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Slide 1 ELT2035 Signals & Systems Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi Lesson 2 Introduction to systems Last lesson review ❑[.]

ELT2035 Signals & Systems

Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi

Lesson 2: Introduction to systems

Last lesson review

❑ Fundamental concepts:

signals, data, information, systems

❑ Classification of signals

CT/DT, Periodic/Nonperiodic, Causal/Anti-causal/Noncausal

❑ Energy and power of signals

Energy signal and power signal

❑ Basic operations on signals

Amplitude scaling, addition, multiplication, differentiation, integration Time scaling, reflection, time shifting

❑ Elementary signals

Step, impulse, ramp, sinusoidal, real/complex exponential signals

Fundamental concepts of systems

❑ Systems are used to process input signals in order to obtain the

desired output signals

➢ A system may consists of physical components or may consists of an

algorithm that computes the output signal from the input signal.

➢ A system is characterized by its inputs, its outputs (or responses), and the

rules of operation (or laws) that governs its behavior.

❑ The mathematical description of the system’s rules of operation

are called the model of the system

➢ Usually expressed as an operator H[.] that relates the outputs to the inputs,

and commonly illustrated by the below “black box” concept.

❑ The study of systems includes 3 major areas: mathematical

modelling, system analysis, and system synthesis (design)

➢ Analysis: determine the output given the system and the input.

➢ Synthesis/design: construct a system which will produce the desired output for the given input.

System H[.]

Input signal Output signal

Systems as interconnections of

operations

❑ The operator H[.] in the “black box” approach can also be viewed

as a combination of basic operations performed on the input to yield the output

➢ Example: A 3-point discrete-time moving average system with input-output relationship 𝑦 𝑛 = 1

3 𝑥 𝑛 + 𝑥[𝑛 − 1] + 𝑥[𝑛 − 2] can be represented as follows

❑ In practice, systems are subject to unwanted signals that tend to

disturb the operation of the system → noise

❑ Classification by configurations

➢ Single-input single-output (SISO)

➢ Single-input multiple-output (SIMO)

➢ Multiple-input single-input (MISO)

➢ Multiple-input multiple-output (MIMO)

❑ Classification by properties of the operator H[.] that represents

the system

➢ Continuous time/discrete time

➢ Stable/unstable

➢ Instantaneous (memoryless)/dynamic (with memory)

➢ Causal/noncausal

➢ Invertible/non-invertible

➢ Time variant/time invariant

➢ Linear/non-linear

❑ A system may simultaneously possess several properties above,

e.g Linear time invariant (LTI) systems

Classification of systems

❑ Stability: a system is BIBO stable iff every bounded input

results in a bounded output

❑ Memory: a system is said to possess memory if the output

depends on past or future values of the input

➢ A system is memoryless if its output depends only on the current value of the input.

❑ Causality: a system is causal if the present value of the output

depends only on the present or past values of the input

➢ Noncausal system: the output depends on at least one future value of the input → NOT capable of operating in real time.

❑ Invertibility: a system is invertible if the input can be recovered

from the output

➢ The process of inverting the output is characterized by the operator H inv

such that H inv H = I (identity operator) The system associated with H inv is called the inverse system (for example: network equalizer).

Important properties of systems (1)

❑ Time invariance: a system is time invariant if a time shift in

the input leads to an identical time shift in the output

➢ The characteristics/behaviours of a time invariant system do not change with time

❑ Linearity: a system is said to be linear in terms of the system

input 𝑥(𝑡) and the system output 𝑦(𝑡) if it satisfies the following two properties

➢ Superposition: if the system respectively produces outputs 𝑦1(𝑡) and

𝑦2(𝑡) to the input 𝑥1(𝑡) and 𝑥2(𝑡), then the composite input 𝑥1 𝑡 + 𝑥2(𝑡) must yield the corresponding output 𝑦1 𝑡 + 𝑦2(𝑡).

➢ Homogeneity: whenever the input 𝑥(𝑡) is scaled by a factor 𝑎, the output 𝑦(𝑡) must be scaled by the same constant factor 𝑎.

❑ In this course, we will only work with linear time invariant (LTI)

systems

Important properties of systems (2)

Exercise #1

Consider x(t) defined by x(t) = 0.5t, for 0 ≤ t ≤ 3 and x(t) = 0, for t < 0 and t ≥

3 Plot x(t) for t in [0, 5] Is x(t) a signal? If not, modify the definition to make

it a signal

Exercise #2

Is it periodic or aperiodic?

