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Unformatted text preview: ENEE 241 02* HOMEWORK ASSIGNMENT 8 Due Tue 04/05 The MATLAB functions OSHIFT, OFLIP and FDIAG found in the Assignment 8 folder can be used
for circular shifts, circular reversals and products with Fourier sinusoids (by means of F). You may
use these functions to verify your results for Problem 8A. Copy these Mﬁles to a local directory
and include that directory in the MATLAB path.
Problem 8A
Suppose the signal
s=
of Problem 7C has DFT
S= 4 abcdef 1 + 2j 5 3j gh −7 −3j T
5 1 − 2j T Without computing any DFTs or inverse DFTs, determine (in numerical form) the DFTs of the
signal vectors s(1) through s(7) constructed in that problem.
Problem 8B
Let x be a realvalued vector of length N = 64 whose DFT X satisﬁes
X [k ] = 0 for k = 0 : 13 and X [k ] = 0 for k = 14 : 32 Explain your answers to (i)–(v) below.
(i) (4 pts.) For what values of k between 33 and 63 does X [k ] equal zero?
(ii) (4 pts.) Let
x(1) [n] = x[n] cos(3π n/8) , n = 0 : 63 For what values of k does X(1) [k ] equal zero?
(iii) (4 pts.) Let
x(2) = P4 x + P−4 x
For what values of k does X(2) [k ] equal zero?
(iv) (4 pts.) Let
y = x + Rx
What is the imaginary part of the vector Y? Is it true that Y = RY?
(v) (4 pts.) Determine all the values of ω such that the vector y(1) deﬁned by
y (1) [n] = y [n] cos(ω n) , n = 0 : 63 is certain to have a realvalued DFT Y(1) .
Problem 8C
(i) (6 pts.) Determine the circular convolution s = x y of
x= 4 −3 1 −1 2 T and
y= 5 2 3 0 T −2 Using the FFT function in MATLAB, verify that the DFTs X, Y and S satisfy X Y = S.
(ii) (3 pts.) Without any numerical computation, determine the circular convolution of
a=
and
b=
(iii) (6 pts.) If 1 −1 2 4 −3 2 3 0 −2 5 T T 2 −2
3 −3
5 −1
c0
−1 c1 −7
−1
2 −2
3 −3
5 5 −1
2 −2
3 −3 c2 = 9 c3 2
−3
5 −1
2 −2
3 c4 12
3 −3
5 −1
2 −2
−2
3 −3
5 −1
2
c5
−7 , explain how the vector c can be obtained using DFTs (as opposed to a conventional solution of a
6 × 6 system of equations). Implement this solution in MATLAB to obtain c.
(iv) (5 pts.) Let x and y be time domain vectors of length N such that for every n = 0, . . . , N − 1,
x[n] = 1
= 0
y [n] If X and Y are the respective DFTs, determine the circular convolution vector X Y.
Solved Examples
S 8.1 (P 3.11 in textbook). The timedomain signal
x=
has DFT
X= abcdef T ABCDEF Using the given parameters, and deﬁning
1
λ=
2 and µ= T √ 3
2 for convenience, write out the components of the DFT vector X(r) for each of the following timedomain signals x(r) .
T
a −b c −d e −f
x(1) =
T
a0c0e0
x(2) =
T
afedcb
x(3) = x(4) =
x(5) = x(8)
def abc baf edc T
T f +b a+c b+d c+e d+f
T
0 µb µc 0 −µe −µf
=
T
ABCDEF
= x(6) =
x(7) e+a T S 8.2 (P 3.15 in textbook). Consider the twelvepoint vectors x, y and s shown in the ﬁgure. If
the DFT X is given by
T
X = X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
express the DFT’s Y and S in terms of the entries of X.
x
1 0 1 2 3 4 5 6 7 8 9 10 11 7 8 9 10 11 y
1 0 1 2 3 4 5 6
s 1 0 1 2 3 5 7 9 10 11 1 S 8.3 (P 3.13 in textbook). An eightpoint signal x has DFT
T
X = 0 0 0 −1 2 −1 0 0 Without inverting X, compute the DFT Y of y, which is given by the equation
πn
y [n] = x[n] · cos
,
n = 0, . . . , 7
4 S 8.4 (P 3.14 in textbook). Consider the signal x shown in the ﬁgure (on the left). Its spectrum
is given by
T
X= A B C D E F G H
(i) The DFT vector shown above contains duplicate values. What are those values?
(ii) Express the DFT Y of the signal y (shown on the right) in terms of the entries of X.
S 8.5 (P 3.16 in textbook). Run the MATLAB script x y 4 4
3 3
2 3 2
1 0 2 1 3
2 1 0 1 n = (0:63)’;
X =[ones(11,1); zeros(43,1); ones(10,1)];
bar(X), axis tight
max(imag(ifft(X)))
% See (i) below
x = real(ifft(X));
bar(x)
cs = cos(3*pi*n/4);
% See (ii) below
y = x.*cs;
bar(y);
max(imag(fft(y)))
% See (iii) below
Y = real(fft(y));
bar(Y)
% See (iv) below
(i) Why is this value so small?
(ii) Is this a Fourier sinusoid for this problem?
(iii) Why is this value so small?
(iv) Derive the answer for Y analytically, i.e., based on known properties of the DFT.
S 8.6 (P 3.20 in textbook). The timedomain signals x and y have DFT’s
X=
and
Y=
(i) Is either x or y realvalued? 1 0 1 −1
3 5 8 −4 T T (ii) Does either x or y have circular conjugate symmetry?
(iii) Without inverting X or Y, determine the DFT of the signal s(1) deﬁned by
s(1) [n] = x[n]y [n] , n = 0, 1, 2, 3 (iv) Without inverting X or Y, determine the DFT of the signal s(2) deﬁned by
s(2) = x y
S 8.7 (P 3.21 in textbook). The timedomain signals
x=
and
y= 2013 T 1 −1 2 −4 T have DFT’s X and Y given by
X=
and Y= X0 X1 X2 X3
Y0 Y1 Y2 Y3 T T Determine the timedomain signal s whose DFT is given by
S= X0 Y2 X1 Y3 X2 Y0 X3 Y1 T ...
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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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