CE 40763 Digital Signal Processing Fall 1992 Amplitude

CE 40763 Digital Signal Processing Fall 1992 Amplitude and Phase Properties of Ideal Filters Hossein Sameti Department of Computer Engineering Sharif University of Technology

Frequency response of LTI Systems x(n) LTI System h(n) y(n) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 2

Example of Calculating Frequency Response Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 3

LTI Systems as filters We can thus view an LTI system as a filter for sinusoids of different frequencies. Hence, the basic digital filter design problem involves determining the parameters of an LTI system to achieve a desired H(ω). Note that the output of an LTI system cannot contain frequency components that are not contained in the input signals. For that to happen, the system should be either timevariant or non-linear. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 4

Characteristics of Ideal Filters (amplitude) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 5

Characteristics of Ideal Filters (amplitude) - Bandwidth is the range of frequencies over which the spectrum (the frequency content) of the signal is concentrated. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 6

Ideal Filter Characteristics: Linear-phase property • Observations: 1. The magnitude of the frequency response is 1 for all ω. 2. The phase is linear in ω. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 7

Group delay Group delay: For pure delay: Desirable, since pure delay is tolerable. Group delay is thus constant: -All the frequencies are thus delayed by the same amount when they pass through this system. Thus, no distortion is added to the signal. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 8

Example (Effect of Group Delay) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 9

Generalized Linear-phase (GLP) FIR Filters x(n) y(n) LTI System h(n) GLP filters Linear-phase filters • Example: pure delay Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 10

Necessary condition for h(n) to become GLP Real (eq. 1) (eq. 2) If we equate (eq. 1) and (eq. 2), we get GLP. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 11

Necessary condition for h(n) to become GLP • Case 1: The above equation is satisfied. Symmetry Condition • Case 2: The above equation is satisfied. Anti-symmetry Condition N: the length of h(n) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 12

Examples of symmetry property Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 13

Examples of anti-symmetry property Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 14

Summary of symmetry properties • Case 1: odd N even • Case 2: odd N even Hossein Sameti, ECE, UBC, Summer 2012 Originally Prepared by: Mehrdad Fatourechi, 15

Types of GLP FIR filters Type II Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 16

Types of GLP FIR filters Type III Type IV Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 17

Types of GLP FIR filters Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 18

Relationship with Frequency Response Symmetry N Constraint Type III Type IV Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 19

Digital Filter Design Using GLP Systems Type II Type IV Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 20

Digital Filter Design Using GLP Systems Type III Type IV Type I Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 21

Digital Filter Design Using GLP Systems Low-pass High-pass Band-stop Type I √ √ Type III √ Type IV √ √ Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 22

Summary Linear-phase is desirable for filters as it leads to a fixed delay for all input frequencies (i. e. , no distortion in the output of the filter). If we impose symmetry or anti-symmetry on h(n), we can have linear-phase property. Type I FIR filter can be used to design all filters (lowpass, high-pass, bandpass and bandstop). Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 23