Ch02 1 - Western Michigan University

© 2010 The McGraw-Hill Companies

Communication Systems, 5e

Chapter 2: Signals and Spectra

A. Bruce Carlson

Paul B. Crilly


Chapter 2: Signals and Spectra

• Line spectra and fourier series

• Fourier transforms

• Time and frequency relations

• Convolution

• Impulses and transforms in the limit

• Discrete Fourier Transform (new in 5th ed.)

What kind of math do we do?• Transmit

• Receive

3

𝑠 𝑡 = 𝐴 ⋅ 1 + 𝜇 ⋅ 𝑚 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 + 𝜃 + 𝜑 ⋅ 𝑚 𝑡 + 2𝜋 ⋅ Δ ⋅ 𝑚 𝜆 ⋅ 𝑑𝜆

𝑥 𝑡 = 𝐴 ⋅ 𝑠 𝑡 + 𝜏 + 𝑛 𝑡 ∗ ℎ 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 + 𝜗 ∗ ℎ 𝑡

𝑥 𝑡 = 𝑠 𝑡 + 𝜏 + 𝑛 𝑡There is a lot of mixing/multiplying by sin/cos and there will be convolutions (filtering), integration and decimation!

Conceptual Domains• Time Domain (Signals)

– time based waveforms

– some consistent elements

– some “random” elements (time varying, but in an understandable way – like a message)

• Frequency Domain (Spectra)– long term and short term frequency content

– consistent – Fourier transforms (Bode Plots)

– random/changing in time – Power Spectral Densities

4

Phasors and line spectra


0 0

0 0

0 0

0

( ) cos( ) cos(2 )

where 2 frequency in radians/second 1/ period in seconds cyclical frequency in Hz phase in radians

v t A t A f t

fT ff

jtfjAtv 02expRe

Phasor representation


(a) Phasor diagram, (b) spectrum Caution: frequencies can be positive or negative – they “rotate” in opposite directions

Euler’s theorem


0 0( )0

cos sin

where 1

( ) cos( ) Re[ ] Re[ ]

phasor representation

j

j t j tj

e j

j

v t A t A e Ae e

Two sided spectrum


It is often more useful to describe a signal using its two sided spectrum

0 02 20cos(2 )

2 2 pair of conjugate phasors

j f t j f tj jA AA f t e e e e

This is similar to the Fourier Transform real and imaginary frequency spectrum plot.

Usually plot “power spectrum” in dB, not amplitude or magnitude

9

Complex Phasor Notation• Complex Signal Representation (can be easier)

• Math: Euler’s theorem and related equalities

tf2cosAtv 0

exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 = cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 + 𝑗 ⋅ sin 2𝜋 ⋅ 𝑓 ⋅ 𝑡

jtf2jexpARetv 0

10

Two Dimensional Visualization• Real-Imaginary Plot of Phasor Rotation

– All real signals can be composed as the real part of a complex signal

– The real axis component will be the real output signal

tftfjtfj 000 2cos22exp2exp

11

Textbook Convention(not required for Dr. Bazuin and your solutions)

• Positive cosine representations– Convert sin to cosine

– Convert negative to 180° rotation

– The complex phasors are identical; therefore, equal

1802cos2cos 00 tfAtfA

90tf2cosAtf2sinA 00

1exp j ij

2exp

12

Classification of signals …• Periodic and non-periodic signals

• Analog and discrete signals

A discrete signal

Analog signals

A non-periodic signalA periodic signal

13

Classification of signals ..• Energy and power signals

– A signal is an energy signal if, and only if, it has nonzero but finite energy for all time:

– A signal is a power signal if, and only if, it has finite but nonzero power for all time:

• General rule: – Periodic and random signals are power signals. – Signals that are both deterministic and non-periodic are energy

signals.

𝐸 = lim→

𝑥 𝑡 ⋅ 𝑑𝑡 ⇒ 𝑥 𝑡 ⋅ 𝑑𝑡 xE0

𝑃 = lim→

1

𝑇⋅ 𝑥 𝑡 ⋅ 𝑑𝑡 xP0

14

Fourier and Laplace

• f vs. s

• The Laplace Transform (typically) deals with single sided time signals (causal)

• The Fourier Transform is one particular special case of the Laplace Transform (and usually uses two-sided time)

