Ch02 2 - Western Michigan University

© 2010 The McGraw-Hill Companies

Communication Systems, 5e

Chapter 2: Signals and Spectra

A. Bruce Carlson

Paul B. Crilly

© 2010 The McGraw-Hill Companies

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.)

3

Fourier Transform

• Time to frequency domain

• Frequency to time domain

• Condition …

dttf2jexptvfV

dftf2jexpfVtv

dttvE02

Table T.1 on pages 780-780

Table T.1: Fourier Transform Pairs

RC Filter

5

Fourier Transform Properties

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


Section 2.3, pp. 44-52

Section 2.4, pp. 52-58

Fourier Transform Properties

6

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

– Time delaylinear phase

– Convolution Multiplication

7

Why Know Properties• Properties may allow mathematical short cuts!

– Pre-derived simplification.

• When properties exist, the solution must obey the properties– Only incorrect solutions do not have the properties (e.g.

purely real or imaginary, symmetry, etc.)

– Check your result to see if it makes sense!

8

Two Most Important Properties

• Convolution

• Mixing

9

fWfVtwtv

fWfVtwtv *

Convolution

• Convolution in the time domain

• The Fourier Transform Pair is:

10

dtvwdtwvtwtv

fWfVtwtv

fWfVdevfW

dvefWdvdtetw

dtedtwvtwtvF

fj

fjtfj

tfj

2

22

2

Mixing

• Convolution in the frequency domain

• The Fourier Transform Pair is:

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dfVWdfWVfWfV

fWfVtwtv *

twtvdeVtw

dvetwdVdfefW

dfedfWVfWfVF

fj

fjtfj

tfj

2

22

21

Continuing

• Continuation from previous Chapter 2-1 notes …

12

13

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

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 ( u(t) step function)

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!

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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:

15

dtxhtz

T

0

dtxhtz

16

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

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Homework 2.2-8

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

18

0t,0

0t,tbexpAtv

fjb

AfV

2

b

AEE tv

2

2

fBW=b/2π

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

– fBW= b/2π and fBW = 4b/2π = 2b/π

𝑩𝑾

𝒇𝑩𝑾

𝑩𝑾

𝒇𝑩𝑾𝑩𝑾

𝐸 =𝐴

𝑗 ⋅ 2𝜋 ⋅ 𝑓 + 𝑏⋅

𝐴

−𝑗 ⋅ 2𝜋 ⋅ 𝑓 + 𝑏⋅ 𝑑𝑓

𝒇𝑩𝑾

𝒇𝑩𝑾

= 2 ⋅𝐴

2𝜋 ⋅ 𝑓 + 𝑏⋅ 𝑑𝑓

𝒇𝑩𝑾

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Homework 2.2-8 (cont)

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

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

𝒇𝑩𝑾%𝑏2𝜋 =

2

𝜋⋅ tan 1 =

2

𝜋⋅𝜋

4=1

2

𝒇𝑩𝑾%4 ⋅ 𝑏

2𝜋 =2

𝜋⋅ tan 4 =

2

𝜋⋅ 𝜋 ⋅ 0.422 = 0.844

RCwb co1

𝒇𝑩𝑾%10 ⋅ 𝑏

2𝜋 =2

𝜋⋅ tan 10 =

2

𝜋⋅ 𝜋 ⋅ 0.468 = 0.937

RCA 1

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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=fco1/2πRC

• 84.4% received at ffilter=4 x fco 4/2πRC

• 93.7% received at ffilter=10 x fco 10/2πRC

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

• This also has implication for digital sampling rates!

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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=?

23

24

Modulation (Mixing)

• 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

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Modulation (Mixing)

• 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

27

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

Frequency Translation of a Bandlimited Spectrum

tfjtvts c 2exp

cffVfS

(a) Initial Signal Spectrum (b) Output Spectrum

Multiplication-Convolution

• Convolution in the time domain

• The Fourier Transform Pairs are:

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dtvwdtwvtwtv

fWfVtwtv

fWfVtwtv *

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Frequency Translation: Complex Mixing tfjtvts c 2exp

cffjVfS *

(a) Initial Signal Spectrum (b) Output Spectrum

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(b) Magnitude spectrum

RF Pulse Mixing tf

tAts c

2cos

cc ffA

ffA

fS sinc2

sinc2

Is convolution easier?

