Hossein Sameti Department of Computer Engineering Sharif University of Technology

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

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 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 time- variant or non-linear. 4 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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

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

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

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

11 (eq.1) (eq.2) If we equate (eq.1) and (eq.2), we get GLP. Real Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

12 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

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Case 1: Case 2: N odd even N odd even 15 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Type I Type II 16 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Type III Type IV 17 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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SymmetryNConstraint Type I Type II Type III Type IV 19 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Type I Type II Type IV 20 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Type I Type II Type III Type IV 21 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Low-passHigh-passBand-passBand-stop Type I√√√√ Type II√  √  Type III  √  Type IV  √√  22 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

 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 (low- pass, high-pass, bandpass and bandstop). 23 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology