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:
x团n DT System Now suppose the system is LTI, and define the unit sample response hn n2→hn From Ti m-k→bn-k From lti =∑6m一刷→列=∑mh Convolution sum
x[n] DT System y[n] • Now suppose the system is LTI, and define the unit sample response h[n]: From LTI: From TI:
Convolution Sum representation of Response of lti systems k-0 Interpretation 6[n] hn [kSIn-k k]h[n-k k n Sum up responses over all k
Convolution Sum Representation of Response of LTI Systems Interpretation n n n n
Visualizing the calculation of gn=.n* hm Choose value of n and consider it fixed n=∑xkhn-刚 k=o View as functions ofk with n fixed 0 {k] y]=∑ prod of 12 0 k overlap n=0 h[n-k yl=∑ prod of overlap for k n=1 7+1
Visualizing the calculation of y[0] = ∑ prod of overlap for n = 0 y[1] = ∑ prod of overlap for n = 1 Choose value of n and consider it fixed View as functions of k with n fixed
Calculating Successive Values: Shift, Multiply, Sum hIn-k yn=0 for n 4
Calculating Successive Values: Shift, Multiply, Sum -1 1 × 1 = 1 (-1) × 2 + 0 × (-1) + 1 × (-1) = -3 (-1) × (-1) + 0 × (-1) = 1 (-1) × (-1) = 1 4 0 × 1 + 1 × 2 = 2 (-1) × 1 + 0 × 2 + 1 × (-1) = -2
Properties of Convolution and dt ltI Systems d A dT LTI System is completely characterized by its unit sample response Ex#1: hn]=8n-nol There are many systems with this repsonse to d[n There is only one LTI System with this response to 8[n an-00
Properties of Convolution and DT LTI Systems 1) A DT LTI System is completely characterized by its unit sample response
