Lecture 4 Signals & Systems.pdf

EC2610:Fundamentals of Signals and Systems
By
Sadananda Behera
Assistant Professor
Department of Electronics and Communication Engineering
NIT, Rourkela
1
CHAPTER-1 : SIGNALS AND SYSTEMS-Part 4

Systems
 A system can be viewed as a process in which input signals are transformed
by the system or cause the system to respond in some way, resulting in other
signals as outputs.
 A system is an entity that process one or more input signals in order to
produce one or more output signals.
● E.g.: An image-enhancement system transforms an input image into an
output image that has some desired properties, such as improved contrast.

Continuous-Time and Discrete-Time Systems
 A continuous-time system is a system in which continuous-time input signals are applied
and result in continuous-time output signals.
 A discrete-time system is a system that transforms discrete-time inputs into discrete-time
outputs.

Interconnection of Systems
 Describing a system in terms of an interconnection of simpler subsystems, we may in fact
be able to define useful ways in which to synthesize complex systems out of simpler basic
building blocks.
• Series Interconnection (Cascade Interconnection):
• Parallel Interconnection:

• Series-Parallel Interconnection:
• Feedback Interconnection:

Basic System Properties
(1)Systems with and without memory:
• A system is said to be memoryless if its output for each value of the independent
variable at a given time is dependent only on the input at that same time.
• Examples of memoryless (static) system:
 𝑦 𝑛 = (2𝑥 𝑛 − 𝑥2
[𝑛])2
 𝑦(𝑡) = 𝑅𝑥(𝑡), [𝑅 : Resistance, 𝑥(𝑡) : Current, 𝑦(𝑡) : Voltage.]
 𝑦(𝑡) = 𝑥(𝑡), [Identity System.]
 𝑦 𝑛 = 𝑥 𝑛 , [Identity System.]

• Examples of systems with memory (dynamic systems):
 𝑦 𝑛 = 𝑥[𝑘]
𝑛
𝑘=−∞ , [Accumulator/Summer.]
 𝑦 𝑛 = 𝑥[𝑛 − 1], [Delay.]
 Output voltage across a capacitor:
𝑦(𝑡) =
1
𝐶
𝑥 𝛾 𝑑𝛾
𝑡
−∞
, 𝐶 : Capacitance, 𝑥(𝑡) : Current through the capacitor,
𝑦(𝑡) : Voltage.

(2)Invertibility and Inverse Systems:
• A system is said to be invertible if distinct inputs lead to distinct outputs.
• E.g.: An invertible continuous-time system is 𝑦(𝑡) = 2𝑥(𝑡),
The inverse system is 𝑤(𝑡) =
1
2
𝑦(𝑡).

• Accumulator:
𝑦 𝑛 = 𝑥[𝑘]
𝑛
𝑘=−∞
= 𝑥 𝑘 + 𝑥 𝑛 = 𝑦 𝑛 − 1 + 𝑥[𝑛]
𝑛−1
𝑘=−∞
The inverse system is 𝑤 𝑛 = 𝑦 𝑛 − 𝑦[𝑛 − 1].
• Examples of Non-Invertible System:
 𝑦 𝑛 = 0
 𝑦 𝑡 = 𝑥2(𝑡)

(3) Causality:
• A system is causal if the output at any time depends only on values of the input at
the present time and in the past.
• Such a system is often referred to as being nonanticipative, as the system output
does not anticipate future values of the input.
• Examples of Causal System:
 𝑦 𝑡 = 𝑥 𝑡 − 1
 𝑦 𝑛 = 𝑥 𝑛 − 𝑥[𝑛 − 1]
 𝑦 𝑛 = 𝑎𝑥 𝑛
 𝑦 𝑛 = 𝑥[𝑘]
𝑛
𝑘=−∞

• Examples of Non-Causal System:
 𝑦 𝑛 = 𝑥 𝑛 − 𝑥[𝑛 + 1]
 𝑦 𝑡 = 𝑥 𝑡 + 1
 𝑦 𝑛 =𝑥 𝑛2
 𝑦 𝑛 =𝑥 2𝑛
 𝑦 𝑛 =𝑥 −𝑛
 𝑦 𝑛 =
1
(2𝑀+1)
𝑥[𝑛 − 𝑘]
𝑀
𝑘=−𝑀 , [Non-causal averaging system]

● All memoryless systems are causal, since the output responds only to the current value of
the input.
● In real time signal processing applications we can not observe future values of the signal
and hence a non-causal system is not physically realizable (can not be implemented).
● Causality is not often an essential constraint in applications in which the independent
variable is not time.
● If the signal is recorded so that the processing is done offline (non-real time), it is
possible to implement a non-causal system, since all values of the signal are available at
the time of processing. This is often the case in processing of audio, image, video,
geographical and metrological signals.

