6.02 lecture 4 – signals and noise signals and...
TRANSCRIPT
6.02 Lecture 4 – Signals and Noise
• Signals and Noise– Sources of Noise– Small Eye more noise sensitivity
• Analyzing Noise– Decompose into Noisefree plus Noise– Noise Metrics (Sample Mean and Variance)
• Probability Density Functions– Connection to Histograms– Use in estimating bit error rates
6.02 Fall 2009
Noise Can Be Due to Fundamental Processes
6.02 Fall 2009
http://www4.nau.edu/meteorite/Meteorite/Images/Sodium_chloride_crystal.png
e-
Randomized path leads to noisy current flow
Electron Moving Through Crystal with Vibrating Atoms
Bits in
Effect of Many Interactions Can Be Modeled as Noise
6.02 Fall 2009
http://www.imageteck.net/PCB%20Board%203.JPG
Xmit
Many components connected by thin wires (that have inductance and resistance) to single power supply – 1000’s of devices switching on and off creates “noisy” power supply.
6.02 IR Transceiver Noise Source – Room Light
20 Samples/bit
http://www.clker.com/cliparts/0/6/b/6/11954240371585298002ryanlerch_lamp_outline.svg.med.png
Receiver Waveform
Eye Diagram
Threshold
6.02 Fall 2009
Slow channel eye diagram (40 samples/bit)
6.02 Fall 2009
Eye diagram generated from 40 samples per bit and using a 200 bit long random sequence.
Why is a small eye bad?
6.02 Fall 2009
Threshold
Ones
Zeros
Noisy Signal Can Cause Bit Errors
6.02 Fall 2009
Decompose Into Noisefree + Noise
6.02 Fall 2009
6.02 Fall 2009 Slide thanks to C. Sodini and M. Perrott
Experiment to see Statistical Distribution
• Create histograms of sample values from trials of increasing lengths
• Assumption of independence and stationarity implies histogram should converge to a shape known as a probability density function (PDF)
The Probability Density Function PDF
• Define x as a random variable whose PDF has the same shape as the histogram we just obtained
• Denote PDF of x as fX(x)– Scale fX(x) such that
its overall area is 1
This shape is referredto as a Gaussian PDF
Slide thanks to C. Sodini and M. Perrott6.02 Fall 2009
Formalizing Probability• The probability that random variable x takes on a
value in the range of x1 to x2 is calculated from the PDF of x as:
• Note that probability values are always in the range of 0 to 1– Higher probability values imply greater likelihood that
the event will occur
Slide thanks to C. Sodini and M. Perrott6.02 Fall 2009
Example Probability Calculation
• Verify that overall area is 1:
• Probability that x takes on a value between 0.5 and 1.0:
This shape is referred to as a uniform PDF
Slide thanks to C. Sodini and M. Perrott6.02 Fall 2009
Mean and Variance
• The mean of random variable x, , corresponds to its average value– Computed as
• The variance of random variable x, , gives an indication of its variability– Computed as
• The standard deviation of a random variable x, is denoted σ x
σ x2
μ x
Slide thanks to C. Sodini and M. Perrott6.02 Fall 2009
Visualizing Mean and Variance from PDF
• Changes in mean shift the center of mass of PDF• Changes in variance narrow or broaden the PDF
– Note that area of PDF must always remain equal to one
σ x
Slide thanks to C. Sodini and M. Perrott6.02 Fall 2009
Example Mean and Variance Calculation
• Mean:
• Variance:
Slide thanks to C. Sodini and M. Perrott6.02 Fall 2009
Summary - Estimating Bit Errors
• Assume Gaussian PDF for noise and no inter-symbol interference (ISI next time) yields the following picture:
• We can estimate the bit-error rate (BER) as
0 VV/2
p(xmit =0) = p0μ = 0σ = σNOISE
p(xmit=1) = p1μ = Vσ = σNOISE
p(rcv=1|xmit=0)=p01p(rcv=0|xmit=1) =p10
6.02 Fall 2009