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

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EE361: SIGNALS AND SYSTEMS II

REVIEW SIGNALS AND SYSTEMS I

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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.

2 Physical system T 𝑤𝑡(𝑢) 𝑤𝑑 𝑢 𝑗(𝑢) Abstract system

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

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 Memoryless

 Output does not depend on

past/future values

 Invertible

 Another system exists that accepts

𝑧(𝑢) as input and returns 𝑦(𝑢)

 Causal

 Output only depends on past or

present values

 Realizable system since it does not

need future values

 Implement non-causal systems with

delays

 Stable

 BIBO criterion: bounded input

results in bounded output

 Linear

 Given 𝑈 𝑦 𝑢

→ 𝑧(𝑢)

 𝑏𝑦1 𝑢 + 𝑐𝑦2 𝑢 → 𝑏𝑧1 𝑢 + 𝑐𝑧2(𝑢)

 Time Invariant

 Time shift on input results in

same time shift on output

 𝑈 𝑦 𝑢 − 𝑢0

→ 𝑧 𝑢 − 𝑢0

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BASIC SYSTEM PROPERTIES

LTI SYSTEM

 Linear and time-invariant systems  Impulse response ℎ 𝑜 completely specifies

input/output relationship

5 LTI 𝑦[𝑜] 𝑧[𝑜] 𝜀[𝑜] ℎ[𝑜] ℎ[𝑜] 𝑦[𝑜] 𝑧 𝑜 = 𝑦 𝑜 ∗ ℎ 𝑜 = ෍

𝑙=−∞ ∞

𝑦 𝑙 ℎ[𝑜 − 𝑙 = ෍

𝑙=−∞ ∞

ℎ 𝑙 𝑦[𝑜 − 𝑙]

LTI PROPERTIES

 Memoryless

 ℎ 𝑢 = 𝑏𝜀(𝑢), where 𝑏 is a constant

 Invertible

 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 < ∞

6 ℎ[𝑜] 𝑦[𝑜] 𝑧[𝑜] 𝑕[𝑜] 𝑥 𝑜 = 𝑦[𝑜]

EIGEN PROPERTY

 Eigen function (signal) for an LTI system is a signal for

which the output is the input times a (complex) constant

 CT: 𝑓𝑡𝑢 → 𝐼 𝑡 𝑓𝑡𝑢

 𝐼(𝑡) – eigenvalue from Laplace Transform (system/transfer

function)  DT: 𝑨𝑜 → 𝐼 𝑨 𝑨𝑜

 𝐼 𝑨 - eigenvalue from Z-transform (system/transfer

function)

7 ℎ(𝑢) 𝑦𝜇(𝑢) 𝑧 𝑢 = 𝜇𝑦𝜇(𝑢) eigenvalue