This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE 3250 PERIODIC SIGNALS AND FOURIER SERIES Fall 2008 1. Periodic signals You all know intuitively what a periodic signal is. A signal is periodic if it cycles repeti tively through a set of values. Here’s the formal definition. As usual, F means R or C and F R means the set of all continuoustime Fvalued signals. Definition 1: A signal x ∈ F R is periodic if and only if there exists T > 0 such that Shift T ( x ) = x . In this case, we say that T is a period of x . Every constant signal is trivially periodic, and every T > 0 is a period of such a signal. Sines and cosines are in a sense the prototypical periodic signals. Given Ω o > 0, the signals t 7→ cosΩ o t and t 7→ sinΩ o t are periodic, and every T of the form m 2 π/ Ω o , with m a positive integer, is a period of both of these. Another paradigmatic periodic signal is t 7→ e j Ω o t , which has the same periods as the cosine and sine above. Other periodic signals we encounter frequently are periodic square waves, triangle waves, and sawtooth waves. If T is a period of x , then any integer multiple of T is also a period of x . It follows that every periodic signal has arbitrarily large periods. Note, however, that all the examples in the preceding paragraph except the constant signal have some “smallest period.” A nonconstant signal lacking a “smallest period” is the signal x with specification x ( t ) = if t is rational 1 if t is irrational . Every rational number T > 0 is a period of x because the sum of two rational numbers is rational and the sum of a rational number and an irrational number is irrational. This x , familiar from the handout on continuoustime signals and convolution, is not a decent signal. As it happens, every decent nonconstant periodic signal has a “smallest period.” Fact 1: If x is a decent nonconstant periodic signal, then there exists a smallest T o > that is a period of x . Proof: I will show that if x has arbitrarily small periods, then x must be constant. Given an arbitrary T > 0, we can find a sequence { T 1 ,T 2 ,... } of periods of x such that T n increases to T in the limit as n → ∞ . To construct this sequence, form T n +1 from T n by adding on a sufficiently small period of x so that T n +1 < T . For any t ∈ R and n > 0, we have x ( t T n ) = x ( t ) since T n is a period of x . If t T is a point of continuity of x , it follows that x ( t T ) = x ( t ) because lim n →∞ x ( t T n ) = x ( t T ). What if t T is a jump point for x ? As it happens, x can’t have any jumps. If t o were a jump point of x , then for some > 0 we would have x ( t o δ ) 6 = x ( t o + δ ) for every δ < . But x has a period τ < , by assumption, which implies that x ( t o τ/ 2) = x ( t o + τ/ 2), a contradiction. Accordingly, x must be continuous since it is a decent signal with no jumps, and x ( t T ) = x ( t ) for every t ∈ R , which implies that x is constant since T was arbitrary. The bottom line is that if x is decent and not constant, it can’t have arbitrarily small periods.is decent and not constant, it can’t have arbitrarily small periods....
View
Full Document
 '04
 LIPSON
 Fourier Series, wk, Hilbert space, Sn, ωo, XTo

Click to edit the document details