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Application Estimation Techniques to Sonar Data Lofargram

Justin M. Bell

1Portland State University

The use of a Lofargram or a Spectrogram is common place for its simplicity to view the Power Spectral Density of a signal in Slowtime

vs Frequency. It is common to simply use the built-in Matlab function to perform this action, but by using nonparametric estimation

techniques for smoothing and averaging a stronger result may be extracted.

I. INTRODUCTION

PECTROGRAMS are an important function in Acoustics as a

tool to visualized the change in frequency of a signal over

time. The spectrogram calculates the Power Spectral Density

for each frequency at each segment of time. Time segments can

be specified to overlap, but are not averaged or smoothed.

Utilizing estimation techniques to smooth and average the

Power Spectral Density estimates can visually highlight

features on the spectrogram and reveal signal patterns that may

otherwise go unnoticed.

The data used in this report is acoustic passive sonar data

taken off the coast of Florida as a part of the Shallow Water

Array Performance (SWAP) project. The small subset of data

was taken from an array of 32 hydrophones over a length of 900

seconds with a sampling frequency of 1 KHz.

II. SIGNAL ESTIMATOR

A. Periodogram

Real world data is typically continuous but the measurements

of it are limited in number and size of data making it discrete.

To analyze this data an inherent window must be applied when

performing a Discrete Fourier Transform to characterize the

power versus frequency distribution of a stochastic process.

The “natural estimate” would be to take the magnitude

squared of the windowed signal segment v(n).

�̂�𝑥(𝑒𝑗𝜔) ≜

1

𝑁|∑ 𝑣(𝑛)𝑒−𝑗𝜔𝑛𝑁−1

𝑛=0

|

2

This is called a Periodogram, it is a biased estimate and is non-

negative for all frequencies. This estimate in not consistent,

and a poor estimator due to the fact that it has excessive

variance on the order of Rx^2(e^jw) which does not converge

to the true PSD as the number of samples N goes to infinity.

There are two approaches to correcting the Periodogram,

either average Periodograms across multiple realizations, or

perform averaging across contiguous values to smooth the

variance.

B. Welch-Bartlett Method

The Welch-Bartlett Method of averaging multiple

Periodograms is achieved by subdividing the signal into K

overlapping windowed data segments of length L.

�̂�𝑥(𝑃𝐴)(𝑒𝑗𝜔) ≜

1

𝐾𝐿∑|𝑋𝑖(𝑛)𝑒

−𝑗𝜔𝑛|

𝐾−1

𝑖=0

2

As the number of segments K increases the variance is

asymptotically unbiased and tends toward zero on the order of

Rx^2(e^jw)/K. Also by taking smaller window lengths the

variance is reduced, but at the cost of additional smoothing

which reduces frequency resolution. Overlapping the signals

also reduces the variance, but additional reduction saturates

beyond 50% overlap.

The averaging of the Welch-Bartlett Method can be applied

to the PSD estimates within a spectrogram.

III. SPECTROGRAMS

A. Matlab Spectrogram

The built-in Matlab function “spectrogram” splits an input

signal into equal overlapping segments and calculates the

modified Periodogram of each segment using a Hamming

window.

Each segment is interpreted as a “Snapshot” of time of a

length determined by the length of the signal segment and

sampling frequency. The resultant image of stacking the

Modified Periodogram Snapshots next to each other gives an

estimate of how the intensity and spectral content of the Power

Spectral Density of the signal changes slowly in time.

The features of the Modified Periodogram have a very high

frequency resolution but the narrow peaks of concentrated

power make visualizing the signal difficult due to the large

dynamic range. Strong signals at exact frequencies may also

create delta functions that may not visible as they can be as

narrow as a fraction of a pixel in the window.

B. Welch-Bartlett Spectrogram

The Welch-Bartlett Spectrogram similarly splits the signal

into overlapping equal length Snapshot segments. The Welch’s

estimate for the Power Spectral Density additionally splits each

segment with an overlapping averaging window. The length of

the averaging window must be shorter than the signal segment,

and the shorter the window is the more averaging performed.

S

2

Due to mainlobe width and sidelobe leakage in the averaging

window the narrow peaks of concentrated power are smoothed

and widened, creating a strong visual of the signals Power

Spectral Density as it shifts slowly in time. The width of the

mainlobe is wider with a shorter averaging window, therefore

there is a tradeoff between variance reduction and frequency

resolution.

High frequency resolution and high variance reduction can

both be achieved by using longer signal Snapshots, though this

comes at a cost of fewer Snapshots and therefore a reduction in

time resolution. If the time resolution is too coarse the signal

snapshots may no longer be “Locally Stationary”, at which

point the estimation of the autocorrelation between signal

segments becomes a poor estimate of the True Power Spectral

Density.

IV. RESULTS

A. Spectrogram Setup

The Acoustic Data contained a matrix of 32 hydrophones

with 900,000 samples each. These 900,000 samples were taken

at a sampling frequency of 1 kHz over 900 seconds. Only data

from the first hydrophone was used in this comparison

experiment.

