communications ii lecture 7: performance of digital modulationkkleung/communications2...psk can be...
TRANSCRIPT
![Page 1: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/1.jpg)
Communications II
Lecture 7: Performance of digital modulation
Professor Kin K. LeungEEE and Computing Departments
Imperial College London© Copyright reserved
![Page 2: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/2.jpg)
© KKL 2
Outline
• Digital modulation and demodulation
• Error probability of ASK
• Error probability of FSK
• Error probability of PSK
• Noncoherent demodulation
• Reference: Lathi, Chap. 13.
![Page 3: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/3.jpg)
© KKL 3
Bandpass Data Transmission
How do we transmit symbols 1 and 0?
• Amplitude Shift Keying (ASK)
• Frequency Shift Keying (FSK)
• Phase Shift Keying (PSK)
![Page 4: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/4.jpg)
© KKL 4
ASK, FSK and PSK
![Page 5: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/5.jpg)
© KKL 5
How do we recover the transmitted symbol?
• Coherent (synchronous) detection Use a BPF to reject out-of-band noiseMultiply the incoming waveform with a cosine of the carrier frequency
Use a LPF Requires carrier regeneration (both frequency and phase synchronization by using a phase-lock loop)
• Noncoherent detection (envelope detection etc.) Makes no explicit efforts to estimate the phase
![Page 6: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/6.jpg)
© KKL 6
Amplitude shift keying (ASK) = on-off keying (OOK)
s0(t) = 0 s1(t) = A cos(2 fct)
Ors(t) = A(t) cos(2 fct), A(t) ∈ {0, A}
![Page 7: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/7.jpg)
© KKL 7
Coherent Detection
Assume an ideal band-pass filter with unit gain on [fc −W, fc +W ]. For a practical band-pass filter, 2W should be interpreted as the equivalent bandwidth.
![Page 8: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/8.jpg)
© KKL 8
Pre-detection signal:
)2sin()()2cos()]()([)2sin()()2cos()()2cos(
)()()(
tftntftntAtftntftntfA
tntstx
CSCC
CSCCC
![Page 9: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/9.jpg)
© KKL 9
• After multiplication with 2 cos(2 fct):
• After low-pass filtering:
)4sin()())4cos(1)](()([)2cos()2sin(2)()2(cos2)]()([)( 2
tftntftntAtftftntftntAty
CSCC
CCSCC
)()()(~ tntAty C (166)
![Page 10: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/10.jpg)
© KKL 10
Reminder: The in-phase noise component nc(t) has the same variance as the original band-pass noise n(t) The received signal (166) is identical to the received signal (147) for baseband digital transmission
The sample values of will have PDFs that are identical to those of the baseband case
For ASK the statistics of the receiver signal are identical to those of a baseband system
)(~ ty
![Page 11: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/11.jpg)
© KKL 11
The probability of error for ASK is the same as for the baseband case
Assume equiprobable transmission of 0s and 1s.Then the decision threshold must be A/2 and the probability of error is given by:
2,AQP ASKe
![Page 12: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/12.jpg)
© KKL 12
Phase shift keying (PSK)
s(t) = A(t) cos(2 fct), A(t) ∈ {−A, A}
Use coherent detection again, to eventually get the detection signal:
)()()(~ tntAty C
![Page 13: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/13.jpg)
© KKL 13
Probability density functions for PSK for equiprobable 0s and 1s in noise:
(a): symbol 0 transmitted(b): symbol 1 transmitted
![Page 14: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/14.jpg)
© KKL 14
Use threshold 0 for detection
Conditional error probabilities:
dnAnPe
0 2
2
0 2)(exp
21
dnAnPe
0
2
2
1 2)(exp
21
![Page 15: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/15.jpg)
© KKL 15
In the first set
In the second set
AnnnddnAnn ~,0whenand~~
ndnPAe
~2
~exp
21
2
2
0
:~,,~,0whenand~)(~
nnwhenandAnn
nddnAnAnn
ndn
ndnP
A
A
e
~2
~exp
21
~)1(2
~exp
21
2
2
2
2
1
![Page 16: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/16.jpg)
© KKL 16
So:
Change variable of integration to z ≡ n/σ ⇒ dn = σdz and when n = A, z = A/σ. Then:
Remember that
dnnPPPAPSKeee
2
2
,10 2exp
21
AQdzeP Az
PSKe2/
,
2
21
dttxQx
)2/exp(21 2
(172)
![Page 17: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/17.jpg)
© KKL 17
Frequency Shift Keying (FSK)
s0(t) = A cos(2 f0t), if symbol 0 is transmitted s1(t) = A cos(2 f1t), if symbol 1 is transmitted
Symbol recovery:Use two sets of coherent detectors, one operating at a frequency f0 and the other at f1.
![Page 18: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/18.jpg)
© KKL 18
Coherent detector for FSK
The two BPF’s are non-overlapping in frequency spectrum
![Page 19: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/19.jpg)
© KKL 19
Each branch = an ASK detector
presentnot symbol if noise
present symbol if noiseA branch each onoutput LPF
![Page 20: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/20.jpg)
© KKL 20
n0(t): the noise output of the top branchn1(t): the noise output of the bottom branch
Each of these noise terms has identical statistics to n(t).
