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Flashcards in Signals and Systems Deck (161)
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1
Q

What are samples?

A

Points of data

2
Q

What are sampling intervals?

A

The distance between the sample points

3
Q

Discrete-time signals can be defined in 2 ways…

A
  1. As a function
  2. As a list of values of a sequence
4
Q

In this picture, the arrow indicates….?

And if the arrow is not shown, where is the ____ point?

A

The n=0 point

The first value in the sequence is the n=0 point

5
Q

Define analog signals

A

Analog Signals: An analog signal is a continuous-time signal x(t) that can take on any value in the continuous interval (a, b), where a may be - ∞ and b may be +∞.

6
Q

Define Digital Signals

A

Digital Signals: A digital signal is a discrete-time signal x[n] that can only take on only a finite number of distinct values.

7
Q

What is a real signal?

A

Real Signals: A signal x(t) is said to be a real signal if its value is a real

number

8
Q

What is a complex signal?

A

Complex Signals: A signal x(t) is said to be a complex signal if its value is a complex number

9
Q

A general (continuous- or discrete-time) complex signal x(t) is a function of the form

A

where x1(t) and x2(t) are real signals and j = √−1

10
Q

Define Deterministic Signals

A

Deterministic Signals: Those signals whose values are exactly specified for any given time in the time span of interest are called deterministic signals. Such signals may be represented by a known function of time t.

11
Q

Define Random Signals

A

Random Signals: Those signals that take random values at any given time are called random or non-deterministic signals. Such signals must be characterized statistically (that is, represented in probabilistic terms)

12
Q

Any signal x(t) or x[n] can be expressed as a sum of ____ ____, one of which is ____ and one of which is ____

A

Any signal x(t) or x[n] can be expressed as a sum of two signals, one of which is even and one of which is odd

13
Q

The product of two even signals or of two odd signals is a…?

A

…even signal

14
Q

The product of an even signal and an odd signal is an…?

A

…odd signal

15
Q

Define Periodic continuous-time signals

A

Periodic Continuous-time Signals: A continuous-time signal x(t) is said to be a periodic signal with period T if there is a unique positive nonzero value of T for which the following expression is valid

16
Q

A dc signal is a signal in which x(t) is ____; such a signal is not periodic since its fundamental period is ____

A

A dc signal is a signal in which x(t) is constant; such a signal is not periodic since its fundamental period is undefined

17
Q

Define Aperiodic Continuous-time Signals

A

Aperiodic Continuous-time Signals: Any continuous-time signal which is not periodic is called a non-periodic signal or an aperiodic signal

18
Q

Define Periodic Discrete-time Signals

A

Periodic Discrete-time Signals: A sequence (discrete-time signal) x[n] is periodic with period N if there is a unique positive integer N for which the following expression is valid

19
Q

Define Aperiodic Discrete-time Signals

A

Aperiodic Discrete-time Signals: Any sequence which is not periodic is called a non-periodic sequence or an aperiodic sequence

20
Q

A sequence obtained by uniform sampling of a periodic continuous-time signal may…

A

…not be periodic

21
Q

The sum of two continuous-time periodic signals may…

A

… not be periodic

22
Q

The sum of two periodic sequences is…

A

…always periodic

23
Q

For any continuous-time signal x(t), the total energy of the signal over the interval t1 ≤ t ≤ t2 is defined as

A
24
Q

The average power of the signal x(t) over the interval t1 ≤ t ≤ t2 is defined as

A
25
Q

|x(t)| denotes the ____ of x(t) and the signal x(t) could be a ____ or ____ signal.

A

|x(t)| denotes the magnitude of x(t) and the signal x(t) could be a real or complex signal.

26
Q

For any discrete-time signal x[n], the total energy of the signal over the interval n1 ≤ n ≤ n2 is defined as

A
27
Q

The average power of the signal x[n] over the interval n1 ≤ n ≤ n2 is defined as

A
28
Q

A signal x(t) (or x[n]) is said to be an ____ ____ (or sequence) if and only if 0 < E∞ < ∞, in which case, P∞ = 0.

