University of Amsterdam
Institute of Physics: Van der Waals-Zeeman Institute
Soft Matter Group
Non-destructive imaging of the internal
properties of Wood using Fourier Space
Analysis
Author:
Espen Tierolff
10445889
Supervisor:
Dr. R Sprik
Dr. N.F. Shahidzadeh
Bachelorproject Natuur- en Sterrenkunde: 15 EC
April 1, 2015 - July 31, 2015
Populaire Samenvatting
Het toepassen van geluid bij het vaststellen van de kwaliteit van een voorwerp is een eeuwenoud principe. Zo kan een pottenbakker bepalen of zijn potten wel goed zijn door er een tik op te geven, waardoor de pot begint te trillen. Dit gebeurt met een bepaalde frequentie, de natuurlijke frequentie van de pot. Door te luisteren naar het geluid kan de pottenbakker bepalen of er een scheur in zit of niet, het geluid zal namelijk vervormd klinken als er een scheur aanwezig is. Deze methode heeft wel een nadeel, de klank wordt alleen veranderd door grote fouten in het voor-werp en is dus niet erg nauwkeurig. De nauwkeurigheid hangt af van de golflengte en daarmee van de frequentie. De golflengtes van de hoorbare trillingen zijn te groot om kleine onregelmatigheden waar te nemen. Dit wordt opgelost door gebruik te maken van ultrasoon geluid, dat bestaat uit trillingen met een zeer hoge frequentie.
Dit proces kunnen we ook toepassen op andere materialen, zoals hout. We laten de ultrasone golven door hout heengaan en we kijken of deze veranderen. Aan de hand daarvan proberen we te bepalen of er misschien foutjes in het hout aanwezig zijn, zoals knoesten of gaten. Ook proberen we te bepalen of het hout vochtig is. Dit is vooral van waarde voor de kunstwereld, aangezien de restauratie van vele werken afhangt van de aanwezigheid van eventuele fouten. Een kunstwerk mag echter niet aangeraakt worden. Daarom sturen we de ultrasone golven door de lucht naar het hout.
Om het onderscheid te kunnen maken tussen de verschillende fouten, meten we het effect van hun aawezigheid op het geluid. Zo bepalen we de invloed van water in het hout, van gaten in het hout en de richting van de nerf. Dit wordt gedaan door de voortplantingssnelheid en de amplitude te meten. Ook worden de effecten van de fouten op de verschillende frequenties bestudeerd. Dit vereist een ontbinding van de functie in de verschillende frequentiecomponenten. Deze ontbinding noemen we de Fourier Transformatie. Na deze transformatie kunnen we precies zien welke frequenties worden beinvloed door de verschillende fouten. Het blijkt dat de aanwezigheid van vocht alleen maar invloed heeft op de amplitude. Een gat in het hout veroorzaakt een afname van de amplitude en de snelheid. Ook zorgt het ervoor dat bepaalde frequenties niet meer uit het hout komen. Als de richting van de nerf maar een klein beetje verschilt met de richting van de meting, neemt de snelheid en de amplitude af. Als de nerf een grote hoek maakt met de richting van de meting, zien we dezelfde effecten als bij een gat.
Deze observaties kunnen gebruikt worden om te bepalen welke fouten in een onbekend stuk hout aanwezig zijn. Op elk punt van het hout wordt de snelheid, amplitude en de Fourier ontbinding van het geluid berekend en aan de hand van de eerdere conclusies wordt de locatie van de verschillende fouten bepaalt.
Abstract
Introduction Wood has been used for centuries. Due to its wide availability and strength, it is used in numerous fields. A large number of invaluable paintings and sculptures have been made using wood, which caused great interest in the field of preservation and restoration of these works of art. The ability to detect structural defects and moisture in wood would be of great use to this field. The detection has to be non invasive and nondestructive. Recent research has focused on the use of ultrasonic transducers to measure the damping and velocity. We suspect that the frequency representation of the signal can be exploited to obtain additional details. Background The mechanical properties of wood change greatly during the process of drying. The ultrasonic transducers produce ultrasonic waves that travel through a coupling agent to the sample. To achieve nondestructive testing, air-coupling is used. Transmission of the ultrasonic waves is not very efficient due to the large disparity of the acoustic impedances of wood and air. The specific properties have different effects on the ultrasonic waves. Method 21 mm thick samples of Pinewood are scanned. 1D scans are performed on a drying sample and a sample with holes to simulated structural defects. A sample is rotated to determine the effects of a change in grain direction. A 2D scan is made of a pinewood sample containing a knot and several other defects. The Discrete Fourier Transform is used to improve the SNR and compute the frequency representation. The damping, kurtosis, modulation and time of flight are calculated. The Teager Energy Operator is used an alternative to compute the damping. Results The drying sample showed a shift in damping. The sample with holes showed a large shift in modulation index, kurtosis and damping. The effect on the velocity was less significant. A small change of the angle (< 25°) between the grain and the measurement (< 25°) showed a decrease in damping and velocity. A large change of the angle (> 25°) showed an increase of kurtosis and modulation index and a decrease of damping and velocity. Discussion The observations could be used to determine the different properties in the 2D sample. A change in velocity and damping corresponded to a small change in grain direction (< 25°). A change in kurtosis, modulation index and damping corresponded to a structural defect or a large change in grain direction (> 25°) and a change in damping with no change of the other parameters implies a localized difference in moisture content. These conclusions rely heavily on the assumption of a flat tangential surface of growth rings, which limits the use of this method. More research is needed to achieve a more complete method.
