ece 4710: lecture #16 1 bandpass spectrum spectrum of bandpass signal is directly related to...

ECE 4710: Lecture #16 1

Bandpass Spectrum

Spectrum of bandpass signal is directly related to spectrum of complex envelope

We have already shown that

tfj cetgtv 2)(Re)(

)]()([)]([)( 21

cc ffGffGtvfV F

f

|)(| fV

fc-B fc fc+B

2A

-fc-B -fc -fc+B

2A

-B 0 +Bf

A |)(| fG


Bandpass PSD

PSD of bandpass signal is

Derivation is in book but this is intuitively correct since FT V/Hz so that PSD | FT |

2 W/Hz Average normalized power of bandpass waveform is

Bandpass power found from baseband signal representation g(t) if desired, otherwise from PSD

)]()([)( 41

cgcgv fffff PPP

2212 |)(|)0()()( tgRdfftvP vvv

P


Peak Envelope Power

Peak Envelope Power (PEP) is the average power that would be obtained if | g(t) | were held constant at its peak value Useful measure of power for high power Tx specifications

» AM Broadcast Radio, TV, etc.

Transmitters must be able to handle instantaneous signal power, e.g. peak, without saturating or being damaged» Average power does not provide any measure of what the worst-

case peak power may be

PEP given by 221 |])(|[max tgPPEP


AM Signal

General meaning of amplitude modulation the time variation of the amplitude of the carrier signal contains/represents the source information signal m(t) There are many types that meet the general definition

Amplitude Modulated (AM) signal is a specific case of the general class of amplitude modulated signals where This is used for AM broadcast radio and is also called

Double Side Band – Large Carrier DSB-LC

)](1[)( tmAtg c


AM Signal

AM Baseband Signal AM Bandpass Signal

AM signal g(t) is purely real since m(t) only represents amplitude information so

Using Euler’s Identity

So tftfjtfe ccc

tfj c 2cos2sin2cosRe Re 2

)](1[)( tmAtg c

tfj cetgts 2)(Re)(

XjXe jX sincos tfjtfj cc etgetgts 22 Re)()(Re)(

tftmAtftgts ccc 2cos)](1[2cos)()(


AM Signal Spectrum

AM Baseband Spectrum Table 2-2, pg. 64 : 1 (f) so

Ac represents DC power and constant carrier such that even if m(t) = 0 the carrier signal s(t) = Ac cos 2 fc t is always present DSB-LC

)]([)]([)( tmAAtgfG cc FF

)()()]([)( fMAfAtmAAfG cccc F

-B 0 +Bf

|)(| fM

0 f

)( fAc

-B 0 +Bf

|)(| fG


AM Signal Spectrum

AM Bandpass Spectrum )]([)( tsfS F

)]()()()([)( 21

ccccc ffMffffMffAfS

LSB + USB = DSB

LC


AM Signal Power

Using baseband signal g(t)

If DC power in source waveform m(t) is zero then 2m(t) = 0 No delta function in M(f )

Signal power is “wasted” on carrier does not contribute to S/N at Rx of the recovered information waveform LC enables extremely simple Rx circuit but AM is power ineffecient

)()(21)()(21

|)(1[||)(|)(

222122

21

2212

21

tmtmAtmtmA

tmAtgdffP

cc

cvAM

P

]1[])(1[)()(21 22122

2122

21

mcccAM PAtmAtmtmAP

)( signal modulatingin power andpower carrier 221 tmPA mc


Communication System

Goal: Design system to transmit information, m(t), with as little deterioration as possible within design constraints of signal power, signal bandwidth, and system cost

˜

Information Source

BasebandSignal

Processing

Modulation & CarrierCircuits

TransmissionChannel

Demodulation & CarrierCircuits

BasebandSignal

Processing

Information Sink

Noisen (t)

m (t) s (t) r (t) m (t)

Transmitter (Tx) Receiver (Rx)


Rx S+N

Model for received signal plus noise s(t) is signal out of transmitter

» Spectral response may be modified by channel» Noise added in channel

Thus the signal at Rx input is

If channel is distortion free (a big IF!!) then

» Constant amplitude (A) and linear phase (2f Tg)

tfj cetgts 2)(Re)(

input Rx @ noise )(

response impulse channel )( where)()()()(

tn

thtnthtstr c

c

gTfjj eAefH 20)(


Rx S+N

Distortion free received signal + noise is

Tg and (fc) must be estimated by Rx for digital signals Accomplished by bit synchronizer for digital signals Not necessarily required for analog signals (e.g. AM)

where)]()(Re[)( 2)( tneTtgAetr tfjg

fj cc

channelby causedshift phasecarrier )(

delay (envelope) group channel

1) A (normally gain channel

c

g

f

T

A


Rx S+N

A high performance Rx is designed to correct for Channel attenuation amplify signal Channel delay synchronization circuits Channel frequency distortion equalizing filter

If channel effects are largely corrected then

Uncompensated effects of channel spectral response can be included in g(t) if needed

This is a best-case approach and is not valid for some applications wireless mobile radio

)(])(Re[)( 2 tnetgtr tfj c


Analog Filters

Filters modify the spectral characteristics of an input signal to produce desired output signal

Variety of needs and applications Pulse shaping for minimizing BW Correcting for distortion caused by channel Selection of desired signals from specific frequencies Rejection of undesired signals and noise outside of desired signal BW

Filters classified by type of construction (LC, SAW, etc.) and by spectral response characteristics (Butterworth, Chebyshev, etc.)

Elements used to construct filter should have high Q


Analog Filters

Two Q types to describe filter quality Energy Storage Q

» LC circuit elements are imperfect and have some resistance which leads to energy dissipation via heat

» Desire high Q for individual circuit elements

Frequency Selective Q

» fo is resonant frequency (design center frequency) & B is 3-dB BW

» Measure of the filter’s overall ability to select desired frequency band

» Higher selectivity means narrower band filter on a % basis

cycleper dissipatedenergy cycle 1in storedenergy

2EQ

dB3

0B

fQFS


Analog Filter Types


Filter Responses

Butterworth Maximally flat response in passband Modest rolloff for attenuation response

Chebyshev Sharpest rolloff for minimum number of circuit elements 1-3 dB amplitude variation in passband ripple

Bessel Linear phase response in passband Distortion-free filter to preserve pulse shape

Raised Cosine Pulse shaping to minimize signal BW and no ISI

ece 4710: lecture #16 1 bandpass spectrum spectrum of bandpass signal is directly related to...

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