ORE 654Applications of Ocean Acoustics

Lecture 6aSignal processing

Bruce HoweOcean and Resources Engineering

School of Ocean and Earth Science and TechnologyUniversity of Hawai’i at Manoa

Fall Semester 2014

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Signal processing• Sampling rules• Fourier Representation and Transform• Filtering• Signal energy• Pings• Noise• Signal-to-noise ratio

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Sampling rules

• Acoustic signals in the water are converted to electrical signals in transducers, which are then sampled at discrete times. The transducers can be spatially distributed. They can be TX or RX.

• When going from analog to digital (or vice versa) must sample correctly

• Enough bits to represent amplitude (gain, …)• Time and space – Nyquist criteria / sampling

rules

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Amplitude - 1

• Analog-to-digital converters – volts to quanta• Number of “quantum levels” 2n, n is typically

8, 12, 16, 24, 30 • Dynamic range 2n

• Resolution and least count

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Amplitude - 2

• Resolution and least count – volts/quanta• For 1 V p-p

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Time• Sample clock jitter• Error in measured voltage• Affects coherence• If normalized Δt < 0.001,

then coherence between two time series > 0.98

• If Δt = 0.01, coherence drops to 0.80

• Need simultaneous measurements

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Sampling rules• Shannon-Nyquist sampling theorem• Nyquist criteria / Time and space sampling rules• Space: the spatial sampling interval must be less than half the

minimum wavelength• Time: the temporal sampling interval must be less than half the

minimum period• Nyquist frequency = fN = 1/2Δt; maximum resolved frequency

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Spatial and temporal sampling

• Fisheries example – net samples – loose most information

• Acoustic arrays – element spacing relative to wavelengths (beam patterns with side lobes indication of less than perfect sampling)

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Temporal sampling

• At discrete times • Signal b is sampled

adequately in c, but undersampled in d

• Produces a low frequency signal that was not in original

• ALIASING• High frequencies “aliased

” into low frequencies, folded at fN

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Fourier representation

• Any signal can be represented by sum of sines and cosines

• Figure – sum of 3 cosines

• Fourier series• Repeat with period T,

frequency f1

• Frequency resolution ~ 1/T (1/record length)

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Periodic signals

• g(t) repeats every T s• Often need Energy

averaged over a period

• Project g on exp to get G

• Here g and G are in volts (say)

• Frequency representation of g

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Transient signals• Finite energy – volts2-

s• Fourier transform pair• G(f) is spectrum of

signal g(t) - Units volts/Hz or volts-s

• Sum now integral• Same for spatial –

replace 2πf with k and t with x

• Note - normalizations

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Signal energy

• Power spectral density• (units g)2/Hz• If g has units Pa, spectral density has units (Pa)2/Hz• Integral over freq gives (Pa)2

• True intensity spectral density - divide by acoustic impedance to get (Pa2/ρAc)/Hz = (watts/m2)/Hz

• Intensity spectrum level ISL• Signal can be “noise”• Parceval’s theorem

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Filters• Separate signal from noise or from other unwanted

signals• Analog filters, Digital filters• a/d, d/a• Assume linear output = linear(input)• Frequency response function

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Filters• dd

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Anti-aliasing low pass filter in front of a/d

ACO hydrophone

transfer function

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Time gated pings

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• Same duration but different frequencies

• Envelop – cosine taper, here 0.1 s long

• Same spectral width (~ 1/tp)

Time gated pings

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• Same frequency but different duration

• Width of spectra decreases as signal duration increases, = ~ 1/T

• Td minimum duration necessary to have bandwidth Δf

Temporal resolution

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• How close can two signals be in time and we can still resolve?

• Gaussian envelope function, fc carrier frequency, Ping width Δt

• Resolution does not depend on fc

• Difference in travel time for ΔR• If Δt << Δt’ resolved; Barely resolved Δt’=Δt• Spectrum (also Gaussian) • System bandwidth 1/e width of spectrum; need

this TB product to resolve. Equate 2 spectra forms• Conservative > 1; Minimum BW to resolve objects

separated by Δt

Temporal resolution

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• What bandwidth signal do we need to resolve fish 0.3 m apart?