ECEN4523 Commo Theory Lecture #10 9 September 2015 Dr. George Scheets n Read Chapter 3.6 – n Problems: 3.4-1, 3.5-3, n Quiz #3, 18 September n Quiz #2 Results u Hi = 10, Low = 0.8, Ave = 4.88, σ = 3.82

ECEN4523 Commo Theory Lecture #11 11 September 2015 Dr. George Scheets n Read Chapter thru 3.8 n Problems: , 4, & 5 n Quiz #3, 18 September

100 KHz Sinusoids n A cosine (or sine) at any phase angle and amplitude can be constructed by adding together an appropriately sized cosine and sine of the same frequency. u Left image: volt peak cosine delayed by degrees u Right image: volt peak cosine volt peak sine

Autocorrelation n Autocorrelations deal with predictability over time. I.E. given an arbitrary point x(t1), how predictable is x(t1+τ)? time Volts t1 τ

Autocorrelation n Given an some waveform x(t), how similar is a shifted version x(t+τ)? time x(t) time x(t+τ)

Discrete Time Autocorrelation R x (τ) = 1/(N-|τ|) x(i)x(i+τ) R x (τ) = 1/(N-|τ|) x(i)x(i+τ) x(1)x(2)x(3)x(100) x(1)x(2)x(3)x(100) Example: R x (0) for discrete time signal Sum up x(1) 2 + x(2) x(100) 2, Then take a 100 point average. x(i) x(i+0) ∑ i=1 N-τ...

Discrete Time Autocorrelation x(1)x(2)x(3)x(100) x(i) x(1)x(2)x(3)x(100) x(i+1) x(99) Example: R x (1) for discrete time signal Sum up x(1)x(2) + x(2)x(3) x(99)x(100), Then take a 99 point average....

Discrete Time Autocorrelation x(1)x(2)x(3) x(100) x(i) x(1)x(2)x(3)x(100) x(i-1) x(99) Example: R x (-1) for discrete time signal Sum up x(1)x(2) + x(2)x(3) x(99)x(100), Then take a 99 point average....

Discrete Time Autocorrelation x(1)x(2)x(3)x(100) x(1)x(2)x(99)x(98) Example: R x (2) for discrete time signal Sum up x(1)x(3) + x(2)x(4) x(98)x(100), Then take a 98 point average. x(i) x(i+2)...

Autocorrelation Characteristics n If the average is positive... u Then x(t) and x(t+τ) tend to be alike Both positive or both negative n If the average is negative u Then x(t) and x(t+τ) tend to be opposites If one is positive the other tends to be negative n If the average is zero u There is no predictability

255 point discrete time Noise waveform (Adjacent points are independent) time Volts 0 V dc = 0 v, Normalized Power = 1 watt If true continuous time White Noise, no predictability.

Autocorrelation Estimate of Discrete Time White Noise tau (samples) Rxx 0

255 point Noise Waveform (Low Pass Filtered White Noise) Time Volts 23 points 0

Autocorrelation Estimate of Low Pass Filtered White Noise tau samples Rxx 0 23

R xx (0) n The sequence x(n) x(1) x(2) x(3)... x(255) n multiply it by the unshifted sequence x(n+0) x(1) x(2) x(3)... x(255) n to get the squared sequence x(1) 2 x(2) 2 x(3) 2... x(255) 2 n Then take the time average [x(1) 2 +x(2) 2 +x(3) x(255) 2 ]/255 n Estimate of waveform's POWER

R xx (1) n The sequence x(n) x(1) x(2) x(3)... x(254) x(255) n multiply it by the shifted sequence x(n+1) x(2) x(3) x(4)... x(255) n to get the sequence x(1)x(2) x(2)x(3) x(3)x(4)... x(254)x(255) n Then take the time average [x(1)x(2) +x(2)x(3) x(254)x(255)]/254

Examples of Amplified Noise Radio Static (Thermal Noise) Radio Static (Thermal Noise) Analog TV "snow" Analog TV "snow" 2 seconds of White Noise