Chapter 4 Chapter 4 The Fourier Series and The Fourier Series and - PDF document

Chapter 4 Chapter 4 The Fourier Series and The Fourier Series and Fourier Transform Fourier Transform Fourier Series Representation of Fourier Series Representation of Periodic Signals Periodic Signals • Let x ( t ) be a CT periodic signal with period + = ∀ ∈ ( ) ( ), x t T x t t R T, i.e., • Example: the rectangular pulse train rectangular pulse train

The Fourier Series The Fourier Series • Then, x ( t ) can be expressed as ∞ ∑ ω = ∈ � jk t ( ) , x t c e t 0 k =−∞ k ω = π 2 / T where is the fundamental fundamental 0 frequency ( rad/sec ) of the signal and frequency / 2 T 1 ∫ − ω = = ± ± … jk t ( ) , 0, 1, 2, c x t e dt k o k T − / 2 T c is called the constant or dc component of x ( t ) 0 Dirichlet Conditions Dirichlet Conditions • A periodic signal x ( t ), has a Fourier series if it satisfies the following conditions: 1. x ( t ) is absolutely absolutely integrable integrable over any period, namely + a T ∫ < ∞ ∀ ∈ � | ( ) | , x t dt a a 2. x ( t ) has only a finite number of maxima finite number of maxima and minima over any period and minima 3. x ( t ) has only a finite number of finite number of discontinuities over any period discontinuities

Example: The Rectangular Pulse Train Example: The Rectangular Pulse Train ω = π = π T = 2 / 2 2 • From figure , so 0 • Clearly x ( t ) satisfies the Dirichlet conditions and thus has a Fourier series representation Example: The Rectangular Pulse Example: The Rectangular Pulse Train – Cont’d Train – Cont’d ∞ 1 1 ∑ − π = + − ∈ |( 1)/ 2| � k jk t ( ) ( 1) , x t e t π 2 k =−∞ k k odd

Trigonometric Fourier Series Trigonometric Fourier Series • By using Euler’s formula, we can rewrite ∞ ∑ ω = ∈ jk t � ( ) , x t c e t 0 k as =−∞ k ∞ ∑ = + ω + ∠ ∈ � ( ) 2 | |cos( ), x t c c k t c t �� � ��� � 0 0 k k = 1 k dc component dc component -th th harmonic harmonic k - k as long as x(t) is real • This expression is called the trigonometric trigonometric Fourier series of x ( t ) Fourier series Example: Trigonometric Fourier Example: Trigonometric Fourier Series of the Rectangular Pulse Train Series of the Rectangular Pulse Train • The expression ∞ 1 1 ∑ − π = + − ∈ |( 1)/ 2| � k jk t ( ) ( 1) , x t e t π 2 k =−∞ k k odd can be rewritten as ∞ π ⎛ ⎞ 1 2 ∑ ⎡ ⎤ = + π + − − − ∈ ( k 1)/ 2 � ( ) cos ⎜ ( 1) 1 ⎟ , x t k t t ⎣ ⎦ π ⎝ ⎠ 2 2 k = k 1 k odd

Gibbs Phenomenon Gibbs Phenomenon • Given an odd positive integer N, define the N -th partial sum of the previous series π ⎛ ⎞ N 1 2 ∑ ⎡ ⎤ = + π + − − − ∈ ( 1)/ 2 k � ( ) cos ( 1) 1 , ⎜ ⎟ x t k t t ⎣ ⎦ π N ⎝ ⎠ 2 2 k = 1 k k odd • According to Fourier Fourier’ ’s theorem s theorem, it should be − = lim | ( ) ( ) | 0 x t x t N →∞ N Gibbs Phenomenon – Cont’d Gibbs Phenomenon – Cont’d 3 ( ) x t 9 ( ) x t

Gibbs Phenomenon – Cont’d Gibbs Phenomenon – Cont’d 21 ( ) x t 45 ( ) x t overshoot: about 9 % of the signal magnitude overshoot N → ∞ (present even if ) Parseval’s Theorem Parseval’s Theorem • Let x ( t ) be a periodic signal with period T • The average power average power P of the signal is defined as / 2 T 1 ∫ = 2 ( ) P x t dt T − / 2 T ∞ ∑ ω = ∈ � jk t • Expressing the signal as ( ) , x t c e t 0 k =−∞ k it is also ∞ = ∑ 2 | | P c k =−∞ k

Fourier Transform Fourier Transform • We have seen that periodic signals can be represented with the Fourier series • Can aperiodic aperiodic signals signals be analyzed in terms of frequency components? • Yes, and the Fourier transform provides the tool for this analysis • The major difference w.r.t. the line spectra of periodic signals is that the spectra of spectra of aperiodic signals signals are defined for all real aperiodic ω values of the frequency variable not just for a discrete set of values Frequency Content of the Frequency Content of the Rectangular Pulse Rectangular Pulse ( ) x t ( ) x t T = ( ) lim ( ) x t x t T →∞ T

