continuous time signals basic signals – singularity functions transformations of continuous time...

Continuous Time Signals

•Basic Signals – Singularity Functions•Transformations of Continuous Time Signals•Signal Characteristics•Common Signals

April 19, 2023 Veton Këpuska 2

Continuous-Time Signals

Assumptions: Functions, x(t), are of the one

independent variable that typically represents time, t.

Time t can assume all real values: -∞ < t < ∞,

Function x(t) is typically a real function.

Singularity Functions


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

u(t)

time [sec]

Unit Sample Signal

Unit Step Function

Unit step function definition:

0,0

0,1

t

ttu


Unit Step Function Properties Scaling:

Unit step function can be scaled by a real constant K (positive or negative)

Multiplication: Multiplication of any

function, say x(t), by a unit step function u(t) is equivalent to defining the signal x(t) for t≥0.


tKutf

0, ttxtutx

Unit Ramp Function

Unit Ramp Function is defined as:


0,0

0,

t

tttr

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

r(t)

time [sec]

Unit Sample Signal

Unit Ramp Function Properties Scaling:

Unit step function can be scaled by a real constant K (positive or negative)

Integral of the unit step function is equal to the ramp function:

Derivative of the unit ramp function is the unit step function.


tKrtf

Slope of the straight line

t

dutr

dt

tdrtu

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

(t)

time [sec]

Unit Sample Signal

Unit Impulse Function

Unit Impulse Function, also know as Dirac delta function, is defined as:


&01

0,0

0,

d

t

tt

Unit Impulse Function Properties Scaling:

Unit impulse function can be scaled by a real constant K (positive or negative)

Delta function can be approximated by a pulse centered at the origin


)(lim tdtA

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

(t)

time [sec]

Unit Sample Signal

A2

1

A2

1

A

Unit Impulse Function Properties Unit impulse function is

related to unit step function:

Conversely:


dt

tdut

0&

ttdtut

Proof:1. t<0

2. t>0

0 since ,00

ttdtut

0,1 since ,01

dtdtut

Time Transformation of Signals



Time Reversal:

)()(

&

)()(

,

)()(

00

00

0

txty

txty

tt

txty

1

2

0-1-2

-1

1 2 t

1

2

0-1-2

-1

1 2 t

y(t)=x(-t)

x(t)


Time Scaling

)()(

&

)()(

,

0

&

)()(

00

000

txa

ty

atxty

a

tt

atxty1

2

0-1-2

-1

1 2 t

1

2

0-1-2

-1

1 2 t

y(t)=x(2t)

x(t)

1

2

0-1-2

-1

1 2 t

y(t)=x(0.5t)

3 4

|a| > 1 – Speed Up

|a| < 1 – Slow Down


Time Shifting

)0()(

:Note

constant a is ,

)()(

: signal aGiven

0

0

0

xty

t

ttxty

x(t)

1

2

0-1-2

-1

1 2 t

x(t)

1

2

0-1-2

-1

1 2 t

x(t-2)

3 4

1

2

0-1-2

-1

1 2 t

x(t+1)


Example 1

233cos

23cos

)()(

23cos)(

0

0

0

0

0

ttee

tte

ttxty

tetx

tt

tt

t


Independent Variable Transformations

a

b

a

tytx

batxtytt

a

b

atbat

babatxty

00

00

0 )(

)()(

, ),()(


Example 2

2

3

232

)32()(

tt

txty


Example 3

2

0-1-2

-1

1 2

x()

1

024 -2 t=2-2

2

0-2-4

-1

2 4

y(t)=x(1-t/2)

1

2

1

2

11)(

)0()212()1()(1

2212

21)(

0

0

0 xxty

yyxtxt

tt

txty


Independent Variable Transformations

1. Replace t with , on the original plot of the signal.

2. Given the time transformation:

Solve for

3. Draw the transformed t-axis directly below the -axis.

4. Plot y(t) on the t-axis.

bat

a

b

at


Amplitude Transformations

constants real,

)()(

BA

BtAxty


Example 4 Consider signal in the

figure. Suppose the signal is applied to an amplifier with the gain of 3 and introduces a bias (a DC value) of -1. That is:

1)(3)( txty

2

0-1-2

-1

1 2 t

x(t)

1

3 +-1

x(t) 3x(t) 3x(t)-1

2

0-1-2

-1

1 2 t

3x(t)-1

1


Example 5

2

0-1-2

-1

1 2

3x()-1

1

246 0 -2 t

1 t/2 => t=-22

2

-1

3x(1 t/2 )-1

1

26 0 -2 t4

222

1

12

13)(

tt

txty


Transformations of Signals

Name y(t)

