DiscreteTime Systems

1
Discrete-Time Systems Filters

• Trac D. Tran
• ECE Department
• The Johns Hopkins University
• Baltimore, MD 21218

2
Outline

• Linear time-invariant (LTI) systems
• Discrete-time filters
• Impulse response
• Convolution
• Low-pass filtering averaging
• High-pass filtering differencing
• Frequency representation
• Examples of filtering in practical applications
• Signal sub-sampling or decimation
• Filtering followed by down-sampling
• Signal interpolation
• Up-sampling followed by filtering

3
Discrete-time LTI Systems
output
input
H
Operator or System or Transfer Function

• Linear Property or Superposition Principle
• Scaled linear summation of inputs leads to scaled
  linear summation of outputs
• Time-Invariant or Shift-Invariant Property
• Shifted input leads to shifted output

4
Impulse Response
output
input
H
LTI System

• Unit Impulse
• Impulse Response

1
n
0
1
2
2
1
5
Filtering via Convolution
H
LTI System

• Convolution
• Neighborhood operation
• Each output sample yn is a weighted sum of a
  local input neighborhood xn1, xn, xn1,
  xn2
• Can be thought of as a series of dot products
  between h and x

6
Convolution Properties

• Commutative
• Linear
• Associative

7
Convolution A Demonstration
H
7
5
3
1/2
1/2
2

1
0
n
n
6
0
1
1
2
2
3
1
2
0
1
3
4
5

• Flip filter hn around 0
• Slide flipped window through xn
• Weight samples in window then sum to produce
  each yn sample

n
7
1
2
0
1
3
4
5
6
8
Low-Pass Filter Averaging Operator

• Low-pass filter
• Smoothing operator, softening operator,
  trend-showing
• Noise reduction, reduction of details, blurring
• Simple moving average
• Weighted moving average

9
Filtering in Stock Market Analysis
Apple
Google
10
LP Image FilteringSmoothingBlurring
H
2D Averaging Window
11
High-Pass Filter Difference Operator

• High-pass filter
• Difference operator, sharpening operator
• Details emphasizing, discontinuity detector,
  gradient
• Simple moving difference
• Weighted moving difference

12
HP Image Filtering Edge Detection
Horizontal Filtering
Vertical Filtering
13
Fourier Transform
H
LTI System or Operator

• Fourier transform ? Frequency representation of
  signals and systems
• Continuous complex function with period 2pi
• Uniform sampling of the continuous Fourier
  transform yields the Discrete Fourier Transform
  (DFT)
• DFT has numerous fast algorithms called FFT

14
Convolution Theorem
Time Domain
Frequency Domain
Frequency Representation
15
Down-Sampling
2
Just keep all even-indexed samples throw away
all odd-indexed ones
Linear Time-Variant Lossy Operator

• Equivalent to not sampling fast enough ?
  aliasing error!

16
Signal Decimation
Anti-aliasing Filter
2
H

• Filter first to mitigate aliasing, then
  down-sample!

17
Up-Sampling
2
Insert a zero between two existing samples

• Simplest way to increase the resolution of the
  signal xn

18
Signal Interpolation
2
H