Acknowledgements
I would like to express my gratitude to my supervisor dr. Rudolf Sprik, who has helped me tremendously to overcome the challenges that are ever prevalent in the field of experimental physics. I would also like to thank dr. Erik van Heumen for pointing me in the right direction during my search for a compelling subject, and dr. Noushine Shahidzadeh for taking the time to be the second reader of this thesis.
Contents
1 Introduction 5
2 Background 7
2.1 Structural features of wood . . . 7
2.1.1 General Features . . . 7 2.1.2 Softwood . . . 8 2.1.3 Hardwood . . . 8 2.1.4 Moisture Relations . . . 8 2.2 Ultrasonic Inspection . . . 9 3 Method 10 3.1 Experimental Setup . . . 10 3.2 Data Analysis . . . 11
3.2.1 Discrete Fourier Transform . . . 13
3.2.2 Method of Moments . . . 15
3.2.3 Teager Energy Operator . . . 16
4 Results 17 4.1 DFT filtering and envelope approximation . . . 17
4.2 Moisture Content . . . 18 4.3 Holes . . . 20 4.4 Grain Direction . . . 21 4.5 2D Scans . . . 22 4.6 Margin of error . . . 25 5 Discussion 26 6 Conclusion 29 7 Appendix 32 7.1 A. The Rectangular Function . . . 32
1
Introduction
Wood has been used for centuries. Its wide availability and strength have made it one of the most commonly used materials around the world. It has been argued that wood played an indispensable role in human survival due to its widespread application. This was not limited to construction, a significant amount of paintings have wood as a background and a lot of sculpting has been done with wood. This has been done for centuries. There is significant interest in the preservation of these works of art due to their importance to our cultural heritage ([Dardes and Rothe, 1995]). The preservation proves to be challenging; Firstly, wood is neither homogeneous nor isotropic. This means that there can be large differences between pieces of wood that have been cut from the same tree. Secondly, wood is hygroscopic, which means that it tends to take on moisture from its surroundings ([Glass et al., 2010]). This is problematic since the mechanical properties of wood dependent heavily on its moisture content ([Green et al., 1999]). All these traits make wood a material that is hard to control.
The ability of observing and locating these structural differences might be of use to the process of preservation. If we are able to see how the properties of the wood differ within the sample, we can use this knowledge to decide what actions are needed to maximize the preservation of the sample. Since the 1990’s, a number of studies have been performed on the imaging of the internal structures of wood. The current research mainly focuses on the detection of cracks, splits and other signs of weakening. X-ray imaging has been used to determine the absorption of ammonia, and various groups have used ultrasonic waves to map the internal structures ([Xu et al., 2014], [Sandoz et al., 2000], [Gan et al., 2005]). This is requires a means of transporting the ultrasonic waves from the transducers to the sample, usually achieved by either water immersion or the use of contact trans-ducers ([Dzbe´nski and Wiktorski, 2007], [Berndt et al., 1999]). This coupling of the transducers and the sample are problematic for the case at hand. A wood panel painting cannot be touched or immersed in water without doing any damage. This calls for the development of a non-destructive imaging process. The most obvious choice is to take a look at air-coupling. where the waves are transported through air to the sample. This way, the wood can be examined without having to touch it. Structural flaws are not the only factors that determine if wood needs to be treated. The determination of the moisture levels inside the wood could be of great value to the process of restoration and preservation. The majority of the present research focuses on the damping and/or the time of flight and lacks focus on the moisture content of the sample ([Berndt et al., 1999], [Eshaghi et al., 2012], [Grimberg et al., 2005]). We believe that the addition of a closer look at the frequency representation of the signal will enable us determine the presence of both dif-ferent structural defects and a change of moisture content in the sample by providing more details. In this thesis, the possibilities of both air coupling and Fourier space analysis are explored. We aim to develop a measuring and analysis method that enables us to simultaneously determine any differences in the internal moisture content along the sample and the prevalence of cracks, splits and knots, based on the damping, time of flight and frequency representation of the signal.
The second section describes the experimental setup and the process of data analysis. The data analysis will be performed using the Discrete Fourier Transform (DFT) and the Teager Energy Operator (TEO), a small time window operator. The next section discusses the results and the usefulness of the mentioned operations. Further research is suggested. The well informed reader on the subject of the wood properties, the DFT and the TEO could skip directly to the last section of this thesis.
2
Background
2.1
Structural features of wood
It is important to remember that wood is a natural resource, harvested from more than ten thou-sand different species of trees. We can regard trees as living things, made up of cells. These cells determine the physical structures of the wood. Hence, it makes sense to look at the a cellular level when trying to comprehend the different structures inside the wood ([Dardes and Rothe, 1995]). We can differentiate between the cells by looking at their respective applications. Different cells serve different needs, which requires different physical properties. A tree needs to be sturdy and resistant to the influences of the weather, which calls for strong cells forming beams and columns. There is also a need for ways of transportation, since food and sap are needed throughout the tree. Both of these cells are typically situated parallel to axis of the stem. The alignment of the stretched and fiber like cells determine the direction of the grain. Combined with the cylindrical growth of the stem and the axial direction of the layers, it is clear what makes wood such a com-plex structure ([Dardes and Rothe, 1995]). This becomes very comcom-plex when coupled with the variations among different species and different types of trees. Therefor we will mostly be looking at the differences between hard- and softwood.
