1. EEE 461 1 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

2. EEE 461 2 Homework Assignments Return date: November 14, 2005. Assignments: Problem 6-15 Problem 6-17 Problem 6-22 Problem 6-25 Problem 6-26

3. EEE 461 3 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

4. EEE 461 4 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:

5. EEE 461 5 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

6. EEE 461 6 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:

7. EEE 461 7 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:

8. EEE 461 8 Simplify using: –Poisson Sum Formula –and PSD of Unipolar NRZ Pulses

9. EEE 461 9 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:

10. EEE 461 10 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() 

11. EEE 461 11 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 )

12. EEE 461 12 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:

13. EEE 461 13 Example RC Low Pass Filter Input is thermal white noise. R C x(t)=n(t) y(t)y(t)

14. EEE 461 14 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

15. EEE 461 15 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

16. EEE 461 16 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

17. EEE 461 17 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)

18. EEE 461 18 Noise Equivalent Bandwidth Ideal LP Filter B H(0) LP Filter B H(f)H(f) Noise Equivalent Bandwidth