Time reversal x(-t)

Time scaling x(at)

Time shifting x(t-t0)

Amplitude reversal -x(t)

Amplitude scaling Ax(t)

Amplitude shifting x(t)+B

Signal Characteristics


Even and Odd Signals

Even Functions xe(t)=xe(-t)

Odd Functions xo(t)=-xo(-t)

2A

0-1-2

-A

1 2 t

xe(t)

A

2A

0-1-2

-A

1 2 t

xo(t)

A


Even and Odd Signals

Any signal can be expressed as the sum of even part and on odd part:

)()(2

1)()()(

2

1)(

)()()(

)()()(

)()()(

txtxtxtxtxtx

txtxtx

txtxtx

txtxtx

oe

oe

oe

oe


Average Value Average Value of the signal x(t) over a period of time [-T, T] is

defined as:

T

T

eT

x

T

T

T

T

oeT

x

T

T

oeT

x

oe

T

TT

x

dttxT

A

dttxdttxT

A

dttxtxT

A

txtxtx

dttxT

A

)(2

1lim

})()({2

1lim

)]()([2

1lim

)]()([)(

)(2

1lim

The average value of a signal is contained in its even function (why?).


Properties of even and odd functions

1. The sum of two even functions is even.2. The sum of two odd functions is odd.3. The sum of an even function and an odd

function is neither even nor odd.4. The product of two even functions is even.5. The product of two odd functions is even.6. The product of an even function and an

odd function is odd.

Periodic Signals



Periodic Signals

tTTtxtx & 0),()(

Continuous-time signal x(t) is periodic if:

T is period of the signal.

A signal that is not periodic is said to be aperiodic.


Periodic Signals

If constant T is a period of of a function x(t) than nT is also its period, where T>0 and n is any positive integer.

)()(

)2()(

nTtxtx

TtxTtx

The minimal value of the constant T >0 is a that satisfies the definition x(t)= x(t+ T) is called a fundamental period of a signal and it is denoted by T0.


Examples of Periodic Signals

0 1 2 3 4 5 6 7 8 9 10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y(t)=

cos(

wt)

Time [s]

)cos()( tty

Sinusoidal Signal Properties

ttAtx cos


• A – Amplitude of the signal

• - is the frequency in rad/sec

• - is phase in radians

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Periodic Signal

cos(t+ )

Angle [rad]

cos(-t+0.00)

cos(-t+0.52)

Sinusoidal Function Properties

Note:


sincoscossinsin

sinsincoscoscos

][ ,

rad

sin2sin

cos2cos

][

rad

Periodicity of Sinusoidal Signal


fT

Ttx

TtA

tA

tA

tAtx

22

cos

2cos

2cos

cos


Example: Sawtooth Periodic Waveform

-3 -2 -1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

y(t)


Period and Frequency

Fundamental Period T0 – Measured in seconds.

Fundamental Frequency f0 – Measured in Hz – number of periods (cycles) per second or equivalently in radian frequency rad/s.

s

rad

Tf

HzT

f

000

00

122

][1


Testing for Periodicity

1.

)()(

2for )sin()sin(

)(

?)(

)sin(

)sin(

)sin(

txeTtx

TtTt

eTtx

etx

t

Tt

t


Testing for Periodicity

2.

)(

)(

)()(

2for )sin()sin(

)()(

?)(

)sin(

)sin()sin(

)sin(

)sin(

)sin(

tx

Tetx

Tete

eTtTtx

TtTt

eTtTtx

tetx

t

tt

t

Tt

t


Composite Signals Each signal can be decomposed into a sum of series of

pure periodic signals (Taylor Series Expansion/Fourier Series Expansion)

The sum of continuous-time periodic signals is periodic if and only if the ratios of the periods of the individual signals are ratios of integers.

Composite Signals If a sum of N periodic signals is periodic, the

fundamental period can be found as follows:

1. Convert each period ratio, To1/Toi≤ i ≤ N , to a ratio of integers, where To1 is the period of the first signal considered and Toi is the period of one of the other N-1 signals. If one or more of these ratios is not rational, the sum of signals is not periodic.

2. Eliminate common factors from the numerator and denominator of each ratio of integers.

3. The fundamental period of the sum of signals is To=koTo1 ; kois the least common multiple of the denominators of the individual ratios of integers.


