[한양대 기계공학부[ 동역학제어실험 실험3 Auto-correlation 과 Spectral density A+ 자료

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1. Random Signal

불규칙 신호(random signal)는 일정한 형태를 갖지 않는 신호를 의미한다. 이러한 불규칙 신호는 통계적인 방법으로만 그 특성을 묘사할 수 있다. 진동 해석에 있어서 불규칙성은 동일한 방법, 동일한 환경에서 수행한 실험 결과가 각각 다른 응답을 갖는 것으로 생각할 수 있다. 이러한 경우 응답 x(t)는 하나의 신호로써가 아닌 동일한 조건에서 얻어지는 time history들의 일련의 집합(ensemble)으로써 고려하여야 한다.
2. Stationarity

불규칙 신호에 있어서 stationary라 함은 통계적인 특성들(보통 평균)이 시간에 따라 변하지 않는 것을 의미한다. 불규칙 신호는 먼저 stationary인지 아닌지를 구분해야 한다.
3. Auto-correlation

Auto-correlation 함수는 신호 x(t)가 어느 정도 빠르게 변화하고 있는지 그 정도를 나타내는 함수이다. Auto-correlation 함수를 Fourier transform한 것을 power spectral density(PSD)라 한다.
4. Power Spectral Density (PSD)

PSD는 불규칙 신호의 주파수 특성을 나타내는 함수이다. PSD는 input force의 PSD와 system의 dynamic 및 response의 PSD 사이의 중요한 관계를 나타내고 있다.
5. Sine Wave Auto-correlation and PSD

Sine wave의 auto-correlation과 PSD를 계산하면 cos 함수 형태의 그래프를 가지게 된다.
6. Square Wave Auto-correlation and PSD

Square wave의 auto-correlation과 PSD를 계산하면 sin 함수 형태의 그래프를 가지게 된다.
7. Cosine Wave Auto-correlation and PSD

Cosine wave의 auto-correlation과 PSD는 sine wave의 auto-correlation과 PSD와 동일한 형태를 가진다.
8. Pitch-Catch Cross-correlation

Pitch-Catch 실험에서 input data와 receive data를 cross-correlation하면 두 음파 사이의 시간차와 상관계수를 알 수 있다. 이를 통해 음파의 속도 유사성을 분석할 수 있다.
9. Experimental Results and Analysis

실험 결과 분석을 통해 sine wave와 square wave의 auto-correlation과 PSD 특성을 이해할 수 있었다. 또한 cosine wave의 auto-correlation과 PSD가 sine wave와 동일한 형태를 가지는 것을 확인하였다. Pitch-Catch 실험에서는 샘플 가로 부분의 cross-correlation 결과가 신뢰할 만한 반면, 세로 부분의 결과는 신뢰성이 낮았다.
10. Conclusion

이번 실험을 통해 불규칙 신호 분석 방법, 다양한 파형의 auto-correlation과 PSD 특성, Pitch-Catch 실험을 통한 음파 정보 추출 방법 등을 학습할 수 있었다. 이를 통해 실제 상황에서 음파를 이용한 정보 추출 기술을 적용할 수 있는 기반을 마련하였다.

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1. Random Signal

Random signals are ubiquitous in various fields, including communication systems, signal processing, and control engineering. Understanding the properties and characteristics of random signals is crucial for effectively analyzing, processing, and utilizing them. Random signals are inherently unpredictable and exhibit a high degree of variability, making them challenging to study and model. However, the development of statistical and probabilistic tools has enabled researchers and engineers to gain valuable insights into the behavior of random signals. By analyzing the statistical properties, such as mean, variance, and probability distributions, we can better understand the underlying patterns and trends within random signals. This knowledge is essential for designing robust and efficient systems that can effectively handle and leverage the inherent randomness present in various applications.
2. Stationarity

Stationarity is a fundamental concept in the analysis of random signals, as it determines the statistical properties of the signal over time. A stationary signal is one whose statistical properties, such as mean and variance, do not change over time. This property is crucial for the application of various signal processing techniques, as it allows for the use of powerful analytical tools and simplifies the modeling and prediction of the signal's behavior. Understanding stationarity is particularly important in areas like time series analysis, spectral analysis, and signal filtering, where the assumption of stationarity is often necessary for the validity of the employed methods. Identifying and characterizing the stationarity of a signal can provide valuable insights into the underlying processes generating the signal, enabling more accurate modeling, prediction, and decision-making in a wide range of applications, from communication systems to financial time series analysis.
3. Auto-correlation

Auto-correlation is a fundamental concept in signal processing and time series analysis, as it provides a measure of the similarity between a signal and a delayed version of itself. This metric is particularly useful for identifying patterns, periodicities, and dependencies within a signal, which can have important implications in various applications. By analyzing the auto-correlation function of a signal, we can gain insights into the signal's underlying structure, the presence of trends or seasonality, and the degree of predictability. Auto-correlation is a powerful tool for tasks such as signal filtering, system identification, and time series forecasting. It can also be used to detect the presence of non-stationarity in a signal, which is crucial for the appropriate selection and application of signal processing techniques. Understanding and effectively utilizing auto-correlation is essential for a wide range of fields, from communication systems and control engineering to econometrics and bioinformatics.
4. Power Spectral Density (PSD)