(a) RF Pulse

Another Interpretation

• A limited time duration cosine waveform– window sample of an infinite periodic signal

– As the window becomes longer, the sinc gets narrower … going to an impulse as τ∞

• This is critically important when we talk about finite time sample lengths of signals.

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tft

Ats c

2cos

cc ffA

ffA

fS sinc2

sinc2

32

Mixing

RF Input IF Output

LocalOscillator

tLOtRFtIF

LOLOLO tfAtLO 2cos

RFRFRF tfAtRF 2cos

LOLOLORFRFRF tfAtfAtIF 2cos2cos

LOLORFRFRFLO tftfAAtIF 2cos2cos

33

Trigonometry Identities

sincoscossinsin

sincoscossinsin

sinsincoscoscos

sinsincoscoscos

cos2

1cos

2

1sinsin

sin2

1sin

2

1cossin

cos2

1cos

2

1coscos

sin2

1sin

2

1sincos


34

Mixing (2)

• Restating

• Using an Identity

• Ideal Low Pass Filtering

35

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

Real Signal Spectrum

Mixing Cos Signal Spectrum

Convolved Signal Spectrum

Low Pass Spectrum

36

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

Real Signal Spectrum

Mixing Exp Signal Spectrum

Convolved Signal Spectrum

Low Pass Spectrum

37

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

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Convolution

• Filtering of unwanted spectral components is performed by filtering. – Convolution in the time domain

– Multiplication in the frequency domain

39


Graphical interpretation of convolution

dtwvtwtv

Sketching a ConvolutionWhere does it start?Where does it change?Where does it end?What is the general shape entering a region?What is the shape in the region?What is the shape leaving the region?

tueAtv t

T

Ttrect

T

ttw 2

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Result of the convolution

Sketching a ConvolutionWhere does it start? 0Where does it change? T, triangle overlaps exponentialWhere does it end? neverWhat is the general shape entering a region? More than linear increaseWhat is the shape in the region? Exponential decreaseWhat is the shape leaving the region? Always inside, doesn’t happen

Now derive 2 equations: entering and decreasingAnd one value: the maximum at T

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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.

42

Delta Function Approximations

• Rect

• Sinc

• Gaussian

t

rectt1

t

t sinc1

2

2

exp1

tt

43


Rect and Sinc impulses as 0

44

Impulse Properties

• Continuous sampling is equivalent to discrete samples

• Scaling

ddd tttvtttv

dd tvdttttv

𝛿 𝑎 ⋅ 𝑡 =1

𝑎⋅ 𝛿 𝑡

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Impulses in Frequency Domain

• Transform pairs

ℑ 𝑣 𝑡 = 𝐴 ⋅ 𝛿 𝑓 = lim→

𝐴

2 ⋅ 𝑊⋅ ∏

𝑓

2 ⋅ 𝑊

𝟏

→

𝐴 ⋅ sinc 2 ⋅ 𝑊 ⋅ 𝑡 ↔𝐴

2 ⋅ 𝑊⋅ ∏

𝑓

2 ⋅ 𝑊

fAAF

AtAF

46

Signal Smoothing

• Signal approximations that provide rounding or smoothing of rapid transitions in time …– Effects of low pass filtering or attenuation of

higher frequency components.

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

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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 finding the order of “filter roll-off” is provided in the text.

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

Figure 2.5-7

Using Rect to Make a Filter Shape in the Time Domain

• Any continuous time signal can have a “possibly desirable” part isolated to create a filter.– Raised Cosine

– Cosine squares

– Sine single period (for odd-symmetric signals)

– Gaussian

– Main-Lobe of Sinc

• With computer tools, people can try anything!If it works, great, if not, you tried.

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Ch02 2 - Western Michigan University

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