EE361: SIGNALS AND SYSTEMS II REVIEW SIGNALS AND SYSTEMS I - PowerPoint PPT Presentation

1 EE361: SIGNALS AND SYSTEMS II REVIEW SIGNALS AND SYSTEMS I http://www.ee.unlv.edu/~b1morris/ee361

2 SIGNALS AND SYSTEMS I RECAP  Signals – quantitative descriptions of physical phenomena  Represent a pattern of variation  System – quantitative description of a physical process to transform an input signal to an output signal  The system is a “black box”  E.g. Physical system Abstract system 𝑤 𝑑 𝑢 T 𝑤 𝑡 (𝑢) 𝑗(𝑢)

3 SIGNALS  This course deals with signals that are a function of one variable  Most often called “time”  Continuous time (CT) signal  𝑦 𝑢 , 𝑢 ∈ ℝ  Time is a real valued (e.g. 1.23 seconds)  Discrete time (DT) signal  𝑦 𝑜 , 𝑜 ∈ ℤ  Time is discrete (e.g. 1 or 5)  Signal is a sequence and 𝑜 is the location within the sequence

4 BASIC SYSTEM PROPERTIES  Memoryless  Stable  Output does not depend on  BIBO criterion: bounded input past/future values results in bounded output  Invertible  Linear  Another system exists that accepts 𝑧(𝑢) as input and returns 𝑦(𝑢)  Given 𝑈 𝑦 𝑢 → 𝑧(𝑢)  Causal  𝑏𝑦 1 𝑢 + 𝑐𝑦 2 𝑢 → 𝑏𝑧 1 𝑢 + 𝑐𝑧 2 (𝑢)  Output only depends on past or present values  Time Invariant  Realizable system since it does not need future values  Time shift on input results in same time shift on output  Implement non-causal systems with delays  𝑈 𝑦 𝑢 − 𝑢 0 → 𝑧 𝑢 − 𝑢 0

5 LTI SYSTEM  Linear and time-invariant systems LTI 𝑦[𝑜] 𝑧[𝑜] 𝜀[𝑜] ℎ[𝑜]  Impulse response ℎ 𝑜 completely specifies input/output relationship 𝑧 𝑜 = 𝑦 𝑜 ∗ ℎ 𝑜 𝑦[𝑜] ℎ[𝑜] ∞ 𝑦 𝑙 ℎ[𝑜 − 𝑙 = ෍ 𝑙=−∞ ∞ = ෍ ℎ 𝑙 𝑦[𝑜 − 𝑙] 𝑙=−∞

6 LTI PROPERTIES  Memoryless  ℎ 𝑢 = 𝑏𝜀(𝑢) , where 𝑏 is a constant  Invertible 𝑦[𝑜] ℎ[𝑜] 𝑧[𝑜] 𝑕[𝑜] 𝑥 𝑜 = 𝑦[𝑜]  ℎ 𝑜 ∗ 𝑕 𝑜 = 𝜀 𝑜  Causal  ℎ 𝑢 = 0, 𝑢 < 0  Does not depend on future input – see convolution integral  Stable  Absolutely integrable/summable ∞ ℎ 𝜐 𝑒𝜐 < ∞  ׬ −∞

7 EIGEN PROPERTY  Eigen function (signal) for an LTI system is a signal for which the output is the input times a (complex) constant 𝑦 𝜇 (𝑢) ℎ(𝑢) 𝑧 𝑢 = 𝜇𝑦 𝜇 (𝑢) eigenvalue  CT: 𝑓 𝑡𝑢 → 𝐼 𝑡 𝑓 𝑡𝑢  𝐼(𝑡) – eigenvalue from Laplace Transform (system/transfer function)  DT: 𝑨 𝑜 → 𝐼 𝑨 𝑨 𝑜  𝐼 𝑨 - eigenvalue from Z-transform (system/transfer function)