Signals and Systems Flashcards
What are samples?
Points of data
What are sampling intervals?
The distance between the sample points
Discrete-time signals can be defined in 2 ways…
- As a function
- As a list of values of a sequence
In this picture, the arrow indicates….?
And if the arrow is not shown, where is the ____ point?
The n=0 point
The first value in the sequence is the n=0 point
Define analog signals
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 +∞.
Define Digital Signals
Digital Signals: A digital signal is a discrete-time signal x[n] that can only take on only a finite number of distinct values.
What is a real signal?
Real Signals: A signal x(t) is said to be a real signal if its value is a real
number
What is a complex signal?
Complex Signals: A signal x(t) is said to be a complex signal if its value is a complex number
A general (continuous- or discrete-time) complex signal x(t) is a function of the form
where x1(t) and x2(t) are real signals and j = √−1
Define Deterministic Signals
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.
Define Random Signals
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)
Any signal x(t) or x[n] can be expressed as a sum of ____ ____, one of which is ____ and one of which is ____
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
The product of two even signals or of two odd signals is a…?
…even signal
The product of an even signal and an odd signal is an…?
…odd signal
Define Periodic continuous-time signals
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
A dc signal is a signal in which x(t) is ____; such a signal is not periodic since its fundamental period is ____
A dc signal is a signal in which x(t) is constant; such a signal is not periodic since its fundamental period is undefined
Define Aperiodic Continuous-time Signals
Aperiodic Continuous-time Signals: Any continuous-time signal which is not periodic is called a non-periodic signal or an aperiodic signal
Define Periodic Discrete-time Signals
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
Define Aperiodic Discrete-time Signals
Aperiodic Discrete-time Signals: Any sequence which is not periodic is called a non-periodic sequence or an aperiodic sequence
A sequence obtained by uniform sampling of a periodic continuous-time signal may…
…not be periodic
The sum of two continuous-time periodic signals may…
… not be periodic
The sum of two periodic sequences is…
…always periodic
For any continuous-time signal x(t), the total energy of the signal over the interval t1 ≤ t ≤ t2 is defined as
The average power of the signal x(t) over the interval t1 ≤ t ≤ t2 is defined as
|x(t)| denotes the ____ of x(t) and the signal x(t) could be a ____ or ____ signal.
|x(t)| denotes the magnitude of x(t) and the signal x(t) could be a real or complex signal.
For any discrete-time signal x[n], the total energy of the signal over the interval n1 ≤ n ≤ n2 is defined as
The average power of the signal x[n] over the interval n1 ≤ n ≤ n2 is defined as
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 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.
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 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∞ = ∞
A signal that does not satisfy 0 < E∞ < ∞ and 0 < P∞ < ∞ is not an ____ or a ____ signal
A signal that does not satisfy 0 < E∞ < ∞ and 0 < P∞ < ∞ is not an energy or a power signal
Define a periodic signal
A periodic signal is a power signal if its energy content per period is finite
If a > 1, the time scaling operation is a ‘____ __’ process (it looks like a ‘____’ process)
If a > 1, the time scaling operation is a ‘speed up’ process (it looks like a ‘compression’ process)
If 0 < a < 1, the operation is a ‘____ __’ process (it looks like an ‘____’ process
If 0 < a < 1, the operation is a ‘slow down’ process (it looks like an ‘expansion’ process
What would this sketch look like if it goes from x(t) to x(3t-5)?
Define a system (long)
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)
What is the input of a system?
An excitation or a stimulus that is applied to the system from an external source
What is the output of a system?
The actual response of the system due to the application of an input signal
What is a system (short)
A system can be considered a unit that transforms an input signal into an output signal using a well-defined rule or mathematical operation.
y = T(x) where T is the ____ ____ that maps _ onto _
y = T(x) where T is the mathematical operation that maps x onto y
What does SISO systems stand for and what does it look like?
What does MIMO system stand for and what does it look like?
A system whose output at any time depends on only the input at that same time is called a…?
…memoryless system; otherwise, the system is said to have memory.
Example of a memoryless system?
A continuous-time signal
A discrete-time signal turns into a…?
…Continuous signal
What is a system with aid and memory?
A capacitor
System with memory are…
Systems without memory are…
…Dynamic
…Static
A causal system is one whose output 𝑦(𝑡) at time 𝑡 = 𝑡0 depends on only the inputs 𝑥(𝑡) for…?
…𝑡 ≤ 𝑡
If a system is not casual, it is a…?
…non-casual system
All memoryless systems are ____, but not all causal systems are ____
All memoryless systems are casual, but not all causal systems are memoryless
The system is a linear system if it satisfies the following two conditions…?
- Superposition (or Additivity)
- Homogeneity (or Scaling)
For the system to satisfy the superposition condition,
y1+y2 = ?
y1+y2 = T(x1+x2)
For the system 𝑦 = 𝑇(𝑥) to satisfy the homogeneity condition, it behaviour must satisfy…
𝛼∙𝑦 = 𝑇(𝛼∙𝑥)
A time-varying, or time-variant, system is one in which…?
…one or more of the parameters of the system vary as a function of time
Example of time-varying system
y = tx
A time-invariant system is one whose parameter(s)…?
…does not vary as a function of time
A time-invariant system: Mathematically…
𝑦 = 𝑇(𝑥)
A time-variant system: Mathematically
𝑦 = 𝑇(𝑥, 𝑡)
For time-invariant systems, a time-shift in the input signal leads to…
Although this is not true for…?
…the same time-shift in the output signal
…time-variant systems in general
- A linear system that is also time-invariant is called a…
- A nonlinear system that is time-invariant is called a…
- A linear system that is not time-invariant is called a…
- A nonlinear system that is not time-invariant is called a…
- Static or dynamic systems can be ____ or ____ and can be ____- ____ or ____-____
- Linear Time - Invariant System
- Nonlinear Time - Invariant System
- Linear Time - Varying System
- Nonlinear Time - Varying Systems.
- Static or dynamic systems can be linear or nonlinear and can be time-variant or time-invariant
In terms of impulse response, a system is ____ if its impulse response approaches ____ as time approaches ____
In terms of impulse response, a system is stable if its impulse response approaches zero as time approaches infinity
In terms of response to bounded input, a system is stable if…
…every bounded input produces a bounded output
Control systems can be classified into two general categories
- Open-loop control systems
- Closed-loop control system
A control system is a system whose function is to ____, ____, or ____ itself or another system
A control system is a system whose function is to command, direct, or regulate itself or another system
Define open-loop control systems
A control system whose input signal is independent of the output signal
Define closed-loop systems
This class of systems are often called…?
A control system whose input signal is somehow dependent of the output signal
…feedback control systems
In a feedback system, the actual output signal of the system being controlled is…
…fed back and compared with the desired output signal to determine the appropriate control action or signal
If a system satisfies both superposition and homogeneity, it is ____, if neither then it’s ___-____
Linear, non-linear
If a system is linear, then x1+x2 → ? + ?
x1+x2 → x12+x22
What does LTI stand for?
Linear Time Invarient
LTI systems;
∂(t) = ?
x(t) = ? = ? x ? = ?
∂(t) = h(t)
x(t) = y(t) = x(t) x h(t) = ∞∫-∞ x(J) h(t-J)dT
When f(t) = 0 it’s ?
If f(t) = u it’s ?
Homogeneous
Non-homogeneous
Define transient
point of y(t) that dies down as t → ∞
Define steady state
point of y(t) that does not die down as t→∞
The degree of the differential equation is the one to the highest…?
power
In y(t) = est y(0)
s = ? for Laplace
s= ? for Fourei
Laplace s = ∂ + jw
Fourei s = jw
BIBO stands for…?
Bounded Input Bounded Output
x(t) = est
= est Y(gamma)
What is the eigen function and the eigen value?
est = eigen function
est Y(gamma) = eigen value
If it’s constant it’s ____ ____
If it’s not constant it’s ____ ____
Linearly dependant
Linearly independant
Method that can solve diff equations? (Involves D, r and y)
(D+r)y = 0
D = -r
y = CeDt
= Ce-rt
What is the discriminant and if it’s > 0 what does that mean?
b2 - 4ac >0 means real and distinct
Both terms have to be less than 0 to be…?