– See Table T.1 on p. 847-848 for Fourier Transform pairs

fjwjs 2

0

dtetfsF ts

dtetffF tfj 2

Table T.1

16

Fourier Transform• Time to frequency domain

• Frequency to time domain

• Condition … finite energy (usually assumed)(finite power signals use “Fourier Series”)

dttf2jexptvfV

dftf2jexpfVtv

dttvE02

FT – Rectangular Pulse

17

2

2

2

22

2exp2exp

fj

tfjAdttfjAfV

fj

fjA

fj

fjA

fj

fjAfV

222sin2

222exp

222exp

fAf

fAfV sinc

sin

t

rectAtv

18

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Rectangular pulse spectrum

V(ƒ) = A sinc ƒv(t) = A rect t/

Note 1: I hate this phase plot, but it is required when the absolute value is used.Note 2: Is there really a difference between +180 degrees and -180 degrees on a circle?But … the frequency domain must be conjugate symmetric!

19

Periodic Signals• Relationship

– for m an integer and To the period

• Average/mean (computed from one period)

• Average Power (computed from one period)

0Tmtvtv

01

10

1Tt

t

dttvT

tv

01

1

2

0

2 1Tt

t

dttvT

tvP

20

Fourier Series (Periodic Signals with Finite Power)

• Fourier Series Coefficients

• Signal Representation

dttfn2jexptvT

1c 0

T

0n

0

n

0n tfn2jexpctv

00 T

1fwhere

n

n fnfcfV 0

21


FS - Rectangular pulse train

dttfnjtvT

c

T

Tn

0

2

20

2exp1

0

0

2

2

00

2exp dttfnjT

Acn

The previous math at f0

00

1

Tf

FS - Rectangular pulse train (2)• Continuing

22

2

2

00

2exp dttfnjT

Acn

00

sinc fnT

Acn

Table T.2

Fourier Series Table

24


Spectrum of rectangular pulse train

with ƒ0 = 1/4 (a) Amplitude (b) Phase

00 T

1fwhere

Note: Envelope is defined from 1 period, impulse spacing defined by periodicity

Fourier Series:The Fourier Transform of Periodic Signals

• The convolution of one period with a comb function

25

t

rectAtv fAfV sinc

n T

nf

TTfcomb 1

k

TktrectAtv

n

fT

nf

T

AfV

sinc

Time Domain Frequency Domain

k

TktT

tcomb

Fourier Transforms and Series• For a periodic signal:

– Take the Fourier Transform of one period of the waveform and plot the spectrum

• This is the “envelope” of the Fourier Series spectrum

– The Fourier Series spectrum is then a “line spectrum” with coefficients spaced at n x f0

26

Fourier Transforms and Series

27

Fourier Transform of one rect pulse

Fourier Series of an infinite rect pulse train

Fourier Series to Time Signal• If you know the Fourier Transform or Fourier Series, you

can find the time waveform … approximately.

– Inverse Fourier Transforms and series do not correctly represent discontinuities. If the original time domain curve makes a discrete change,

• The inverse Fourier Curve gives the mid-point between the two curves

• The “inverse Fourier Series” has the Gibbs Phenomenon

– Infinite energy and infinite power limitations means that the functions are approximated, but not directly solved.

28

29

Fourier-series reconstruction of a rectangular pulse train


N

Nn0n tf2jexpctv

Matlab Gibbs Phenomenon

30


Fourier-series reconstruction of a rectangular pulse train

See MATLAB example

31


0dttf2jexpctvlimte0T

2N

Nn0n

N

Gibbs phenomenon at a step discontinuity

32

Useful Signal Property Definitions: Symmetry

• Symmetry (even)

• Anti-Symmetry (odd)

• Conjugate Symmetry (real is even, imag is odd)

tvtv

tvtv

*tvtv

Properties allow you to use “math tricks”to simplify computations, if you recognize them

33

Symmetry Examples (2)

• Symmetry (even)

• Anti-Symmetry (odd)

• Conjugate Symmetry (real is even, imag is odd)

tftf 2cos2cos

tftf 2sin2sin

tfconjtfj 2exp2exp

Real Signal Transformation• The Fourier Transform becomes

– The real part of V(f) has even symmetric

– The imag. Part of V(f) has odd symmetric

– For Imaginary Signals (property)• Real part of V(f) has odd symmetry

• Imag. part of V(f) has even symmetry

34

dttftvjdttftv

dtetvfV tfj

2sin2cos

2

Symmetry in Transforms• In working with transforms, there are many

properties of time signals (and spectra) that can simplify taking forward and inverse transforms– Purely real signals

– Purely imaginary signals

– Even symmetric signals

– Odd symmetric signals

– Conjugate symmetric signals

35

The properties can be used to check your results!If the result doesn’t have the property, you did the math wrong!