2.1.1 General Features
Magnification is needed to detect the majority of the structures of wood. Since we are interested in the cellular level, most of the important structures are visible with low-level magnification. The most obvious feature of wood are its circular growth rings. This growth is influenced by the season. Hence we can make the differentiation between late- and earlywood. The first is significantly darker, which enables us to determine the age of the tree. We call this the figure of the wood. More structures can be seen using tenfold magnification. The majority of the cells are elongated; their diameter is significantly smaller than their length. This varies from cylinder like cells to needle like cells. The axis of these cells is often parallel to the stem. We also see groups of horizontal oriented cells, called rays, made up of raycells. The rays are perpendicular the growth rings, and therefor point in radial direction. These rays resemble the shape of ribbons. The flattened side of the rays point along the grain. Rays make up 10 % of the volume of the wood ([Dardes and Rothe, 1995]). Most branches originated from the the center of the stem outward in radial direction. These inner structures are roughly conically shaped and have a different grain direction due to the radial orientation relative to the stem. Even if the branches die, the fibers inside the stem remain. Newly formed layers on the stem are laid over the remaining structure, resulting in the formation of a knot ([Record, 1914]). These appear as dark circles on longitudinal sawn boards. The discontinuity between the subsequent layers makes these areas more prone to checking during drying ([Green et al., 1999]).
2.1.2 Softwood
Softwood can be classified as the wood from trees belonging to the group of gymnosperms. This name originates from the Greek words for ’naked’ and ’seeds’, derived from the unenclosed state of the unfertilized seeds. A well known example is the pine along with the redwood, two trees belonging to the division coniferae ([Campbell and Reece, 2005]).
The majority of the volume of softwoods is made up by tracheids, fiber like cells. Tracheids resemble needles, with typical lengths ranging from 2-6 mm. The diameter is generally up to a hundred times smaller, ranging from 20-60 µm. This ratio is used to classify different species of the coniferae since this influences the porosity ([Dardes and Rothe, 1995]). We can also observe resin canals. These canals transport resin and pitch, and are lined with epithelial cells. These tubular pathways are most prevalent in pines. The canals are usually very small and magnification is needed in order to observe them. They can be identified as light lines across of the radial surfaces. Softwood is further characterized by the ratio of the respective densities of the early- and latewood ([Campbell and Reece, 2005]. This ratio can be as high as three ([Dardes and Rothe, 1995]). 2.1.3 Hardwood
In general, hardwoods can be characterized by an increased amount of complexity compared to softwoods. There is an increase in cell types, as well as more variation of the arrangement and ratios of these cells.
Hardwoods use vessels for sap transportation. Vessels form long continuous tubes that run along the axis of the stem. They have thin cell walls and are relatively small, although size differs greatly among different species. These vessels can be observed as circular pores in an end grain slab. Because the diameter of the pores differs greatly (from 40-300µm), it is used as means of classification of the structure. Wood with large pores is coarse textured, while wood with small pores is smoothly textured. These pores can be concentrated in the tangential oriented earlywood. We call this ring porous. Mechanical and physical behaviour varies among hardwoods. This is largely due to the rays. These form planes of weakness, which are more prone to cracks, shrinkage and separations. Radial direction of the rays is also problematic, due its non isotropic shrinkage ([Dardes and Rothe, 1995]).
2.1.4 Moisture Relations
Wood is sensitive to moisture content, caused by its hygroscopic properties. The exchange of moisture with its surroundings is dependent on a number of factors. In practice this means that wood can shrink and swell by absorbing and releasing water. Water can be acquired by water vapor saturation and liquid water absorption. This first one is dependent on the relative humidity and the temperature. The changes due to this type of absorption are generally gradual. The latter is called capillary action and can induced changes in a short time frame. The rate of absorption differences along the tree for both types of absorption. This causes stress in the wood, since some parts are shrinking/swelling at a different rate. The shrinkage is the highest along the growth rings and the lowest along the axis of the stem ([Reijnen, 2012]).
Wood becomes a dimensionally stable material when the moisture content reaches the fiber sat-uration point. At this point, the cell walls are completely saturated but there is no water in the lumina. When the lumina are completely filled that wood reaches the maximum water content ([Glass et al., 2010]).
In hardwoods, checking due to drying generally starts at high moisture contents. These cracks can be superficial or internal ([Eshaghi et al., 2012]).
2.2
Ultrasonic Inspection
The application of sound as a means of inspection is a very old principle. This is supported by the use of expressions like ”sound as a bell”. Apparently the sound of an object can be used to determine if the item is without any flaws. The sound is generally produced by lightly tapping the item, which produces sound waves. These waves are the vibration of the particles in the material through which the sound propagates. The human ear has a limited range, which requires the use of audible sound waves. The drawback to this method is the fact that audible frequencies have a relatively low resolution, which is dependent on the wavelength λ = v
f. This means that only
large flaws can be detected. This is overcome by the use of ultrasonic sound waves, which have a frequency that is higher than the upper limit of the human ear, which is 20kHz. The resulting wavelength of the waves is smaller which increases the resolution ([Hellier, 2003]). The analysis of these inaudible sound waves requires a more sophisticated approach. These ultrasonic waves are produced using transducers, which convert electric signals to acoustic waves and vice versa. The electrical signals can be studied using software, to reveal the effect of the material on the acoustic waves. These waves propagate through a certain medium to overcome the distance between the transducers and the material. We call this the coupling of the transducers and the material. We use air-coupling to achieve nondestructive testing ([Xu et al., 2014]).
The air-coupling between the transducer and the wood is not very efficient. The transmission coefficient is inversely proportional to the density of the wood and the ratio between the acoustic impedance’s of the media ([Kinsler et al., 1999]:
T = 2r1 r2sin(k2)L
(1) with L the length of the sample, rithe respective impedance’s and ki= ωv. The acoustic impedance
is dependent on the density and the velocity. The velocity is dependent on the bulk modulus, E, and the density.
ri= ρivi with vi= Eiρi (2)
Evidently, a large percentage of the signal is reflected due to the difference in acoustic impedance ([Castaings and Cawley, 1996]). This complicates the application of air-coupled ultrasonic trans-ducers as a means of nondestructive testing.
