LTI Systems, Probability

1
LTI Systems, Probability

• Analog and Digital Communications
• Autumn 2005-2006

2
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
LTI Systems

• Linearity
• If
• then
• Time invariance
• Linearity and Time invariance

4
Response of LTI Systems
Impulse response
5
Exponentials and LTI Systems
Exponentials are eigenfunctions of LTI systems!
LTI Systems can not generate new frequencies!
6
Hilbert Transformer

• A filter introducing a constant delay of 90
  degrees to the input signal
• Hilbert transform does not change the domain
  Its merely a convolution

7
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
Random Variables

• Probability mass function (discrete RV)
• Probability density function (cont. RV)
• Joint pdf of independent RVs
• Mean
• Variance
• Characteristic function
• (IFT of pdf)