Even Symmetric Real Signals• The Fourier Transform becomes

36

00

22

0

'2

0

2

0'2

0

2

02

0

22

2cos2

''

''

dttftvdteetv

dtetvdtetv

dtetvdtetv

dtetvdtetvdtetvfV

tfjtfj

tfjtfj

tfjtfj

tfjtfjtfj

Conjugate Symmetric Signals• The Fourier Transform becomes

37

00

0

*

0

*

0

'2*

0

2

0'2

0

2

02

0

22

2sinIm22cosRe2

2sin2cos

''

''

dttftvjdttftv

dttftvtvjdttftvtv

dtetvdtetv

dtetvdtetv

dtetvdtetvdtetvfV

tfjtfj

tfjtfj

tfjtfjtfj

Note that the result is purely real!

38

Fourier Transform Properties• Linearity• Superposition• Time Shifting• Scale Change• Conjugation• Duality• Frequency Translation• Convolution• Multiplication• Modulation

Table T.1 on pages 847-848

Section 2.3, pp. 54-62

Section 2.4, pp. 62-68

Table T.1

Note on Fourier Transforms• Find a table

• Learn to use the table

• Yes, Calculators can do it too …but after a while you should “Know” the easy or continually repeated transformations– RectSync

– Sin/Cosdelta functions at +/- f

– Exp(i*2*pi*f*t)single delta function at f

– Convolution Multiplication

40

41

Rayleigh’s Energy Theorem• Energy in the time domain is equal to energy in

the frequency domain! (E↔v2)

dttvtvdttvE *2

dffVdtetvdttvdfefVE tfjtfj 2**2

dffVfVdffVdtetvconjE tfj *2

dffVdttvE22

42

Important Signal Property: Causality

• Signals that haven’t happened yet are not known!

• Usual application– For a single signal analysis

• Signals start at time t=0, and v(t)=0 for t<0

• Laplace transform signals

• Filters are typically defined as starting at t=0

or

– For signal processing• The signal exist up to a time t0=0, and

• v(t)=0 for t > t0

• We don’t know what comes next … but we know the history!

43

Causality and Filtering

• Convolution Form

– The filter, h, can be defined for positive time only

– The signal, x, is defined for all past time up to time t

• Then, when limited by the filter impulse response:

dtxhtz

T

0

dtxhtz

44

The RC Filter: 1st order Butterworth Low Pass Filter

• Using Rayleigh’s Energy

y(t) v(t)

RC1sRC

1

sC1R

sC1

sHsY

sV

0t,RCtexp

RC

1th

dttvdffVE22

45

Homework 2.2-8

• Percent of total Energy based on the bandwidth of an “exponential” time signal impulse response at – WBW= b/2π and WBW = 4b/2π = 2b/π

0t,0

0t,tbexpAtv

b

A

b

tbAdttbAEE tv

22

2expexp

2

0

2

0

2

fjb

AfV

2

• Total Energy is:

The exponential time signal

• Time Response

• Frequency Response

46

0t,0

0t,tbexpAtv

fjb

AfV

2

b

AEE tv

2

2

WBW=b/2π

47

Homework 2.2-8 (cont)• Percent of total Energy at various “Bandwidths” WBW

– WBW= b/2π and WBW = 4b/2π = 2b/π

WW

W

fV dfbf

Adf

bfj

A

bfj

AE

022

2

22

22

48

Homework 2.2-8 (cont)

• Percent of total Energy at– WBW= b/2π and WBW = 4b/2π = 2b/π

• For a simple RC filter – wco is the 50% power point (w in radians/sec)

RCwb co1 RCA 1

49

Realizable Filters, RC Network

Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,

Prentice Hall PTR, Second Edition, 2001.

The Use of Percent Total Energy

• If you want to receive a finite time signal (finite energy) signal, what bandwidth “perfect” filter should you use?– For a decaying exponential signal

• 50% of the energy received at ffilter=fco1/RC

• 84.4% received at ffilter=4 x fco 4/RC

• 93.7% received at ffilter=10 x fco 10/RC

• We usually want 90%-99% of a “pulses” energy

• This also has implication for digital sampling rates!

50

End Lecture

Another Example

• What bandwidth “ideal” filter should be use if we want to filter a bipolar square wave and receive 90% of the power?

• Use the approach just shown …– Percent energy in frequency for one period of the

periodic square wave

– The general result is the integral of the sinc^2 function from f=0 to f=?

51

52

Modulation

• Frequency translation due to real or complex mixing products (multiply in time domain)

jtfjjtfjtx

tftxtz

00

0

2exp2

12exp

2

1

2cos

0

j

0

j

ff2

eff

2

efXfZ

• Using trig functions, try cosine x cosine mixingTable T.3 on pages 851-853

Convolution in freq. domain

Table T.3

54


Frequency translation of a bandlimited spectrum

tfjtvts c 2exp cffVfS

Baseband Complex Modulated

55


(a) RF pulse (b) Amplitude spectrum

RF Pulse Mixing tf

tAts c

2cos

cc ffA

ffA

fS sinc2

sinc2

56

Mixing

RF Input IF Output

LocalOscillator

tLOtRFtIF

LOLOLO tfAtLO 2cos

RFRFRF tfAtRF 2cos

LOLOLORFRFRF tfAtfAtIF 2cos2cos

LOLORFRFRFLO tftfAAtIF 2cos2cos

57

Trigonometry Identities

sincoscossinsin

sincoscossinsin

sinsincoscoscos

sinsincoscoscos

cos2

1cos

2

1sinsin

sin2

1sin

2

1cossin

cos2

1cos

2

1coscos

sin2

1sin

2

1sincos

Table T.3 on pages 851-853

• Restating

• Using an Identity

• After Low Pass Filtering

58

Mixing (2)

LOLORFRFRFLO tftfAAtIF 2cos2cos

LORFLORFRFLO

LORFLORFRFLO

tffAA

tffAAtIF

2cos2

1

2cos2

1

LORFLORFRFLO tffAAtIF 2cos2

1

59

Spectral Equivalent – Real Mixing

• The mixing of a real RF input with a real Cosine local oscillator– Real Signal and Cosine LO spectrum

– Post mixer sum and difference spectrum

– Post Low Pass Filter (LPF) result

Real Signal

Cosine

Mixing Products

LPF

60

Spectral Equivalent – Complex Mixing

• The mixing of a real RF input with a Complex local oscillator– Real Signal and Complex LO spectrum

– Post mixer sum spectrum (convolution in freq.)

– Post Low Pass Filter (LPF) result

Real Signal

Complex Oscillator

Mixing Products

LPF

61

Higher Order Mixing

Mixers in Microwave Systems (Part 1)Author: Bert C. Henderson WJ Tech-note

http://www.rfcafe.com/references/articles/wj-tech-notes/Mixers_in_systems_part1.pdf

Part of WJ Comm. Technical Publicationshttps://www.rfcafe.com/references/articles/wj-tech-notes/watkins_johnson_tech-notes.htm

Watkins-Johnson (1957-2000) and later WJ Communications (2000-2008) was acquired by Triquint (1985-2014) which merged with RF Micro Devices and is

now called Qorvo (RF Solutions for 5G and beyond)https://www.qorvo.com/

https://www.qorvo.com/products

62

Convolution• Filtering of unwanted spectral components

is performed by filtering. – Convolution in the time domain

– Multiplication in the frequency domain

63


Graphical interpretation of convolution

dtwvtwtv

64


Result of the convolution

65

Impulses and Transformsin the Limit

• When dealing with discrete, inherently discontinuous message data we require appropriate mathematical methods to derive and describe the modulated waveforms.

• Signal descriptions for impulses (in time and frequency), step functions, etc. are required.– Define a continuous time, parameterized function that

approaches an impulse/step function as one of the parameters approaches infinity or zero.

– What are some of these functions.

66

Delta Function Approximations• Rect

• Sinc

• Gaussian

67


Two functions that become impulses as 0

68

Impulse Properties• Continuous sampling is equivalent to discrete

samples

• Scaling

ddd tttvtttv

dd tvdttttv

ta

ta 1

69

Impulses in Frequency

• Transform pairs

W

f

W

Aftv

W 22lim

0

AtWAtvW

2sinclim0

W

f

WtW

F

22

12sinc

fAAF

AtAF

70

Signal Smoothing• Signal approximations that provide rounding or

smoothing of rapid transitions in time ….

• Inherent in transmitted signals due to component and channel effects …

• Usually consider are low-pass filter functions … like an R-C filter previously discussed.

71

Smoothing the Edges

• A more practical frequency domain filter:The raised Cosine filter– Cosine band edge roll-off is often used

– Easy to implement in MATLAB

• A nice explanation of “filter roll-off” is provided

72


Raised cosine pulse. (a) Waveform (b) Derivatives(c) Amplitude spectrum

Figure 2.5-7

Ch02 1 - Western Michigan University

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