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Today’s Schedule. Reading: Lathi 11.2-11.4 (also 7.2), Mini-Lecture 1: Power spectral densities Mini-Lecture 2: Random processes and linear systems Activity: First Project Group Meeting Project Discussion and Questions. Autocorrelation of Random Process.
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Today’s Schedule • Reading: Lathi 11.2-11.4 (also 7.2), • Mini-Lecture 1: • Power spectral densities • Mini-Lecture 2: • Random processes and linear systems • Activity: First Project Group Meeting • Project Discussion and Questions Dickerson EE422
Autocorrelation of Random Process • Correlation or second moment of a real random process with itself at two times: Dickerson EE422
Wide Sense Stationary • Usually, t1=t and t2=t+tso that t1- t2 =t • A random process is Wide-Sense Stationary if: • Properties Dickerson EE422
Power Spectral Density • Definition • Relationship to Time Autocorrelation • Power of a Random Process Dickerson EE422
Properties of PSD • Px(f) is always real • Px(f)> 0 • When x(t) is real, Px(-f)= Px(f) • If x(t) is WSS, Dickerson EE422
General Expression PSD • General expression for the PSD of a Digital Signal: • F(f) is the Fourier Transform of the Pulse Shape f(t) • Ts is the sampling interval • R(k) is the autocorrelation of the data: Dickerson EE422
Autocorrelation Term • an and ak+n are the levels of the nth and (n+k)th symbol positions • Pi is the probability of having the ith anan+k product Dickerson EE422
Digital PSD Discussion • PSD only depends on the • Pulse shape (f(t)) • Statistical properties of the data Dickerson EE422
Example: Unipolar NRZ • Possible levels are +A and 0 • Square pulses of width Tb • Find the PSD: • Find the spectrum of pulse: Dickerson EE422
Evaluate Autocorrelation • For k = 0: there are 2 possibilities, an=A or an=0: • For k >0: there are 4 possibilities, an=0 orA and an+k=0 or A: Dickerson EE422
Putting it all together • Simplify using: • Poisson Sum Formula Dickerson EE422
Activity • Find the PSD for BiPolar NRZ Signaling • Find F.T. of pulse • Find Autocorrelation of signal Dickerson EE422
Solution • Find the spectrum of pulse: • Autocorrelation • For k = 0: an=A or an= -A: • For k >0: an=-A or A and an+k=-A or A: Dickerson EE422
White Noise Process • A random process is said to be a white noise process if the PSD is constant over all frequencies: R(t) P(f) N0/2 N0/2 t f Dickerson EE422
Linear Systems • Recall: • This is still valid if x and y are random processes, x might be signal plus noise or just noise • What is the autocorrelation and PSD for y(t) when x(t) is known? y(t) Y(f) Ry(t) Py(f) x(t) X(f) Rx(t) Px(f) Linear Network h(t) H(f) Dickerson EE422
Key Theorem • IF a WSS random process x(t) is applied to a LTI system with impulse response h(t), the output autocorrelation is: • And the output PSD is: Dickerson EE422
Next Time • Reading: Lathi 11.2-11.4 (also 7.2), • Mini-Lecture 1: • Equivalent Noise Bandwidth • Activity • Mini-Lecture 2: • Quiz 2 • Mini-Lecture 3: Dickerson EE422