2.2 Fourier Series: 2.2.1 Fourier Series and Its Properties

1. 2.2 Fourier Series: 2.2.1 Fourier Series and Its Properties • FourierSeries • for some arbitrary  • x(t) is absolutely integrable over its period, i.e., • → xn값이 존재함 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2. Fourier Series and Its Properties • Observations concerning Fourier series • The coefficients xn are called the Fourier-series coefficients of the signal x(t) • These are generally complex numbers (even when x(t) is a real signal) • The parameter  in the limits of the integral is arbitrary • It can be chosen to simplify the computation of the integral. • Usually =0 or = -T0/2 • The power signal condition is only sufficient conditions for the existence of the Fourier series expansion • For some signals that do not satisfy these conditions, we can still find the Fourier series expansion • The quantity f0 = 1/T0is called the fundamental frequency of the signal x(t) • The frequencies of the complex exponential signals are multiples of this fundamental frequency • The n-th multiple of f0 is called the n-th harmonic Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

3. Fourier Series and Its Properties • Observations concerning Fourier series • The periodic signal x(t) can be described by the period T0 (or the fundamental frequency f0) and the sequence of complex numbers {xn} • To describe x(t), we may specify a countable set of complex numbers • This considerably reduces the complexity of describing x(t), since to define x(t) for all values of t, we have to specify its values on an uncountable set of points • The Fourier series expansion in terms of the angular frequency 0=2f0 • xn = | xn|ejxn • | xn| : Magnitude of the n-th harmonic • xn: Phase • Figure 2.24 : Discrete spectrum of the periodic signal x(t) Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

4. Fourier Series and Its Properties • Proofof Fourier Series • Plug (1) in (2) Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

5. Example 2.2.1: • xn 구하기: 방법 1) 직접 적분 계산 • x(t) : Periodic signal depicted in Figure 2.25 and described analytically by •  : A given positive constant (pulse width) • Determine the Fourier series expansion Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

6. Example 2.2.1 • Solution • Period of the signal is T0 and where we have used the relation • For n = 0, the integration is very simple and yields • Therefore • Figure 2.26 : Graph of these Fourier series coefficients Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

7. Example 2.2.3: • xn 구하기: 방법 1) 직접 적분 계산 • x(t) : Impulse train → Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

8. Example Kwon: • xn 구하기: 방법 2) x(t) 를 complex exponential 식으로 전개하여 xn 을 뽑아냄 • 방법 1)로는 어려움. 이 신호의 주기는? Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

9. 기출문제: Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

10. Positive and Negative Frequencies • Fourier series expansion of a periodic signal x(t) • All positive and negative multiples of the fundamental frequency 1/T0 are present • Positive frequency : • Phasor rotating counterclockwise at an angular frequency  • Negative frequency : • Phasor rotating clockwise at an angular frequency  • Figure 2.29 • Real signals • Positive and negative frequency pairs with amplitudes that are conjugates Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

11. Fourier Series for Real Signals • Real signal x(t) • The positive and negative coefficients are conjugates • |xn| : Even symmetry (|xn| = |x-n| ) • xn: Odd symmetry (xn = - x-n) with respect to the n = 0 axis Figure 2.30 Discrete spectrum of a real-valued signal. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

12. Fourier Series for Real Signals • For a real periodic signal x(t) , we have three alternatives to represent the Fourier-series expansion Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

13. Fourier-Series Expansion for Even and Odd Signals • For real and even x(t) • Since x(t) sin(2nt/T0)is the product of an even and an odd signal, it will be odd and its integral will be zero. Therefore, every xn, is real. • For even signals, the Fourier-series expansion has only cosine terms • For real and odd signals • In a similar way that every an, vanishes • Fourier-series expansion only contains the sine terms, or equivalently, every xn, is imaginary. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

14. 기출문제: • x(t) is real, T0periodic, and power signal. x3 = 2+j4. Let y(t)=[x(t) - x(-t)]/2, then y3 = ? Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

15. 2.2.2 Response of LTI Systems to Periodic Signals • If h(t)is the impulse response of the system, that the response to the exponential ej2f0tis H( f0) ej2f0t • x(t) ,the input to the LTI system, is periodic with period Toand has a Fourier-series representation • Response of LTI systems Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

16. Response of LTI Systems to Periodic Signals • Conclusions • If the input to an LTI system is periodic with period To, then the output is also periodic. (What is the period of the output?) The output has a Fourier-series expansion given by • Only the frequency components that are present at the input can be present at the output. This means that an LTI system cannot introduce new frequency components in the output, if these component are different from those already present at the input. In other words, all systems capable of introducing new frequency components are either nonlinear and/or time varying • The amount of change in amplitude |H(n/T0)| and phase H(n/T0) are functions of n, the harmonic order, and h(t), the impulse response of the system. The function is called the frequency response or frequency characteristics of the LTI system. In general, H( f ) is a complex function that can be described by its magnitude |H(n/T0)| and phase H(n/T0) . The function H ( f ), or equivalently h ( t ) , is the only information needed to find the output of an LTI system for a given periodic input. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

17. 2.2.3 Parseval's Relation • The power content of a periodic signal is the sum of the power contents of its components in the Fourier-series representation of that signal • The left-hand side of this relation is Px, the power content of the signal x(t) • |xn|2is the power content of , the n-th harmonic • Parseval's relation says that the power content of the periodic signal is the sum of the power contents of its harmonics Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

18. Parseval's Relation • Proof) • If x(t) is real, Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/