Dalgacık dönüşümünün fourier dönüşümü ile karşılaştırılması ve uygulama
Comparison of wavelet transform and fourier transform and its applications
Yazar: ESRA TÜZEL
Danışman: PROF. DR. KENAN YÜCE ŞANLITÜRK
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı
Derece: Yüksek Lisans, 2015
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Dalgacık dönüşüm tekniği = Wavelet transforms technique ; Fourier dönüşümü = Fourier transformation ; Heisenberg denklemi = Heisenberg equation ; Mekanik titreşim = Mechanical vibration ; Motor gürültüsü = Motor noise ; Ortogonal dönüşümler = Orthogonal transformation ; Riesz dönüşümü = Riesz transformation ; Titreşim sinyal analizi = Vibration signal analysis ; Vuru gürültüsü = Impulse noise ; Zaman-frekans analizi = Time-frequency analysis

Özet:
Günümüzde, mühendislikte ilgilenilen gürültü ve titreşim sinyalleri içerisinde, karmaşık ve değişken yapılı sinyal tipleri oldukça geniş bir yer kaplamaktadır. Otomotiv gürültü ve titreşim alanında yapılan araştırma ve geliştirme çalışmaları çerçevesinde incelenen sinyallerin de çoğu bu tiptedir. Otomotiv sektöründeki gürültü ve titreşim sinyallerinin incelenmesi için pratikte gerçekleştirilen uygulamalarda Fourier temelli dönüşüm yöntemleri tercih edilmektedir.
 Fourier  dönüşümü, yüzyılı aşkın süredir kullanım alanı bulan, sinyal işleme dünyasının temel taşı niteliğinde bir dönüşümdür. Bu dönüşüm, sinyallerin frekans içeriklerini ayrıştırmaya yarayan bir dönüşüm tipidir. Dönüşüm sonucu elde edilen frekans içeriği bilgisi, dönen parçaların oluşturduğu çeşitli dinamiklerin bulunduğu motor sistemlerinin gürültü ve titreşim özelliklerini incelemek için çok değerlidir. Ancak bu sistemlerde, durağan sinyallerin yanında ani değişimler gösteren sinyal tipleri de mevcuttur ve bu tip sinyallerin incelenebilmesi için, hem zaman hem frekans bilgisi veren dönüşümlere ihtiyaç duyulmaktadır.
Fourier temelli dönüşümlerden Kısa Süreli Fourier Dönüşümü, Fourier dönüşümünden farklı olarak zaman-frekans analizi yapabilmektedir. Ancak bu dönüşüm, yalnızca sabit frekans çözünürlüğünde gerçekleşebilmekte, aynı anda hem yüksek frekans çözünürlüğü, hem de yüksek zaman çözünürlüğünü sağlayamamaktadır. Buna rağmen, motor sistemlerinde, Kısa Süreli Fourier Dönüşümü'nün bir çok durumda sinyallerden gerekli bilgiyi almak için yeterli olduğu bilinmektedir. Ancak, yüksek genlikli patlamalar gibi ani değişimlerin birbirine olan farklarının önem kazandığı veya ani değişiklikler ile sürekli olayların bir arada gerçekleştiği bazı durumlarda, Kısa Süreli Fourier Dönüşümü'nün sabit çözünürlükte çalışması, dezavantaj haline gelmektedir.
Hem ani değişimlerin, hem de durağan bileşenlerin birarada olduğu sinyallerin, hem yüksek frekans çözünürlüğü hem de yüksek zaman çözünüğü ile incelenebilmesini sağlayabilecek bir sinyal işleme yöntemine gereksinim olduğu görülmüştür. Bu sebeple bu çalışmada, frekans-zaman analizi yapan bir diğer yöntem olan "Dalgacık Dönüşümleri" araştırılmış ve otomotiv gürültü ve titreşim sinyallerine uygulanabilirliği örnekler üzerinden incelenmiştir.
Dalgacık dönüşümü yaklaşık son 30 yılda popüler olmuş bir dönüşüm yöntemidir ve konu ile ilgili varolan kaynakların çoğu, makine mühendisliği dallarındaki pratik kullanıma yönelik değildir. Çalışma, bu açığın giderilmesini sağlamak üzere, dalgacık dönüşümlerinin teorik alt yapısını temellerden başlayarak açıklamak ve dalgacık dönüşümlerinin pratik mühendislik uygulamalarında etkili bir şekilde kullanılabileceğini göstermek amacını da taşımaktadır.

Abstract:
Today, majority of the signals dealt in engineering noise and vibration studies are observed to be complex, non-stationary signals sometimes with discontinuities. In the field of automotive noise, vibration and harshness (NVH) development studies, most of the signals are of this type as well. For the investigation of the signals in the NVH field, most common practice is using Fourier transform based techniques for post processing algorithms.