3
Method
3.1
Experimental Setup
We used the following experimental setup:
The Panametrics 5072PR pulse generator (6) produces a one pulse 40dB square wave with a 200 Pulse Repetition Frequency. The NCG200-D19 Transducer (3) transforms the electrical energy to acoustic energy. The transmitted ultrasonic signal is registered by a second NCG200-D19 transducer (3) which transforms it to an electrical signal. A Panametrics 5670PREAMP Ultrasonic amplifier (8) and a DHPVA-100 (7) Voltage Amplifier are used and the resulting signal is fed to the PicoScope 5442B Digital (5) oscilloscope. To achieve non destructive imaging, the transducers and the wood are air-coupled. A diaphragm is used to minimize the contribution of waves that bend around the sample. The program PicoScope is used to display the signal and save the data. The data is saved as multidimensional arrays, using a Matlab extension. Afterwards, Matlab is used to analyze the data using the methods described below.
The transducers work at 200kHz, with a 35-45% bandwidth. The active area is 25 mm x 25 mm. Two Standa Translational Tables (1) are used to move the transducers. This enables us to make a 2D scan of the board. A Standa Rotational Motor is used to change the angle between the grain direction and the measurements. The motors are connected to the computer using two Standa 85MC1-USBh (2). The tables are controlled using LabView. The whole setup is covered by a Faraday Cage to minimize any prevalent outside electrical interference. A Voltcraft MF-100 (4) and a Denver Instruments PSU25C-14E scale are used to determine the relative moisture content of the wood. Using the multiple measurements of the moisture content at different times, a fit is made to approximate the change of moisture content during the measurements.
(a) (b)
Figure 1: The experimental setup
We differentiate between three orthogonal orientations within the wood. The transverse plane or the endgrain surface is perpendicular to the direction of the stem. This is usually the end of a
sawed-off log. The radial surface is the plane that passes through the pith of the tree. This is surface we end up with after manually splitting the wood. The tangential plane follows the growth rings, hence it is circular. If the piece of wood is small enough, we can approximate this curved surface as a flat one ([Dardes and Rothe, 1995]). The different orientation are shown in figure (2). Measurements are performed in radial direction on 21 mm thick pieces of Pinewood. Pines belong to the division of coniferae. Evidently, pinewood is a softwood.
Four different experiments are performed. First, a measurement of one point on the board over the course of the drying process is performed. Next a 1D scan of the board containing three tangential holes is performed. Their respective diameters are 4 mm, 5 mm, and 6 mm. The third measurement is a 66 mm x 50 mm scan of a separate piece of Scots Pine. The fourth measurement consists of the determination of the effects of a change in grain direction by varying the angle between the grain and the direction of the measurements. Finally, the error of the experiments is determined by performing 40 sequential measurements without altering the experimental setup or the sample.
Figure 2: The different orientations in a block of wood. Endgrain surface (X), radial direction (R) and tangential direction (T). From Dardes and Roth in ’The Chemical and Physical Properties of Wood’ (1995).
3.2
Data Analysis
The transmitted signal must be analyzed in order to determine the influence of the sample on the reference signal. The transmitted signal has a certain voltage. The voltage is continuous, i.e. for every imaginable time within our domain, there is a corresponding value of the voltage. The PicoScope Digital oscilloscope compiles the data by performing measurements. The measurements determine the value of the voltage at a certain time t1. The next measurement is performed at
time t2. This means that a ∆t separates the two measurements. Evidently, the data is a discrete
time signal, it is only defined at certain values ti.
sequence consists of n numbers, whit n corresponding to the nth number of the sequence. x = {x[n]}, −∞ < n < ∞ (3) This sequence arises from the periodic sampling of the continuous signal xcont(t). So the nth
number in the sequence corresponds to the continuous signal at time nT . We call T the sampling frequency. Formally:
x[n] = xcont(nT ), −∞ < n < ∞ (4)
This can be expressed in a more general way: x[n] =
∞
X
k=−∞
x[k]δ[n − k] (5) We can do operations on this sequence to analyze it. We call this a Discrete Time System. We perform a transformation O of the input sequence x[n], and generate and output sequence y[n].
y[n] = O{x[n]} (6) O can be viewed as a manual that tells us what kind of transformation to perform and how to do it. It determines the rule of formula which relates x[n] to y[n] ([Oppenheim et al., 2007]). These different operations O are used to manipulate the data in such a way that it can be used as a means of inspection. On of the these operations is the averaging of the signal. This is used to improve the signal to noise ratio, which results in a V, t diagram that shows what part of the initial pulse was transmitted through the wood.
We can identify multiple transmitted pulses. The first pulse is due to direct transmission. The second, third and other pulses are due to reflections upon the many surfaces of the wood and the transducers. All the needed information is contained within the first pulse, so we can discard the remaining pulses.