The estimates for the Modified Spectrogram and Welch’s

Method both used the same number of zero padding FFT points,

Snapshot signal length, and Blackman window tapering. . A

Blackman Window was selected for the Welch’s PSD estimate

for the additional sidelobe suppression. The Spectrogram in

Matlab defaults to the use of a Hamming window, but in this

experiment the use of a Blackman Window was used to match

the window used for the Welch’s PSD estimate.

B. Power Spectral Density Magnitude

The Modified Periodogram estimate of the PSD resulted in

much weaker power magnitudes than the Welch’s Method.

Sharing the same scale (figure 1 and figure 3) the signal in the

Welch’s estimate stands out strongly from a visual perspective.

Many of the features beyond 400 seconds in the Periodogram

cannot be distinguished from the background on a linear scale,

while the Welch’s has strong signal features throughout the 900

seconds for frequencies between 0 – 200 Hz.

The striations which shift slowly in frequency over time are

due to the Doppler shift of noise generated by moving targets

within the sensitivity range of the hydrophone. Much of this

noise is generated from bladed propellers spinning through the

water to drive craft, which is why there is a larger amount of

noise around 60 Hz which you might expect from electronic

noise in the engine.

There is also a large amount of energy at 13 Hz and 20 Hz

and fundamental harmonics of those frequencies. These do not

shift in time, but are also likely due to propeller noise. Doppler

shifts measure a radial velocity which is zero when targets

maintain equal range, as they move parallel.

Fig. 1. For a Spectrogram Snapshot of length 2 seconds, the time varying features of the signal are present up to a frequency of around 200 Hz, but the

total power is low with a high dynamic range.

Fig. 2. For a Spectrogram Snapshot of length 20 seconds, the time varying

features of the signal are present only up to a frequency of around 100 Hz, but

the total power is higher, though a high dynamic range remains.

Fig. 3. The Welch’s PSD for Snapshot of length 2 seconds, the time varying

features of the signal are present up to a frequency around 200 Hz, the total

power is much higher.

Fig. 4. The Welch’s PSD for Snapshot of length 20 seconds, the time varying features of the signal are nearly not present due to the non-Stationarity of the

estimate segments.

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Using beamforming techniques across the whole array of

hydrophone sensors would be able to show the bearing angle of

multiple targets, some of which change appear to be radially

stationary while others move across the face of the array.

C. Snapshot Length

The length of the Snapshot must be higher than the sampling

rate in order to have enough points to accurately represent the

signal therefore test lengths of 2 seconds, 10 seconds, and 20

seconds were analyzed.

At a length of 2 seconds the Periodogram (figure 1) was able

to accurately show time varying features up to a frequency of

200 Hz. The Welch’s Estimate (figure 3) also could show

features up to 200 Hz but had an excessive amount of

smoothing due to a wide mainlobe from a short window.

At a length of 20 seconds the Periodogram (figure 2) showed

features out to 100 Hz but had much lower power estimate.

While the Welch’s Estimate had a lot of signal power, at a lag

of 20 seconds the signal had lost most of its Stationarity and

time varying features could no longer be distinguished.

At a length of 10 seconds the Periodogram (figure 5)

maintained the 200 Hz worth of time varying frequency features

and had a middling power level. Though not as sharp, the

Welch’s estimate (figure 6) was able to maintain some of the

time varying features as the Non-Stationarity was less of a

hindrance and the window was long enough to maintain a

narrow mainlobe.

D. Window Length

The length of the Welch’s window was nominally set to be

9/10ths the length of the Snapshot segment at 9000 samples to

represent a minimal amount of Periodogram averaging (figure

5). As the lag window decreases in length there is consecutively

greater smoothing and greater reduction in the variance of the

background noise.

At a window length of only 1000 samples the averaging is

over smoothing. The narrow features close together on the

frequency axis blur into one another and become harder to

identify.

Conversely it becomes easier to highlight some features that

change in time, such as the pulsed natured of the low frequency

harmonics, which seem to have a high duty cycle.

V. CONCLUSION

The skillful application of Power Spectral Density

Estimation techniques can be used to magnify differences seen

in Lofargrams of hydrophone sensor data. The total power

estimated is greater and reduces the variance of the signal and

background noise at the cost of widening features and reduced

frequency resolution.

REFERENCES

[1] D.G. Manolakis, V. Ingle and S. Kogon, “Statistical and Adaptive Signal

Processing”, Boston: Artech House, Inc., 2005

Fig. 5. On a linear scale the Spectrogram signal becomes difficult to identify out of the background noise at long time lags as the target moves away.

Fig. 6. The Welch’s PSD estimate at a window length of 9000. Very clear on a linear scale, the signal is easy to identify out of the background noise.

Fig. 7. The Welch’s PSD for a window length of 2250. The Narrow signal

peaks have widened and are beginning to blur together.

Fig. 8. The Welch’s PSD for a window length of 1000. Over smoothing has

merged signals together making individual frequencies indistinguishable.

Pulsed nature of low frequency content is now clearly identifiable.