Fundamento Conceptual

Signal analysis is an important part of any ultrasound based experiment. A simple form of signal analysis is the use of a real time lock-in amplifier, which compares an input signal,𝑎(𝑡)to a reference signal (i.e. an oscillating signal at a set reference

frequency,𝑓𝑟𝑒𝑓). A two phase lock-in amplifier does this at two phases of the reference

signal, separated by 90∘_{. Mathematically the lock-in amplifier operates according to}

𝑋(𝑡′) = 1 𝑡𝑎𝑣 ∫︁ 𝑡′ 𝑡′_{−𝑡𝑎𝑣} sin(2𝜋𝑓𝑟𝑒𝑓𝑡)𝑎(𝑡)d𝑡= 1 2𝐴𝑟𝑒𝑓cos(𝜑), and 𝑌(𝑡′) = 1 𝑡𝑎𝑣 ∫︁ 𝑡′ 𝑡′_{−𝑡𝑎𝑣} cos(2𝜋𝑓𝑟𝑒𝑓𝑡)𝑎(𝑡)d𝑡= 1 2𝐴𝑟𝑒𝑓sin(𝜑), (1.28)

where𝐴𝑟𝑒𝑓 is the amplitude of the signal at the reference frequency, 𝜑is the phase

difference between the input signal and the reference, and𝑡𝑎𝑣 is the averaging time,

which needs to be much longer than the reference period to avoid noise and harmonic effects [94, 95]. The time averaging used here is a simple implementation of a low pass filter and is often instead implemented using a different filtering technique, such as a finite impulse response (FIR) filter. The amplitude and phase can be returned from these signals as

𝐴𝑟𝑒𝑓 = √︀ 𝑋2_+𝑌2_{, and} 𝜑= arctan (︂ 𝑌 𝑋 )︂ . (1.29)

This method is useful for analysing signals in real time where there is a single fre- quency of interest that needs to be constantly monitored. As such it is commonly used in AFM techniques, though does not provide much use for NDT techniques where pulsed ultrasound is often used, giving a broadband wave, and where the waves can be dispersive.

Another way of analysing ultrasonic signals is using a direct visualisation of them. In NDT these are often referred to as A-, B-, and C-scans [96]. An A-scan is a trace from a single or averaged collection pulse, i.e. a graph of displacement (or velocity) against time. This can be used to determine, for example, the travel time and amplitude of pulses. A B-scan considers then the spatial positioning of the test probe, either showing analysed data from an A-scan at each point (such as a time delay or amplitude) against position, or an image combining multiple A-scans, displayed as time and position axes with a colour scale corresponding to displacement (or velocity). The latter is useful as parts of the signal can be indicated to be moving in time on the image, e.g. a travelling wave will be seen as a series of diagonal lines on the B-scan, corresponding to the velocity and direction of the wave. Interference effects can also be seen in this way. Image type B-scans may become less useful when dealing with highly dispersive waves, as the different signal components can become difficult to distinguish, and over small spatial ranges the time differences of the waves will be small and thus not easy to see [97,98]. A C-scan is an extension to two dimensional spatial mapping, showing an image with a colour scale that corresponds to analysed data from the A-scans at each point. These allow for the spatial extent of defects to be visualised.

It can also be useful to look at the frequency content of a signal [99,100]. This can be achieved through a Fourier transform of the A-scan, which converts from time space to frequency space, showing the magnitude of each frequency contained within the signal [95]. The infinite time continuous Fourier transform is given by

ˆ

𝑎(𝑓) =ℱ[𝑎(𝑡)] =

∫︁ ∞

−∞

𝑎(𝑡)𝑒−2𝜋𝑖𝑓 𝑡d𝑡 , (1.30)

whereˆ𝑎(𝑓)is the frequency space representation of𝑎(𝑡),𝑖=√−1, andℱ represents the Fourier transform operation. In practice, this form of the Fourier transform can- not be performed on the finite and discrete data collected experimentally. Therefore, the discrete Fourier transform (DFT) is used, given by

ˆ 𝑎(𝑓) =ℱ[𝑎(𝑡)] = 𝑁−1 ∑︁ 𝑛=0 𝑎(𝑛∆𝑡)𝑒−2𝜋𝑖𝑓 𝑛Δ𝑡, (1.31)

where 𝑁 is the number of points in 𝑎(𝑡) and ∆𝑡 is their separation. The DFT is

typically performed computationally using the fast Fourier transform (FFT) algo- rithm [101–103]. While this gives some useful information about the signal, it is not particularly useful for understanding the full behaviour of the wave as it removes all time information. A solution to this is the short time Fourier transform, in which the

signal is windowed and transformed at multiple time values, producing a represen- tation of𝑎in time-frequency space, known as a spectrogram or sonogram [95,104].

The windowing technique is known as the Gabor method [105]. In this a window function is defined that falls off to 0 symmetrically around 𝑡 = 0 and is

normalised (i.e. the integral over all time is equal to 1) is defined. An example of such a function, and the one used in these experiments, is the Gaussian function, given by ℎ(𝑡) = 1 𝑡𝑊 √ 𝜋𝑒 −(︁ 𝑡 𝑡𝑊 )︁2 , (1.32)

where 𝑡𝑊 is the width of the window. This is then used in a Fourier transform

according to

ˆ

𝑎(𝑓, 𝑡′) =ℱ[ℎ(𝑡−𝑡′)𝑎(𝑡)]. (1.33)

Typically, this result will be given as the magnitude or square of ˆ𝑎 as the sign

of the result is not always relevant. The result will then be a full breakdown of the time delay of each component of the waveform and their frequencies. Such a representation is advantageous as it allows for broadband dispersive waves to be analysed, including the overlay of dispersion curves if the sample properties and distance travelled by the pulse are known. Regions (in time and frequency) on these sonograms can then be chosen and averaged to analyse how a particular signal (frequency of a single mode and path) changes with spatial position in a B- or C- scan representation [98, 106]. One disadvantage to this technique is that resolution is inherently lost by the windowing process — a narrow window in time will, by the nature of the Fourier transform, result in very broad frequency representation. Likewise, time resolution can be sacrificed to gain frequency resolution, with the spread of the signal in time inversely related to the spread in frequency.