The shape of the pulse is dependent on the properties of the transducer. The transducers works at 200kHz, with a 35-45% bandwidth. This acts as a filter on the frequencies of the square wave (appendix A). What remains is a signal with a central frequency of 200kHz with small contributions of other harmonic frequencies within the 35-45% range of the central frequency. These frequencies are called the sidebands of the central frequency. We call such a signal an amplitude modulated signal ([Kvedalen, 2003]). The contribution of the sidebands is called the baseband signal. The central frequency is called the carrier frequency (ωc). This can be expressed mathematically as:
sAM = scarrier(t) × (C + sbase(t)) (7)
with scarrier = s0sin(ωct + φ). We then define the modulation index, the ratio between the
amplitudes of the baseband signal and the carrier ([Federation of American Scientists]): M = max(sbase(t))
s0
In the simplest case, the baseband signal consists of a single frequency, sbase = sin(ωbt) with
ωb< ωc. By Simpson’s Rule:
sAM = Csin(ωct + φ) + sin(ωct + φ)sin(ωbt) (9)
= Csin(ωct + φ) +
C
2(cos(ωc− ωb) + cos(ωc+ ωb)) (10) We retrieve the baseband signal by multiplying sAM by the carrier frequency and filtering out the
high frequency contributions with a different operator O. sAM × scarrier= Csin2(ωct) + C 2 cos((ωc− ωb)t) + cos((ωc+ ωb)t) sin(ωct) (11) = Csin2(ωct) + C cos(ωbt) + cos((2ωc− ωb)t) (12)
The speed of sound within the wood can be obtained by determination of the time of arrival of the first pulse. This can be compared to the reference signal. Pulse edge detection is achieved using a simple threshold function depending on the maximum of the signal. Attenuation of the ampli-tude implies the effect of damping. Previous research suggests that the velocity and the damping are dependent on the internal structures of the wood ([Xu et al., 2014], [Murray et al., 1996]). A discontinuity will slow the propagation of the waves and allow for extra reflection ([Berndt et al., 1999]). The velocity of sound is influenced by the direction of the grain and ranges from 0.6 − 5 km/s. The velocity increases if the sound propagates along the grain . The grain also affects the damping by skewing the waves ([Murray et al., 1996], [Eshaghi et al., 2012]). Evidently, we should focus on the amplitude and time of arrival of the first pulse in order to map the inter-nal structures inside the wood. The corresponding wavelengths will be in the order centimeters, allowing for a maximum resolution in the order of millimeters.
The instantaneous power of a signal is :
p(t) = |s(t)|2 (13) |s(t)| is the voltage of the signal at a time T. Then, the total energy is:
E = Z T
t=−T
|s(t)|2dt (14)
In order to differentiate between the attenuation effects of internal flaws and moisture content, we have a closer look at the signal in Fourier space. We expect that the a changing moisture content will have a different effect on the frequency spectrum, compared to a change in grain direction or the presence of a structural defect. This calls for the use of the Discrete Fourier Transform. This operation is explained in more detail below.
3.2.1 Discrete Fourier Transform
poses a problem, as our data is a Discrete Time Signal, and can not be transformed using the FT. To find all the frequency components in a discrete time signal we use the Discrete Fourier Transform (DFT). We can see this transformation as a discrete time systems:
y[n] = DF T {x[n]} (15) The DFT can be derived using the Fourier Series. We will follow the derivation of Oppenheim et al. from Discrete Time Signal Processing.
A sequence ˜x[n] with period N . This sequence has the following property: ˜
x[n] = ˜x[n + rN ] (16) with n and r arbitrary integer values. This sequence can presented as a sum of harmonically related complex exponential sequences. Their respective frequencies are integer multiples of the fundamental frequency 2π N: ˜ x[n] = 1 N X k ˜ X[k]ei(2π/N )kn (17) We need an infinite number of harmonic functions to represent a continuous time signal. However, the Fourier Series of a Discrete Signal is finite, due to the discrete periodicity’s of the harmonic functions.
ek[n] = ej(2π/N )kn→ ek+lN[n] = ei(2π/N )kne2iπln= ei(2π/N )kn= ek[n] (18)
Consequently, we only need N period exponential functions to represent the discrete time signal. Then, equation 16 becomes
˜ x[n] = 1 N N −1 X k=0 ˜ X[k]ei(2π/N )kn (19) This is called the synthesis equation. Conclusively, the analysis equation is
˜ X[n] = N −1 X k=0 ˜ x[k]e−i(2π/N )kn (20) However, this is the frequency domain representation of a periodic signal. We solve this by assuming that the aperiodic sequence x[n] with corresponding FT X(ejω) can be transformed into
a periodic sequence. This done by sampling X(ejω) at multiples of the fundamental frequency to
obtain ˜X[n], the FT of a periodic sequence: ˜
X[k] = X(ei(2π/N )k) (21) These are the coefficients of a periodic sequence ˜x[n]:
˜ x[n] = N −1 X k=0 ˜ X[k]ei(2π/N )kn (22) We made no assumptions about x[n], so we set the limits of its Fourier Transform to inifnity:
X(eiω) =
∞
X
m=−∞
Plugging this into equation 20 yields and substituting the results into equation 21 yields: ˜ x[n] = ∞ X m=−∞ x[m] 1 N N −1 X k=o ei(2π/N )k(n−m) (24) The term inside the large brackets turns out to be the Fourier representation of the impulse train, which ultimately gives us:
˜ x[n] = ∞ X r=−∞ x[n − rN ] (25) This transformation provides multiples of the original aperiodic sequence x[n]. These multiples are spaced in such a way that they do not overlap, which gives us the periodic sequence ˜x[n]. The original aperiodic sequence can extracted by sampling a single period, if we take enough equally spaced samples. In summary, we can take an aperiodic sequence x[n], use equation 24 to form the periodic sequence ˜x[n] which in turn can be represented as a DFS. This whole process is called the Discrete Fourier Transform ([Oppenheim et al., 2007]).
This process is used as a means of the filtering. The absolute value of each of the numbers in the sequence ˜X[n] presents the ’weighing factor’ of the corresponding harmonic function. To get rid of the high frequencies contributions, we simply set the corresponding values in ˜X[n] to zero. Then we can use the Inverse Discrete Fourier Transform to find our signal without any high frequency contributions. We can use this process to achieve the actual signal with a significantly improved SNR.
