Exam 1 Review

Dr. Bradley J. BazuinAssociate Professor

Department of Electrical and Computer EngineeringCollege of Engineering and Applied Sciences

Department of Electrical and Computer Engineering

In Class Exam

• Wednesday 6 October– Open book, open notes – paper only, print what you need.

– Calculators allowed. No cell phones, no laptops, no wireless connections

– The exam is long, you will not have time to “discover” something you have not studied.

• Exam #1 from 2020 is on the password web site

• Likely problems:– Sketch a time limited waveform, perform the Fourier transform, and

sketch the resulting spectrum

– Communication path gains and losses• Free space path lost and Friis equation (watch linear vs. dB)

– Butterworth Filter passband and stopband specification

– Sketch message and then the AM and DSB waveforms

– One or two of the exam prep homework problems (chap. 4)

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Take-home Exam

• To be posted Wednesday on the web site– MATLAB Filter Design

– LT Spice Filter

– things that take more time than an in-class exam allows

• Due Oct. 13 at the beginning of class – 10:30 am

• It must be your own work, no help or collaboration.

• Submit– Written solution with Matlab code and LTspice printouts

• Document successful operations and summarize results.

– E-mail of MATLAB AND MATLAB functions used (*.m files)

– E-mail of LTspice circuit file (*.asc)

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Exam #1 Material (1)

• Chapter 1: Introduction – Elements and limitations

– Modulation and coding

– Electromagnetic wave propagation over a wireless channel

– Emerging developments

– Societal impact and historical perspective

• 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

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Exam #1 Material (2)

• Chapter 3: Signal Transmission and Filtering– Response of LTI systems

– Signal distortion

– Transmission Loss and decibels

– Filters and filtering

– Quadrature filters and Hilbert transform

– Correlation and spectral density

• Chapter 4: Linear CW Modulation – Bandpass signals and systems

– Double-sideband amplitude modulation

– Modulation and transmitters

– Suppressed-sideband amplitude modulation

– Frequency conversion and demodulation

• Exam #1ECE 4600/5640

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Chapter 1: Introduction

• Chapter 1: Introduction– Elements and limitations

– Modulation and coding

– Electromagnetic wave propagation over a wireless channel

– Emerging developments

– Societal impact and historical perspective

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Sinusoidal Signal Modulation

• Message 1: Amplitude Modulation (AM)

• Message 2: Phase Modulation (PM)

• Message 3: Frequency Modulation (FM)

t

3f2p01 dm2tmtf2costm1Ats

“Carrier” waveforms besides a sinusoid may be used– Pulse Train, Pulse Position, Chaos, etc.

Complex Modulation

• Using a complex phasor instead of a real sin or cos.– Perform signal processing in the complex domain (real and

imaginary or in-phase and quadrature-phase).

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t

fp dmjtmjjtfjtmAts 3201 22exp1

Note: Both real and complex signals and signal notations exist and will be used in this class!

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Received Signal

• The receiver must extract the original message as best possible!

• The RF channel(s) must be allocated and efficiently utilized. – Frequency band assignments and regulations (power, direction, etc.)

– Signal modulation structures have different characteristics

tntgtstgtstgtstr NN22

Signal of interestInterference or

other signalsNoise

Signals and Spectra(Reviewing mathematics relevant to communications)

• 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

10Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Complex Phasor Notation

• Complex Signal Representation

• Euler’s theorem and related equalities

tf2cosAtv 0

tf2sinjtf2costf2jexp

jtf2jexpARetv 0

tf2jexp2

1tf2jexp

2

1tf2cos

tf2jexp2

1jtf2jexp

2

1jtf2sin

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Classification of signals …

• Periodic and non-periodic signals

• Analog and discrete signals

A discrete signal

Analog signals

A non-periodic signalA periodic signal

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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.

dttxdttxE

T

TT

x

22

2

2lim xE0

2

2

21lim

T

TT

x dttxT

P xP0

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Fourier Transform

• Time to frequency domain

• Frequency to time domain

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

dttf2jexptvfV

dftf2jexpfVtv

dttvE02

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Fourier Transform Properties

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

Table T.1 on pages 780-780

Section 2.3, pp. 44-52

Section 2.4, pp. 52-58

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FT – Rectangular Pulse

16

2

2

2

22

2exp2exp

fj

tfjAdttfjAfV

fj

fjA

fj

fjA

fj

fjAfV

222sin2

222exp

222exp

fAf

fAfV sinc

sin

t

rectAtv

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Periodic Signals

• Relationship

– for m an integer and To the period

• Average/mean (computed from one period)

