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7/31/2019 Bandpass Signalling
1/16
Eeng 360 1
Chapter4
Bandpass Signalling Bandpass Filtering and Linear Distortion
Bandpass Sampling Theorem Bandpass Dimensionality Theorem
Amplifiers and Nonlinear Distortion
Total Harmonic Distortion (THD)
Intermodulation Distortion (IMD)
Huseyin Bilgekul
Eeng360 Communication Systems IDepartment of Electrical and Electronic Engineering
Eastern Mediterranean University
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Bandpass Filtering and Linear DistortionEquivalent Low-pass filter: Modeling a bandpass filter by using an equivalent lowpass filter (complex impulse response)
])(Re[)( 11tjwcetgtv ])(Re[)( 22
tjwcetgtv
])(Re[)( 11tjwcetkth
*1 1( ) ( ) ( )
2 2
c cH f K f f K f f
cc ffGffGfV *2
1)( tj cetgtv )(Re
)(1 tv
)(2 tv
)(1 th
)( fH
Input bandpass waveform
Output bandpass waveform
Impulse response of the bandpass filter
Frequency response of the bandpass filter
H(f) = Y(f)/X(f)
Bandpass filter
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Eeng 360 3
Bandpass Filtering
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Bandpass Filtering
;2
1
2
1
2
112 tktgtg
fKfGfG2
1
2
1
2
112
fHfVfV 12
cc ffGffG *222
1
cccc ffKffKffGffG **
112
1
2
1
cccc
cccc
ffKffGffKffG
ffKffGffKffG
**
1
*
1
*
11
4
1
,0*1 cc ffKffG
Theorem:
g1(t)complex envelope of inputk(t)complex envelope of impulse response
Also,
Proof: Spectrum of the output is
Spectra of bandpass waveforms are related to that of their complex enveloped
But
.0*1
cc ffKffG
cccccc ffKffGffKffGffGffG
**
11
*
22
2
1
2
1
2
1
2
1
2
1
2
1
The complex envelopes for the input, output, and impulse response of
a bandpass filter are related by
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Bandpass Filtering
fKfGfG2
1
2
1
2
112
Taking inverse fourier transform on both sides
;21
2
1
2
112 tktgtg
Thus, we see that
Any bandpass filter may be described and analyzed by using an equivalent low-pass
filter.
Equations for equivalent LPF are usually much less complicated than those for
bandpass filters & so the equivalent LPF system model is very useful.
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Linear Distortion
AfH
gT
df
fd
2
1
0
2 gfTf
For distortionless transmission of bandpass signals, the channel transfer function
H(f) should satisfy the following requirements: fjefHfH
The amplitude response is constant
A- positive constant
The derivative of the phase response is constant
Tgcomplex envelope delay dfTfHf 2)( )(fHf
Integrating the above equation, we get
Are these requirements sufficient for distortionless transmission?
constantshiftphase0
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Linear Distortion
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gg fTjjfTj eAeAefH 22 00
ttyttxtv cc sincos1 gfTje
2
002 sincos gcggcg TtTtAyTtTtAxtv
ccgccg ftTtAyftTtAxtv sincos2
dcgcc TfTf 20
dcgdcg TtTtAyTtTtAxtv sincos2
delayphaseTshiftphasecarrier;2 d cdcc fTff
Linear Distortion
The channel transfer function is
fjefHfH
If the input to the bandpass channel is
Then the output to the channel (considering the delay Tg due to ) is
Using 00
2 gg TfTf
dfTfHf 2)(
Modulation on the carrier is delayed by Tg & carrier by Td
Bandpass filterdelays input info by
Tg , whereas the
carrier by Td
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Bandpass Sampling Theorem
Ts Bf 2
ttyttxtvcc
sincos
212 fffc
n
n bb
bbc
b
c
b fntf
fntft
f
nyt
f
nxtv
sinsincos
bfnx bfny
If a waveform has a non-zero spectrum only over the interval , where
the transmission bandwidth BT
is taken to be same as absolute BW, BT
=f2
-f1
, then
the waveform may be reproduced by its sample values if the sampling rate is
21 fff
Theorem:
Quadrature bandpass representation
Let fc be center of the bandpass:
x(t) andy(t) are absolutely bandlimited to B=BT/2
The sampling rate required to represent the baseband signal is Tb BBf 2
Quadrature bandpass representation now becomes
Where and samples are independent , two sample values
are obtained for each value of n
Overall sampling rate for v(t): Tbs Bff 22
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Bandpass Dimensionality Theorem
02 TN B T
Assume that a bandpass waveform has a nonzero spectrum only over a frequency
interval , where the transmission bandwidth BT is taken to be the absolute
bandwidth given by BT=f2-f1 and BT
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Received Signal Pulse
tj cetgts Re
tnthtstr
tneTtAgtr cc ftjg Re
tnetgtr tj c Re
The signal out of the transmitter
Transmission
medium(Channel)
Carrier
circuits
Signal
processing
Carrier
circuits
Signal
processing
Information
minput m~)(
~ tg)(tr)(ts)(tg
g(t)Complex envelope ofv(t)
If the channel is LTI , then received signal + noise
n(t)Noise at the receiver input
Signal + noise at the receiver input
Signal + noise at the receiver input
- carrier phase shift caused by the channel, Tgchannel group delay.)( cf
Again of the channel
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Amplifiers
Non-linear Linear
Circuits with memory and circuits with no memory
Memory - Present output value ~ function of present input + previous input values
- contain L & C
No memory - Present output values ~ function only of its present input values.
Circuits : linear + no memoryresistive ciruits
- linear + memoryRLC ciruits (Transfer function)
Nonlinear Distortion
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Nonlinear DistortionAssume no memoryPresent output as a function of present input in t domain
tKvtv i0 K- voltage gain of the amplifier
If the amplifier is linear
In practice, amplifier output becomes saturated as the amplitude of
the input signal is increased.
0
2
2100
n
n
inii vKvKvKKv
output-to-input characteristic (Taylors expansion):
Where
0
0
!
1
iv
n
i
n
ndv
vd
nK
0K
ivK1
2
2 ivK
- output dc offset level
- 1st order (linear) term
- 2nd order (square law) term
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tAtvi 00
sin
tAK
tAK 0
2
022
002 2cos1
2
sin
3032021010 3cos2cos)cos( tVtVtVVtvout
100V
THD(%)1
2n
2
nV
Nonlinear Distortion
Let the input test tone be represented by
Harmonic Distortion associated with the amplifier output:
Then the second-order output term is
In general, for a single-tone input, the output will be
Vnpeak value of the output at the frequency nf0
2
2 ivK =
To the amplifier input
The Percentage Total Harmonic Distortion (THD) of an amplifier is defined by
2nd Harmonic
Distortion with2
2
02AK
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Nonlinear Distortion
Intermodulation distortion (IMD) of the amplifier:
If the input (tone) signals are tAtAtvi 2211 sinsin
Then the second-order output term is
tAKttAAKtAK
tAttAAtAKtsnAtAK
222
2221212122
12
2
22
221211
22
12
2
22112
sinsinsin2sin
sinsinsin2sinsin
IMDHarmonic distortion at 2f1 & 2f2
Second-order IMD is:
212121221212 coscossinsin2 tAAKttAAK
l
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Nonlinear Distortion
Third order term is
)sinsinsin3
sinsin3sin(
sinsin
2
33
22
2
1
2
21
21
2
2
2
11
33
13
3
22113
3
3
tAttAA
ttAAtAK
tAtAKvK i
ttAAKttAAK 1222
1321
2
2
2
13 2cos1sin2
3sinsin3
tttAAK 212122213 2sin2sin
21sin
23
tttAAK
ttAAK
12121
2
213
2
2
1
2
213
2sin2sin2
1sin
2
3
sinsin3
The third term is
The second term (cross-product) is
Intermodulation terms at nonharmonic frequencies
For bandpass amplifiers, wheref1 &f2 are within the pasband,f1 close tof2,
the distortion products at 2f1+f2 and 2f2+f1 ~ outside the passband
Main Distortion Products