● Examples:
 𝑦 𝑛 = 𝑥[−𝑛]
𝑦 2 = 𝑥[−2]
𝑦 −2 = 𝑥[2], hence non-causal.
 𝑦(𝑡) = 𝑥(𝑡) cos(𝑡 + 1)
 𝑦(𝑡) can be written as 𝑦 𝑡 = 𝑥 𝑡 𝑔 𝑡 , where 𝑔 𝑡 = cos(𝑡 + 1) is a time-
varying function.
 So, only the current value of the input 𝑥(𝑡) influences the current value of the
output 𝑦 𝑡 . Therefore the system is causal.

(4)Stability:
 A system is said to be bounded input-bounded output (BIBO) stable if every
bounded input produces a bounded output.
 Mathematically, if 𝑥[𝑛] ≤ 𝑀𝑥 < ∞, then 𝑦[𝑛] ≤ 𝑀𝑦 < ∞ for all 𝑛 where 𝑀𝑥
and 𝑀𝑦 are finite numbers.
 Examples:
i. 𝑦(𝑡) = 𝑡𝑥 𝑡
Let 𝑥 𝑡 = 1, 𝑦 𝑡 = 𝑡, which is unbounded. Hence it is unstable.
ii. 𝑦 𝑡 = 𝑒𝑥 𝑡 , 𝑥 𝑡 < 𝐵
As −𝐵 < 𝑥 𝑡 < 𝐵, 𝑒−𝐵 < 𝑦 𝑡 < 𝑒𝐵. 𝑦 𝑡 is bounded, so the system is stable.

(5)Time Invariance:
 A system is time invariant if the behavior and characteristics of the system are fixed
over time.
 A system is time invariant if a time shift in the input signal results in an identical
time shift in the output signal.
 If 𝑦[𝑛] is the output of a discrete-time, time-invariant system when 𝑥[𝑛] is the input,
then 𝑦 [𝑛 − 𝑛0] is the output when 𝑥 [𝑛 − 𝑛0] is applied.
𝑥 𝑛 → 𝑦 𝑛
𝑥 [𝑛 − 𝑛0] → 𝑦 [𝑛 − 𝑛0]
In continuous time,
𝑥(𝑡) → 𝑦(𝑡)
𝑥 (𝑡 − 𝑡0) → 𝑦 (𝑡 − 𝑡0)

 Examples: 𝑦 𝑡 = sin[𝑥(𝑡)]
 𝑦1 𝑡 = sin[𝑥1(𝑡)], where 𝑥1 𝑡 is any arbitrary input to the system.
 𝑥2 𝑡 =𝑥1 𝑡 − 𝑡0
 The output corresponding to input 𝑥2 𝑡 is
𝑦2 𝑡 = sin 𝑥2 𝑡 = sin 𝑥1 𝑡 − 𝑡0 … (1)
 Time delayed version of 𝑦1 𝑡 is
𝑦1 𝑡 − 𝑡0 = sin 𝑥1 𝑡 − 𝑡0 … (2)
 As (1) and (2) are same, therefore this system is time invariant.

 Examples: 𝑦 𝑡 = 𝑥(2𝑡)
 Consider 𝑥1 𝑡 =
1, 𝑡 < 2
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 𝑦1 𝑡 =
1, 𝑡 < 1
 Take 𝑥2 𝑡 = 𝑥1 𝑡 − 2 =
1, 0 < 𝑡 < 4
 𝑦2 𝑡 =
1,0 < 𝑡 < 2
and 𝑦1 𝑡 − 2 =
1, 1 < 𝑡 < 3
 AS 𝑦1 𝑡 − 2 ≠ 𝑦2 𝑡 or Fig. (d) ≠ Fig. (e), the system is time variant.

 Procedure to determine time variance/invariance:
For time-invariance 𝑦1 𝑡 = 𝑦2 𝑡
System Delay
Delay System
𝑥 𝑡
𝑥 𝑡
𝑦1 𝑡
𝑦2 𝑡

 Example: 𝑦 𝑡 = 𝑥(2𝑡)
As 𝑦1 𝑡 ≠ 𝑦2 𝑡 , the system is time variant.
System
Delay
𝑘
Delay
𝑘
System
𝑥 𝑡
𝑥 𝑡
𝑦1 𝑡 = 𝑥 2 𝑡 − 𝑘 = 𝑥(2𝑡 − 2𝑘)
𝑦2 𝑡 = 𝑥(2𝑡 − 𝑘)
𝑥 2𝑡
𝑥 𝑡 − 𝑘

 Example: 𝑦[𝑛] = 𝑥[−𝑛]
As 𝑦1[𝑛] ≠ 𝑦2[𝑛], the system is time variant.
System
Delay
𝑘
Delay
𝑘
System
𝑥[𝑛] 𝑦1 𝑛 = 𝑥[−𝑛 + 𝑘]
𝑥[−𝑛]
𝑥[𝑛] 𝑥[𝑛 − 𝑘] 𝑦2 𝑛 = 𝑥[−𝑛 − 𝑘]

 Examples of time-invariant system:
i. 𝑦 𝑛 = 𝑎𝑥 𝑛
ii. 𝑦 𝑛 = 𝑥2
[𝑛]
iii. 𝑦 𝑡 = 𝑥(𝑡 + 1)
 Examples of time variant system:
i. 𝑦 𝑛 = 𝑛𝑥 𝑛
ii. 𝑦 𝑛 = 𝑥[−𝑛]
iii. 𝑦 𝑛 = 𝑥 [𝑛2]

(6)Linearity:
 A linear system, in continuous time or discrete time, is a system that possesses the
important property of superposition: If an input consists of the weighted sum of
several signals, then the output is the superposition-that is, the weighted sum-of the
responses of the system to each of those signals.
𝑥1(𝑡) → 𝑦1(𝑡), 𝑥2(𝑡) → 𝑦2(𝑡)
The system is linear if
i. The response to 𝑥1 𝑡 + 𝑥2(𝑡) is 𝑦1 𝑡 + 𝑦2(𝑡), i.e.
𝑥1 𝑡 + 𝑥2(𝑡) → 𝑦1 𝑡 + 𝑦2 𝑡 , [Additivity Property]
ii. The response to 𝑎𝑥1 𝑡 is 𝑎𝑦1 𝑡 , where a is any complex constant, i.e.
𝑎𝑥1 𝑡 → 𝑎𝑦1 𝑡 , [Scaling/Homogeneity Property]

 The two properties defining a linear system can be combined into a single
statement:
i. continuous time: 𝑎𝑥1 𝑡 + 𝑏𝑥2(𝑡) → 𝑎𝑦1 𝑡 + 𝑏𝑦2(𝑡),
ii. discrete time: 𝑎𝑥1 𝑛 + 𝑏𝑥2 𝑛 → 𝑎𝑦1 𝑛 + 𝑏𝑦2 𝑛 ,
where 𝑎 and 𝑏 are any complex constants.
 In general for linear system
if 𝑥 𝑛 = 𝑎𝑘𝑥𝑘[𝑛]
𝑘 = 𝑎1𝑥1[𝑛]+𝑎2𝑥2[𝑛]+⋯
then 𝑦 𝑛 = 𝑎𝑘𝑦𝑘[𝑛]
𝑘 = 𝑎1𝑦1[𝑛]+𝑎2𝑦2[𝑛]+⋯
 For a linear system: zero input yields zero output.

 Examples:
𝑦 𝑡 = 𝑡𝑥(𝑡)
• 𝑥1 𝑡 → 𝑦1 𝑡 = 𝑡𝑥1 𝑡
• 𝑥2 𝑡 → 𝑦2 𝑡 = 𝑡𝑥2 𝑡
• Let 𝑥3 𝑡 be a linear combination of 𝑥1 𝑡 and 𝑥2 𝑡 , i.e. 𝑥3 𝑡 = 𝑎𝑥1 𝑡 +
𝑏𝑥2 𝑡
• If 𝑥3 𝑡 is input to system , then the corresponding output is
𝑦3 𝑡 = 𝑡𝑥3 𝑡 = 𝑡 𝑎𝑥1 𝑡 + 𝑏𝑥2 𝑡 = 𝑎𝑡𝑥1 𝑡 + 𝑏𝑡𝑥2 𝑡 = 𝑎𝑦1 𝑡 + 𝑏𝑦2 𝑡
• So the system is linear.

System
𝑦[𝑛]
𝑥2[𝑛]
𝑥1[𝑛] 𝑎
𝑏
System
System
𝑎
𝑏
𝑦′[𝑛]
𝑥1[𝑛]
𝑥2[𝑛]
● Graphical representation of superposition principle. The system is linear if
and only if 𝑦 𝑛 = 𝑦′[𝑛].

 Examples:
𝑦 𝑛 = 2𝑥 𝑛 + 3
• It is not a linear system.
• If 𝑥 𝑛 = 0,then 𝑦 𝑛 = 3.
• This type of system is called incrementally linear system.
• 𝑦0(𝑡) is zero input response of the system.
Linear
System
𝑥(𝑡) 𝑦(𝑡)
𝑦0(𝑡)

Summary
● A number of basic concepts related to continuous-time and discrete-time signals and
systems have been developed.
● Graphical and mathematical representations of signals are introduced and used these
representations in performing transformations of the independent variable.
● Several basic signals, both in continuous-time and discrete-time, are also defined and
examined. These included complex exponential signals, sinusoidal signals, and unit
impulse and step functions.
● The concept of periodicity for continuous-time and discrete-time signals have been
investigated.
● Some of the elementary ideas related to systems have been developed and block
diagrams are introduced to facilitate the discussions concerning the interconnection of
systems
● A number of important properties of systems are discussed including causality, stability,
time invariance, and linearity.

Lecture 4 Signals & Systems.pdf

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