises are 2 (bit) and 10 (Hartley).

Note: the implementation is computationally intensive; an improved less intensive implementation is possible. If there is an application of the method, the author will provide this improved implementation.

Example

The standard delay-model; a common signal and independent noises added to the x- resp. the y-signal. As common signal we have choosen for 2nd order AR-filtered noise.

``` r=31/32; theta=pi/16; a=[1,(-2 * r * cos(theta)), r^2]; b=; signal=(filter(b,a,normrnd(0,1,1,100))); x=signal+8*normrnd(0,1,1,100); y=8*normrnd(0,1,1,100); y(4:100)=y(4:100)+signal(1:97); [lag,criterion]=delay(x,y); lag = 3 criterion = Columns 1 through 7 0.2026 0.2178 0.2413 0.3085 0.3504 0.2992 0.3209 Columns 8 through 14 0.3368 0.3331 0.3514 0.2804 0.2794 0.2333 0.1915 Columns 15 through 21 0.1934 0.2351 0.3108 0.4260 0.5890 0.7917 0.9486 Columns 22 through 28 1.2224 1.4508 1.4732 1.2399 1.0223 0.7930 0.6299 Columns 29 through 35 0.4734 0.3699 0.3041 0.2302 0.2114 0.2082 0.2017 Columns 36 through 41 0.2354 0.2721 0.2491 0.2363 0.2434 0.2193 ```

none

Literature

Moddemeijer, R., An information theoretical delay estimator, Ninth Symp. on Information Theory in the Benelux, May 26-27, 1988, Mierlo (NL), pp. 121-128, Ed. K.A. Schouwhamer Immink, Werkgemeenschap Informatie- en Communicatietheorie, Enschede (NL), ISBN: 90-71048-04-7, abstract , BibTeX ,

Moddemeijer, R., Delay-Estimation with Application to Electroencephalograms in Epilepsy (Phd-thesis), Universiteit Twente, 1989, Enschede (NL), ISBN: 90-9002668-1, abstract , BibTeX ,

Knapp, C.H. and Carter, C., The generalized correlation method for estimation of time delay, IEEE Transactions on ASSP, 24, pp 320-327,

Source code

 MATLAB Function Reference