6.02 Fall 2014 Lecture #8 • Noise • PDF’s, means, variances, Gaussian noise • Bit error rate for bipolar signaling
6.02 Fall 2014 Lecture 8, Slide #1 Single Link Communication Model End-host Original source computers Receiving app/user
Digitize Render/display, (if needed) etc. Source binary digits (“message bits”) Source coding Source decoding Bit stream Bit stream Channel Channel Mapper Recv Decoding Coding Bits + samples Bits Signals (reducing or (bit error Xmit (Voltages) + removing correction) samples over Demapper physical link bit errors)
6.02 Fall 2014 Lecture 8, Slide #2 From Baseband to Modulated Signal, and Back codeword bits in generate x[n] 1001110101 digitized modulate DAC symbols
NOISY & DISTORTING ANALOG CHANNEL
demodulate sample & ADC 1001110101 & filter y[n] threshold codeword bits out
6.02 Fall 2014 Lecture 8, Slide #3 Mapping Bits to Samples at Transmitter codeword bits in generate x[n] 1001110101 digitized L=16 samples per bit symbols
On-Off Signaling ‘0’à0 volts ‘1’àV volts
1 0 0 1 1 1 0 1 0 1 6.02 Fall 2014 Lecture 8, Slide #4 Samples after Processing at Receiver codeword bits in generate 1001110101 digitized modulate DAC symbols x[n]
NOISY & DISTORTING ANALOG CHANNEL
y[n] Assuming no noise, ADC demodulate filter only end-to-end distortion
y[n]
6.02 Fall 2014 Lecture 8, Slide #5 Mapping Samples to Bits at Receiver codeword bits in generate 1001110101 digitized modulate DAC symbols n = sample index x[n] j = bit index NOISY & DISTORTING ANALOG CHANNEL y[n] b[j]
demodulate sample & 1001110101 ADC & filter threshold codeword bits out 16 samples per bit b[j]
1 0 0 1 1 1 0 1 0 1 Threshold
6.02 Fall 2014 Lecture 8, Slide #6 For this lecture and next, assume no distortion, only additive zero-mean noise
• Received signal: y[n] = x[n] + w[n] i.e., received sample y[n] = the transmitted sample x[n] + zero-mean noise w[n] on that sample Assume noise is iid (independent and identically distributed at each n) • Signal-to-Noise Ratio (SNR) – usually denotes the ratio of (time-averaged or peak) signal power, i.e., squared amplitude of x[n] to noise variance, i.e., expected squared amplitude of w[n]
6.02 Fall 2014 Lecture 8, Slide #7 Noise Characterization: From Histogram to PDF
Experiment: create histograms of sample values from independent trials of increasing lengths. Histogram typically “converges” to a shape that is known – after normalization to unit area – as a probability density function (PDF)
6.02 Fall 2014 Lecture 8, Slide #8 Using the PDF in Probability Calculations
We say that W is a continuous random variable governed by the PDF ≥ fW(w) 0 if W takes on a numerical value in the range of w1 to w2 with a probability calculated from the PDF of W as:
w p(w < W < w ) = 2 f (w)dw 1 2 ∫ w W 1
fW(w)
w w1 w2 A PDF is not a probability – its associated integrals are. Note that the PDF is nonnegative, and the area under it is 1.
