Chapter 6 Random Processes and LTI

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

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. Power Spectral Density • Definition: The PSD Px(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. Properties of PSD • Some properties of PSD are: • Px(f ) is always real • Px(f ) > 0 • When x(t) is real, Px(-f )= Px(f ) • If x(t) is WSS, • PSD at zero frequency is:

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) • Ts is the sampling interval • R(k) is the autocorrelation of the data: • 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 • PSD only depends on the • Pulse shape f(t) • Statistical properties of the data

6. Example: PSD of Unipolar NRZ Pulses • Possible levels are +A and 0 • Square pulses of width Tb • Find the PSD: 1) Find the spectrum of pulse:

7. PSD of Unipolar NRZ Pulses 2)Evaluate the autocorrelation function • For k = 0: there are 2 possibilities, an=A or an=0: • For k >0: there are 4 possibilities, an=0 or A and an+k=0 or A:

8. PSD of Unipolar NRZ Pulses • Simplify using: • Poisson Sum Formula • and

9. Example: PSD for Bipolar NRZ Signalling • Find the spectrum of pulse: • Find the Autocorrelation • For k = 0: an=A or an= -A: • For k >0: an=-A or A and an+k=-A or A:

10. R(t) P(f) N0 /2 t f White Noise Process • A random process is said to be a white noise process if the PSD is constant over all frequencies: N0/2

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? x(t) X(f ) Rx(t) Px(f ) Linear Network h(t) H(f )

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

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. SNR at the Output of a RC LPF • Same RC LPF as before, assume: x(t)=si(t)+ni(t) • si(t) =A cos(w0t + q0),deterministic. • ni(t) is white noise, flat PSD over all frequencies. • ergodic noise (time avg=statistical avg). • Input SNR (SNRi) is zero: • Signal Power: A2/2 • Noise Power is infinity.

16. SNR at the Output of a RC LPF • Output is y(t)=so(t)+no(t) • Output Signal Power • Output Noise Power (from previous example)

17. Noise Equivalent Bandwidth • For a WSS process x(t), the equivalent bandwidth is: • Input: white noise with a PSD of No/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. LP Filter Ideal LP Filter H(f) H(0) B B Noise Equivalent Bandwidth Noise Equivalent Bandwidth