Secondly, the DFT is used to determine the carrier and the baseband signal in the measurement. This is used to determine the modulation index, this is simply the ratio between the amplitude of the carrier frequency and the sidebands in Fourier space. We are also interested in the kurtosis of the signal in Fourier space. A high kurtosis value implies a peaked signal, which means that there is large contribution of the carrier frequency compared to the entire sideband. The kurtosis value can be determined using the method of moments ([Barlow, 1989]).
3.2.2 Method of Moments
The method of moments is used as an estimation tool on data sets. In general, it is used to describe the shape of the given data set. The n-th moment is defined as:
µn =
Z ∞
i∞
(x − c)P (x)dx (26) This is the moment about the value c, where P (x) is the probability of the measurement s. Hence
Z ∞
∞
P (x)dx = 1 (27) We are dealing with discrete data, so this simplifies to
µn= ∞
X
with
∞
X
i=−∞
P [xi] = 1 (29)
In our case, P [x] corresponds to ˜X[n] and xicorresponds to the frequency. The moment about the
weighted mean is called the central moment. The second central moment is defined as the standard deviation. The fourth central moment is a measure of the kurtosis ([Kenney and Keeping, 1943]). 3.2.3 Teager Energy Operator
We are still interested in another operator that can be used to determine the energy of the signal without numerical integration. This is where the Teager Operator comes in. The Teager Energy Operator is defined as
y(t) = ˙x2(t) − x(t)¨x(t) (30) where ˙x(t) and ¨x(t) are the first and second derivative of x(t) to the time. It was originally used to analyze the non-linearity of speech signals ([Kaiser, 1993]). It is also used as a signal peak tracker. The proposed definition is a continuous operator, it acts on a continuous function to produce another continuous function. Thus, we need the discrete time version of the Teager Operator, which acts on a sequence of numbers instead. If the values in the sequence x[n] are equally spaced along the time axis, we can approximate dx/dt as dx/dn, which in turn is either x[n + 1] − x[n] or x[n − 1] + x[n]. Both definitions are equally right. Similarly, d2x/dt2 becomes x[n − 1] − 2x[n] + x[n + 1]. Plugging this into equation 27 yields us
y[n] = x2[n] − x[n − 1]x[n + 1] (31) This is the discrete version of the Teager Energy Operator. It estimates the instantaneous energy of the signal at position n in the sequence x[n]. If we integrate over the output of the Teager Operator we find the total energy of the pulse, which is a measure of the damping. The TEO is an interesting operator because it acts on a very small time window, which makes it very efficient. It also acts as a demodulator on AM signals, producing the envelope of said signal. Because of its small time window, the AM demodulation is heavily affected by noise in the original signal. The difference in energy is equal to the variance of the noise ([Kvedalen, 2003]). In order to achieve the most accurate results possible, we use the the DFT to filter out most the higher frequencies before using the Teager Operator to approximate the envelope and the energy.
4
Results
4.1
DFT filtering and envelope approximation
The approximation of the original signal and the envelope is acquired using the DFT and the TEO. The normalized reference signal is shown in figure (3a). The transmitted signal of an example measurement is shown in below in figure (3b). The frequency spectra of said signals are displayed in figure (3c) and (3d). The transmitted signal has a very high SNR. The frequency spectrum shows a large contribution of high frequency oscillations, which was not present in the reference signal. The needed filtering window is outlined.
(a) (b)
(c) (d)
Figure 3: The normalized reference signal(a), the transmitted signal (b), the frequency represen-tation of the reference signal (c) and the transmitted signal (d). The filtering window is outlined in red.
The DFT filtering yields a signal with a significantly reduced SNR. The TEG is used to determine the envelope. The result of these operations is shown in figure (4). Their respective colors are red and green.
Figure 4: The filtered signal (red), the approximation of the evelope (green) and the original signal (blue)
4.2
Moisture Content
Using the filtered signal, the velocity, the damping, the modulation index and NSD of the frequency representation are calculated. Figure (8) shows the development of the mentioned parameters during the drying process of a regular wooden board, without any defects, over the course of 4 hours and 45 minutes. A total of 120 measurements are performed during this interval. The board is measured in radial direction. The parameters are plotted on the y-axis, as a function of the relative moisture content. The NSD of the parameters over the course of the measurement are shown in the table below.
Parameter
Standard Deviation
Normalized Standard Deviation
Damping 7.5198 ×10−7 0.0421 Modulation Index 0.0363 0.0423 Kurtosis 417.8828 0.1040 Velocity 9.8445 0.0095
Figure 5: The standard deviation and the normalized standard deviation of the parameters during 4 hours and 45 minutes of drying.
(a) (b)
(c) (d)
4.3
Holes
Two 1D scans are made of the second piece of 21 mm thick pinewood. The first scan is performed on the unaltered sample. A total of 40 measurements are performed along 60 mm of the sample. After the first scan, 3 holes are drilled at 12mm, 25mm and 45mm. Their respective diameters are 4mm, 5mm and 6mm. Then, the measurement is repeated. The first scan serves as a reference. The kurtosis, damping, velocity and modulation index are determined before and after the introduction of the holes. The results are shown in the figure below. The blue line is the reference measurement, the red line is the measurement of the altered sample. Figure (8a) shows the damping, figure (8b) shows the velocity, figure (8c) shows the kurtosis and figure (8d) shows the modulation index. The position of the holes is shown using the black lines. The standard deviation and normalized standard deviation of both the measurements are calculated and displayed in figure (7).
Parameter Reference SD Reference NSD Altered SD Altered NSD Damping 8.7965 ×10−5 0.0737 2.8539 ×10−5 0.3052
Modulation Index 0.1023 0.1521 0.2615 0.3602 Kurtosis 264.1966 0.1115 699.7841 0.3081 Velocity 56.1409 0.0458 73.7479 0.0592 Figure 7: The standard deviation (SD) and the normalized standard deviation (NSD) of the parameters in the unaltered and the altered sample. A total of 40 measurements were performed.