• Average Power

0Tmtvtv

01

10

1Tt

t

dttvT

tv

01

1

2

0

2 1Tt

t

dttvT

tvP

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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

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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

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Useful Signal Properties: Symmetry

• Symmetry (even)

• Anti-Symmetry (odd)

• Conjugate (real is even, imag is odd)

tvtv

tvtv

*tvtv

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Symmetry Examples (2)

• Symmetry (even)

• Anti-Symmetry (odd)

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

tftf 2cos2cos

tftf 2sin2sin

tfconjtfj 2exp2exp

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Rayleigh’s Energy Theorem

• Energy in the time domain is equal to energy in the frequency domain!

dttvtvdttvE *2

dffVdtetvdttvdfefVE tfjtfj 2**2

dffVfVdffVdtetvconjE tfj *2

dffVdttvE22

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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:

dtxhtz

T

0

dtxhtz

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Mixing

RF Input IF Output

LocalOscillator

tLOtRFtIF

LOLOLO tfAtLO 2cos

RFRFRF tfAtRF 2cos

LOLOLORFRFRF tfAtfAtIF 2cos2cos

LOLORFRFRFLO tftfAAtIF 2cos2cosECE 4600/5640

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Mixing (2)

LOLORFRFRFLO tftfAAtIF 2cos2cos

LORFLORFRFLO

LORFLORFRFLO

tffAA

tffAAtIF

2cos2

1

2cos2

1

LORFLORFRFLO tffAAtIF 2cos2

1

• Restating

• Using an Identity

• Low Pass Filtering

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Signal Transmission and Filtering

• Chapter 3: Signal Transmission and Filtering– Response of LTI systems

– Signal distortion

– Transmission Loss and decibels

– Filters and filtering

– Quadrature filters and Hilbert transform

– Correlation and spectral density

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Linear Time Invariant (LTI) Systems

• Principle of Superposition– Summed input create transformed summed outputs

• Time-invariant– invariant result if shifted in time

• Homogeneity– Constant multiplication of input results in identical constant

multiplication of the output

• Transfer Functions – input to output relationship

• Expected for “lumped-parameter” elements (R, L, C)

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Transfer Function Methods

• For continuous-time systems, the Laplace transform is used to define the filter transfer function

– Frequency Response

– or

28

11

11

011

1

sss

sss

sX

sYsH

nn

nn

nn

mm

1222

222

11

1

011

1

fjfjfj

fjfjfj

fX

fYfH n

nn

n

nn

mm

fXj

fYj

efX

efY

fX

fYfH

arg

arg

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Power Gain

• Gain is defined as the power out divided by the power in

2

in

out 0HP

Pg

• Cascaded gain is the gain of the receiver chain

in

out10dB P

Plog10g

321 GGGG 1in

3out

3in

3out

2in

2out

1in

1out

P

P

P

P

P

P

P

PG

dB3dB2dB1dB GGGG ECE 4600/5640

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Noise Figure

• The Noise Figure Described the expected output noise power reflected to the input of the device. Thus, it defines the additional “equivalent input” noise contribution.

0

device

0

device0

N

N1

N

NNF

• Cascaded noise figure is effected by the gain

21

3

1

21 GG

1F

G

1FFF

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Free-Space Loss

• As an RF signal propagates, there is path loss.

tPtG rG

f

R

rP

𝐿𝑜𝑠𝑠 =4𝜋 ⋅ 𝑅

𝜆=

4𝜋 ⋅ 𝑓 ⋅ 𝑅

𝑐

• As shown above

𝑐 = 𝜆 ⋅ 𝑓

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1st Order RF Range Estimate

• Friis Transmission Formula– Direct, line-of-sight range-power equation

– No real-world effects taken into account

where: rP is the received (or transmitted)

tG is the effective transmitter (or receiver) antenna gain

R is the distance between the transmitter and receiver, and is the wavelength f is the frequency

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

6 GHz4 GHz

dB13.1997e8.9048e3

6e369e64

c

Rf4L 2

22

u

dB60.1957e18.6038e3

6e369e44

c

Rf4L 2

22

d

Satellite relay system Ex. 3.3-1 (1 of 2)

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

dBWdBdBdBdBWP rcvsat 1.89201.1995535_

dBW6.110dB51dB6.195dB16dBW18Pout

22

rtt

rttr

R4

GGP

L

GGPP

dB1.107dBW1.89dBW18gamp

Satellite relay system (2 of 2)

Pt=35 dBW

Pt=18 dBW

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Frequency Selective Filtering

• Filters are intended to remove noise and interference– Maximize the amount that is removed

• Filters will modified the spectrum of the signal of interest in some way– Minimize any signal distortion that may be caused by the filter.