Output if a symbol 1were transmittedy1(t) = A + [n1(t) − n0(t)]
Output if a symbol 0 were transmittedy0(t) = -A + [n1(t) − n0(t)]
![Page 21: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/21.jpg)
© KKL 21
Set detection threshold to 0
Difference from PSK: the noise term is now n1(t) − n0(t).
The noises in the two channels are independent because their spectra are non-overlapping.
the variances add
the noise variance has doubled!
![Page 22: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/22.jpg)
© KKL 22
Probability of error for FSK:
Replace σ2 in (172) by 2σ2 (or σ by 2 σ )
2,AQP FSKe
![Page 23: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/23.jpg)
© KKL 23
Noise of the sum or difference of two independent zero mean random variables:
x1: a random variable with variance σ12
x2: a random variables with variance σ22
What is the variance of y ≡ x1 ± x2?
![Page 24: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/24.jpg)
© KKL 24
By definition, the variance of y ≡ x1 ± x2 is
For independent variables: E{x1x2} = E{x1}E{x2} For zero-mean random variables:E{x1} = E{x2} = 0 ⇒ E{x1x2} = 0
So
}{}{}{}2{
}){(
}{}{
2221
21
2221
21
221
222
xExxExExxxxE
xxE
yEyEy
22
21
22
21
2 }{}{
xExEy
![Page 25: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/25.jpg)
© KKL 25
Comparison of the three schemes
![Page 26: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/26.jpg)
© KKL 26
To achieve the same error probability (fixed Pe):
PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in amplitude)
FSK can be reduced by 3 dB compared with a baseband or ASK (a factor of 2 reduction in amplitude)
Caution: The comparison is based on peak SNR. In terms of average SNR, PSK only has a 3 dB improvement over ASK, and FSK has the same performance as ASK
![Page 27: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/27.jpg)
© KKL 27
Properties of the Q-function
Upper bounds and good approximations:
Becomes tighter for large x.
A better upper bound for small x.
0,21)( 2/2
xex
xQ x
0,21)( 2/2
xexQ x
![Page 28: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/28.jpg)
© KKL 28
![Page 29: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/29.jpg)
© KKL 29
Non-coherent detection
Accurate phase synchronization may be difficult in a dynamic channel.
Phase synchronization error is due to frequency drift, instability of the local oscillator, effects of strong noise ...
When the carrier phase is unknown, one must rely on non-coherent detection.
The phase θ is assumed to be uniformly distributed on [0, 2 ].
Circuitry is simpler, but analysis is more difficult!
![Page 30: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/30.jpg)
© KKL 30
ASK
![Page 31: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/31.jpg)
© KKL 31
Error analysis
When symbol 0 is sent, the envelope (noise alone) has Rayleigh distribution
When symbol 1 is sent, the envelope (signal + noise) has Riciandistribution
(180) dominates the error probability when A/σ >> 1.
0,)( )2/(2
22
rerrf r
0,)( 20)2/()(
2
222
rArIerrf Ar
(180)
![Page 32: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/32.jpg)
© KKL 32
Let the threshold be A/2 for simplicity.
The error probability (dominated by symbol 0) is given by
Cf. coherent demodulation
)8/(
2/
)2/(2
22
22
21
0,21
A
A
re
e
rdrerP
)8/(,
22
21
2
A
ASKe eAQP
![Page 33: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/33.jpg)
© KKL 33
FSK
![Page 34: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/34.jpg)
© KKL 34
Error probability
When a symbol 1 is sent, one branch has Rayleigh distribution, the other has Rice distribution.Error occurs if Rice < Rayleigh, and it can be shown that
Cf. coherent demodulation
)4/( 22
21 A
e eP
)4/(,
22
21
2
A
FSKe eAQP
![Page 35: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/35.jpg)
© KKL 35
DPSK: Differential PSK
It is impossible to demodulate PSK with an envelop detector, since PSK signals have the same frequency and amplitude.
We can demodulate PSK differentially, where phase reference is provided by a delayed version of the signal in the previous interval.
Differential encoding is essential: bn = bn−1 an, where an , bn ∈ 1.
![Page 36: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/36.jpg)
© KKL 36
Differential detection
Error probability
Cf. coherent demodulation
)2/(,
22
21 A
DPSKe eP
)2/(,
22
21
A
PSKe eAQP
![Page 37: Communications II Lecture 7: Performance of digital modulationkkleung/Communications2...PSK can be reduced by 6 dB compared with a baseband or ASK system (a factor of 2 reduction in](https://reader035.vdocuments.us/reader035/viewer/2022071604/61408e1c2e263e64232a2390/html5/thumbnails/37.jpg)
© KKL 37
Conclusions
Non-coherent demodulation retains the hierarchy of performance.
Non-coherent demodulation has error performance slightly worse than coherent demodulation, but approaches coherent performance at high SNR.
Non-coherent demodulators are considerably easier to build.