EEE Chapter 6 Random Processes and LTI Huseyin Bilgekul EEE 461 Communication Systems II Department of Electrical and Electronic Engineering Eastern Mediterranean University  Power Spectral Density  White Noise Process  Random Processes in LTI Systems

EEE Homework Assignments Return date: November 14, Assignments: Problem 6-15 Problem 6-17 Problem 6-22 Problem 6-25 Problem 6-26

EEE Power Spectral Density Definition: The PSD P x (f) of a random process is defined by, where the subscript T denotes the truncated version of the signal Relationship to Time Autocorrelation, Wiener-Khintchine Theorem: When x(t) is a wide sense stationary process, the PSD is defined as: Average Power of a Random Process

EEE Properties of PSD  Some properties of PSD are: P x (f ) is always real P x (f ) > 0 When x(t) is real, P x (-f )= P x (f ) If x(t) is WSS, PSD at zero frequency is:

EEE General Expression PSD of a Digital Signal General expression for the PSD of a Digital Signal: F(f ) is the Fourier Transform of the Pulse Shape f(t) – T s is the sampling interval –R(k) is the autocorrelation of the data: –a n and a k+n are the levels of the nth and (n+k)th symbol positions –P i is the probability of having the ith a n a n+k product PSD only depends on the –Pulse shape f(t) –Statistical properties of the data

EEE Example: PSD of Unipolar NRZ Pulses Possible levels are +A and 0 Square pulses of width T b Find the PSD: 1) Find the spectrum of pulse:

EEE PSD of Unipolar NRZ Pulses 2) Evaluate the autocorrelation function For k = 0: there are 2 possibilities, a n =A or a n =0: For k >0: there are 4 possibilities, a n =0 or A and a n+k =0 or A:

EEE Simplify using: –Poisson Sum Formula –and PSD of Unipolar NRZ Pulses

EEE Example: PSD for Bipolar NRZ Signalling 1.Find the spectrum of pulse: 2.Find the Autocorrelation For k = 0: a n =A or a n = -A: For k >0: a n =-A or A and a n+k =-A or A:

EEE White Noise Process A random process is said to be a white noise process if the PSD is constant over all frequencies: N 0 /2 f P(f)P(f) R()R() 

EEE Linear Systems Recall that for LTI systems: 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? Linear Network h(t) H(f ) x(t) X(f ) R x (  ) P x (f )

EEE Output of an LTI System 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: The power transfer function is:

EEE Example RC Low Pass Filter Input is thermal white noise. R C x(t)=n(t) y(t)y(t)

EEE SNR at the Output of a RC LPF Input SNR is ratio of the input signal to input noise Output SNR is ratio of the output signal to output noise

EEE Same RC LPF as before, assume: x(t)=s i (t)+n i (t) –s i (t) =A cos(   t    deterministic. –n i (t) is white noise, flat PSD over all frequencies. –ergodic noise (time avg=statistical avg). Input SNR (SNR i ) is zero: –Signal Power: A 2 /2 –Noise Power is infinity. SNR at the Output of a RC LPF

EEE Output is y(t)=s o (t)+n o (t) Output Signal Power Output Noise Power (from previous example) SNR at the Output of a RC LPF

EEE Noise Equivalent Bandwidth For a WSS process x(t), the equivalent bandwidth is: Input: white noise with a PSD of N o /2 to a low pass filter: The Noise Equivalent Bandwidth is the filter bandwidth of H(f ) that gives the same average noise power as an ideal low pass filter of DC gain H(0)

EEE Noise Equivalent Bandwidth Ideal LP Filter B H(0) LP Filter B H(f)H(f) Noise Equivalent Bandwidth