A

A signal x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E∞ < ∞, in which case, P∞ = 0.

29
Q

A signal x(t) (or x[n]) is said to be a ____ signal (or sequence) if and only if 0 < P∞ < ∞, in which case, E∞ = ∞

A

A signal x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P∞ < ∞, in which case, E∞ = ∞

30
Q

A signal that does not satisfy 0 < E∞ < ∞ and 0 < P∞ < ∞ is not an ____ or a ____ signal

A

A signal that does not satisfy 0 < E∞ < ∞ and 0 < P∞ < ∞ is not an energy or a power signal

31
Q

Define a periodic signal

A

A periodic signal is a power signal if its energy content per period is finite

32
Q

If a > 1, the time scaling operation is a ‘____ __’ process (it looks like a ‘____’ process)

A

If a > 1, the time scaling operation is a ‘speed up’ process (it looks like a ‘compression’ process)

33
Q

If 0 < a < 1, the operation is a ‘____ __’ process (it looks like an ‘____’ process

A

If 0 < a < 1, the operation is a ‘slow down’ process (it looks like an ‘expansion’ process

34
Q

What would this sketch look like if it goes from x(t) to x(3t-5)?

A
35
Q

Define a system (long)

A

A SYSTEM is a combination, collection or set of things or physical components connected or related together in such a manner to form a whole unit in order to achieve a certain task (or set of tasks) or objective(s). A system achieves its objective or set of objectives by transforming input signal(s) into output signal(s)

36
Q

What is the input of a system?

A

An excitation or a stimulus that is applied to the system from an external source

37
Q

What is the output of a system?

A

The actual response of the system due to the application of an input signal

38
Q

What is a system (short)

A

A system can be considered a unit that transforms an input signal into an output signal using a well-defined rule or mathematical operation.

39
Q

y = T(x) where T is the ____ ____ that maps _ onto _

A

y = T(x) where T is the mathematical operation that maps x onto y

40
Q

What does SISO systems stand for and what does it look like?

A
41
Q

What does MIMO system stand for and what does it look like?

A
42
Q

A system whose output at any time depends on only the input at that same time is called a…?

A

…memoryless system; otherwise, the system is said to have memory.

43
Q

Example of a memoryless system?

A

A continuous-time signal

44
Q

A discrete-time signal turns into a…?

A

…Continuous signal

45
Q

What is a system with aid and memory?

A

A capacitor

46
Q

System with memory are…

Systems without memory are…

A

…Dynamic

…Static

47
Q

A causal system is one whose output 𝑦(𝑡) at time 𝑡 = 𝑡0 depends on only the inputs 𝑥(𝑡) for…?

A

…𝑡 ≤ 𝑡

48
Q

If a system is not casual, it is a…?

A

…non-casual system

49
Q

All memoryless systems are ____, but not all causal systems are ____

A

All memoryless systems are casual, but not all causal systems are memoryless

50
Q

The system is a linear system if it satisfies the following two conditions…?

A
  1. Superposition (or Additivity)
  2. Homogeneity (or Scaling)
51
Q

For the system to satisfy the superposition condition,

y1+y2 = ?

A

y1+y2 = T(x1+x2)

52
Q

For the system 𝑦 = 𝑇(𝑥) to satisfy the homogeneity condition, it behaviour must satisfy…

A

𝛼∙𝑦 = 𝑇(𝛼∙𝑥)

53
Q

A time-varying, or time-variant, system is one in which…?

A

…one or more of the parameters of the system vary as a function of time

54
Q

Example of time-varying system

A

y = tx

55
Q

A time-invariant system is one whose parameter(s)…?

A

…does not vary as a function of time

56
Q

A time-invariant system: Mathematically…

A

𝑦 = 𝑇(𝑥)

57
Q

A time-variant system: Mathematically

A

𝑦 = 𝑇(𝑥, 𝑡)

58
Q

For time-invariant systems, a time-shift in the input signal leads to…

Although this is not true for…?

