FIR filter design - Continuous and Discrete Time Signals and Systems - page 711

stop-band edge frequencies = 0 . 2 π and 0 . 8 π ;
maximum pass-band ripple < 0 . 02;
maximum stop-band ripple < 0 . 009 .

Use M ATLAB to confirm that the designed bandpass FIR filter satisfies
the given specifications.

15.12 Using the Kaiser window, design a bandstop FIR filter based on the
following specifications:

stop-band edge frequencies = 0 . 3 π and 0 . 7 π ;
pass-band edge frequencies = 0 . 4 π and 0 . 6 π ;
maximum pass-band ripple < 0 . 05;
maximum stop-band ripple < 0 . 05 .

Use M ATLAB to confirm that the designed bandstop FIR filter satisfies
the given specifications.

15.13 Equation (15.44) defines the expression for the normalized weighting
function used in the design of a lowpass filter using the Parks-McClellan
algorithm Derive the expressions for the normalized weighting functions
for highpass, bandpass, and bandstop filters.

15.14 For a type I FIR filter of length N , show that the degree L of the error
function ε ( Ω ) defined in Eq. (15.42) is given by ( N

− 1) / 2.

15.15 Repeat Problem 15.14 for a type II FIR filter of length N by showing
that the degree L of the error function ε ( Ω ) = Ŵ (cos( Ω )) defined in Eq.
(15.42) is given by ( N − 2) / 2.

15.16 Repeat Problem 15.14 for a type III FIR filter of length N by showing
that the degree L of the error function ε ( Ω ) defined in Eq. (15.42) is
givenby( N

− 3) / 2.

15.17 Repeat Problem 15.14 for a type IV FIR filter of length N by showing
that the degree L of the error function ε ( Ω ) defined in Eq. (15.42) is
givenby( N

− 2) / 2.

15.18 Truncate the impulse response of an ideal bandstop FIR filter with edge
frequencies of 0.25 π and 0.75 π with a 20-tap rectangular window filter.
Plot the magnitude response of the resulting FIR filter and compare the
frequency characteristics with a 40-tap FIR filter.

15.19 Using M ATLAB , determine the impulse response of the FIR filters
designed in Problems 15.4 and 15.5. Sketch the magnitude response and
ensure that the FIR filters satisfy the given specifications. Comment on
the complexity and frequency characteristics of the designed filters.