3226170 Fourier Series Representation Of Periodic Signals

Fourier Series Representation of Periodic Signals

Fourier Analysis background • • • •

Development of Fourier analysis dates in old times. In in 1948 Euler analyzed the motion of a vibrating string. The normal modes of string. If we consider the vertical deflection f(t,x) at time t and distance x along the string than for a fixed instant of time the normal modes are harmonically related sinusoidal functions of x. • During 1807 Fourier found that a series of harmonically related sinusoids are useful in representing the temperature distribution through in a body. • Any periodic signal can be represented in a series of harmonically related sinusoids.

Why do We Need Fourier Analysis? • The essence of Fourier analysis is to represent signals in terms of complex exponentials (or trigonometric functions) x(t ) =

∞

jkω 0t − 2 jω 0 t − jω 0 t jω 0 t 2 jω 0 t a e = ... + a e + a e + a + a e + a e + ... ∑ k −2 −1 0 1 2

k = −∞

• Many reasons: – Almost any signal can be represented as a series of complex exponentials – Response of an LTI system to a complex exponential is also a complex exponential with a scaled magnitude. – A compact way of approximating several signals.

Harmonically Related Complex Exponentials • We will use linear combinations (add, scale, time-shift) of basic periodic signal:

x(t ) = e jω 0t • This signal x(t ) is periodic with fundamental frequency ω 0 and fundamental period . • The set of harmonically related complex exponentials: T0 = 2π / ω 0

φk (t ) = e jkω 0t = e jk ( 2π / T )t , k = 0,1,2,3,... • Each of the signals in φ k (t ) are periodic with T, because their fundamental frequencies are multiple of ω 0.

Harmonically Related Complex Exponentials • Thus, a linear combination of them is also periodic!: x(t ) =

∞

jkω 0t a e ∑ k

k = −∞

• So, that means it should be possible to split a periodic signal into a set of periodic signals with same fundamental frequency. • This above representation of a periodic signal is referred as Fourier Series representation of that signal. • In the Fourier series above, the terms for k=1 and k=-1 are referred as the fundamental components or the first harmonic components of x(t).

Fourier Series Coefficients at ContinuousTime • How can we find the coefficients ak for a given continuous-time periodic signal x(t)? – First, determine the period T of x(t). – Then, find the fundamental frequency ω0=2π/T. – Finally, evaluate the Fourier coefficients by the 1 formula: ak = ∫ x(t )e − jkω 0t dt TT

∫

– where T represents integration over any interval of length T. – What happens when k=0?

Find Fourier series coefficients of x(t) = sinω0t By expanding the signal sinω0t = (1/2j)ejω0t - (1/2j)e-jω0t Comparing with previous equation a1 = 1/2j,

Ex 3.4

a-1 = -1/2j

Find Fourier Series coefficients of

Where T = 4

Convergence of Fourier Series • It may not be possible to represent a periodic signal as a Fourier series, if: – The signal is not integratable over any period – Over a finite interval of time, the signal has infinite number of variations – Over a finite interval of time, the signal has infinite number of discontinuities.

• However, such signals are not realistic. So, convergence is not an important issue.

Dirichlet Conditions

GIBBS PHENOMENON •Start by taking a signal with a finite number of discontinuities (like a square pulse) and finding its Fourier Series representation. •We then attempt to reconstruct it from these Fourier coefficients. •What we find is that the more coefficients we use, the more the signal begins to resemble the original. However, around the discontinuities, we observe rippling that does not seem to subside

GIBBS cont’d •As we consider even more coefficients, we notice that the ripples narrow, but do not shorten. • As we approach an infinite number of coefficients, this rippling still does not go away. This is when we apply the idea of almost everywhere. •While these ripples remain (never dropping below 9% of the pulse height), the area inside them tends to zero, meaning that the energy of this ripple goes to zero

Fourier series approximation of a square wave

Gibbs Phenomena •

Properties of CT Fourier Series • Given two periodic signals with same period T and fundamental frequency ω0=2π/T: x(t ) ↔ ak

y (t ) ↔ bk • Linearity:

z (t ) = Ax(t ) + By (t ) ↔ Aak + Bbk

• Time-Shifting:

z (t ) = x(t − t0 ) ↔ ak e − jω 0t0

• Time-Reversal (Flip): • Time-Scaling:

z (t ) = x(−t ) ↔ a− k

z (t ) = x(αt ) ↔ ak , α > 0

Properties of Fourier Series (cont’d) dx(t ) ↔ jkω 0 ak • Differentiation:z (t ) = dt t 1 z (t ) = ∫ x(t )dt ↔ a k , a0 = 0 jkω 0 • Integration: −∞

• Even-Odd Decomposition of Real Signals: z (t ) = Even{x(t )} ↔ ℜe{ak }

z (t ) = Odd{x(t )} ↔ jℑm{ak }

• Multiplication:

z (t ) = x(t ) y (t ) ↔ ak * bk =

∞

∑a b

l = −∞

l k −l