A

…the same time-shift in the output signal

…time-variant systems in general

59
Q
  1. A linear system that is also time-invariant is called a…
  2. A nonlinear system that is time-invariant is called a…
  3. A linear system that is not time-invariant is called a…
  4. A nonlinear system that is not time-invariant is called a…
  5. Static or dynamic systems can be ____ or ____ and can be ____- ____ or ____-____
A
  1. Linear Time - Invariant System
  2. Nonlinear Time - Invariant System
  3. Linear Time - Varying System
  4. Nonlinear Time - Varying Systems.
  5. Static or dynamic systems can be linear or nonlinear and can be time-variant or time-invariant
60
Q

In terms of impulse response, a system is ____ if its impulse response approaches ____ as time approaches ____

A

In terms of impulse response, a system is stable if its impulse response approaches zero as time approaches infinity

61
Q

In terms of response to bounded input, a system is stable if…

A

…every bounded input produces a bounded output

62
Q

Control systems can be classified into two general categories

A
  1. Open-loop control systems
  2. Closed-loop control system
63
Q

A control system is a system whose function is to ____, ____, or ____ itself or another system

A

A control system is a system whose function is to command, direct, or regulate itself or another system

64
Q

Define open-loop control systems

A

A control system whose input signal is independent of the output signal

65
Q

Define closed-loop systems

This class of systems are often called…?

A

A control system whose input signal is somehow dependent of the output signal

feedback control systems

66
Q

In a feedback system, the actual output signal of the system being controlled is…

A

…fed back and compared with the desired output signal to determine the appropriate control action or signal

67
Q

If a system satisfies both superposition and homogeneity, it is ____, if neither then it’s ___-____

A

Linear, non-linear

68
Q

If a system is linear, then x1+x2 → ? + ?

A

x1+x2 → x12+x22

69
Q

What does LTI stand for?

A

Linear Time Invarient

70
Q

LTI systems;

∂(t) = ?

x(t) = ? = ? x ? = ?

A

∂(t) = h(t)

x(t) = y(t) = x(t) x h(t) = ∞∫-∞ x(J) h(t-J)dT

71
Q

When f(t) = 0 it’s ?

If f(t) = u it’s ?

A

Homogeneous

Non-homogeneous

72
Q

Define transient

A

point of y(t) that dies down as t → ∞

73
Q

Define steady state

A

point of y(t) that does not die down as t→∞

74
Q

The degree of the differential equation is the one to the highest…?

A

power

75
Q

In y(t) = est y(0)

s = ? for Laplace

s= ? for Fourei

A

Laplace s = ∂ + jw

Fourei s = jw

76
Q

BIBO stands for…?

A

Bounded Input Bounded Output

77
Q

x(t) = est

= est Y(gamma)

What is the eigen function and the eigen value?

A

est = eigen function

est Y(gamma) = eigen value

78
Q

If it’s constant it’s ____ ____

If it’s not constant it’s ____ ____

A

Linearly dependant

Linearly independant

79
Q

Method that can solve diff equations? (Involves D, r and y)

A

(D+r)y = 0

D = -r

y = CeDt

= Ce-rt

80
Q

What is the discriminant and if it’s > 0 what does that mean?

A

b2 - 4ac >0 means real and distinct

81
Q

Both terms have to be less than 0 to be…?

A

…Stable

82
Q

The unit step function __, or the ____ step function, is a function with a value of 0 for ____ and _ for 𝑡 > 0

A

The unit step function 𝑢(𝑡), or the Heaviside step function, is a function with a value of 0 for 𝑡 < 0 and 1 for 𝑡 > 0

83
Q

The unit step function is ____ and may be represented mathematically as;

A

The unit step function is ____ and may be represented mathematically as;

84
Q

Draw a Diagrammatic Representation of the unit step function

A
85
Q

What is the Time Shift Operation on 𝑢(𝑡) mathematically?