A triangular wave is depicted in the below figure

Periodic

What is its fundamental frequency? 1/0.2=5Hz

Is it an energy signal or a power signal? Power

What is the average power of this signal? 1/3

Exercise #3

Prove that every signal 𝑓 𝑡 can be expressed as a sum of even and odd signals Find the even and odd components of 𝑓 𝑡 = 𝑒𝑗𝑡

Solution:

1 Rewrite 𝑓 𝑡 as 𝑓 𝑡 = 1

2 𝑓 𝑡 + 𝑓(−𝑡)

𝑒𝑣𝑒𝑛

+ 1

2 𝑓 𝑡 + 𝑓(−𝑡)

𝑜𝑑𝑑

2 Applying previous results for 𝑓 𝑡 = 𝑒𝑗𝑡, we obtain

𝑓𝑒 𝑡 = 1

2 𝑒

𝑗𝑡 + 𝑒−𝑗𝑡 = cos(𝑡)

𝑓𝑜 𝑡 = 1

2 𝑒

𝑗𝑡 − 𝑒−𝑗𝑡 = 𝑗 sin(𝑡)

Exercise #5 Consider the rectangular pulse 𝑥 𝑡 of unit amplitude and a duration of 2 units Sketch 𝑦 𝑡 = 𝑥(2𝑡 + 3).

Exercise #6

Show that ׬−∞∞ 𝑒−2 𝑥−𝑡 𝛿 2 − 𝑡 𝑑𝑡 = 𝑒−2 𝑥−2

Solution:

Applying the sampling property of the Dirac function:

Hence ׬−∞∞ 𝑒−2 𝑥−𝑡 𝛿 2 − 𝑡 𝑑𝑡 = ׬−∞∞ 𝑒−2 𝑥−2 𝛿 2 − 𝑡 𝑑𝑡 =

Dirac function

Exercise #7

a Consider a CT system whose input x(t) and output y(t) are

related by 𝑦 𝑡 = ׬𝜏=0𝑡+1𝑥 𝜏 𝑑𝜏 for 𝑡 > 0 Is the system

memoryless? stable? causal?

b Consider a DT system whose input and output are related by

𝑦 𝑛 = 3𝑥 𝑛 − 2 − 0.5𝑥 𝑛 + 𝑥 𝑛 + 1 Is the system

memoryless? stable? causal?

c Consider 𝑦 𝑡 = cos 𝜔𝑐𝑡 𝑥(𝑡) Is the system memoryless? linear? time-invariant?

d Consider 𝑦 𝑛 = 2𝑛 + 1 𝑥 𝑛 Is the system memoryless? linear? time-invariant?

Solution:

a Dynamic, stable, noncausal

b Dynamic, stable, noncausal

c Memoryless, linear, time varying

d Memoryless, linear, time varying

Exercise #8

The output of a discrete-time system is related to its input as

follows 𝑦 𝑛 = 𝑎0𝑥 𝑛 + 𝑎1𝑥 𝑛 − 1 + 𝑎2𝑥 𝑛 − 2 + 𝑎3𝑥 𝑛 − 3 Let the operator 𝑆𝑘 denote a system that shifts 𝑥 𝑛 by 𝑘 time units to produce 𝑥[𝑛 − 𝑘]

1 Find the operator H for the system relating 𝑦 𝑛 to 𝑥 𝑛

2 Sketch the block diagram for H

Solution:

1 Using operator 𝑆𝑘, we can rewrite 𝑦 𝑛 as 𝑦 𝑛 = 𝑎0𝑥 𝑛 +

𝑎1𝑆1 𝑥 𝑛 + 𝑎2𝑆2 𝑥 𝑛 + 𝑎3𝑆2 𝑥 𝑛 = (

)

𝑎0 + 𝑎1𝑆1 + 𝑎2𝑆2 +

𝑎3𝑆3 𝑥 𝑛 = 𝐻 𝑥 𝑛 Thus 𝐻 = 𝑎0 + 𝑎1𝑆1 + 𝑎2𝑆2 + 𝑎3𝑆3

2 See the next figure

Exercise #9

Consider the DT system given in exercise #8: 𝑦 𝑛 = 𝑎0𝑥 𝑛 +

𝑎1𝑥 𝑛 − 1 + 𝑎2𝑥 𝑛 − 2 + 𝑎3𝑥 𝑛 − 3 with constant coefficients

𝑎0, ⋯ , 𝑎3 are finite Is system BIBO stable? Why?

Solution:

Using the given input-output relation 𝑦 𝑛 = 𝑎0𝑥 𝑛 + 𝑎1𝑥[

]

𝑛 −

1 + 𝑎2𝑥 𝑛 − 2 + 𝑎3𝑥 𝑛 − 3 we may write

𝑦 𝑛 = 𝑎0𝑥 𝑛 + 𝑎1𝑥 𝑛 − 1 + 𝑎2𝑥 𝑛 − 2 + 𝑎3𝑥 𝑛 − 3

≤ 𝑎0𝑥 𝑛 + 𝑎1𝑥 𝑛 − 1 + 𝑎2𝑥 𝑛 − 2 + 𝑎3𝑥 𝑛 − 3

≤ 𝑎0 𝑀𝑥 + 𝑎1 𝑀𝑥 + 𝑎2 𝑀𝑥 + 𝑎3 𝑀𝑥,

where 𝑀𝑥 = 𝑥 𝑛 Hence, provided that 𝑀𝑥 is finite, the absolute value of the output will always be finite It follows therefore that the system is BIBO stable