Unformatted text preview: to ”ﬂip and drag.” One will
generally be easier than the other. EE102A:Signal Processing and Linear Systems I; Win 0910, 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 0910, 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 0910, 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 0910, 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 0910, 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 timeinvariant: 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 0910, 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 0910, 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.
 Fall '13
 Mukamel
 Signal Processing

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