A
86
Q

Draw a diagrammatic representation of the time shift operation

A
87
Q

Define the unit impulse function

A

The unit impulse function 𝛿(𝑡), or the Dirac delta function, is a generalized function on the real number line with a value of 0 everywhere except at t = 0

88
Q

The unit impulse function has an ____ __ _ over the entire real line

A

The unit impulse function has an integral of 1 over the entire real line

89
Q

The unit impulse function may be represented mathematically as:

A
90
Q

Draw a Diagrammatic Representation of the unit impulse function

A
91
Q

What does the Time Shift Operation on 𝛿(𝑡) look like mathematically?

A
92
Q

It can be concluded that any continuous-time signal 𝑥(𝑡) can be expressed as,

x(t) = ∫…

A
93
Q

The relationship between unit step and unit impulse functions can be expressed as

A
94
Q

𝑒𝑗𝑞 =𝑒𝑗𝜔𝑡 = ?

A

𝑒𝑗𝑞 =𝑒𝑗𝜔𝑡=cos𝜔𝑡+𝑗sin𝜔𝑡

What is this known as?

95
Q

𝑒𝑗𝑞 =𝑒𝑗𝜔𝑡=cos𝜔𝑡+𝑗sin𝜔𝑡

What is this known as?

A

The complex exponential signal

96
Q

To summarize, a general complex exponential signal 𝑥 𝑡 = 𝑒𝑠𝑡 where s=𝜎+𝑗𝜔 can be written as:

A

𝑥 (𝑡) = 𝑒𝑠𝑡 =𝑒(𝜎+𝑗𝜔)𝑡 =𝑒𝜎𝑡(cos𝜔𝑡+𝑗sin𝜔𝑡)

97
Q

𝑥1(𝑡) =𝑒𝜎𝑡 (cos𝜔𝑡) = ?

𝑥2(𝑡) =𝑒𝜎𝑡 (sin𝜔𝑡) = ?

A

𝑥1(𝑡) =𝑒𝜎𝑡 (cos𝜔𝑡) =𝑅𝑒[𝑥(𝑡)]

𝑥2(𝑡) =𝑒𝜎𝑡 (sin𝜔𝑡) =𝐼𝑚[𝑥(𝑡)]

98
Q
  1. If 𝜎 < 0, it is a ____ sinusoidal function
  2. If 𝜎 > 0, it is an ____ sinusoidal function
  3. If 𝜎 = 0 ____
A
  1. If 𝜎 < 0, it is a decreasing sinusoidal function
  2. If 𝜎 > 0, it is an increasing sinusoidal function
  3. If 𝜎 = 0 constant
99
Q

Equation for the fundamental period

A
100
Q

Equation for the fundamental frequency

A
101
Q

Equation for the fudamental angular frequency

A
102
Q

Linear differential equation of the form:

is called a…?

A

homogeneous nth-order linear differential equation if f(t) = 0; otherwise it is non homogeneous

103
Q

dy/dt + ry = ? in terms of D

and so D = ?

And then this solution can be wirtten as;

y = ?

A

(D+r)y = 0

D = -r

y = CeDt

104
Q

If D <0, the response of the system naturally ____ as t…?

Therefore, the system is ____

A

If D <0, the response of the system naturally decays as t…tends to infinity.

Therefore, the system is stable

105
Q

If D<0, the response of the system naturally ____ unboundelly as t…?

Therefore, the system is ____

A

If D<0, the response of the system naturally increases unboundelly as t tends to infinity.

Therefore, the system is unstable

106
Q

If the (b2-4ac) > 0, the roots of the characteristic equation D1 and D2 are ____ and ____ (i.e. ____)

With the general solution being;

y = ? + ?

A

If the (b2-4ac) > 0, the roots of the characteristic equation D1 and D2 are real and distinct (i.e. unequal)

With the general solution being;

y = c1eD1t + c2eD2t

107
Q

If (b2-4ac) = 0 then the roots of the characteristic are ____ and ____

The general solution is

y = ? + ?

A

If (b2-4ac) = 0 then the roots of the characteristic are real and equal

The general solution is

y = c1eD1t + c2teD2t

108
Q

If (b2-4ac) < 0 then the roots of the characterstic are ____ ____

The general solution is

y = ? + ?