Composite Signals

If x1(t) is periodic with period T1, and

x2(t) is periodic with period T2, Then

x1(t)+x2(t) is periodic with period equal to the least common multiple (T1, T2) if the ratio of the two periods is a rational number, where k1 and k2 are integers:


22111

2

2

1 TkTkk

k

T

T

Composite Signals

Let T’= k1T1 = k2T2

y(t) = x1(t)+x2(t)

Then y(t+T’) = x1(t+T’)+x2(t+T’)= x1(t+ k1T1)+x2(t+ k2T2)=

x1(t)+x2(t) = y(t)


Example 2.7 a)

Assume that v(t) is a sum of periodic signals given below. Determine if the signal is periodic and what its periodicity?


)6

7cos()(

)2cos()(

)5.3cos()(

)()()()(

3

2

1

321

ttx

ttx

ttx

txtxtxtv

0 10 20 30 40 50 60-4

-2

0

2

4

v(t)

time

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

x1(t)

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

x2(t)

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

x3(t)

Solution

Determine whether v(t) constituent signals have periods with ratios that are integers (rational numbers):


67

22

2

22

5.3

22

303

202

101

T

T

T

21

7

5.367

67

25.3

2

7

4

5.3

2

22

5.32

03

01

02

01

T

T

T

T

Solution

Ratios of periods are rational numbers thus the composite signal v(t) is periodic.

Elimination of common factors: T01/T02 = 4/7

T01/T03 = 7/21=1/3

Least common multiple of the denominator ratios: n1= 3*7=21

Fundamental period of v(t) is: T0= n1 T01 = 21*2/3.5=12


Example 2.7 b)

Assume that to v(t) is added a periodic signal x4(t) given below. Determine if the signal is periodic and what its periodicity?


)5cos(3)(

)6

7cos()(

)2cos()(

)5.3cos()(

)()()()()(

4

3

2

1

4321

ttx

ttx

ttx

ttx

txtxtxtxtv

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

v(t)

time

Solution

Since ratio of the x1(t) and x4(t) periods is not a rational number the v(t) is not periodic.


5

22

404 T

7

10

5.3

5

52

5.32

04

01 T

T

Homework #1:


1. For x(t)=Acos(t+) find What are its maximum and minimum values?

What are corresponding times when they occur?

What is the value of the function when it crosses vertical y- axis (ordinate) and horizontal x-axis (abscissa)?

At what time instances the function becomes zero?

Indicate all the above point values in a plot.

Homework #1 Use the following MATLAB

script to test your calculations and plot the function:

function pfunc(A, f, th1, th2)%% Periodic Sine Function% A - gain (1)% f - frequency (1)% th1 - phase of the first signal (0)% th2 - phase of the second signal

(pi/6)% w = 2.*pi.*f; % radial frequencyfs = 0.0001*f;mint = -pi*f/2;maxt = pi*f/2;miny = -1.2*A;maxy = 1.2*A;

t = mint:fs:maxt; % time axisy = A*cos(w*t+th1); plot(t, y, 'b', 'LineWidth',2);

title('Periodic Signal'); grid on; hold; axis([mint maxt miny

maxy]);y = A*cos(w*t+th2);plot(t, y, 'r', 'LineWidth',2);ylabel('cos(\omegat+\theta)');

xlabel('Angle x\pi [rad]'); grid on; hold; axis([mint maxt miny

maxy]);x=-0.8; text(x,A*cos(w*x+th1),sprintf('%s+

%3.2f)','\leftarrow cos(-\pit',th1),...

'HorizontalAlignment','left',... 'BackgroundColor','b');x=-0.6; text(x,A*cos(w*x+th2),sprintf('%s+

%3.2f)','\leftarrow cos(-\pit',th2),...

'HorizontalAlignment','left',... 'BackgroundColor','r');



Homework #1

Problems 2.1, 2.2, 2.9, 2.10, 2.13, 2.14, 2.20.


Example

2

0-1-2

-1

1 2 t

x(t)

1

2

0-1-2

-1

1 2 t

x(-t)

1

2

0-1-2

-1

1 2 t

xe(t)

1

2

0-1-2 1 2 t

xo(t)

1

-1

Consider the signal to the left and its time reversed version. The signal is decomposed into its even and odd functions:

)()(2

1)()()(

2

1)( txtxtxtxtxtx oe

Common Signals in Engineering


Common Signals

Continuous-time physical systems are typically modeled with ordinary linear differential equations with constant coefficients.