• The above restrictions can usually not both be performed!– Compromise between the two!!

See Figure 3.1-7 on p. 85.

See Matlab Example35ECE

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Filter Terminology

• Passband– Frequencies where signal is

meant to pass

• Stopband– Frequencies where some defined

level of attenuation is desired

• Transition-band– The transitions frequencies

between the passband and the stopband

• Filter Shape Factor– The ratio of the stopband

bandwidth to the passband bandwidth

PB

SB

BW

BWSF

PBBW

SBBW

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

Typical amplitude ratio of a real bandpass filter

Real Bandpass Filter

The -3 dB or half-power bandwidth is shown

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Butterworth Low Pass Filter

• Maximally Flat, Smooth Roll-off, Constant 3dB point for all orders

𝐻 𝑗𝑤 ⋅ 𝐻 −𝑗𝑤 = 𝐻𝑗𝑤 =1

1 + 𝑤𝑤

𝐻 𝑠 =1

1 + 𝑠𝑗 ⋅ 𝑤

=1

1 + −𝑗 ⋅ 𝑠𝑤

=1

1 + −1 𝑠𝑤

M.E. Van Valkenburg, Analog Filter Design, Oxford Univ. Press, 1982. SBN: 0-19-510734-9

10-1

100

101

102

103

-120

-100

-80

-60

-40

-20

0

Butterworth Filter Family

Frequency (normalized)

Att

enua

tion

(dB

)

1st order

2nd order

3rd order4th order

5th order

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Pulse Response and Risetime

• Low Pass Filters cause sharp signal edges to be smoothed.

• The amount of smoothing is based on the bandwidth of the filter– More smoothing smaller bandwidth

• Fourier relationship:– a narrow rect function in time results in a broad (wide bandwidth)

sinc function in frequency

– a wide rect function in time results in a narrow (small bandwidth) sinc function in frequency

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Hilbert Transform

• It is a useful mathematical tool to describe the complex envelope of a real-valued carrier modulated signal in communication theory.

• The precise definition is as follows:

http://en.wikipedia.org/wiki/Hilbert_transform

dt

x1

t

1txtx̂

t

1thQ

f0j

0f,0

0f,j

fsgnjfHQ

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Hilbert Transform Real to Complex Conversion

• Original Real

• Hilbert Transform Complex

fXtx

fXfjjfXtxjtxthtc sgnˆ

fXffXtxjtxthtc sgnˆ

0,0

0,2ˆ

ffor

fforfXtxjtxthtc

The Hilbert Transform can be used to create a single sided spectrum! The complex representation of a real signal.ECE

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Linear CW Modulation

• Chapter 4: Linear CW Modulation– Bandpass signals and systems

– Double-sideband amplitude modulation

– Modulation and transmitters

– Suppressed-sideband amplitude modulation

– Frequency conversion and demodulation

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MATLAB AM

• AM= Ac * (1+μ*m(t))*cos(2π*fc*t)

• DSB= Ac * m(t)*cos(2π*fc*t)

43

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Message

Time (sec)

Am

plitu

de

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-1.5

-1

-0.5

0

0.5

1

1.5AM Waveform

Time (sec)

Am

plitu

de

message AMECE 4600/5640

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A family of AM Waveforms

• Double-Sideband, Carrier (AM in text, AM radios)

• Double-Sideband, Suppressed Carrier (DSB in text)

tf2costAtv c

tmAtA c

tm1AtA c

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Conventional AM

• Baseband

• Bandpass

• Fourier Domain

tm1AtA c

tf2costmAtf2cosA

tf2costm1Atv

cccc

cc

ccc

ccc ffMffM

2

Affff

2

AfV

carrier message

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AM Sidebands

• Assume that the message is a cosine wave– messages are typically bounded by +/- 1.0

