1. 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

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

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

4. 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).

5. 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

6. 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

7. Impulse Response 2B H (f ) H0(f ) 1 1 -fc B fc Impulse Responsefc
-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

8. 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

9. PSD of BP Random Processes

10. PSD of BP Random Processes

11. 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)

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

13. 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:

14. 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:

15. 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