0,)0()(

:Solution

constant),()(

textx

ataxdt

tdx

at

Exponential Signals

Useful Complex Exponential Relations


partImaginary

part Real

constantscomplex becan &,)(

ja

aCCetx at

Euler’s Formula


j

eeee

je

je

jjjj

j

j

2sin&

2cos

sinsin& coscos

sincos

sincos

Example of Exponential Functions

1. C and a real, x(t)=Ceat a= Increasing Exponential:

Chemical Reactions, Uninhibited growth of bacteria, human population?

a= Decaying Exponential: Radioactive decay, response of an RC circuit, damped mechanical

system. a= Constant (DC) signal.


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

Aet

Ae+/

-t

Time [sec]

Aet, >0Ae

t, <0

Aet, =0

Time Constant of the Exponential Function

The constant parameter t is called the time constant in of the exponential function presented below.

To relate to the time constant the following is necessary:

0,

t

at CeCetx

CbC

abaty

Ce

CCxt

eC

Cedt

d

dt

tdx

t

t

tt

;

0,0

0,

0

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

Aet

Ae

t

Time [sec]

0.368

Example of Exponential Functions

2. C complex, a imaginary, x(t)=Ceat a=j C=Aej – A and are real:

For C – real (=0)

2. x(t) is periodic:

3. Why x(t) is periodic?


tjAtAAeeAeCetx tjtjjat sincos

tjAtAAeCetx tjat sincos

period theis TTtxtx

Periodicity of Complex Exponential


txtxeTtx

jje

jTjT

Tff

T

TjTe

txeeCeCeTtx

Cetx

Tj

Tj

Tj

TjTjtjTtj

tj

~

1012sin2cos

2sin

2cossincos

2 2&

1

sincos

~

Example of Complex Exponentials

3. C complex, a complex, x(t)=Ceat C=Aej A and are real; a=j are also real.


ImRe

Factor Damping

sincos

sincos

tAejtAe

tjtAe

eAeeAeAe

eAeCetx

tt

t

tjtjtjtjtjt

tjjat

Time-Shifted Signals

Time-Shifted Impulse Function Shift (t) by t0:

“Sifting” Property of the Impulse Function:


0

00 0

,1

tt

tttt

00 tfdtf

Examples of Impulse Functions


Properties of the Unit Impulse Function


tt

dta

tt

adttat

t

ttdtttu

ttudt

dtt

tttftttftttf

tttftfdttttf

tttftfdttttf

t

.7

1.6

0,0

,1.5

.4

at continuous .3

at continuous ,.2

at continuous ,.1

-

0

-

0

-

000

00

0000

0

-

00

0

-

00

Shifted Unit Step Function


-10 -8 -6 -4 -2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

u(t) & u(t-td)

u(t)

& u

(t-t

d)

Time [sec]

u(t) u(t-td)

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

u(t) & u(t-td)

u(t)

& u

(t-t

d)

Time [sec]

u(t)u(t-td)

4& tutu

5& tutu

0

00 0

,1

tt

ttttu

Continuous & Piece-wise Continuous Functions

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

p(t)

p (t)

Time [sec]

p(t)

-/2 +/2

Rectangular Function


elsewhere

ttptp0

22,1,

Rectangular Function


-10 -8 -6 -4 -2 0 2 4 6 8 10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

p(t)=u(t+td)-u(t-td)

p (t)

, u(t)

& u

(t-t

d)

Time [sec]

u(t)

u(t-td)

p(t)

-/2 +/2

22,

tututptp

Triangular Pulse Function


201

2

02

,12

tt

tt

tp tr

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

0.8

1

ptr(t)

p tr(t)

Time [sec]

ptr(t), t<0 p

tr(t), t>0

ptr(t)

-/2 +/2

Straight Line Equation

00 xxmyy

[x0,y0] - a point on the line



m = (y2-y1)/(x2-x1) - slope


Composite Signal from Straight Lines


3222111100

322222

211100

211111

10

10000

,

,

,

,

tttxttuttmtxttuttmtxtx

tttttuttmtx

tttxttuttmtxtx

tttttuttmtx

tttxtx

ttttuttmtx

Example of Composite Signal


25113 tututttututf

-10 -8 -6 -4 -2 0 2 4 6 8 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

f(t)=

3u(t)

+tu(

t)-(t-

1)u(

t-1)-5

u(t-2

)

Time [sec]

u(t)

tu(t)

-(t-1)u(t-1)

-5u(t-2)

f(t)=3u(t)+tu(t)-(t-1)u(t-1)-5u(t-2)

continuous time signals basic signals – singularity functions transformations of continuous time...

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