A

If (b2-4ac) < 0 then the roots of the characterstic are complex numbers

The general solution is

y = K1eD1t + K2e<span>D2t</span>

Using Eulier’s identity the last expression can be simplified to

y = eøt(c1 cosßt + c2 sinßt)

Where c1 = K1 + K2

c2 = j(K1 - K2)

109
Q

2nd-order systems have _ inital conditions, an nth-order system will have _ inital conditions

A

2nd-order systems have 2 inital conditions, an nth-order system will have n inital conditions

110
Q

y’’’ - y’ = 0

It’s characteristic equation is…?

A

D3 - D = 0

111
Q

Define steady state response

A

The point of the total response that does not approach zero as time approaches infinity

112
Q

Define transient response

A

The point of the total response that approaches zero as time approaches infinity

113
Q

y(t) = ya(t) + yb(t)

What is the forced response and what is the free response?

A

y(t) = ya(t) + yb(t)

ya is the free response

yb is the forced response

114
Q

yb(t) = ∫ w(t-J)

What is w(t-J) called?

A

The weighting function or the kernal of the differential equation

115
Q

wn = √(c/a)

What is this called?

A

The (undamped) natural frequency of the system

116
Q

b/(2qw) = b/(2√(ac))

What is this called?

A

The damping ratio of the system

117
Q

What is this called?

A

The damping coefficient

118
Q

What is this called?

A

(The inverse of the damping coefficient) is called the time-constant of the system

119
Q

This is provided in the data booklet, but what would the Laplace transform of a single-sided or unilateral look like?

Where would it be useful?

A

The change in ∞ to 0t

This would be particularly useful for finding the Laplace transform of functions that are discontinuous at t=0

120
Q

In Laplace transform, L is called the…?

A

Laplace transform operator

121
Q

What is Euler’s Identity?

A

eajt = cos(at) + j sin(at)

122
Q

Time scaling: If the Laplace transform of a function x(t) is x(s), then the Laplace transform of the function x(at) is given by…

A

L{x(at)] = 1/a . X(s/a)

123
Q

Division by t: If the Laplace transform of a function x(t) is X(s), then the Laplace transform of x(t)/t is given by…

A

L{x(t)/t}= ∞∫s X(u)du

124
Q

If you have an equation like this, what should you do?

A
  1. Factorise
  2. Partial fraction
125
Q

The first term of on the right of the equation is the ____ ____ and the second term is the ____ ____ of the system

A

The first term of on the right of the equation is the forced response and the second term is the free response of the system

126
Q

The transfer function of a LTI system is the point of the first term in the right side of the equation multiplying U(s)

A
127
Q

Define the transfer function of a LTI system

A

The ratio of the Laplace transform of the output variable Y(s) to the Laplace transform of the input variable U(s), with all initial conditions assumed to be zero

128
Q

The output of any continuous-time LTI system is the…?

A

…convolution of the input with the impulse response of the system

129
Q

What are the roots of the characteristic equation called?

A

Poles

130
Q

What are the roots of numerator polynomial of the transfer function called?

A

Zeros

131
Q

When is the system stable?

A

If all of the roots of the characteristic equation (that is the system poles) have negative real point/parts

132
Q

The transfer function, G(s) = ?

A

G(s) = Y(s) / U(s)

133
Q

What is the characteristic equation in this equation?

A

Characteristic equation: s2+4s+3

134
Q

The characteristic equation is the…?

A

…denominator

135
Q

What equation has very significant/ important mathematical implication?

A

𝑦(𝑡)=𝑇{𝑥(𝑡)} =𝑇{𝑒𝑠𝑡} =𝜆𝑒𝑠𝑡​

136
Q

In this equation 𝑦(𝑡) =𝜆𝑒𝑠𝑡

What is the eigenvalue and what is the eigenfunction?

A

𝜆 is the eigenvalue

𝑒𝑠𝑡 is the eigenfunction

137
Q

y(0) = ?

A

𝑦 (0) = 𝜆 = 𝐻(𝑠)

138
Q

Define frequency response

A

Frequency response is the steady-state response of a system to a sinusoidal input signal

139
Q

What is 𝜙 and ∠?