(a) (b)
(c) (d)
Figure 8: The damping (a), the velocity (b), the kurtosis (c) and the modulation index (d) of the board without any drilled holes (blue) and with holes (red). 40 measurements in radial direction were performed along 60 mm of the pinewood sample. The position of the holes is indicated by the black lines.
4.4
Grain Direction
The effect of the angle between the grain direction and the direction of measurement is studied by performing 15 measurements on the same sample with differing angles of measurement, ranging from 0° to 45°. The velocity, kurtosis, modulation index and damping are calculated and displayed in figure (9a), (9b), (9c) and (9d).
(a) (b)
(c) (d)
Figure 9: The velocity (red), the damping (magenta), the kurtosis (blue) and the modulation (green) for different angles between the grain direction and the direction of measurement. A total of 15 measurements were performed.
4.5
2D Scans
A 66 mm x 40 mm scan is made. A total of 630 measurements are performed in radial direction. The value of each parameter is calculated for every pixel. The results are shown in figure (11). The first subfigure (a) shows the damping, subfigure (b) shows the velocity, subfigure (c) shows the kurtosis and subfigure (d) shows the modulation index. The pinewood sample is shown in the figure below. The scanned area is outlined in black. Subfigure (a) shows the front of sample, subfigure (b) shows the back of the sample.
(a) (b)
(a) (b) (c) (d) Figure 11: The v elo cit y (a), the mo dulation index (b), the kurtosis (c) and the damping (d) in the [XSIZE]x[YSIZE] pinew o o d sample. A total of 630 measureme n ts w ere p e rformed in radial direction.
The damping of the scanned sample is also calculated using the TEO. The result is shown in figure (12).
Figure 12: The damping in the pinewood sample calculated using numerical integration of the output of the Teager Energy Operator. The measurements were performed in radial direction.
4.6
Margin of error
The errors of the measurements are determined by performing 40 sequential measurements on the same sample using the same experimental setup. The damping, velocity, kurtosis and skewness are calculated. The mean, the standard deviation, the normalized standard deviation, the maximum deviation and the maximum deviation as a percentage of the mean are calculated and shown in the table below.
Parameter Mean SD NSD MD MD in % Damping 6.9627 ×10−4 9.4120 ×10−6 0.0135 1.8894 ×10−5 2.7137 Modulation Index 0.8404 0.0451 0.1521 0.1018 12.1182
Kurtosis 3.7563 ×103 62.7532 0.0167 155.0661 4.1282
Velocity 2.2104 ×103 13.2690 0.0060 28.4834 1.2886
Figure 13: The mean, standard deviation (SD), the normalized standard deviation (NSD), the maximum deviation from the mean (MD) of the parameters in the unaltered and the altered
5
Discussion
The DFT filtering process was reasonably effective. It was very effective in blocking any noise outside of the desired frequency window. However, DFT filtering was not very useful if the noise had frequency components inside our filtering window. The contribution of noise was minimized by maximizing the averaging time, which decreased the contribution of the present noise.
The Teager Energy Operator did not seem to be very effective. Although the calculated envelope had the general shape of the first oscillations of the signal, it did not provide a good approximation of the total transmitted pulse. This can be explained by the fact that the transmitted signal was not a perfect AM signal. The frequency representation only showed a single sideband peak next to the carrier frequency, instead of two identical sideband peaks next to the carrier frequency. This is the reason that envelope calculation using the TEO was of no use to this thesis, which became even more evident in the 2D graph of the pinewood sample, that showed no useful information. The determination of the margin of error provided us with the possibility to determine if an ob-served change of the parameters was caused by a real effect or random variation. The modulation index had the largest margin of error, followed by the kurtosis and the damping. The velocity had the lowest margin of error.
The changing moisture content inside the wood had no significant effect on the majority of the parameters. The velocity, kurtosis and modulation index remained largely unaffected during the drying process. This was unexpected, as a large number of mechanical properties of waterlogged wood change heavily upon drying. The damping was the only parameter that seemed to be signif-icantly affected by the changing moisture content. The presence of moisture had a positive effect on the attenuation of the signal. This seems reasonable, as the viscosity of water is larger than the viscosity of air.
The presence of holes had a large effect on most of the parameters, supported by the change in NSD between the reference measurement and the measurement on the altered sample. The damping, kurtosis and the modulation index were heavily affected by the presence of the holes, and their respective position could easily be determined in figure (8a), figure (8c) and figure (8d). This was probably due to acoustic scattering on the holes, which is dependent on the wavelength. As a result, the frequency representation and the damping are altered.
The velocity was affected to a lesser extent by the 4 mm and 5 mm holes, only showing a slight shift when compared to the reference measurement. This can be explained by taking a closer look at the properties of the transducer. The active area of the transducer is significantly larger than the surface area of the drilled holes. This means that there was still transmission with unaltered velocity. The 6 mm hole appeared to be large enough to prevent this phenomena, and could easily be identified as a large peak in the velocity graph.
The velocity, kurtosis and modulation index were influenced by a change in grain direction. The kurtosis and the modulation index remained relatively constant when the angle between the
di-rection of measurement and grain was < 25°. The velocity decreased slightly. When the angle exceeded the mentioned limit, the modulation index and kurtosis increased heavily, while the ve-locity was significantly reduced. The response of the modulation index and kurtosis was probably caused by the skewing effect of the grain direction on the pulse. This effect could very well be dependent on the wavelength, which would explain the change of the signal in Fourier Space. This would also explain the change in damping. However, we could not determine if the change in damping was solely caused by this effect. The increase in distance travelled through the wood could very well have affected the damping as well.
