Lesson 5Discrete Filters

Lesson 2 Recap

Lesson 3 and 4 Recap

Transfer functionInputOutput

Transfer function

y[n] = 0.5 y[n-1] + x[n] y[n] = (x[n-1]+x[n]+x[n+1])/3

Transfer function y[n] = 0.5 y[n-1] + x[n]

Transfer function

y[n] = (x[n-1]+x[n]+x[n+1])/3

IIR and FIR

Infinite Impulse Response (IIR)

Finite Impulse Response (FIR)recursive

Linear Phase

Time shifting

Linear Phase

Example

Inverse Filter

How to remove the effect of a filter with transfer function H(z)?

Frequency Scales

0.5 fsfst fp0

0

0

0

Ωp Ωst 0.5Ωs

f(Hz)

Ω=2πf (rad/sec)

ωp ωst π ω=ΩTs (rad)

ωp/π ωst/π 1 ω/π

Loss Function

Loss Function

Example

Advantages of FIR

Stability (BIBO: Bounded Input Bounded Output) Possible linear phase Efficient implementation (convolution sum via FFT) Minor disadvantage compared with IIR: a larger

number of coefficients results in slightly larger storage

FIR Filter Design

FIR Filter Design

Windowing

FIR Filter Design

The effect of windowing

FIR Filter Design

Shifting

FIR Filter Design

The effect of shifting

Linear Phase

FIR Filter Design

Different Window Functions Rectangular Triangular or Bartlett Hamming Kaiser Etc.

FIR Filter Design

Example

FIR Filter Design

High-pass filter

Properties of DTFT

Frequency shifting

FIR Filter Design

High-pass filter

FIR Filter Design

Example

Design a high-pass filter of order 14 and a cut-off frequency 0.2π using the Kaiser window.