Chapter 6 Bandpass Random Processes PSD of bandpass random processes BP Filtered White Noise Sinusoids in Gaussian Noise Huseyin Bilgekul EEE 461 Communication Systems II Department of Electrical and Electronic Engineering Eastern Mediterranean University

Homework Assignments Return date: December 13, 2005. Assignments: Problem 6-38 Problem 6-41 Problem 6-45 Problem 6-48 Problem 6-50

Equivalent Representations of Bandpass Signals Remind: Equivalent representations of a bandpass signal Px(f) f fc -fc

Bandpass Random Process If x(t) and y(t) are jointly WSS processes, the real bandpass process Will be WSS stationary if and only if: If v(t) is a Gaussian random process then g(t), x(t) and y(t) are Gaussian processes since they are linear functions v(t). However R(t) θ(t) are NOT Gaussian because they are NONLIEAR functions of v(t).

Bandpass Random Process What happens to a signal at a receiver? How does the PSD of the signal after a BPF correspond to the signal before the BPF? Remind: Equivalent representations of a bandpass signal Px(f) f fc -fc

BPF System Bandpass random process can be written as: With the impulse response: cos(wct+q) 2cos(wct+q) Baseband x(t) Inphase Ideal LPF H0(f) x x v(t) BP Process v(t) BP Process + 2sin(wct+q) sin(wct+q) Ideal LPF H0(f) x x Baseband y(t) Quadrature

Impulse Response 2B H (f ) H0(f ) 1 1 -fc B fc Impulse Response fc -fc 1 B H0(f ) Impulse Response Transfer Function So, x(t) and y(t) are low-pass random processes, what else can be deduced? Assume theta is uniformly distributed phase noise

PSD of BP Random Processes PSD of x(t) and y(t) Pv(f ) f fc -fc Pv(f-fc) Pv(f+fc) LPF -2fc 2fc f Px(f) or Py(f) -2fc f 2fc

PSD of BP Random Processes

PSD of BP Random Processes

Properties of WSS BP Processes If the narrowband noise is Gaussian, then the in-phase x(t) and quadrature y(t) components are jointly Gaussian If the narrow band noise is wide-sense stationary (WSS), then the in-phase and quadrature components are jointly WSS. In-phase and quadrature components have the same PSD. In-phase and quadrature components of narrowband noise are zero-mean Noise comes original signal being passed through a narrowband linear filter Variance of the processes is the same (area under PSD same)

Properties of WSS BP Processes Continued Bandpass PSD from baseband PSD. PSD of I and Q from bandpass PSD

BP White Noise Process The PSD of a BP white noise process is No/2. What is the PSD and variance of the in-phase and quadrature components? From the SNR calculations, it is clear that the variance of the white noise is:

Sinusoids in Gaussian Noise Signal is a sinusoid mixed with narrow-band additive white Gaussian noise (AWGN) Can be written in terms of ENVELOPE and PHASE terms as:

Sinusoids in Gaussian Noise In-phase and Quadrature terms of noise Gaussian with variance s2. Similar transformation to that used for calculating the dart board example, the joint density can be found in polar coordinates. Marginal density of the Envelope is Rician type. Approaches a Gaussian if A >> s