The previous research was combined with the previous observations. As a result, a number of conclusions could be drawn that allowed us to identify the differing properties inside the sample. • A change in damping and velocity and no change in kurtosis and modulation index implies
a small change of grain direction (< 25°)
• A change of damping and no change of kurtosis, velocity and modulation index implies a local difference of moisture content
• A change in damping, kurtosis and modulation index implies a structural defect or a large change in grain direction (> 25°)
As expected, we saw a significant increases in velocity at the position of the knot. This was due to the fact that the grain direction in the knot was relatively perpendicular to the board, so the grain of the knot pointed along the direction of the measurement. The increase of damping was caused by the increased density of the knot. The kurtosis and modulation index were relatively uniform within the knot, which implied the absence of structural defects or a uniform grain direction. The latter seemed to be the case. There were however some pixels that showed a sharp increase in modulation index, kurtosis and a change in velocity, which implies a discontinuity or a significant change of grain direction. Closer inspection of the board showed a small radial crack. The lower right corner showed a sharp increases in damping, with constant kurtosis, modulation index and velocity. The middle of the damping plot showed a sharp localized increase in damping, with little change of the velocity, kurtosis and modulation index. These regions could be linked to a local difference in moisture content according to our observations. This seemed dubious, as there was no reason to assume that a wooden board that had been dried for months would not have a relatively uniform moisture content. Closer inspection of the second region revealed an irregular patch of material, which appeared to be porous. The increase of the velocity in the first region could have been caused by the curvature of the tangential surface, which meant that our approximation of a flat tangential surface was no longer valid.
This method was not exempt of drawbacks. The mentioned process could only be used to provide qualitative information about the differences inside the wood. This had to do with the fact that the measurements were very dependent on the precise setup of the experiment. The exact placement of the board on the translational tables and the setup of the transducers had a large effect on the
of the board caused small changes in the alignment of the transducers, which made it impossible to achieve the same damping after reattachment of the board. Another problem that this method encountered was the measurement itself. Due to the large SNR, a significant period of time was needed to average the signal. This caused the total required time to skyrocket. The 2D scan of the pine wood sample took up to 11 hours. This limited the application of this method to determination of the moisture content inside a drying board. Due to the prolonged measurement interval, there is no way to tell if the difference in observed moisture content would be caused by an actual difference or by the time difference between the measurements. Also, the drying process was only measured in a regular piece of pine without any defects. It could be possible that the drying process around structural defects is an entirely different process, which means that our developed method of localized moisture content is of little value. Also, the measurements were performed on relatively small samples, so that the curved tangential surface could be approximated as a flat one. The evaluation of larger samples could be influenced by the fact that this assumptions no longer holds. However, the achieved sensibility to structural defects was significantly better than expected. The active area of the transducers was 25 mm x 25 mm but we were able to observe radial cracks with a width of only 1 mm.
The drawbacks do not take away from the value of applying Fourier Space analysis to the process of ultrasonic imaging. Further research needs to be conducted to determine the exact role of Fourier space analysis in situations that are not dealt with in this thesis.
6
Conclusion
We set out to develop a new measuring and analysis method that incorporates both damp-ing/velocity determination and Fourier space analysis. During the drying process, the sample only showed a change in damping. The addition of holes showed a stark shift in damping, mod-ulation index and kurtosis. The velocity was only significantly affected by the largest hole. A change in grain direction influenced the damping and the velocity. A small change in grain di-rection (< 25°) caused the velocity and damping to increase. A large change in grain direction (> 25 °) showed the same characteristics as a structural defect. These observations allowed us to make a distinction between different structural defects, grain direction and moisture content. The drawbacks of this method are the dependency on a low SNR, and the assumption of a flat tangential surface of the growth rings. This limits the application of the developed method to the proposed case. More studies on the effects of different structural defects on the mentioned parameters need to be done in order to achieve a more complete method of analysis.
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7
Appendix
7.1
A. The Rectangular Function
The rectangular function is defined as:
rect(t) = 0, if |t| > 12 1 2, if |t| = 1 2 1, if |t| < 12 (32)
This block function consists of a large number of periodic functions with differing frequencies. We can use the Fourier Transformation to find all these periodic functions. This is why we call the Fourier Transform the frequency domain representation of a function([Kreyszig, 2010]), we generate a frequency domain spectrum from a function of time. The Fourier Transformation is defined as: rect(f ) = Z ∞ −∞ rect(t)e−2iπf tdt = Z ∞ −∞ rect(t)e−iωt (33) with f for the frequency. Each of the frequency components contributes differently to the rect-angular function. Since the rectrect-angular function is zero outside of the interval |t| < 1
2, we cam
determine these contribution using the Fourier Series: cn=
Z 1/2
−1/2
e−2iπntdt − ∞ < n < ∞ (34)
If we plug formula (1) into formula (2), we find that these expressions are equal with ω = 2πn Solving this is straightforward and yields us:
cn=
2sin(ω/2)
ω (35)
7.2
B. Linear Time Independent Systems
Linear Time Independent Systems, LTI’s, are a special subclass of the Discrete Time Systems. LTI’s are linear, which means that they are additive and homogeneous. We can mathematically summarize this as:
T {ax1[n] + bx2[n]} = aT {x1[n]} + bT {x2[n]} (36)
where a and b are arbitrary constants. This means that x[n] consists of a superposition of all its inputs:
x[n] =X
k
akxk[n] (37)
Conclusively, the output of will be
y[n] =X
k
akyK[n] (38)
The time independency dictates that that if there is a time shift or delay in the input sequence, that there has to be a corresponding time shift in the output sequence. Mathematically, this means that
corresponds to
y1[n] = y[n − n0] (40)
