1. 2.2 Fourier Series: 2.2.1 Fourier Series and Its Properties
for some arbitrary 
x(t) is absolutely integrable over its period, i.e.,
→ xn 값이 존재함
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:

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/T0 is 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:

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:

4. Fourier Series and Its Properties
Proof of Fourier Series
Plug (1) in (2)
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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:

6. Example 2.2.1 Solution Period of the signal is T0 and Therefore
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:

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:

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:

9. 기출문제:
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:

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:

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:

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:

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:

14. 기출문제: x(t) is real, T0 periodic, 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:

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f0t is H( f0) ej2f0t
x(t) , the input to the LTI system, is periodic with period To and has a Fourier-series representation
Response of LTI systems
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:

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:

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|2 is 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:

18. Parseval's Relation Proof) If x(t) is real,
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: