1. Chapter 2. Signals and Linear Systems
Text Book: Essentials of Communication Systems Engineering
- John G. Proakis and Masoud Salehi
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2. 2.0 Pre-requisites: 2.0.1 Sine & Cosine Functions
Definisions: Y y r θ X O x
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3. 2.0 Pre-requisites: 2.0.1 Sine & Cosine Functions
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4. 2.0 Pre-requisites: 2.0.2 Complex Numbers
Notations: Y y r θ X O x
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5. 2.0 Pre-requisites: Exercises
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6. 2.1 Basic Concepts: 2.1.1 Basic Operations on Signals
x(t) : A signal
Time shifting or delaying
y(t) : Time shifted version of x(t) : , t0 : Time shift
When t0 > 0, x(t) is shifted to the right : time delay
When t0 < 0, x(t) is shifted to the left : time advance
Time reversal, or flipping
y(t) : Signal by replacing time t by –t :
Reflected version of x(t) about t = 0
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7. Basic Operations on Signals
Time scaling
Expands (stretches) or contracts (compresses) a signal along the time axis
y(t) : Signal by scaling the independent variable, time t, by a factor a
If a>1, the signal y(t) is a compressed version of x(t)
If 0<a<1, the signal y(t) is an expanded version of x(t)
Combination of these operation
x(-2t) = Combination of flipping the signal then contracting it by a factor of 2
x(2t-3) : x[2(t-1.5)] = Contract the signal by a factor of 2 and then shifting it to the right by 1.5
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8. 2.1.2 Classification of Signals: Real and Complex Signals
Real signals
Takes its values in the set of real numbers
x(t)  R, t  R
Complex signals
Takes its values in the set of complex numbers
x(t)  C , t  R
In communications
Used to model signals
Convey amplitude and phase information
Represented by two real signals
Real and imaginary parts
Absolute value (or modulus or magnitude) and phase
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9. Representation of Complex signals – Example 2.1.3
The signal :
Complex signals
From Euler’s relation :
Its real part :
Its imaginary part :
Absolute value of x(t) :
Its phase :
Figure 2.8
Representation of complex signals
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10. Periodic and Nonperiodic Signals
The signal x(t) is periodic if and if only (iff) there exists a T0 >0 such that
T0 : period of the signals (positive real number)
the minimum value satisfying the condion
A signal that does not satisfy the conditions of periodicity is called non-periodic
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11. Even and Odd Signals Even Signals Odd Signals
A signal x(t) is said to be an even signals if
Odd Signals
A signal x(t) is said to be an odd signals if
In general, any signal x(t) can be written as the sum of its even and odd part as
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12. Hermitian Symmetry for Complex Signals
Another form of symmetry for complex signals
A complex signal x(t) is called Hermitian if its real part is even and its imaginary part is odd
Its magnitude is even and its phase is odd
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13. Energy-Type and Power-Type Signals
Energy of the signal
For any signal x(t), the energy of the signal is defined by
Power of the signal
For any signal x(t), the power of the signal is defined by
For real signal, is replaced by
A signal is an energy-type signal if and only if Ex is finite
A signal is an power-type signal if and only if Px satisfies
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14. Example 2.1.9
The energy of
Therefore, this signal is an energy-type signal.
Hence, x(t) is a power-type signal and its power is
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15. Example 2.1.10 The energy content of
Therefore, this signal is not an energy-type signal
However, the power of this signal is
Hence, x(t) is a power-type signal and its power is
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16. Example 2.1.11 For any periodic signal with period T0, the energy is
Therefore, periodic signals are not typically energy type
The power of any periodic signal is
This means that the power content of a periodic signal is equal to the average power in one period
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17. 2.1.3 Some Important Signals & Their properties Sinusoidal Signal & Complex Exponential Signal
Sinusoidal signals
Definition :
Real Part of complex exponential signal
A : Amplitude
f0 : Frequency
 : Phase
Period : T0 = 1/f0
Figure 2.6
Complex exponential signal
Figure 2.8
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18. Unit Step, Rectangular & Triangular Signal
Unit step signal
Definition
The unit step multiplied by any signal produces a “causal version” of the signal
Note that for positive a, we have
Figure 2.9
Rectangular pulse, Square signal, П signal
Figure 2.13
Triangular Signal
Figure 2.15
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19. Example 2.1.12, 13, 14 Plot the signal . Figure 2.12
Represent x(t) in Figure 2.14 using rectangular pulses.
Plot the signal
Figure 2.16
Therefore, periodic signals are not typically energy type
The power content of any periodic signal is
This means that the power content of a periodic signal is equal to the average power in one period
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20. Convolution of Two Signals
Definition
Example: p35
Example: p39, property 10
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21. Sinc & Sign or Signum Signal
Sinc signal
Definition
The sinc signal achieves its maximum of 1 at t = 0.
The zeros of the sinc signal are at t = 1, 2, 3, 
Figure 2.17
Sign or Signum signal
Definition :
Sign of the independent variable t
Can be expressed as the limit of the signal xn(t) when n  
Figure 2.18
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22. Impulse, Delta, Sampling Signal
일반 함수가 아님, 특이함수임
defined in terms of its effect on another function (usually called the “test function”) under the integral sign.
Definition of the impulse distribution (or signal) by the relation
Helpful to visualize (t) as the limit of certain known signals
Figure 2.19
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23. Impulse Signal – Properties
Figure 2.20 : Schematic representation of the impulse signal
(t) = 0 for all t  0 and (0) = 
x(t)(t-t0) = x(t0)(t-t0)
For any (t) continuous at t0, → 다른 책에서는 (t) 의 정의식임, Sampling Signal
For any (t) continuous at t0,
For all a  0,
Let all a = -1, , even signal
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24. Impulse Signal – Properties
The result of the convolution of any signal with the impulse signal is the signal it self :
The unit step signal is the integral of the impulse signal, and the impulse signal is the generalized derivative of the unit step signal
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25. Impulse Signal – Properties
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26. Impulse Signal – Exercisesb) c)
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27. 2.1.4 Classification of Systems
An interconnection of various elements or devices that behave as a whole.
Communication point of view
A entity that is excited by an input signal and , as a result of this excitation, produces an output signal
Communication engineer’s point of view
A law that assigns output signals to various input signals
Its output must be uniquely defined for ant legitimate input
x(t) : Input
y(t) : Output
 : Operation performed by the system
Characteristics
The operation that describes the system : Use the operator  to denote the operation
The set of legitimate input signals : Use the operator H 
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28. Linear and Nonlinear Systems
Systems for which the superposition property is satisfied
The system response to a linear combination of the inputs is the combination of the response to the corresponding inputs
A system  is linear if and only if, for any two input signals x1(t) and x2(t) and for any two scalars  and , we have
A system that does not satisfy this relationship is called nonlinear
Linearity : Additive + homogeneous
Output of a linear system
Decompose the input into a linear combination of some fundamental signals
Linear combination of the corresponding outputs
L : Operation of linear systems
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29. Time-Invariant and Time-Varying Systems
Time-invariant system
A system is called time invariant if its input-output relationship does not change with time
A delayed version of an input results in a delayed version of the output
A system is time-invariant if and only if, for all x(t) and all values of t0, its response to x(t-t0) is y(t-t0) , y(t) is the response of the system to x(t)
Linear time-invariant systems
The response of these systems to inputs can be derived simply by finding the convolution of the input and the impulse response of the system
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:

30. Example 2.1.19-22 Differentiating & integrating systems:
→ Linear & Time Invariant → RLC circuits: LTI
Time delaying system:
→ Linear & Time Invariant
: Nonlinear & Time Invariant
: Linear & Time Variant
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31. Analysis of LTI Systems in the Time Domain
Impulse response
The impulse response h(t) of a system is the response of the system to a unit impulse input (t)
The convolution integral: Output y(t) of an LTI system to any input signal x(t)
Pf)
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32. Example
Let a linear time-invariant system have the impulse response h(t)
Assume this system has a complex exponential signal as input,
The response to the input
The response of an LTI system to the complex exponential with frequency f0 is a complex exponential with the same frequency
Amplitude of the response : Multiplying the amplitude of the input by
Phase response : Adding to the input phase
: A function of the impulse response and the input frequency
Complex exponential
Eigenfunctions of the class of linear time-invariant systems
Eigenfunctions of a system : Set of inputs for which the output is a scaling of the input
Simple to find the response of LTI system to the class of complex exponential signals
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33. HW & QZ #1 Textbook Problems from p105
2.1.1, , , ,
2.5.1,
2.6.2, ,
2.7.1, , ,
2.9.4
2.13.9, ,
2.16.1, , , ,
2.24.1, , , ,
2.34.1, ,
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