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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online