A

𝜙 = ∠𝐺(𝑗𝜔)

∠ = tan-1

140
Q

The magnitude and phase of the output signal differ from those of…

A

…the input signal

141
Q

The output signal differ from the input signal only in…

A

…amplitude and phase angle

142
Q

The amount of difference (in magnitude and phase) is a function of…

A

…the input frequency

143
Q

The output signal and the signals throughout the system is…

A

…in steady-state

144
Q

G (𝑗𝜔) is called the…

A

frequency response function, it is also called the sinusoidal transfer function

145
Q

The frequency response plot of a system is usually represented in two graphical plots…?

A

(i) the plot of 𝐺(𝑗𝜔) versus 𝜔
(ii) the plot of 𝜙(𝜔) versus ω

146
Q

What is so special about a bode magnitude and bode phase plot?

A

The frequency axis is a logarithmic scale, these plots can be graphed over a wide range of frequency

147
Q

Define Bandwidth

A

BANDWIDTH: The bandwidth of a system, 𝜔𝐵, is the frequency at which the magnitude of its frequency response function has declined by 3 dB from its low-frequency value. That is, the frequency 𝜔𝐵 at which

20𝑙𝑜𝑔 𝐺𝑗𝜔 =−3𝑑𝐵

148
Q

Define Gain Margin

A

GAIN MARGIN: The gain margin of a system is the reciprocal of the magnitude of its frequency response function at the frequency at which the phase angle reaches −180°. That is,

𝐺𝑎𝑖𝑛 𝑀𝑎𝑟𝑔𝑖𝑛 = 1/ Magnitude of (𝐺 𝑗𝜔-180°)

149
Q

Define Phase margin

A

PHASE MARGIN: The phase margin of a system is the difference between the phase angle at which the magnitude of its frequency response function is equal to unity and the −180° phase angle. That is,

𝑃h𝑎𝑠𝑒 𝑀𝑎𝑟𝑔𝑖𝑛 = 𝜙𝑃𝑀 = ∠𝐺 𝑗𝜔𝑐 − (−180°)

=∠𝐺𝑗𝜔𝑐 +180°

150
Q

What is 𝜔c called and what does it do????

A

𝜔𝑐 denotes the frequency at which the magnitude of the frequency response function equals 1 and is called the gain crossover frequency

151
Q

In a bode plot like this, where is the phase and gain margin?

A
152
Q

The transfer function is G(s) = Y(s)/U(s) but what is it in terms of words?

A

Transfer function = Laplace transform of the output / Laplace Transorm of the input

Provided that all inital condinos are zero

153
Q

Y = G.U What is G?

A

The impulse response

154
Q

Frquency response is the _______ response and the _______ ____ and the __________

A

Frquency response is the function response and the phase angle and the frequency

155
Q

Define Impulse response

A

Impulse Response: The impulse response h(𝑡) of the continuous-time LTI system 𝑇 is defined as the response of the system when the input is the unit impulse function 𝛿(𝑡). Mathematically, the impulse response can be represented as:

h(𝑡) =𝑇𝛿(𝑡)

156
Q

Mathematically, how can the impulse response be represented?

A

Mathematically, the impulse response can be represented as:

h(𝑡) =𝑇𝛿(𝑡)

157
Q

Any continuous-time signal 𝑥(𝑡) can be expressed as…

A
158
Q

What is the equation known as the convolution integral?

A
159
Q

y(t) = x(t)*h(t) = ?

Whereas

y(t) = h(t)*x(t)=?

A
160
Q

Define the step response

A

The step response 𝑠(𝑡) of the continuous-time LTI system 𝑇 is defined as the response of the system when the input is the unit step function 𝑢(𝑡).

Mathematically, the step response can be represented as:

𝑠(𝑡) =𝑇 {𝑢(𝑡)}

161
Q

How can the step response be represented mathematically?

A

Mathematically, the step response can be represented as:

𝑠(𝑡) =𝑇 {𝑢(𝑡)}