6.02 Fall 2014 Lecture 8, Slide #9 Mean & Variance of Continuous r.v. W
fW(w)
σw
W μ The mean or expected value W is defined and computed as: ∞ µ = w f (w)dw W ∫ −∞ W
2 The variance σW is the expected squared variation or deviation of the random variable around the mean, and is thus computed as:
∞ E[(W − µ )22] ] = σ 2 = (w − µ )2 f (w)dw W W ∫ −∞ W W
The square root of the variance is the standard deviation, σW 6.02 Fall 2014 Lecture 8, Slide #10 Mean of a Function a(W) : ∞ E[a(W )] = a(w) f (w)dw ∫ −∞ W
Mean of a Sum b(V) + c(W) : E[b(V)+ c(W )] = E[b(V)]+ E[c(W )]
Mean of b(V) + c(W) + d(V)e(W) when V, W are independent : E[b(V)+ c(W )+ d(V)e(W )] = E[b(V)]+ E[c(W )]+ E[d(V)]E[e(W )]
6.02 Fall 2014 Lecture 8, Slide #11 The Gaussian Distribution
A Gaussian random variable W with mean μ and variance σ2 has a PDF described by
2 −(w−µ) 1 2σ 2 fW (w) = e 2πσ 2
6.02 Fall 2014 Lecture 8, Slide #12 The Q(.) Function --- Area in the tail of a zero-mean unit-variance Gaussian
1 1 v2/2 Q(t)= e dv p2⇡ Zt Q( t)=1 Q(t)
2 2 t e t /2 1 e t /2
6.02 Fall 2014 Lecture 8, Slide #13 Tail probability of a general Gaussian in terms of the Q(.) function
1 1 (v µ)2/(2 2) e dv 2 p2⇡ t Z t µ = Q ⇣ ⌘
6.02 Fall 2014 Lecture 8, Slide #14 The Ubiquity of Gaussian Noise The net noise observed at the receiver is often the sum of many small, independent random contributions from many factors. Under fairly mild conditions, the Central Limit Theorem says their sum will be a Gaussian. The figure below shows the histograms of the results of 10,000 trials of summing 100 random samples drawn from [-1,1] using two different distributions.
1 Triangular Uniform 0.5 PDF PDF -1 1 -1 1 6.02 Fall 2014 Lecture 8, Slide #15 Distinguishing “1” from “0”
• Assume bipolar signaling:
Transmit L samples x[.] at +Vp (=V1) to signal a “1”
Transmit L samples x[.] at –Vp (=V0) to signal a “0”
• Simple-minded receiver: take a single sample value y[nj] at an appropriately chosen instant nj in the j-th bit interval. Decide between the following two hypotheses:
y[n] = +Vp + w[n] (==> “1”) or
y[n] = –Vp + w[n] (==> “0”) where w[n] is Gaussian, zero-mean, variance σ2
6.02 Fall 2014 Lecture 8, Slide #16 Connecting the SNR and BER
P(“0”)=0.5 P(“1”)=0.5 µ=V V E µ=-Vp p p = S σ= σ σ= σnoise noise 2 2σ = N0 -Vp +Vp
2E BER = P(error) = Q( S ) N0
6.02 Fall 2014 Lecture 8, Slide #17 In terms of erfc (as in notes, kind of ugly!)
P(“0”)=0.5 P(“1”)=0.5 µ=V µ=-Vp p Vp = ES σ= σ σ= σnoise noise 2 2σ = N0 -Vp +Vp
1 E 1 V BER = P(error) = erfc( S ) = erfc( p ) 2 N0 2 σ 2 6.02 Fall 2014 Lecture 8, Slide #18 SNR Example
Changing the amplification factor (gain) A leads to different SNR values: • Lower A → lower SNR • Signal quality degrades with lower SNR
6.02 Fall 2014 Lecture 8, Slide #19 10logX X Signal-to-Noise Ratio (SNR) 100 10000000000 90 1000000000 The Signal-to-Noise ratio (SNR) is useful in 80 100000000 judging the impact of noise on system 70 10000000 performance: 60 1000000 P 50 100000 SNR = signal 40 10000 30 1000 Pnoise 20 100 SNR for power is often measured in 10 10 decibels (dB): 0 1 ! P $ -10 0.1 signal -20 0.01 SNR (dB) =10log10 # & # & -30 0.001 " Pnoise % -40 0.0001 -50 0.000001 Caution: For measuring ratios of amplitudes rather than powers, take -60 0.0000001 -70 0.00000001 20 log10 (ratio). -80 0.000000001 -90 0.0000000001 3.01db is a factor of 2 -100 0.00000000001 6.02 Fall 2014 in power ratio Lecture 8, Slide #20