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 - β and b may be +β.

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 = ββ1

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

**and one of which is**

__even__

__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 β€ t β€ t2 is defined as

The average power of the signal x(t) over the interval t1 β€ t β€ 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

**or**

__real__**signal.**

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

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

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

A signal x(t) (or x[n]) is said to be an __energy__** signal** (or sequence) if and only if 0 < Eβ < β, in which case, Pβ = 0.

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

A signal x(t) (or x[n]) is said to be a ** power** signal (or sequence) if and only if 0 < Pβ < β, in which case, Eβ = β

A signal that does not satisfy 0 < Eβ < β and 0 < Pβ < β is not an ____ or a ____ signal

A signal that does not satisfy 0 < Eβ < β and 0 < Pβ < β is not an ** energy** or a

**signal**

__power__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 β____ __β process (it looks like a β____β process)

If a > 1, the time scaling operation is a β__speed__** up**β process (it looks like a β

**β process)**

__compression__If 0 < a < 1, the operation is a β____ __β process (it looks like an β____β process

If 0 < a < 1, the operation is a β** slow down**β process (it looks like an β

**β process**

__expansion__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

**onto**

__x__

__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 π¦(π‘) at time π‘ = π‘_{0} depends on only the inputs π₯(π‘) for…?

…π‘ β€ π‘

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,

y_{1}+y_{2 }= ?

y_{1}+y_{2} = T(x_{1}+x_{2})

For the system π¦ = π(π₯) to satisfy the homogeneity condition, it behaviour must satisfy…

πΌβπ¦ = π(πΌβπ₯)

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…

π¦ = π(π₯)

A time-variant system: Mathematically

π¦ = π(π₯, π‘)

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
or__linear__and can be__nonlinear__-__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

**as time approaches**

__zero__

__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**,

**, or**

__direct__**itself or another system**

__regulate__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 x_{1}+x_{2} β ? + ?

x_{1}+x_{2} β x_{1}^{2}+x_{2}^{2}

What does LTI stand for?

**L**inear **T**ime **I**nvarient

LTI systems;

β(t) = ?

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

β(t) = h(t)

x(t) = y(t) = x(t) x h(t) = ββ«-β 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 β β

Define **steady state**

point of y(t) that does ** not** die down as tββ

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

power

In y(t) = e^{st }y(0)

s = ? for **Laplace**

s= ? for **Fourei**

**Laplace** s = β + j*w*

**Fourei** s = j*w*

**BIBO** stands for…?

**B**ounded **I**nput **B**ounded **O**utput

x(t) = e^{st}

= e^{st} Y(gamma)

What is the **eigen function** and the **eigen value?**

e^{st }= **eigen function**

e^{st} 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 = Ce^{Dt}

= Ce^{-rt}

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

b^{2} - 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 π‘ > 0

The unit step function ** π’(π‘)**, or the

**step function, is a function with a value of 0 for**

__Heaviside__**and**

__π‘ < 0__**for π‘ > 0**

__1__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 π’(π‘) mathematically?

Draw a diagrammatic representation of the **time shift operation**

Define the **unit impulse function**

The **unit impulse function** πΏ(π‘), 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 πΏ(π‘) look like mathematically?

It can be concluded that any continuous-time signal π₯(π‘) can be expressed as,

x(t) = β«…

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

π^{ππ} =π^{πππ‘} = ?

π^{ππ} =π^{πππ‘}=cosππ‘+πsinππ‘

What is this known as?

πππ =ππππ‘=cosππ‘+πsinππ‘

What is this known as?

The complex exponential signal

To summarize, a general complex exponential signal π₯ π‘ = π^{π π‘} where s=π+ππ can be written as:

π₯ (π‘) = π^{π π‘} =π^{(π+ππ)π‘} =π^{ππ‘}(cosππ‘+πsinππ‘)

π₯_{1}(π‘) =π^{ππ‘} (cosππ‘) = ?

π₯_{2}(π‘) =π^{ππ‘} (sinππ‘) = ?

