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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....
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This note was uploaded on 08/10/2010 for the course ECE 4370 at Cornell University (Engineering School).
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