Definition and Convergence Conditions of CTFS Representation
2.1.1 Definition and Convergence Conditions of CTFS Representation
Let a function x(t ) be periodic with period P in t , that is,
x (t ) = x(t + P) ∀ t
where P [s] and ω 0 = 2π/P [rad/s] are referred to as the fundamental period and fundamental (angular) frequency, respectively, if P is the smallest positive real number to satisfy Eq. (2.1.1) for periodicity. Suppose x(t ) satisfies at least one of the following conditions A and B:
2.1 Continuous-Time Fourier Series (CTFS) of Periodic Signals 63
< Condition A >
(A1) The periodic function x(t ) is square-integrable over the period P, i.e.,
|x(t)| 2 dt < ∞
(2.1.2a)
where P means the integration over any interval of length P. This implies that the signal described by x(t ) has finite power.
< Condition B : Dirichlet condi ti on > (B1) The periodic function x(t ) has only a finite number of extrema and disconti-
nuities in any one period. (B2) These extrema are finite. (B3) The periodic function x(t ) is absolutely-integrable over the period P, i.e.,
|x(t)| dt < ∞
(2.1.2b)
Then the periodic function x(t ) can be represented by the following forms of continuous -time Fourier series (CTFS), each of which is called the Fourier series representation : < Trigonometric form>
x (t ) = a 0 +
t+
a k cos kω 0 b k sin kω 0 t (2.1.3a)
(P : the period of x(t ))
P where the Fourier coefficients a 0 , a k , and b k are
a 0 = x (t ) dt (the integral over one period P) P P
(2.1.3b) P P
x (t ) cos kω 0 t dt
x (t ) sin kω 0 t dt
< Magnitude -and-Phase form> x ∞ (t ) = d
t+φ k ) (2.1.4a) where the Fourier coefficients are
d k cos(kω 0
k=1
a 2 =a 2 = k +b k , φ k = tan −1 (−b k / a k ) (2.1.4b)
64 2 Continuous-Time Fourier Analysis < Complex Exponential form>
j kω (t ) = t c
k e 0 (2.1.5a)
k=−∞
where the Fourier coefficients are
c k x (t ) e − jkω 0 = t dt (the integral over one period P) (2.1.5b)
Here, the k th frequency kω 0 (|k| > 1) with fundamental frequency ω 0 = 2π/P = 2π f 0 [rad/s](P: period) is referred to as the k th harmonic. The above three forms of Fourier series representation are equivalent and their Fourier coefficients are related with each other as follows:
x (t ) dt = Pd 0 = Pa 0 (2.1.6a)
x (t ) e − jkω 0 t dt =
x (t ) (cos kω 0 t − j sin kω 0 t ) dt
= (a k (2.1.6b)
2 − jb k )= d k ∠φ 2 k
c x (t ) e j kω 0 −k t = dt = x (t ) (cos kω 0 t + j sin kω 0 t ) dt
−k −c k 2Im{c k } = P
a +c
−k
2Re{c }
jP =−
P (2.1.6d)
The plot of Fourier coefficients (2.1.4b) or (2.1.5b) against frequency kω 0 is referred to as the spectrum. It can be used to describe the spectral contents of a signal, i.e., depict what frequency components are contained in the signal and how they are distributed over the low/medium/high frequency range. We will mainly use Eqs. (2.1.5a) and (2.1.5b) for spectral analysis.
< Proof of the Complex Exponential Fourier Analysis Formula (2.1.5b)> To show the validity of Eq. (2.1.5b), we substitute the Fourier synthesis formula (2.1.5a) with the (dummy) summation index k replaced by n into Eq. (2.1.5b) as
c j nω k =
c n e 0 t − jkω 0 t
1 j (n−k) 2π t
dt =
=c P P
dt O.K.
n=−∞
n=−∞
2.1 Continuous-Time Fourier Series (CTFS) of Periodic Signals 65 This equality holds since
e (n−k) 2π P t dt P −P/2
1 P/ 2 j
(n−k) P/
=0 for
= −P/2 = δ[n − k] (2.1.8) ⎪ ⎩ ⎪ 1 P/ 2
⎨ P· j(n−k)2π/P
−P/2 dt = 1 for n=k