…Stable
The unit step function __, or the ____ step function, is a function with a value of 0 for ____ and _ for 𝑡 > 0
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
The unit step function is ____ and may be represented mathematically as;
The unit step function is ____ and may be represented mathematically as;
Draw a Diagrammatic Representation of the unit step function
What is the Time Shift Operation on 𝑢(𝑡) mathematically?
Draw a diagrammatic representation of the time shift operation
Define the unit impulse function
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
The unit impulse function has an ____ __ _ over the entire real line
The unit impulse function has an integral of 1 over the entire real line
The unit impulse function may be represented mathematically as:
Draw a Diagrammatic Representation of the unit impulse function
What does the Time Shift Operation on 𝛿(𝑡) look like mathematically?
It can be concluded that any continuous-time signal 𝑥(𝑡) can be expressed as,
x(t) = ∫…
The relationship between unit step and unit impulse functions can be expressed as
𝑒𝑗𝑞 =𝑒𝑗𝜔𝑡 = ?
𝑒𝑗𝑞 =𝑒𝑗𝜔𝑡=cos𝜔𝑡+𝑗sin𝜔𝑡
What is this known as?
𝑒𝑗𝑞 =𝑒𝑗𝜔𝑡=cos𝜔𝑡+𝑗sin𝜔𝑡
What is this known as?
The complex exponential signal
To summarize, a general complex exponential signal 𝑥 𝑡 = 𝑒𝑠𝑡 where s=𝜎+𝑗𝜔 can be written as:
𝑥 (𝑡) = 𝑒𝑠𝑡 =𝑒(𝜎+𝑗𝜔)𝑡 =𝑒𝜎𝑡(cos𝜔𝑡+𝑗sin𝜔𝑡)
𝑥1(𝑡) =𝑒𝜎𝑡 (cos𝜔𝑡) = ?
𝑥2(𝑡) =𝑒𝜎𝑡 (sin𝜔𝑡) = ?
𝑥1(𝑡) =𝑒𝜎𝑡 (cos𝜔𝑡) =𝑅𝑒[𝑥(𝑡)]
𝑥2(𝑡) =𝑒𝜎𝑡 (sin𝜔𝑡) =𝐼𝑚[𝑥(𝑡)]
- If 𝜎 < 0, it is a ____ sinusoidal function
- If 𝜎 > 0, it is an ____ sinusoidal function
- If 𝜎 = 0 ____
- If 𝜎 < 0, it is a decreasing sinusoidal function
- If 𝜎 > 0, it is an increasing sinusoidal function
- If 𝜎 = 0 constant
Equation for the fundamental period
Equation for the fundamental frequency
Equation for the fudamental angular frequency
Linear differential equation of the form:
is called a…?
…homogeneous nth-order linear differential equation if f(t) = 0; otherwise it is non homogeneous
dy/dt + ry = ? in terms of D
and so D = ?
And then this solution can be wirtten as;
y = ?
(D+r)y = 0
D = -r
y = CeDt
If D <0, the response of the system naturally ____ as t…?
Therefore, the system is ____
If D <0, the response of the system naturally decays as t…tends to infinity.
Therefore, the system is stable
If D<0, the response of the system naturally ____ unboundelly as t…?
Therefore, the system is ____
If D<0, the response of the system naturally increases unboundelly as t tends to infinity.
Therefore, the system is unstable
If the (b2-4ac) > 0, the roots of the characteristic equation D1 and D2 are ____ and ____ (i.e. ____)
With the general solution being;
y = ? + ?
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
If (b2-4ac) = 0 then the roots of the characteristic are ____ and ____
The general solution is
y = ? + ?
If (b2-4ac) = 0 then the roots of the characteristic are real and equal
The general solution is
y = c1eD1t + c2teD2t
If (b2-4ac) < 0 then the roots of the characterstic are ____ ____
The general solution is
y = ? + ?
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)
2nd-order systems have _ inital conditions, an nth-order system will have _ inital conditions
2nd-order systems have 2 inital conditions, an nth-order system will have n inital conditions
y’’’ - y’ = 0
It’s characteristic equation is…?
D3 - D = 0
Define steady state response
The point of the total response that does not approach zero as time approaches infinity
Define transient response
The point of the total response that approaches zero as time approaches infinity
y(t) = ya(t) + yb(t)
What is the forced response and what is the free response?
y(t) = ya(t) + yb(t)
ya is the free response
yb is the forced response
yb(t) = ∫ w(t-J)
What is w(t-J) called?
