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6.3we - s> α then(6 L τ a f t s = e-as F s for all s>...

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6.3: STEP FUNCTION KIAM HEONG KWA An important function that occurs in electrical systems is the (de- layed) unit step function (1) u a ( t ) = ( 0 if t < a, 1 if t a, where a 0. It is also known as Heaviside function . It delays its output until t = a and then assumes a constant value of one unit. In the literature, the Heaviside function is also commonly defined as (2) u a ( t ) = ( 0 if t < a, 1 if t > a. It is easy to see that both defintions give rise to the same Laplace transform: (3) L [ u a ( t )]( s ) = e - as s for s > 0. The multiplication of a function f defined for t 0 by u a translates the function f a distance a in the positive t direction in the following sense. For each a 0, let (4) τ a f ( t ) = ( 0 if t < a, f ( t - a ) if t a. The function τ a f represents a translation of f a distance a in the positive t direction. It is easy to check that (5) τ a f ( t ) = u a ( t ) f ( t - a ) . Translating the function f results in multiplying its Laplace trans- form by a corresponding exponential term: Date : February 22, 2011. 1
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2 KIAM HEONG KWA Theorem 1 (First translation theorem) . If F ( s ) = L [ f ( t )]( s ) exists
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Unformatted text preview: s > α , then (6) L [ τ a f ( t )]( s ) = e-as F ( s ) for all s > α . Proof. The proof is computational: L [ τ a f ( t )]( s ) = Z ∞ e-st u a ( t ) f ( t-a ) dt = Z ∞ a e-st f ( t-a ) dt by the definition of u a = Z ∞ e-s ( τ + a ) f ( τ ) dτ by the substitution τ = t-a = e-sa Z ∞ e-sτ f ( τ ) dτ = e-sa F ( s ) for s > α . ± Conversely, multiplying the function f by an exponential term trans-lates its Laplace transform: Theorem 2 (Second translation theorem) . If F ( s ) = L [ f ( t )]( s ) exists for s > α , then for any real constant a , (7) L [ e at f ( t )]( s ) = F ( s-a ) for all s > α + a . Proof. The conclusion follows from the existence of F ( s-a ) = L [ f ( t )]( s-a ) for s-a > α , for F ( s-a ) = Z ∞ e-( s-a ) t f ( t ) dt = Z ∞ e-st e at f ( t ) dt = L [ e at f ( t )]( s ) . ±...
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