1. 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

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

3. 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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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 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

13. 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

14. 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

15. 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 Hz bandpass filter. © inoput 150 Hz ping and a Hz whale song. (d) filtered signal output using the 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

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

17. 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

18. 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

19. 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

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