Means that convolution is commutative ee102asignal

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Unformatted text preview: to ”flip and drag.” One will generally be easier than the other. EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 18 Simple Example (x*h) 2 2 x(!) 1 0 -1 1 3! 2 2 x∗h h(t − !) -1 h(!) 1 0 -1 h∗x x(!) 1 0 1 3! 2 x(t − !) -1 1 3! 2 2 1 0 h(!) 1 3! 2 y(t ) = (x ∗ h)(t ) 2 1 0 -1 1 2 3! EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 19 If we convolve three functions f , g , and h (f ∗ (g ∗ h))(t) = ((f ∗ g ) ∗ h)(t) which means that convolution is associative. To show this, we write out the integrals ￿ (f ∗ (g ∗ h))(t) = ∞ −∞ ￿∞ = f (τ1) [(g ∗ h)(t − τ1)] dτ1 f (τ1) −∞ ￿￿ ∞ −∞ ￿ g (τ2)h(t − τ1 − τ2) dτ2 dτ1 We let τ3 = τ1 + τ2. dτ3 = dτ2 in the inner integral, (f ∗ (g ∗ h))(t) = ￿ ∞ −∞ f (τ1) ￿￿ ∞ −∞ EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly ￿ g (τ3 − τ1)h(t − τ3) dτ3 dτ1 20 Interchanging the order of integration, (f ∗ (g ∗ h))(t) = ￿ ∞ ￿￿ ∞ −∞ ￿ −∞ ￿ f (τ1)g (τ3 − τ1) dτ1 h(t − τ3) dτ3 ￿￿ ￿ (f ∗g )(τ3 ) = ((f ∗ g ) ∗ h)(t) Combining the commutative and associate properties, f ∗ g ∗ h = f ∗ h ∗ g = ··· = h ∗ g ∗ f We can perform the convolutions in any order. EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 21 Convolution is also distributive, f ∗ (g + h) = f ∗ g + f ∗ h which is easily shown by writing out the convolution integral, ￿∞ (f ∗ (g + h))(t) = f (τ ) [g (t − τ ) + h(t − τ )] dτ = −∞ ∞ ￿ −∞ f (τ )g (t − τ ) dτ + = (f ∗ g )(t) + (f ∗ h)(t) ￿ ∞ −∞ f (τ )h(t − τ ) dτ Together, the commutative, associative, and distributive properties mean that there is an “algebra of signals” where • addition is like arithmetic or ordinary algebra, and • multiplication is replaced by convolution. EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 22 Properties of Convolution Systems The properties of the convolution integral have important consequences for systems described by convolution: • Convolution systems are linear: for all signals x1, x2 and all α, β ∈ ￿, h ∗ (αx1 + β x2) = α(h ∗ x1) + β (h ∗ x2) • Convolution systems are time-invariant: if we shift the input signal x by T , i.e., apply the input x1(t) = x(t − T ) to the system, the output is y1(t) = y (t − T ). In other words: convolution systems commute with delay. EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 23 • Composition of convolution systems corresponds to convolution of impulse responses. The cascade connection of two convolution systems y = (x ∗ f ) ∗ g Com posit ion x ∗f w y ∗g is the same as a single system with an impulse response h = f ∗ g x ∗( f ∗ g) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly y 24 Since convolution is commutative, the convolution systems are also commutative. These two cascade conn...
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This note was uploaded on 09/28/2013 for the course EE 108b taught by Professor Mukamel during the Fall '13 term at Singapore Stanford Partnership.

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