The weighting function or the kernal of the differential equation
wn = √(c/a)
What is this called?
The (undamped) natural frequency of the system
b/(2qw) = b/(2√(ac))
What is this called?
The damping ratio of the system
What is this called?
The damping coefficient
What is this called?
(The inverse of the damping coefficient) is called the time-constant of the system
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?
The change in ∞ to 0t
This would be particularly useful for finding the Laplace transform of functions that are discontinuous at t=0
In Laplace transform, L is called the…?
Laplace transform operator
What is Euler’s Identity?
eajt = cos(at) + j sin(at)
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…
L{x(at)] = 1/a . X(s/a)
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…
L{x(t)/t}= ∞∫s X(u)du
If you have an equation like this, what should you do?
- Factorise
- Partial fraction
The first term of on the right of the equation is the ____ ____ and the second term is the ____ ____ of the system
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
The transfer function of a LTI system is the point of the first term in the right side of the equation multiplying U(s)
Define the transfer function of a LTI system
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
The output of any continuous-time LTI system is the…?
…convolution of the input with the impulse response of the system
What are the roots of the characteristic equation called?
Poles
What are the roots of numerator polynomial of the transfer function called?
Zeros
When is the system stable?
If all of the roots of the characteristic equation (that is the system poles) have negative real point/parts
The transfer function, G(s) = ?
G(s) = Y(s) / U(s)
What is the characteristic equation in this equation?
Characteristic equation: s2+4s+3
The characteristic equation is the…?
…denominator
What equation has very significant/ important mathematical implication?
𝑦(𝑡)=𝑇{𝑥(𝑡)} =𝑇{𝑒𝑠𝑡} =𝜆𝑒𝑠𝑡
In this equation 𝑦(𝑡) =𝜆𝑒𝑠𝑡
What is the eigenvalue and what is the eigenfunction?
𝜆 is the eigenvalue
𝑒𝑠𝑡 is the eigenfunction
y(0) = ?
𝑦 (0) = 𝜆 = 𝐻(𝑠)
Define frequency response
Frequency response is the steady-state response of a system to a sinusoidal input signal
What is 𝜙 and ∠?
𝜙 = ∠𝐺(𝑗𝜔)
∠ = tan-1
The magnitude and phase of the output signal differ from those of…
…the input signal
The output signal differ from the input signal only in…
…amplitude and phase angle
The amount of difference (in magnitude and phase) is a function of…
…the input frequency
The output signal and the signals throughout the system is…
…in steady-state
G (𝑗𝜔) is called the…
…frequency response function, it is also called the sinusoidal transfer function
The frequency response plot of a system is usually represented in two graphical plots…?
(i) the plot of 𝐺(𝑗𝜔) versus 𝜔
(ii) the plot of 𝜙(𝜔) versus ω
What is so special about a bode magnitude and bode phase plot?
The frequency axis is a logarithmic scale, these plots can be graphed over a wide range of frequency
Define Bandwidth
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
Define Gain Margin
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°)
Define Phase margin
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°
What is 𝜔c called and what does it do????
𝜔𝑐 denotes the frequency at which the magnitude of the frequency response function equals 1 and is called the gain crossover frequency
In a bode plot like this, where is the phase and gain margin?
The transfer function is G(s) = Y(s)/U(s) but what is it in terms of words?
Transfer function = Laplace transform of the output / Laplace Transorm of the input
Provided that all inital condinos are zero
Y = G.U What is G?
The impulse response
Frquency response is the _______ response and the _______ ____ and the __________
Frquency response is the function response and the phase angle and the frequency
Define Impulse response
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)
Mathematically, how can the impulse response be represented?
Mathematically, the impulse response can be represented as:
h(𝑡) =𝑇𝛿(𝑡)
Any continuous-time signal 𝑥(𝑡) can be expressed as…
What is the equation known as the convolution integral?
y(t) = x(t)*h(t) = ?
Whereas
y(t) = h(t)*x(t)=?
Define the step response
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)}
How can the step response be represented mathematically?
Mathematically, the step response can be represented as:
𝑠(𝑡) =𝑇 {𝑢(𝑡)}
