Signals and systems Fall 2003 Lecture#3 11 September 2003 1) Representation of dt signals in terms of shifted unit samples 2) Convolution sum representation of dt lti systems 3) Examples 4) The unit sample response and properties of dt lti systems
Signals and Systems Fall 2003 Lecture #3 11 September 2003 1) Representation of DT signals in terms of shifted unit samples 2) Convolution sum representation of DT LTI systems 3) Examples 4) The unit sample response and properties of DT LTI systems
Exploiting Superposition and Time-Invariance ∑ Linear system akin 1n=∑am k Question: Are there sets of basic signals so that a) We can represent rich classes of signals as linear combinations of these building block signals b) The response of lti systems to these basic signals are both simple and insightful Fact: For Lti Systems(Ct or dt) there are two natural choices for these building blocks Focus for now: dt Shifted unit samples Ct Shifted unit impulses
Exploiting Superposition and Time-Invariance Question: Are there sets of “basic” signals so that: a) We can represent rich classes of signals as linear combinations of these building block signals. b) The response of LTI Systems to these basic signals are both simple and insightful. Fact: For LTI Systems (CT or DT) there are two natural choices for these building blocks Focus for now: DT Shifted unit samples CT Shifted unit impulses
Representation of DT Signals Using Unit Samples a[n] [0] 08n x[1]6{n-1] 1]6m+1] x{2]6[7
Representation of DT Signals Using Unit Samples
That is +x{-26m+2+x{-1]07+1+x06m]+x16n-1]+ m=∑:k k=-0 Coefficients Basic signals The Sifting Property of the Unit sample
That is ... Coefficients Basic Signals The Sifting Property of the Unit Sample
Xlr DT System Suppose the system is linear, and define hk[n] as the response to an-k 6|7-k→hk] From superposition =∑k6n-→列m=∑ahk团 k= k
x[n] DT System y[n] • Suppose the system is linear, and define hk[n] as the response to δ[n - k]: From superposition: