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Unformatted text preview: ENEE 241 HOMEWORK ASSIGNMENT 12 Due Tue 05/03 Problem 12A
Consider the FIR ﬁlter given by the following inputoutput relationship (note the missing coeﬃcient):
√
√
y [n] = x[n] + 3x[n − 1] − 3x[n − 3] − x[n − 4] ,
n∈Z
(i) (3 pts.) Show that the input sequences deﬁned for all n by x(1) [n] = 1 and x(2) [n] = (−1)n
both result in output sequences which are identically equal to zero. (ii) (3 pts.) Write MATLAB code which computes and plots the amplitude and phase response
of the ﬁlter at 1024 equally spaced frequencies in [0, 2π ). Submit the plots, properly labeled.
(iii) (4 pts.) Express the ﬁlter’s frequency response in the form
H (ej ω ) = je−j (ωM/2) F (ω )
where F (ω ) is a realvalued sum of sines.
(iv) (5 pts.) The amplitude response plotted in (ii) above has six zeros at frequencies other than
ω = 0 and ω = π . Determine the values of these frequencies analytically, using the result of part
(iii) and the identity sin 2θ = 2 sin θ cos θ.
(v) (5 pts.) Determine analytically the exact locations of the local maxima of the amplitude
response H (ej ω ) in the frequency range [0, π /2]. Do so by diﬀerentiating F (ω ) and using the
identity cos 2θ = 2 cos2 θ − 1. Express your answers using cos−1 (ρ1 ) and cos−1 (ρ2 ), where ρ1 and
ρ2 are exact (sums of rational numbers and square roots thereof).
Problem 12B
Consider the FIR ﬁlter with the following inputoutput relationship (note the missing coeﬃcient):
y [n] = x[n] + 3x[n − 1] + 3x[n − 3] + x[n − 4] , n∈Z (i) Determine the response y (i) [ · ] of the ﬁlter to each of the input signals given by the equations
below (valid for all n ∈ Z).
x(1) [n] = (3/4)n
(2) x (3) x (4) x (5) x [n] = (−4/3) (2 pts.)
n −n (2 pts.) [n] = 1 + 3 (2 pts.) [n] = cos(n(π /6) + 2.5) (3 pts.) −n [n] = 2 · cos(nπ /6) (4 pts.) (ii) (5 pts.) The ﬁlter above is connected in series (cascade) with a ﬁlter having inputoutput
relationship
y [n] = x[n] − 2x[n − 1] + x[n − 2] ,
n∈Z
Determine the system function H (z ) of the twoﬁlter cascade. (iii) (2 pts.) Write out the inputoutput relationship of the twoﬁlter cascade. Solved Examples
S 12.1 (P 4.5 in textbook). Consider the FIR ﬁlter
y [n] = x[n] − 3x[n − 1] + x[n − 2] + x[n − 3] − 3x[n − 4] + x[n − 5]
(i) Write MATLAB code which includes the function fft, and which computes the magnitude and
phase response of the ﬁlter at 256 equally spaced frequencies between 0 and 2π (1 − 256−1 ).
(ii) Express the frequency response of the ﬁlter in the form
e−j αω F (ω )
where F (ω ) is a realvalued sum of cosines.
(iii) Determine the response y [n] of the ﬁlter to the exponential input sequence
n
1
x[n] =
,
n∈Z
2
S 12.2 (P 4.6 in textbook). The MATLAB code
a
H
A
q =
=
=
= [ 1 3 5 3 1 ].’ ;
fft(a,500);
abs(H);
angle(H); computes the magnitude response A and phase response q of a FIR ﬁlter over 500 equally spaced
frequencies in the interval [0, 2π ).
(i) If x and y are (respectively) the input and output sequences of that ﬁlter, write an expression
for y [n] in terms of values of x.
(ii) Determine the output y of the ﬁlter when the input x is given by
n
1
x[n] =
,
n∈Z
3
(iii) Express the frequency response of the ﬁlter in the form
e−j αω F (ω )
where F (ω ) is a realvalued sum of cosines.
S 12.3 (P 4.4 in textbook). Consider the FIR ﬁlter whose input x and output y are related by
y [n] = x[n] − x[n − 1] − x[n − 2] + x[n − 3]
(i) Write out an expression for the system function H (z ).
(ii) Express H (ej ω )2 in terms of cosines only. Plot H (ej ω ) as a function of ω . (iii) Determine the output y [n] when the input sequence x is given by each of the following expressions (where n ∈ Z): • x[n] = 1
• x[n] = (−1)n
• x[n] = ej πn/4
• x[n] = cos(π n/4 + φ)
• x[n] = 2−n
• x[n] = 2−n cos(π n/4)
(In all cases except the third, your answer should involve realvalued terms only.)
S 12.4 (P 4.8 in textbook). Consider two FIR ﬁlters with coeﬃcient vectors b and c, where
b=
and
c= 32123
1 −2 2 −1 T T (i) Determine the system function H (z ) of the cascade. Is the cascade also a FIR ﬁlter? If so,
determine its coeﬃcient vector.
(ii) Express the amplitude response of the cascade as a sum of sines or cosines (as appropriate)
with realvalued coeﬃcients.
S 12.5 (P 4.9 in textbook). Consider the FIR ﬁlter with coeﬃcient vector
b= 1111 T Two copies of this ﬁlter are connected in series (cascade). (i) Determine the system function H (z ) of the cascade. Is the cascade also a FIR ﬁlter? If so,
determine its coeﬃcient vector.
(ii) Determine the response y [n] of the cascade to the sinusoidal input sequence
nπ
x[n] = cos
,
n∈Z
2 ...
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This note was uploaded on 09/27/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff

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