π₯_{1}(π‘) =π^{ππ‘} (cosππ‘) =π
π[π₯(π‘)]

π₯_{2}(π‘) =π^{ππ‘} (sinππ‘) =πΌπ[π₯(π‘)]

- If π < 0, it is a ____ sinusoidal function
- If π > 0, it is an ____ sinusoidal function
- If π = 0 ____

- If π < 0, it is a
sinusoidal function__decreasing__ - If π > 0, it is an
sinusoidal function__increasing__ - If π = 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 = Ce^{Dt}

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 (b^{2}-4ac) > 0, the roots of the characteristic equation D_{1} and D_{2} are ____ and ____ (i.e. ____)

With the general solution being;

y = ? + ?

If the (b^{2}-4ac) > 0, the roots of the characteristic equation D_{1} and D_{2} are ** real** and

**(i.e.**

__distinct__**)**

__unequal__With the general solution being;

y = **c _{1}e^{D1t} + c_{2}e^{D2t}**

If (b^{2}-4ac) = 0 then the roots of the characteristic are ____ and ____

The general solution is

y = ? + ?

If (b^{2}-4ac) = 0 then the roots of the characteristic are ** real** and

__equal__The general solution is

y = **c _{1}e^{D1t }+ c_{2}**

__t__

**e**

^{D2t}If (b^{2}-4ac) < 0 then the roots of the characterstic are ____ ____

The general solution is

y = ? + ?

If (b^{2}-4ac) < 0 then the roots of the characterstic are __complex numbers__

The general solution is

y = K_{1}e^{D1t }+ K_{2}e^{<span>D2t</span>}

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

y = e^{ΓΈt}(c_{1 }cosΓt + c_{2} sinΓt)

Where c_{1} = K_{1} + K_{2}

c_{2} = j(K_{1} - K_{2)}

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

**inital conditions**

__n__y’’’ - y’ = 0

It’s characteristic equation is…?

D^{3} - 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) = y_{a}(t) + y_{b}(t)

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

y(t) = y_{a}(t) + y_{b}(t)

y_{a} is the **free response**

y_{b} is the **forced response**

y_{b}(t) = β« w(t-J)

What is w(t-J) called?

The **weighting function** or the **kernal** of the differential equation

w_{n} = β(c/a)

What is this called?

The (undamped) **natural frequency** of the system

b/(2qw) = b/(2β(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 β 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?

e^{ajt} = 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}= ββ«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

**of the system**

__free response__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**: s^{2}+4s+3

The characteristic equation is the…?

…denominator

What equation has very significant/ important mathematical implication?

π¦(π‘)=π{π₯(π‘)} =π{π^{π π‘}} =π^{ππ π‘β}

In this equation π¦(π‘) =ππ^{π π‘}β

What is the **eigenvalue** and what is the **eigenfunction**?

π is the **eigenvalue**

π^{π π‘} is the **eigenfunction**

y(0) = ?

π¦ (0) = π = π»(π )

Define **frequency response**

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

What is π and β ?

π = β πΊ(ππ)

β = 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 (ππ) 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 πΊ(ππ) versus π

(ii) the plot of π(π) versus Ο

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, ππ΅, 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 ππ΅ at which

20πππ πΊππ =β3ππ΅

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 β180Β°. That is,

πΊπππ ππππππ = 1/ Magnitude of (πΊ ππ_{-180Β°})

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 β180Β° phase angle. That is,

πhππ π ππππππ = πππ = β πΊ πππ β (β180Β°)

=β πΊπππ +180Β°

What is π_{c} called and what does it do????

π_{π }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__**and the**

__angle__

__frequency__Define **Impulse response**

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

h(π‘) =ππΏ(π‘)

Mathematically, how can the **impulse response** be represented?

Mathematically, the **impulse response** can be represented as:

h(π‘) =ππΏ(π‘)

Any continuous-time signal π₯(π‘) 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 π (π‘)** of the continuous-time LTI system π is defined as the **response of the system when the input is the unit step function π’(π‘).**

Mathematically, the step response can be represented as:

π (π‘) =π {π’(π‘)}

How can **the step response** be represented mathematically?

Mathematically, **the step response** can be represented as:

π (π‘) =π {π’(π‘)}