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LCS_6_notes - 1 LINEAR CIRCUITS AND SIGNALS Fourier...

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1 LINEAR CIRCUITS AND SIGNALS Fourier Transform Department of Electrical and Computer Engineering University of Miami I. P RELIMINARIES FS gives a frequency domain description of periodic signals. Can it be used with aperiodic signals as well? Let us explore this issue via a simple example. Example Consider the T -periodic square wave signal f ( t ) = V, for t [ - τ, + τ ]; 0 , for t [ - T/ 2 , + T/ 2] \ [ - τ, + τ ] F n = 2 V τ T , for n = 0; V sin[ o τ ] , for n 6 = 0 . Let us study F n further: F n = V ω o τ π sinc [ o τ ] = V 2 τ T sinc [ o τ ] . So, by substituting ω for o , we may write the ‘envelope’ of F n as env [ F n ] = V 2 τ T sinc [ ωτ ] . Now, let us make T → ∞ . Then, the spacing ω o = 2 π/T between frequency components of F n tends to zero and the periodic signal f ( t ) becomes an aperiodic single pulse signal. What happens to the FS coefﬁcients F n of f ( t ) ? F n 0 and env [ F n ] 0 , and they cannot give much information regarding the frequency content of the aperiodic single pulse signal that f ( t ) becomes. On the other hand, TF n = V 2 τ sinc [ o τ ]; env [ TF n ] = V 2 τ sinc [ ωτ ] ,

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2 are independent of T . As T → ∞ , the spacing ω o = 2 π/T between the frequency components of TF n tends to zero, and eventually, the frequencies corresponding to the terms TF n will constitute the whole real axis! II. F OURIER T RANSFORM (FT) With the discussion above in mind, let us deﬁne F ( o ) = TF n = Z T f ( t ) e - jnω o t dt. (1) Note that f ( t ) = + X n = -∞ F n e jnω o t = 1 T + X n = -∞ F ( o ) e jnω o t = 1 2 π + X n = -∞ F ( o ) e jnω o t ω o . (2) To proceed, let us deﬁne ω = o . (3) Since T → ∞ makes the spacing ω o = 2 π/T between the frequency components tend to zero, let us deﬁne Δ ω = ω o . (4) Now, with T → ∞ , we can express (2) as F ( ω ) = Z + t = -∞ f ( t ) e - jωt dt f ( t ) = 1 2 π Z + ω = -∞ F ( ω ) e jωt dω. (5) This pair of equations constitutes the Fourier transform (FT) pair f ( t ) F ( ω ) . The ﬁrst equation is the (forward) FT; the second equation is the inverse FT. Remarks. Existence of a FT is guaranteed for aperiodic functions that satisfy the Dirichlet conditions where the absolute convergence condition that was used with periodic functions is replaced by absolute integrability over ( -∞ , + ) , i.e., Z -∞ | f ( t ) | dt < . (6) This absolute integrability condition is only a sufﬁcient condition for the existence of the FT; it is not necessary. The inverse FT equation describes the aperiodic function f ( t ) as a weighted ‘sum’ of sinusoids over all frequencies; F ( ω ) form the weight at frequency ω .
3 This last remark, when combined with the fact that e jωt LTI H ( ω ) H ( ω ) e jωt , yields u ( t ) = 1 2 π Z U ( ω ) e jωt U

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This note was uploaded on 10/28/2010 for the course EEN 307 taught by Professor Kamalpremeratne during the Spring '10 term at University of Miami.

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LCS_6_notes - 1 LINEAR CIRCUITS AND SIGNALS Fourier...

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