Power Spectral Density (PSD) is a crucial concept in signal processing, as it provides a comprehensive representation of the frequency-domain characteristics of a signal. The PSD describes the distribution of a signal's power across different frequencies, revealing the dominant frequency components and their relative magnitudes. This information is invaluable for a wide range of applications, including communication systems, vibration analysis, and audio processing. By analyzing the PSD of a signal, we can identify the presence of periodic or dominant frequencies, detect the existence of noise or interference, and gain insights into the underlying physical processes that generated the signal. The PSD is also a fundamental tool for spectral analysis, filter design, and system identification. Understanding and effectively utilizing PSD can lead to improved signal processing, enhanced system performance, and more informed decision-making in various fields, from engineering and physics to finance and neuroscience.
5. Sine Wave Auto-correlation and PSD

The analysis of sine wave auto-correlation and power spectral density (PSD) is a crucial topic in signal processing, as it provides a deep understanding of the behavior and characteristics of this fundamental waveform. Sine waves are ubiquitous in various fields, from electrical engineering and communication systems to physics and music. Studying the auto-correlation and PSD of sine waves reveals important properties, such as the presence of a single dominant frequency, the relationship between the signal's amplitude and its power distribution, and the periodic nature of the waveform. This knowledge is essential for tasks like signal filtering, modulation and demodulation, and the design of efficient communication systems. Furthermore, the insights gained from sine wave analysis can be extended to more complex waveforms, enabling a better understanding of the frequency-domain characteristics of various signals. Mastering the concepts of sine wave auto-correlation and PSD is a fundamental step in developing a comprehensive understanding of signal processing principles and their practical applications.
6. Square Wave Auto-correlation and PSD

The analysis of square wave auto-correlation and power spectral density (PSD) is a valuable topic in signal processing, as it provides insights into the behavior and characteristics of this common waveform. Square waves are widely used in digital electronics, communication systems, and control applications due to their distinct on-off transitions and well-defined frequency content. Studying the auto-correlation and PSD of square waves reveals important properties, such as the presence of multiple frequency components, the relationship between the signal's duty cycle and its power distribution, and the periodic nature of the waveform. This knowledge is crucial for tasks like signal filtering, pulse-width modulation, and the design of digital communication systems. Furthermore, the insights gained from square wave analysis can be extended to other periodic waveforms, enabling a better understanding of the frequency-domain characteristics of various signals. Mastering the concepts of square wave auto-correlation and PSD is an essential step in developing a comprehensive understanding of signal processing principles and their practical applications.
7. Cosine Wave Auto-correlation and PSD

The analysis of cosine wave auto-correlation and power spectral density (PSD) is an important topic in signal processing, as it provides insights into the behavior and characteristics of this fundamental waveform. Cosine waves are widely used in various fields, including communication systems, control engineering, and audio processing, due to their periodic nature and well-defined frequency content. Studying the auto-correlation and PSD of cosine waves reveals important properties, such as the presence of a single dominant frequency, the relationship between the signal's amplitude and its power distribution, and the periodic nature of the waveform. This knowledge is crucial for tasks like signal filtering, modulation and demodulation, and the design of efficient communication systems. Furthermore, the insights gained from cosine wave analysis can be extended to more complex waveforms, enabling a better understanding of the frequency-domain characteristics of various signals. Mastering the concepts of cosine wave auto-correlation and PSD is an essential step in developing a comprehensive understanding of signal processing principles and their practical applications.
8. Pitch-Catch Cross-correlation

Pitch-catch cross-correlation is a powerful technique used in various fields, including non-destructive testing, structural health monitoring, and ultrasonic imaging. This method involves the analysis of the cross-correlation between two signals, typically generated by a transmitter (pitch) and received by a receiver (catch). By studying the characteristics of the cross-correlation function, valuable information can be extracted about the propagation of the signal through the medium or structure under investigation. The pitch-catch cross-correlation approach can provide insights into the presence of defects, changes in material properties, or the integrity of a system. This technique is particularly useful in applications where the direct measurement of a parameter is not feasible or where the signal propagation is influenced by complex environmental factors. Understanding the principles and applications of pitch-catch cross-correlation is essential for developing robust and reliable monitoring and diagnostic systems in a wide range of industries, from civil infrastructure to aerospace and medical imaging.
9. Experimental Results and Analysis

The presentation and analysis of experimental results is a crucial step in the scientific and engineering research process. Carefully designed experiments, coupled with rigorous data analysis, can provide valuable insights and validate theoretical models or hypotheses. The experimental results section should present the key findings in a clear and concise manner, highlighting the relevant observations, measurements, and trends. The analysis of these results should be thorough and objective, drawing connections between the experimental data and the underlying principles or theories being investigated. This step is essential for assessing the validity and significance of the research, as well as identifying potential limitations or areas for further exploration. Effective presentation and analysis of experimental results can lead to a deeper understanding of the phenomena under study, inform the development of improved models or techniques, and ultimately contribute to the advancement of knowledge in the field. A well-executed experimental results and analysis section is a hallmark of high-quality research, fostering scientific progress and enabling informed decision-making in various applications.
10. Conclusion

The conclusion of a research or technical work is a crucial component that ties together the key findings, insights, and implications of the study. A well-crafted conclusion should summarize the main objectives, highlight the most significant results, and discuss the broader significance and potential impact of the work. It should provide a clear and concise synthesis of the research, addressing the original problem or question and drawing meaningful conclusions based on the evidence presented. The conclusion should also acknowledge any limitations or uncertainties in the findings, and suggest potential avenues for future research or development. By effectively communicating the overall significance and implications of the study, the conclusion helps the reader to fully appreciate the value and contribution of the work. A strong conclusion not only reinforces the importance of the research but also inspires further exploration and application of the knowledge gained. Crafting a compelling and insightful conclusion is an essential skill for researchers, engineers, and technical professionals, as it solidifies the impact and relevance of their work.