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LCS_8_notes

# LCS_8_notes - 1 LINEAR CIRCUITS AND SIGNALS Convolution and...

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1 LINEAR CIRCUITS AND SIGNALS Convolution and Sampling Department of Electrical and Computer Engineering University of Miami I. P RELIMINARIES Previously we developed a frequency domain description of how LTI systems operate. In this chapter, we look at their time domain behavior. II. C ONVOLUTION Definition 1: We refer to y ( t ) = Z + τ = -∞ f ( τ ) g ( t - τ ) dτ, as the convolution between the pair of signals f ( t ) and g ( t ) ; it is denoted by y ( t ) = ( f * g )( t ) . Example Let us convolve a given signal f ( t ) with the unit step function 1( t ) : ( f * 1)( t ) = Z τ f ( τ ) 1( t - τ ) = Z t τ = -∞ f ( τ ) dτ, because 1( t - τ ) = 1 , for t - τ > 0 τ < t ; 0 , otherwise . Example Let us convolve the unit step with itself: (1 * 1)( t ) = Z τ 1( τ ) 1( t - τ ) = Z t τ =0 = t, for t > 0; 0 , otherwise = t 1( t ) . This is the unit ramp function.

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2 III. P ROPERTIES OF C ONVOLUTION A. Some Basic Properties Claim 1 (Commutativity): Convolution is commutative, i.e., for arbitrary functions f ( t ) and g ( t ) , f * g = g * f. Proof: ( f * g )( t ) = Z + τ = -∞ f ( τ ) g ( t - τ ) = - Z -∞ T =+ f ( t - T ) g ( T ) dT = Z + T = -∞ g ( T ) f ( t - T ) dT = ( g * f )( t ) . Claim 2 (Distributivity): Convolution is distributive (over addition), i.e., for arbitrary functions f ( t ) , g ( t ) , and h ( t ) , f * ( g + h ) = ( f * g ) + ( f * h ) . Claim 3 (Associativity): Convolution is associative, i.e., for arbitrary functions f ( t ) , g ( t ) , and h ( t ) , f * ( g * h ) = ( f * g ) * h. Therefore we may write f * g * h with no ambiguity. B. FT and Convolution One of the most important properties of convolution is the following: Claim 4 (Time Convolution): Suppose f ( t ) F ( ω ) and g ( t ) G ( ω ) . Then ( f * g )( t ) F ( ω ) G ( ω ) . Proof: Let h ( t ) = ( f * g )( t ) = Z τ f ( τ ) g ( t - τ ) dτ.
3 Therefore H ( ω ) = Z t ( f * g )( t ) e - jωt dt = Z t Z τ f ( τ ) g ( t - τ ) dτ e - jωt dt = Z τ f ( τ ) Z t g ( t - τ ) e - jωt dt dτ = Z τ f ( τ ) G ( ω ) e - jωτ = G ( ω ) Z τ f ( τ ) e - jωτ = F ( ω ) G ( ω ) . Similarly, we also have Claim 5 (Frequency Convolution): Suppose f ( t ) F ( ω ) and g ( t ) G ( ω ) . Then f ( t ) g ( t ) 1 2 π ( F * G )( ω ) . Proof: Begin with ( F * G )( ω ) = Z Ω F (Ω) G ( ω - Ω) d Ω , and proceed as above. C. Some Basic Operations Claim 6 (Shift): Let ˜ ( · )( t ) = ( · )( t - t o ) . Then y ( t ) = ( f * g )( t ) = ˜ y ( t ) = ( ˜ f * g )( t ) = ( f * ˜ g )( t ) . Proof: We can use Claims 4 and 5 to prove this. First note that ˜ f ( t ) ˜ F ( ω ) = F ( ω ) e - jωt o . So, ( ˜ f * g )( t ) ˜ F ( ω ) G ( ω ) = F ( ω ) e - jωt o G ( ω ) = F ( ω ) ˜ G ( ω ) ( f * ˜ g )( t ) . Since the inverse FT must generate a unique time domain function, we must have ( ˜ f * g )( t ) = ( f * ˜ g )( t ) . This same relationship yields ( ˜ f * g )( t ) F ( ω ) G ( ω ) e - jωt o = Y ( ω ) e - jωt o ˜ y ( t ) .

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4 Again, this must imply that ( ˜ f * g )( t ) = ˜ y ( t ) . Claim 7 (Derivative): y ( t ) = ( f * g )( t ) = y (1) ( t ) = ( f (1) * g )( t ) = ( f * g (1) )( t ) . Proof: Again, use Claims 4 and 5 to prove this. D. Properties Related to Support Claim 8 (Start-Point): Suppose f ( t ) = 0 , t < t f , and g ( t ) = 0 , t < t g . Then y ( t ) = ( f * g )( t ) = 0 , t < t f + t g . Proof: Note that y ( t ) = ( f * g )( t ) = Z τ f ( τ ) g ( t - τ ) = 0 , whenever τ < t f or t - τ < t g ⇐⇒ τ > t - t g .
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