1. Sep 16, 2005CS477: Analog and Digital Communications1 LTI Systems, Probability Analog and Digital Communications Autumn 2005-2006

2. Sep 16, 2005CS477: Analog and Digital Communications2 Sampling First consider modulation Product with Cosine in time domain Convolution with two impulses in frequency domain Next consider sampling Product with a train of impulses in time domain Convolution with a train of impulses in the frequency domain Nyquist sampling theorem A bandlimited signal [-B, +B] can be characterized by its samples taken every 1/(2B) seconds. i.e., 2B samples per second Undersampling leads to aliasing

3. Sep 16, 2005CS477: Analog and Digital Communications3 LTI Systems Linearity If then Time invariance Linearity and Time invariance

4. Sep 16, 2005CS477: Analog and Digital Communications4 Response of LTI Systems Impulse response:

5. Sep 16, 2005CS477: Analog and Digital Communications5 Exponentials and LTI Systems Exponentials are eigenfunctions of LTI systems! LTI Systems can not generate new frequencies!

6. Sep 16, 2005CS477: Analog and Digital Communications6 Hilbert Transformer A filter introducing a constant delay of 90 degrees to the input signal Hilbert transform does not change the domain; It’s merely a convolution

7. Sep 16, 2005CS477: Analog and Digital Communications7 Random Variables Outcomes and sample space Random Variables Mapping outcomes to: Discrete numbers  Discrete RVs Real line  Continuous RVs Cumulative distribution function One variable Joint cdf

8. Sep 16, 2005CS477: Analog and Digital Communications8 Random Variables Probability mass function (discrete RV) Probability density function (cont. RV) Joint pdf of independent RVs Mean Variance Characteristic function (IFT of pdf)