Fourier transforms have been around longer than a century and since the day they were publicized; they have met a well-deserved, great attention from many disciplines. It owes its fame to its talent of decomposing the signals into their harmonic components and representing the frequency spectrum. For the automotive vibration and noise signals coming from the engine systems, where many rotating components and related complex dynamics takes place, this spectral information is rather valuable. However, spectral information is not sufficient for signals composing of not only stationary components but also non-stationary components with abrupt changes. In order to detect the instantaneous incidents in the signal, a method that provides both frequency and time information is required.
A Fourier transform based approach Short Time Fourier Transform (STFT) method, generates the time-frequency representation of the signals. Major drawback of STFT is that it uses a fixed resolution property. This may result in incomplete information extraction from the signals with both intermittent high frequency content and dominant low frequency content, just like most of the signals in the NVH field. Although the fixed resolution may not reveal all the information simultaneously, in most cases, this is not a crucial loss as it is not required to know all the details in a signal. However, fixed resolution becomes a disadvantage if the abrupt changes from the engine combustion related vibration or noise signals or such phenomenon are to be investigated.
Realization of the need for a signal processing method that can provide both good frequency resolution and good time resolution simultaneously has led to searches to find another method. For this purpose, wavelet transforms and their applicability to the NVH signals are investigated in this study.
Wavelet transforms have become popular only within the last 3 decades and most of the literature resources on this topic are not straightforward for practical usage in mechanical engineering field studies. Thus, this study aims to provide an understanding of basics of the wavelet transform, starting from the fundamentals, and investigate if the wavelet transforms can be used effectively in the engineering applications as well.
In line with the purpose of this study, a section including some mathematical concepts is provided to ensure comprehension of the full text. This section includes vital concepts in wavelet theory such as inner space productions -which are the core of mathematical transforms used in wavelet functions-, orthogonality –the property wavelets should have for unique transformation- and convolution -the operation that enables wavelets to slide over the whole signal duration and gives a comparison coefficient-.
Subsequent to these explanations, a detailed description and properties of Fourier transform is given since the Fourier transform is a good starting point for comprehension of wavelet theory. Fourier transform uses infinite harmonic bases, namely sine and cosine functions. A signal can be represented as a linear sum of sine and cosine functions with this transform. Unless the signal is non-stationary, this transform provides an elegant way of presentation and provides much valuable information about the analysed signal. When it is non-stationary, time information becomes very valuable which can not be extracted though sole Fourier transform. To overcome this disadvantage, an approach using windowing functions to create a sense of time is developed; Short Time Fourier Transform. STFT gives time, frequency and amplitude information at the same time. However, due constraints stated by Heisenberg uncertainty principle, the time resolution property of STFT is limited by frequency resolution. A multiresolution transformation that enables both high time and frequency resolution is required for non-stationary, multi-frequency content and irregular type of signals. 
Wavelet transforms are multiresolution transforms and they provide fine frequency and time resolution with amplitude information. Just like the Fourier transform decomposes signals into harmonic components, wavelet transform decomposes signals into scaled and shifted versions of a chosen mother wavelet. They compare the wavelets with the signal and give wavelet coefficients to reveal the properties of the signal. There are many different types of wavelets and this study introduces some of the most popular ones.
In order to understand the applicability of the wavelet transforms instead of Fourier based transforms, both Fourier transform and wavelet transform is applied to basic functions created in MATLAB software. Results discussed in this thesis include Fourier transforms applicability to stationary signals with success and the presence of lowest and highest frequency dependent limits in scale selection of wavelet transform. The same basic example functions are also subjected to STFT for better comparison of the three mentioned transformation types. Analysis results discussions include the trade-off between time resolution and frequency resolution, the identification of time dependent features in the analysed signal and wavelet transforms superiority in this detection. 
With the deductions gained from observations on basic signals, complex vibration and noise data taken from real automotive systems are analysed. It is also emphasized that the wavelet type should be selected to suit the application. This is done through observing the time-amplitude representation of the analysed signal. Conclusions from the analyses of the NVH signals are found to be parallel to the results seen on previous chapters. Moreover, the time dependent feature detection capability of wavelet transforms are observed to be more efficient than of STFT and Fourier transforms.  Another valuable observation gained from these inspections is that, the wavelet transform can distinguish the source of the feature in the signal if data is taken from multiple locations. This property is especially valid for discrete wavelet transform performed with enough levels of decomposition. This means, if transducers are located in multiple locations in the powertrain and vehicle, the source of a signal can be tracked down by means of comparison of the discrete wavelet transform results of these signals at the sufficient decomposition level. This result is rather valuable for the studies in NVH field as such fault source detection methods are highly desirable for efficient usage of the resources. 
This study inspires the usage of wavelet transform based signal analysis in NVH applications. Further studies are also possible such as the opportunities on software tool coding that is flexible and specialized for NVH usage.