Engineering and PhysicsUniversity of Central Oklahoma

Dr. Mohamed Bingabr

Ch2Signals and Signal Space

ENGR 4323/5323Digital and Analog Communication

Outline

• Size of a Signal• Classification of Signals• Useful Signals and Signal Operations• Signals Versus Vectors• Correlation of Signals• Orthogonal Signal Sets• Trigonometric and Exponential Fourier Series

Signal Energy and Power

Energy Signal

Power Signal

Energy

Power

𝐸𝑔=∫− ∞

∞

𝑔2 (𝑡 )𝑑𝑡 𝐸𝑔=∫− ∞

∞

|𝑔 (𝑡)|2𝑑𝑡

𝑃𝑔= lim𝑇 →∞

1𝑇 ∫

−𝑇 / 2

𝑇 / 2

𝑔2 (𝑡 )𝑑𝑡 𝑃𝑔= lim𝑇 →∞

1𝑇 ∫

−𝑇 / 2

𝑇 / 2

¿𝑔 (𝑡 )∨¿2𝑑𝑡 ¿

Signal Classification

1. Continuous-time and discrete-time signals 2. Analog and digital signals 3. Periodic and aperiodic signals 4. Energy and power signals 5. Deterministic and probabilistic signals 6. Causal and non-causal 7. Even and Odd signals

Analog continuous Digital continuous

Analog Discrete Digital Discrete

PeriodicDeterministic Aperiodic Probabilistic

Useful Signal Operation

Time Shifting Time Scaling Time Inversion

1-1

12

Useful Signals

Unit impulse Signal

Unit step function u(t)𝑑𝑢(𝑡)𝑑𝑡 =𝛿(𝑡 )

Signals Versus Vectors

By sampling, a continuous signal g(t) can be represented as vector g.

g = [ g(t1) g(t2) … g(tn)]

Vector ApproximationTo approximate vector g using another vector x then we need to choose c that will minimize the error e.

g = cx + e

Dot product: <g, x> = ||g||.||x|| cos θ

Signals Versus Vectors

Value of c that minimizes the error

 

Signal Approximationg(t) = cx(t) + e(t)

 

x(t)

 

Correlation of Signals

Two vectors are similar if the angle between them is small.

Correlation coefficient

 

 

Note: Similarity between vectors or signals does not depend on the length of the vectors or the strength of the signals.

Example

1=12=13=-1

4=0.9615=0.6286=0

Which of the signals g1(t), g2(t), …, g6(t) are similar to x(t)?

Cross-Correlation Function

4

g(t-)

Autocorrelation Function

g(t)

g(t-)

Orthogonal Signal Sets

g = c1x1 + c2x2 + c3x3

 

Orthogonal Vector Space

g(t) = c1x1(t)+ c2x2(t) + … + cNxN(t) 

Orthogonal Signal Space

Parseval’s Theorem 

Trigonometric Fourier Series

1 1

000 2sin2cosn n

nn tnfbtnfaatx

∫0

00

2cos2

Tn dttnftx

Ta

∫0

00

2sin2

Tn dttnftx

Tb

∫00

01

T

dttxT

a

Example

• Fundamental periodT0 =

• Fundamental frequencyf0 = 1/T0 = 1/ Hzw0 = 2/T0 = 2 rad/s

. as amplitudein decrease and 1618 504.0 2sin2

1612 504.0 2cos2

504.0121

2sin2cos

202

202

20

20

10

∫

∫

∫

nban

ndtnteb

ndtntea

edtea

ntbntaatf

nn

t

n

t

n

t

nnn

0

1e-t/2

f(t)

122 2sin

16182cos

16121504.0

n

ntn

nntn

tf

To what value does the FS converge at the point of discontinuity?

1 harmonicnth

0

component dc

0 2cosn

nn tnfCCtx

00 aC 22

nnn baC

n

nn a

b1tan

Compact Trigonometric Fourier Series

We can use the trigonometric identity a cos(x) + b sin(x) = c cos(x + )to find the compact trigonometric Fourier series

C0, Cn, and θn are related to the trigonometric coefficients an and bn as:

Role of Amplitude in Shaping Waveform

1

00 2cosn

nn tnfCCtx

Role of the Phase in Shaping a Periodic Signal

1

00 2cosn

nn tnfCCtx

Compact Trigonometric

• Fundamental periodT0 =

• Fundamental frequencyf0 = 1/T0 = 1/ Hzw0 = 2/T0 = 2 rad/s

nab

nbaC

aCn

nb

na

a

ntCCtf

n

nn

nnn

o

n

n

nnn

4tantan

161

2504.0

504.01618 504.0

1612 504.0

504.0

2cos

11

2

22

0

2

2

0

10

0

1e-t/2

f(t)

1

1

24tan2cos

161

2504.0504.0n

nntn

tf

• The amplitude spectrum of x(t) is defined as the plot of the magnitudes |Cn| versus w

• The phase spectrum of x(t) is defined as the plot of the angles versus w

• This results in line spectra• Bandwidth the difference between the

highest and lowest frequencies of the spectral components of a signal.