• The spectral response becomes

tf2costm m

mcmcc

mcmcc

ccc

ffffff4

A

ffffff4

A

ffff2

AfV

mm ffff2

1fM

carrier

sidebands

sidebandsECE 4600/5640

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AM Positive Frequencies

• For the positive frequency segment of the spectrum

mcc

cc

mcc fff

4

Aff

2

Afff

4

AfV

Signal carrier with two sidebands

Lower sideband Upper sidebandCarrier

Ma

gcfmc ff mc ff

2Ac

4Ac

4Ac

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AM Modulation index

• For normalized message signals, the magnitude of the modulation index , , defines whether there is a “phase reversal” of the carrier wave.– =1, 100% modulation index, if the message ever becomes -1, the

baseband signal will become zero

– <1, 0-99.9% modulation index, if the message is bounded by +/-1, the baseband signal is always greater than zero

– >1, the baseband signal will have regions that are negative; thereby, causing a phase reversal of the carrier for the negative values. Envelope distortion will occur.

tm1AtA c

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DSB (with Suppressed Carrier)

• Baseband

• Bandpass

• Fourier Domain

tmAtA c

tf2costmAtv cc

ccc ffMffM

2

AfV

Message only no carrier !ECE 4600/5640

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DSB Positive Frequencies

• For the positive frequency segment of the spectrum

mcc

mcc fff

4

Afff

4

AfV

No signal carrier, two sidebands

Lower sideband Upper sideband

Ma

g

cfmc ff mc ff

4Ac

4Ac

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Phasor Analysis AM

• Given a tone message …

• A positive frequency phasor can be defined and drawn

𝑠 𝑡 =𝐴

2⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 +

𝐴

2⋅ 𝜇 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 − 𝑓 ⋅ 𝑡 +

𝐴

2⋅ 𝜇 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 + 𝑓 ⋅ 𝑡

𝑠 𝑡 = 𝐴 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 +𝐴

2⋅ 𝜇 ⋅ cos 2𝜋 ⋅ 𝑓 − 𝑓 ⋅ 𝑡 +

𝐴

2⋅ 𝜇 ⋅ cos 2𝜋 ⋅ 𝑓 + 𝑓 ⋅ 𝑡

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Phasor Analysis DSB

• Given a tone message …

• A positive frequency phasor can be defined and drawn

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Power in AM

• Assume that the message is an independent, zero mean, random process

2

2

22222

2

22 2coslim2coslim

T

Tc

Tc

T

Tc

Tcs dttftmAdttfAP

Carrier and two

symmetric subbands

𝑆 = 𝑃 = 𝑠 𝑡 = 𝐴 ⋅ 1 + 𝜇 ⋅ 𝑚 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

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DSB Power Levels

• All the power is in the message subbands

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Linear CW Radios

Modulation

• Conventional AM– Multiply and sum

– Nonlinearity

• DSB– Multiply

– AM sum and difference

Demodulation

• Conventional AM– Noncoherent Envelope

Detector

– Coherent Multiply and highpass filter

– Complex Mix & HPF

• DSB– Coherent Multiply

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Frequency Demodulation

• Non-Coherent Demodulation– An unknown phase, find another way …

– Envelope detection, lose ½ the signal power

– Advanced version, use a diode bridge rectifier instead of a single diode

• Coherent Demodulation– A known phase, filter and your done.

– The carrier frequency can be used to provide a local oscillator, after very narrowband filtering

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

(a) Circuit; (b) Waveforms

Envelope detection

In-class example for homework

- Picking R’s and C’s

AM signal spectrum centered at an intermediate frequency

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Coherent AM Demodulator

• Carrier extracted from modulated signal– Filter isolates the carrier or pilot tone

– Mix by phase-coherent sinusoid

– Note: a better method uses a phase-lock loop to lock onto the carrier (coming soon)

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Second MethodWeaver’s SSB modulator

59

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

𝑥 𝑡 = cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ ℎ 𝑡 ∗ 𝑥 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

𝑥 𝑡 = sin 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ ℎ 𝑡 ∗ 𝑥 𝑡 ⋅ sin 2𝜋 ⋅ 𝑓 ⋅ 𝑡

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Equivalent Complex Operation

• Complex Mixing of signal band to “zero” intermediate frequency (IF)

• LPF replaces IF LPF or HPF at IF frequency

• Complex up conversion from “zero” IF to RF

tf2jexp 1

tx

tf2jexp 2

tx c

𝑥 𝑡 = exp −𝑗 ⋅ 2𝜋 ⋅ 𝑓 + 𝑓 ⋅ 𝑡 ⋅ ℎ 𝑡 ∗ 𝑥 𝑡 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡

Weaver Modulator Spectrum

• Complex Mix

• Baseband LPF

• Complex Mix – then keep the real part

61