Frequency Content of the Frequency Content of the Rectangular Pulse – Cont’d Rectangular Pulse – Cont’d ( ) • Since is periodic with period T, we x t T can write ∞ ∑ ω = ∈ jk t � ( ) , x t c e t 0 T k =−∞ k where / 2 T 1 ∫ − ω = = ± ± jk t … ( ) , 0, 1, 2, c x t e dt k o k T − / 2 T Frequency Content of the Frequency Content of the Rectangular Pulse – Cont’d Rectangular Pulse – Cont’d • What happens to the frequency components T → ∞ ( ) of as ? x t T k = 0: • For = 1/ c T 0 k = ± ± … 1, 2, : • For ω ω ⎛ ⎞ ⎛ ⎞ 2 1 k k = = 0 0 sin ⎜ ⎟ sin ⎜ ⎟ c ω π k ⎝ ⎠ ⎝ ⎠ 2 2 k T k 0 ω = π 2 / T 0

Frequency Content of the Frequency Content of the Rectangular Pulse – Cont’d Rectangular Pulse – Cont’d plots of | | T c k ω = ω vs. k 0 T = 2,5,10 for Frequency Content of the Frequency Content of the Rectangular Pulse – Cont’d Rectangular Pulse – Cont’d • It can be easily shown that ω ⎛ ⎞ = ω ∈ � lim sinc ⎜ ⎟ , Tc π k ⎝ ⎠ →∞ 2 T where πλ πλ � sin( ) sin( ) λ λ � sinc( ) sinc( ) πλ πλ

Fourier Transform of the Fourier Transform of the Rectangular Pulse Rectangular Pulse • The Fourier transform of the rectangular pulse x ( t ) is defined to be the limit of Tc k T → ∞ as , i.e., ω ⎛ ⎞ ω = = ω ∈ � ( ) lim sinc ⎜ ⎟ , X Tc π k ⎝ ⎠ →∞ 2 T X ω X ω X ω X ω arg( ( )) arg( ( )) | ( ) | | ( ) | The Fourier Transform in the The Fourier Transform in the General Case General Case • Given a signal x ( t ), its Fourier transform Fourier transform X ω ( ) is defined as ∞ ∫ − ω ω = ω ∈ � j t ( ) ( ) , X x t e dt −∞ • A signal x ( t ) is said to have a Fourier transform in the ordinary sense if the above integral converges

The Fourier Transform in the The Fourier Transform in the General Case – Cont’d General Case – Cont’d • The integral does converge if 1. the signal x ( t ) is “ well behaved ” well- -behaved 2. and x ( t ) is absolutely integrable , namely, absolutely integrable ∞ ∫ < ∞ | ( ) | x t dt −∞ • Note: well behaved well behaved means that the signal has a finite number of discontinuities, maxima, and minima within any finite time interval Example: The DC or Constant Signal Example: The DC or Constant Signal = ∈ � • Consider the signal ( ) 1, x t t • Clearly x ( t ) does not satisfy the first requirement since ∞ ∞ ∫ ∫ = =∞ | ( ) | x t dt dt −∞ −∞ • Therefore, the constant signal does not have a Fourier transform in the ordinary sense Fourier transform in the ordinary sense • Later on, we’ll see that it has however a Fourier transform in a generalized sense Fourier transform in a generalized sense

Example: The Exponential Signal Example: The Exponential Signal − = ∈ � bt ( ) ( ), • Consider the signal x t e u t b • Its Fourier transform is given by ∞ ∫ − − ω ω = bt j t ( ) ( ) X e u t e dt −∞ =∞ t ∞ 1 ∫ ⎡ ⎤ − + ω − + ω = = − ( ) ( ) b j t b j t e dt e ⎣ ⎦ + ω b j = 0 0 t Example: The Exponential Signal – Example: The Exponential Signal – Cont’d Cont’d b < X ω ( ) 0 • If , does not exist = X ω b = ( ) ( ) ( ) • If , and does not 0 x t u t exist either in the ordinary sense b > 0 • If , it is 1 ω = ( ) X + ω b j amplitude spectrum amplitude spectrum phase spectrum phase spectrum 1 ω ⎛ ⎞ ω = | ( ) | ω = − X arg( ( )) arctan ⎜ ⎟ X + ω 2 2 ⎝ ⎠ b b

Example: Amplitude and Phase Example: Amplitude and Phase Spectra of the Exponential Signal Spectra of the Exponential Signal − = − = 10 t 10 t ( ) ( ) x t ( ) e u t ( ) x t e u t Rectangular Form of the Fourier Rectangular Form of the Fourier Transform Transform • Consider ∞ ∫ − ω ω = ω ∈ j t � ( ) ( ) , X x t e dt −∞ X ω ( ) • Since in general is a complex function, by using Euler’s formula ⎛ ⎞ ∞ ∞ ∫ ∫ ω = ω + − ω ⎜ ⎟ ( ) ( )cos( ) ( )sin( ) X x t t dt j x t t dt ⎝ ⎠ ��� −∞ ��� � −∞ ��� � ���� � ω ω ( ) R ( ) I ω = ω + ω ( ) ( ) ( ) X R jI