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Unformatted text preview: ENEE 241 02∗ HOMEWORK ASSIGNMENT 3 Due Tue 02/22 Problem 3A
(i) (4 pts.) Consider the frequency f0 = 180 Hz and the sampling rate fs = 640 samples/sec. List
all the aliases of f0 with respect to fs in the frequency range 0.0 to 2.5 kHz.
(ii) (2 pts.) If we sample, at rate fs = 640 samples/sec, a continuoustime realvalued sinusoid
whose frequency is one of the values found in part (i) above, what is the resulting frequency ω of
the sample sequence? Give your answer in the range [0, π ] (radians/sample).
(iii) (5 pts.) The discretetime sinusoid x[n] = 5.9 cos(0.625π n + 2.1) is obtained by sampling a
continuoustime sinusoid x(t) at a rate of 640 samples per second. If it is known that the frequency
of x(t) is in the range 1,280 to 1,600 Hz, write an equation for x(t).
(iv) (4 pts.) Let x[n] be as in (iii), with the sampling rate unchanged. If, instead, it is known
that the frequency of x(t) is in the range 960 to 1,280 Hz, write a new equation for x(t).
(v) (5 pts.) The continuoustime signals
x1 (t) = cos(144π t − 1.4) , and x3 (t) = cos(256π t + 1.4) are sampled at the same rate fa = 1/Ta to produce the sequences x1 [n] = x1 (nTa ) and x2 [n] =
x2 (nTa ). Determine the highest value of fa such that the two sequences are identical, i.e.,
x1 [n] = x2 [n] (all n) For that value of fa , are the two (identical) sequences periodic, and if so, what is the fundamental
period?
Problem 3B
Let 6
4
A 0 = −2 ,
−3
−5 3
1
A −3 = −5 3
1 and 3
−1
A 3 = −1 0
−4 (i) (2 pts.) What are the dimensions of A?
(ii) (8 pts.) Determine the three columns a(1) , a(2) and a(3) of A (and thus A itself). Hint: Sum
both sides of the last two equalities.
(iii) (4 pts.) In the usual threedimensional Cartesian space (with orthogonal axes), consider the
plane S which contains he origin and has normal vector [ 1 1 1 ]T . Determine the reﬂection of the
point [ 1 0 0 ]T about S . (Hint: Express [ 1 0 0 ]T as a sum of two vectors, one parallel to [ 1 1 1 ]T
and one orthogonal to it.)
(iv) (3 pts.) By symmetry, determine the reﬂections of [ 0 1 0 ]T and [ 0 0 1 ]T about S . Explain why the matrix A represents a reﬂection of an arbitrary point (in threedimensional space) about
S.
(v) (3 pts.) Study the function TOEPLITZ in MATLAB documentation (Help). Give a single row
vector c such that A = toeplitz(c)
produces the matrix A of parts (i)–(iv) above.
Solved Examples
S 3.1 (P 1.21 in textbook). For what frequencies f in the range 0 to 3.0 KHz does the sinusoid
x(t) = cos(2π f t)
yield the signal
x[n] = cos(0.4π n)
when sampled at a rate of fs = 1/Ts = 800 samples/sec?
S 3.2. The continuoustime sinusoid
x(t) = cos(2π f t + 1.8)
is such that 700 < f ≤ 800 (Hz). It is sampled every Ts = 10.0 ms to produce the discretetime
sinusoid x[n] = x(nTs ).
(i) If x[n] = cos(0.7π n + 1.8), what is the value of f ?
(ii) If, on the other hand, x[n] = cos(0.4π n − 1.8), what is the value of f ?
S 3.3 (P 1.22 in textbook). The continuoustime sinusoid
x(t) = cos(300π t)
is sampled every Ts = 2.0 ms, so that
x[n] = x(0.002n)
(i) For what other values of f in the range 0 Hz to 2.0 KHz does the sinusoid
v (t) = cos(2π f t)
produce the same samples as x(t) (i.e., v [·] = x[·]) when sampled at the same rate? (ii) If we increase the sampling period Ts (or equivalently, drop the sampling rate), what is the
least value of Ts greater than 2 ms for which x(t) yields the same sequence of samples (as for Ts = 2
ms)?
S 3.4 (P 2.3 in textbook). If
B
determine 1
0 3
= −2 −1 and B 2
−1 B −1
1 4
= 7 ,
2 What are the dimensions of the matrix B?
S 3.5. (P 2.4 in textbook). Let G be a m × 2 matrix such that
−1
2
G
=u
and
G
=v
1
1
Express
G
in terms of u and v. 3
3 S 3.6. (P 2.1 in textbook). In MATLAB, enter the matrix
A = [1 2 3 4; 5 6 7 8; 9 10 11 12]
(i) Find twoelement arrays I and J such that A(I,J) is a 2 × 2 matrix consisting of the corner
elements of A.
(ii) Suppose that B=A initially. Find twoelement arrays K and L such that
B(:,K) = B(:,L)
swaps the ﬁrst and fourth columns of B.
(iii) Explain the result of
C = A(:)
(iv) Study the function RESHAPE. Use it together with the transpose operator .’ in a single
(oneline) command to generate the matrix
1234 5 6
7 8 9 10 11 12
from A.
S 3.7 (P 2.5 in textbook). The transformation A : R3 → R3 is such that 1
4
0
−2
0
1
A 0 = −1 ,
A 1 = 3 ,
A 0 = 0 0
2
0
−1
1
3
Write out the matrix A and compute Ax, where
T
x = 2 5 −1 (ii) The transformation B : R3 → R2 is such that
1
1
2
3 0 = 1 =
B
,
B
,
−4
−2
0
0
Determine the matrix B.
1 1 = −2
B
1
1 ...
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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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