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
ECE 4710: Lecture #16 2
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
ECE 4710: Lecture #16 3
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
ECE 4710: Lecture #16 4
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
ECE 4710: Lecture #16 5
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)()(
ECE 4710: Lecture #16 6
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
ECE 4710: Lecture #16 7
AM Signal Spectrum
AM Bandpass Spectrum )]([)( tsfS F
)]()()()([)( 21
ccccc ffMffffMffAfS
LSB + USB = DSB
LC
ECE 4710: Lecture #16 8
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
ECE 4710: Lecture #16 9
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)
ECE 4710: Lecture #16 10
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)(
ECE 4710: Lecture #16 11
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
ECE 4710: Lecture #16 12
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
ECE 4710: Lecture #16 13
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
ECE 4710: Lecture #16 14
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
ECE 4710: Lecture #16 15
Analog Filter Types
ECE 4710: Lecture #16 16
Analog Filter Types
ECE 4710: Lecture #16 17
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