ORE 654 Applications of Ocean Acoustics Lecture 6a Signal processing Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology University of Hawai’i at Manoa Fall Semester 2014 4/20/2017 ORE 654 L5

Signal processing Sampling rules Fourier Representation and Transform Filtering Signal energy Pings Noise Signal-to-noise ratio 4/20/2017 ORE 654 L5

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 4/20/2017 ORE 654 L5

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 4/20/2017 ORE 654 L5

Amplitude - 2 Resolution and least count – volts/quanta For 1 V p-p 4/20/2017 ORE 654 L5

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 So if dT is sample interval, jitter is dt. 4/20/2017 ORE 654 L5

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 4/20/2017 ORE 654 L5

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) 4/20/2017 ORE 654 L5

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 4/20/2017 ORE 654 L5

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) 4/20/2017 ORE 654 L5

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 4/20/2017 ORE 654 L5

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 Figure 4.1.2 Examples of signals and their spectra. The same time and frequency scales are used for all g(t) or their spectra. The veritical units are arbitrary. Only positive frequencies are shown. (a) CW; (b), (c) short pings; (d) explosive source, Te = T1. 4/20/2017 ORE 654 L5

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 4/20/2017 ORE 654 L5

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 4/20/2017 ORE 654 L5

Filters dd Anti-aliasing low pass filter in front of a/d RHS – Fig 5.5 Filter operation shown in the time domain. (a) signal input is a 150 Hz ping having a duration of 0.01 sec. (b) signal out of a 50-150 Hz bandpass filter. © inoput 150 Hz ping and a 20 20 Hz whale song. (d) filtered signal output using the 50-150 Hz bandpass filter. Fig 5.6 Frequency domain analuysis. Bandwidts of equivalent bandpass filters are 2 Hz. (b) digitally calculated spectrum of 150 Hz ping. Digitall calculated spectrum of ping plus whale song. Ping 1/40 spectral amp of whalte. 4/20/2017 ORE 654 L5

ACO hydrophone transfer function 4/20/2017 ORE 654 L5

Time gated pings Same duration but different frequencies Envelop – cosine taper, here 0.1 s long Same spectral width (~ 1/tp) Figure 5.7. Signals of same duration but different frequencies. (a) Time domain presentation of pings with carrier frequencies of 50, 100, 150 Hz. (b) Spectral amplitudes of these pings. The spectra were computed using digital algorithms. 4/20/2017 ORE 654 L5

Time gated pings Same frequency but different duration Width of spectra decreases as signal duration increases, = ~ 1/T Td minimum duration necessary to have bandwidth Δf 4/20/2017 ORE 654 L5

Temporal resolution 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 4/20/2017 ORE 654 L5

Temporal resolution What bandwidth signal do we need to resolve fish 0.3 m apart? 4/20/2017 ORE 654 L5