# Systems y x f g com posit ion x f w y g is

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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 09-10, 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 09-10, 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 09-10, 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 09-10, 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 09-10, 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 09-10, Pauly 30...
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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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