ece 4710: lecture #31 1 system performance chapter 7: performance of communication systems...
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ECE 4710: Lecture #31 1
System Performance
Chapter 7: Performance of Communication Systems Corrupted by Noise
Important Practical Considerations: Complexity vs. Cost Coherent vs. Non-Coherent Detection
Important Performance Measures: Signal BW Spectral Efficiency
Probability of Bit Error Pe or BER
Required S / N for given Pe digital systems only
» Analog systems output S / N only (no Pe)
ECE 4710: Lecture #31 2
System Performance
Shannon’s Channel Capacity
Defines S / N & spectral efficiency for specific Pe
Example: a digital modulation method with a S / N = 10 dB yields a 3 bps/Hz spectral efficiency @ Pe = 10-5
For a received signal corrupted by noise (channel + system) how do we determine the specific Pe for a given S / N ?
(bps/Hz) )1(logor )1(log 2BC
2 NS
NSBC
ECE 4710: Lecture #31 3
System Performance
Numerous methods for signal demodulation and detection Coherent vs. Non-Coherent Optimum vs. Sub-optimum
Optimum Maximize S / N and minimize Pe
» Usually coherent demodulation + specialized filtering/processing
Sub-optimum» Often done in order to lower cost practical consideration
Non-coherent Rx has simpler circuitry» Sometimes performance is very close to optimum Rx for practical
systems
ECE 4710: Lecture #31 4
Binary System
)(tm
Bandpass
SuperH LNA, Mixer, IF,
IF Filter + Amp, Detection, etc.
Bit Synch
Binary Decision / Detection
Noise causes bit errors to
occur !!
ECE 4710: Lecture #31 5
BER Evaluation
Develop general technique for determining Bit Error Rate (BER) for binary signaling
Transmitted bandpass (RF) signal over bit period T is
Baseband output signal (after RF/IF processing circuits) is
Baseband analog signal is corrupted by noise
"0"binary for ,)(
"1"binary for ,)()(
2
1
ts
tsts
"0"binary for ,)(
"1"binary for ,)()(
02
010 tr
trtr
ECE 4710: Lecture #31 6
BER Evaluation
Baseband analog waveform is sampled at some time to during bit interval:
For matched filter processing circuits to is usually to= T» End of bit period integration operation to average out signal
fluctuations and reduce impact of noise
For simple processing to is usually to= T/2» Middle of bit period maximum eye opening of line code
is a random variable whose probability density function (PDF) is continuous b/c the signal is corrupted by noise (channel, system, etc.)
"0"binary for ,)(
"1"binary for ,)()(
002
00100 tr
trtr
)( 00 tr
ECE 4710: Lecture #31 7
BER Evaluation
For simplified notation let so
is called the “test statistic”» Random variable with continuous PDF
Probability Density Function PDF
Statistical characterization of random variation For our purposes it is the random variation of received
signal (which contains noise) at sampling point t0
"0"binary for ,
"1"binary for ,)(
02
01000 r
rtrr
)( 000 trr
0r
ECE 4710: Lecture #31 8
Received signal + noise over one bit period
PDF is ensemble average of r0 (t0) values
t0 T
Avg Signal Strength Noise
Variation
t0 T
Avg Signal Strength
NoiseVariation
Avg Signal Strength
Noise Variation
V
Avg Signal Strength
Noise Variation
V
ECE 4710: Lecture #31 9
PDFs
Two PDFs one for each possible state, r01 or r02 , of received signal
Conditional PDFs depend on transmitted state Denote conditional PDFs as:
Functional shape of PDF depends largely on » Channel noise characteristics» Type of detector & filter circuits
"0"binary for ,
"1"binary for ,)(
02
01000 r
rtrr
011 when PDF sent) |( rrsrf oo
022 when PDF sent) |( rrsrf oo
ECE 4710: Lecture #31 10
Gaussian PDFs
Must set threshold voltage VT to detect binary data
r0 > VT “1” r0 < VT “0” Detection Decision :
Binary “1” Binary “0”
ECE 4710: Lecture #31 11
Bit Errors
Signal + Noise at Rx Errors occur in two ways for binary system:
If binary 1 is sent but If binary 0 is sent but
Probability of error is integration of conditional PDF over “tail regions” If binary 1 is sent If binary 0 is sent
r0 > VT “1” r0 < VT “0”
TV
o drsrfsP 011 )|(sent) |error1(
TV o drsrfsP 022 )|(sent) |error2(
ECE 4710: Lecture #31 12
Bit Error Rate
The rate of bit errors is the summation of the error type multiplied by the probability of the bit state
General expression for binary system
& are source statistics Most applications assume all states are equally likely For binary system then
2
1
)()|()(i
ii sPsEPEP
)(sent) |error2()(sent) |error1()(BER 2211 sPsPsPsPEPPe
T
T
V o
V
oe drsrfsPdrsrfsPP 022011 )|()()|()(BER
)( 2sP)( 1sP
21
21 )()( sPsP
ECE 4710: Lecture #31 13
Gaussian Noise
Shape of conditional PDFs depends on Channel noise characteristics Type of detector & filter circuits
In the absence of interference from other signals the channel noise typically has a Gaussian distribution Channel noise is Additive White Gaussian Noise (AWGN)
» Gaussian random noise process n(t) has flat PSD
» “White light” all colors of visible spectrum present
» “White noise” all frequencies (< B) present in noise process
otherwise ,0
||,)( 02
1
f
BfNfnP
ECE 4710: Lecture #31 14
AWGN
Channel noise is typically (not always) AWGN for wireless communication systems when no interference is present Not necessarily true for wired communication systems
Rx circuit acts upon input channel noise Baseband noise will be AWGN if the Rx is linear
(excluding threshold comparator)» SuperH with LNA, mixer, IF stage, & product detector can be linear» Not linear for Rx circuits with AGC, power limiters, non-linear
detectors (envelope detector), etc
ECE 4710: Lecture #31 15
AWGN
For AWGN channel noise + linear Rx circuit the sampled baseband binary signal is
where
s01 & s02 known constants for given Rx type and known input signal waveforms s1(t) and s2(t)
000 nsr
sent being signalon dependshat constant t a is and
variablerandomGaussian mean -zero is
)( , )(
0
0
000000
s
n
tnntrr
"0"binary for ,
"1"binary for ,
02
010 s
ss
AdditiveNoise!!
ECE 4710: Lecture #31 16
Sampled Output
Baseband noise is zero-mean Gaussian random variable
Sampled output r0 is a Gaussian random variable with a mean value of either s01 or s02 depending on whether binary 1 or binary 0 is sent
Gaussian function :
deviation standard is andmean is where
2
1)(
0
2/)(
0
20
2
x
exf xxx
11
)(xf