Documentation

# fillgaps

Fill gaps using autoregressive modeling

## Description

example

y = fillgaps(x) replaces any NaNs present in a signal x with estimates extrapolated from forward and reverse autoregressive fits of the remaining samples. If x is a matrix, then the function treats each column as an independent channel.

example

y = fillgaps(x,maxlen) specifies the maximum number of samples to use in the estimation. Use this argument when your signal is not well characterized throughout its range by a single autoregressive process.

example

y = fillgaps(x,maxlen,order) specifies the order of the autoregressive model used to reconstruct the gaps.

example

fillgaps(___) with no output arguments plots the original samples and the reconstructed signal. This syntax accepts any input arguments from previous syntaxes.

## Examples

collapse all

Load a speech signal sampled at ${F}_{s}=7418\phantom{\rule{0.2777777777777778em}{0ex}}Hz$. The file contains a recording of a female voice saying the word "MATLAB®." Play the sound.

% To hear, type soundsc(mtlb,Fs)

Simulate a situation in which a noisy transmission channel corrupts parts of the signal irretrievably. Introduce gaps of random length roughly every 500 samples. Reset the random number generator for reproducible results.

rng default

gn = 3;

mt = mtlb;

gl = randi([300 600],gn,1);

for kj = 1:gn
mt(kj*1000+randi(100)+(1:gl(kj))) = NaN;
end

Plot the original and corrupted signals. Offset the corrupted signal for ease of display. Play the signal with the gaps.

plot([mtlb mt+4])
legend('Original','Corrupted')

% To hear, type soundsc(mt,Fs)

Reconstruct the signal using an autoregressive process. Use fillgaps with the default settings. Plot the original and reconstructed signals, again using an offset. Play the reconstructed signal.

lb = fillgaps(mt);

plot([mtlb lb+4])
legend('Original','Reconstructed')

% To hear, type soundsc(lb,Fs)

Load a file that contains depth measurements of a mold used to mint a United States penny. The data, taken at the National Institute of Standards and Technology, are sampled on a 128-by-128 grid.

Draw a contour plot with 25 copper-colored contour lines.

nc = 25;

contour(P,nc)
colormap copper
axis ij square

Introduce four 10-by-10 gaps into the data. Draw a contour plot of the corrupted signal.

P(50:60,80:90) = NaN;
P(100:110,20:30) = NaN;
P(100:110,100:110) = NaN;
P(20:30,110:120) = NaN;

contour(P,nc)
colormap copper
axis ij square

Use fillgaps to reconstruct the data, treating each column as an independent channel. Specify an 8th-order autoregressive model extrapolated from 30 samples at each end. Draw a contour plot of the reconstruction.

q = fillgaps(P,30,8);

contour(q,nc)
colormap copper
axis ij square

Generate a function that consists of the sum of two sinusoids and a Lorentzian curve. The function is sampled at 200 Hz for 2 seconds. Plot the result.

x = -1:0.005:1;

f = 1./(1+10*x.^2)+sin(2*pi*3*x)/10+cos(25*pi*x)/10;

plot(x,f)

Insert gaps at intervals (-0.8,-0.6), (-0.2,0.1), and (0.4,0.7).

h = f;

h(x>-0.8 & x<-0.6) = NaN;
h(x>-0.2 & x< 0.1) = NaN;
h(x> 0.4 & x< 0.7) = NaN;

Fill the gaps using the default settings of fillgaps. Plot the original and reconstructed functions.

y = fillgaps(h);

plot(x,f,'.',x,y)

Repeat the computation, but now specify a maximum prediction-sequence length of 3 samples and a model order of 1. Plot the original and reconstructed functions. At its simplest, fillgaps performs a linear fit.

y = fillgaps(h,3,1);

plot(x,f,'.',x,y)

Specify a maximum prediction-sequence length of 80 samples and a model order of 40. Plot the original and reconstructed functions.

y = fillgaps(h,80,40);

plot(x,f,'.',x,y)

Change the model order to 70. Plot the original and reconstructed functions.

y = fillgaps(h,80,70);

plot(x,f,'.',x,y)

The reconstruction is imperfect because very high model orders often have problems with finite precision.

Generate a multichannel signal consisting of two instances of a chirp sampled at 1 kHz for 1 second. The frequency of the chirp is zero at 0.3 seconds and increases linearly to reach a final value of 40 Hz. Each instance has a different DC value.

Fs = 1000;
t = 0:1/Fs:1-1/Fs;
r = chirp(t-0.3,0,0.7,40);
f = 1.1;
q = [r-f;r+f]';

Introduce gaps to the signal. One of the gaps covers the low-frequency region, and the other covers the high-frequency region.

gap = (460:720);
q(gap-300,1) = NaN;
q(gap+200,2) = NaN;

Fill the gaps using the default parameters. Plot the reconstructed signals.

y = fillgaps(q);

plot(t,y)

Fill the gaps by fitting 14th-order autoregressive models to the signal. Limit the models to incorporate 15 samples on the end of each gap. Use the functionality of fillgaps to plot the reconstructions.

fillgaps(q,15,14)

Increase the number of samples to use in the estimation to 150. Increase the model order to 140.

fillgaps(q,150,140)

## Input Arguments

collapse all

Input signal, specified as a vector or matrix. If x is a matrix, then its columns are treated as independent channels. x contains NaNs to represent missing samples.

Example: cos(pi/4*(0:159))+reshape(ones(32,1)*[0 NaN 0 NaN 0],1,160) is a single-channel row-vector signal missing 40% of its samples.

Example: cos(pi./[4;2]*(0:159))'+reshape(ones(64,1)*[0 NaN 0 NaN 0],160,2) is a two-channel signal with large gaps.

Data Types: single | double
Complex Number Support: Yes

Maximum length of prediction sequences, specified as a positive integer. If you leave maxlen unspecified, then fillgaps iteratively fits autoregressive models using all previous points for forward estimation and all future points for backward estimation.

Data Types: single | double

Autoregressive model order, specified as 'aic' or a positive integer. The order is truncated when order is infinite or when there are not enough available samples. If you specify order as 'aic', or leave it unspecified, then fillgaps selects the order that minimizes the Akaike information criterion.

Data Types: single | double | char | string

## Output Arguments

collapse all

Reconstructed signal, returned as a vector or matrix.

## References

[1] Kay, Steven M. Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: Prentice Hall, 1988.

[2] Orfanidis, Sophocles J. Optimum Signal Processing: An Introduction. 2nd Edition. New York: McGraw-Hill, 1996.

[3] Akaike, Hirotugu. “Fitting Autoregressive Models for Prediction.” Annals of the Institute of Statistical Mathematics. Vol. 21, 1969, pp. 243–247.