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Unformatted text preview: ections have the same response x w ∗f x v ∗g y ∗g y ∗f Many operations can be written as convolutions, and these all commute
(integration, diﬀerentiation, delay, ...) EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 25 Example: Measuring the impulse response of an LTI system.
We would like to measure the impulse response of an LTI system, described
by the impulse response h(t)
!(t ) h(t )
t 0 ∗h t 0 This can be practically diﬃcult because input amplitude is often limited. A
very short pulse then has very little energy.
A common alternative is to measure the step response s(t), the response to
a unit step input u(t)
s(t ) u(t )
0 0 t t ∗h EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 26 The impulse response is determined by diﬀerentiating the step response,
s(t ) u(t ) t 0 t 0 t 0 h(t )
d
dt ∗h To show this, commute the convolution system and the diﬀerentiator to
produce a system with the same overall impulse response
!(t ) u(t )
0 t h(t )
t 0 0 d
dt t ∗h EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 27 Convolution Systems with Complex Exponential Inputs
• If we have a convolution system with an impulse response h(t), and and
input est where s = σ + j ω
∞
y (t) =
h(τ )es(t−τ ) dτ
−∞ =e st ∞ h(τ )e−sτ dτ −∞ • We get the complex exponential back, with a complex constant multiplier
∞
H (s) =
h(τ )e−sτ dτ
−∞
st y (t) = e H (s) provided the integral converges.
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 28 • H (s) is the transfer function of the system.
• Putting a complex exponential into an LTI system results in a complex
exponential output with the same frequency multiplied by a complex
constant. The complex exponential is said to be an eigenfunction of the
LTI described by h or H and H (s) is the corresponding eigenvalue.
• If the input is a complex sinusoid ej ωt,
H (j ω ) = ∞ h(τ )e−j ωτ dτ −∞ y (t) = ej ωtH (j ω ) EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 29 Summary • LTI systems can be represented as a the convolution of the input with
an impulse response.
• Convolution has many useful properties (associative, commutative, etc).
• These carry over to LTI systems
– Composition of system blocks
– Order of system blocks
Useful both practically, and for understanding.
• While convolution is conceptually simple, it can be practically diﬃcult.
It can be tedious to convolve your way through a complex system.
• There has to be a better way . . .
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 30...
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 Fall '13
 Mukamel
 Signal Processing

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