Line Spectra of x(t)

)( nn CphaseC

Line Spectra

nn

CC

n

n

4tan161

2504.0 504.0

1

20

0

1e-t/2

f(t)

1

1

24tan2cos

161

2504.0504.0n

nntn

tf

f(t)=0.504 + 0.244 cos(2t-75.96o) + 0.125 cos(4t-82.87o) + 0.084 cos(6t-85.24o) + 0.063 cos(8t-86.24o) + …

0.504

0.244

0.1250.084

0.063

Cn

w0 2 4 6 8 10

w

n

-/2

n

ntfjneDtx 02

D-n = Dn*

,....2,1,0 , 102 ∫ ndtetx

TD

oT

ntfj

on

Exponential Fourier Series

To find Dn multiply both side by and then integrate over a full period, m =1,2,…,n,…

ntfje 02

Dn is a complex quantity in general Dn=|Dn|ej

Even Odd

|Dn|=|D-n| Dn = - D-n

D0 is called the constant or dc component of x(t)

• The line spectra for the exponential form has negative frequencies because of the mathematical nature of the complex exponent.

Line Spectra in the Exponential Form

Cn = 2 |Dn| n 0

Dn = Cn

...)2cos()cos()(

...||||

||||...)(

2021010

2221

012

2

001

0102

ww

ww

ww

tCtCCtx

eeDeeD

DeeDeeDtxtjjtjj

tjjtjj

D0= C0

D-n = - Cn

Example

• Fundamental periodT0 = 2

• Fundamental frequencyf0 = 1/T0 = 1/2 Hzw0 = 2/T0 = 1 rad/s

/2/2

1f(t)

22

Find the exponential Fourier Series for the square-pulse periodic signal.

∫

,15,11,7,3,15,11,7,3 allfor 0

odd /1even 0

21

)2/sinc(5.02/sin

21

0

2/

2/

nn

nnn

D

D

nn

n

dteD

n

n

jntn

|Dn|

Dn

1 1

1 1

Exponential Line Spectra

Example

,15,11,7,3,15,11,7,3 allfor 0

odd 2

even 021

0

nn

nn

nC

C

n

n

/2/2

1f(t)

22

The compact trigonometric Fourier Series coefficients for the square-pulse periodic signal.

1

2/)1(

21)1(cos2

21)(

n

nntn

tf

• Fundamental frequencyf0 = 1/T0 = 1/2 Hzw0 = 2/T0 = 1 rad/s

000

sincos

Im2Re2

cDaCb

CaDDDjb

DDDa

nnn

nnn

nnnk

nnnn

Relationships between the Coefficients of the Different Forms

000

1

22

2

tan

DaCD

DC

ab

baC

nn

nn

n

nn

nnn

000

5.05.0

5.0

5.0

CaDeCCD

jbaDD

jbaD

njnnnn

nnnn

nnn

ExampleFind the exponential Fourier Series and sketch the corresponding spectra for the impulse train shown below. From this result sketch the trigonometric spectrum and write the trigonometric Fourier Series.Solution

10

0

000

0

0

0

)cos(211)(

/1||/2||2

1)(

/1

0

0

0

nT

nn

n

tjnT

n

tnT

t

TDCTDC

eT

t

TD

w

w

2T0T0-T0-2T0

)(0

tT

1- The function g(t) is absolutely integrable over one period.

2- The function g(t) can have only a finite number of maxima, minima, and discontinuities in one period.

Dirichlet Conditions for FS Convergence

• Let x(t) be a periodic signal with period T• The average power P of the signal is defined as

• Expressing the signal as

it is also

1

00 )cos(n

nn tnCCtx w

1

220 2

nnDDP

1

220 5.0

nnCCP

Parseval’s Theorem

∫2/

2/

2)(1 T

Tdttx

TP

Numerical Calculation of Fourier Series

,....2,1,0 , 102 ∫ ndtetg

TD

oT

ntfj

on

𝐷𝑛= lim𝑇 𝑠→ 0

1𝑇 0

∑𝑘=0

𝑁 0 −1

𝑔 (𝑘𝑇 𝑠 )𝑒− 𝑗𝑛𝜔 0𝑘𝑇𝑠𝑇 𝑠

𝐷𝑛= lim𝑇 𝑠→ 0

1𝑁0

∑𝑘=0

𝑁 0 −1

𝑔 (𝑘𝑇 𝑠 )𝑒− 𝑗𝑛Ω 0𝑘

𝐷𝑛=1𝑁0

∑𝑘=0

𝑁 0 −1

𝑔 (𝑘𝑇 𝑠 )𝑒− 𝑗𝑛Ω 0𝑘

Ω 0=𝜔0𝑇 𝑠

𝑁 0=𝑇 0

𝑇 𝑠

Matlab Exercise Page 80

T0 = pi; N0 = 256; Ts = T0/N0;t = 0: Ts : Ts*(N0-1); g = exp(-t/2); g(1) = 0.604;Dn = fft(g)/N0;[Dnangle, Dnmag] = cart2pol( real(Dn), imag(Dn));k =0:length(Dn)-1; k = 2*k;subplot(211), stem(k, Dnmag)subplot(212), stem(k,Dnangle)

0

1e-t/2

g(t)

T0 = f0 = 1/T0 = 1/ Hzw0 = 2/T0 = 2 rad/s