Chapter 2. Characteristics of Signal ※ Signal : transmission of information The quality of the information depends on proper selection of a measurement system and proper interpretation of the measured results Magnitude : the size of the input quantity Frequency : the way the signal changes in time

Chapter 2. Characteristics of Signal Analog : continuous in time Digital : discrete in time Useful when data acquisition and processing are performed using a digital computer

Chapter 2. Characteristics of Signal  Classification of Waveforms

Chapter 2. Characteristics of Signal  Signal Analysis The average or mean value of the signal (dc component or dc offset) The rms value of y(t) over time, t 2 - t 1,

Chapter 2. Characteristics of Signal  Signal Analysis A time-dep. analog signal, y(t), can be represented by a discrete set of N nos. over the time period from t 1 to t 2 y (t) → {y (n δ t)} n = 1, 2,…, N The sampling convolution {y (n δ t)} = y (t) δ (t – n δ t) = {y i } i = 1, 2,…,N δ t : sample time increment N δ t = t 2 - t 1 : the total sample period

Chapter 2. Characteristics of Signal  Signal Analysis ※ Effects of Signal-Averaging period ( 신호 평균주기의 효과 ) 신호해석에 있어 시간 간격 t f = t 2 -t 1 의 선택은 해석하려는 목적에 의존 t f 를 선택하는데 특별한 규칙은 없으며 목적이나 결과의 사용 의도를 고려한 계측계획에 의존 주기를 길게 하면 신호의 특성을 올바르게 표현

Chapter 2. Characteristics of Signal  Signal Analysis ※ DC offset 신호의 교류성분이 주관심사 일 때 직류성분은 제거 될 수 있 다.

Chapter 2. Characteristics of Signal  Signal Amplitude and Frequency Fourier analysis : The method of expressing a complex signal as a series of sines & cosines  A y(t) is a periodic fn. if there is some positive no. T such that y(t + T) = y(t) T : period for y(t) If both y 1 (t) and y 2 (t) have period T, then ay 1 (t) + by 2 (t) also has a period of T

Chapter 2. Characteristics of Signal  Signal Amplitude and Frequency  A trigonometric series is given by A 0 + A 1 cos t + B 1 sin t + A 2 cos 2t + B 2 sin 2t +…+ A n cos nt + B n sin nt Where A n and B n are the coefficients of the series

Chapter 2. Characteristics of Signal  Fourier Coefficients A periodic y(t) with T = 2π y(t) = A 0 + (A n cos nt + B n sin nt) with y(t) known, the coeffs. A n and B n are to be determined

Chapter 2. Characteristics of Signal  Fourier Coefficients (n=m 일 경우만 non zero)

Chapter 2. Characteristics of Signal  Fourier Coefficients Similarly, Euler formula

Chapter 2. Characteristics of Signal  Fourier Coeffs. for fns. having Arbitrary Periods Euler formulas : The coeffs. of a trigonometric series representing a fn. of freq. of W

Chapter 2. Characteristics of Signal  Fourier Coefficients When n=1, the corresponding terms in the Fourier series are called fundamental The fundamental freq. is  = 2n/T Freq. Corresponding to n=2, 3, 4 are known as harmonics

Chapter 2. Characteristics of Signal  Fourier Coefficients Where

Chapter 2. Characteristics of Signal  Even and Odd Functions A fn. g(t) is even if g(-t) = g(t) cos nt : even A fn. h(t) is odd if h(-t) = - h(t) sin nt : odd

Chapter 2. Characteristics of Signal  Fourier Cosine Series If y(t) is even, its Fourier series contains only cosine terms :

Chapter 2. Characteristics of Signal  Fourier Sine Series If y(t) is odd, its Fourier series contains only sine terms :

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum 측정된 동적 신호를 주파수와 진폭의 항으로 나타내기 위한 방법 In practical measurement applications, the input signal may not be known A technique for the decomposition of a measured dynamic signal in terms of amplitude and frequency is described

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum 식 (2.17) If T → ∞, the Fourier series becomes an integral This means that the coeffs. A n and B n become continuous f ns of freq. 진폭 A n 과 B n 은 퓨리에 급수의 n 번째 주파수에 대응

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum To develop the Fourier transform, consider the complex no. defined as Y(w) = A(w) – i B(w)

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum ( f = w/2π = 1/T)

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum If Y(f) is known or measured, we can recover the signal y(t) The inverse Fourier transform of Y(t) Given the amplitude-freq. properties of a signal we can reconstruct the signal y(t)

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum ※퓨리에 변환의 도입은 측정한 신호 y(t) 를 진폭 - 주파수 요소 로 바꾸어 주는 수단을 제공 ; y(t) 의 진폭스펙트럼 (amplitude spectrum)

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum  Power Density Concept The concept that each frequency contributes to the total power of a signal is called the power density ※ 각각의 주파구가 어떤 신호의 전체 파워에서 차치하는 크기 를 나타내는 개념 Power spectrum of y(t)

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum  Discrete Fourier Transform (DFT) y(t) is measured & recorded and it will be stored in the form of a discrete time series The transformation from a continuous to discrete time signal is described by {y (rδt)} = y(t) δ(t-rδ) r = 1, 2,…, N δt : time interval where δ(t-rδt) is the delayed unit impulse f n {y(rδt)} refers to the discrete data set given by y(rδt) for r = 1, 2,…,N

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum  Discrete Fourier Transform (DFT) The DFT is given by one-sided or half transform 으로 데이터 집합이 t=0 에서 t = t f 까지 한쪽 방향으로 존재 t → rδt, f → k/Nδt 로 대체 ( 식 2.30 에서 2.38 유도할때 ) k = 1, 2,…, N/2

Chapter 2. Characteristics of Signal 2.5 Fourier Transform and the Frequency Spectrum  Discrete Fourier Transform (DFT) With the use of this method, a measured discrete signal of unknown functional form can be reconstructed as a Fourier series by using Fourier transform techniques 이산적 신호에 대한 이산 퓨리에 변환을 계산하는 알고리즘은 부록 A 에 이 계산시간은 N 2 의 비율로 증가 (N. 이 큰 집합에 대해서는 비능률 ) 고속 퓨리에 변환 (FFT, Fast Fourier Transform)