ENEE241hw03

# ENEE241hw03 - ENEE 241 02∗ HOMEWORK ASSIGNMENT 3 Due Tue...

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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 continuous-time real-valued 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 discrete-time sinusoid x[n] = 5.9 cos(0.625π n + 2.1) is obtained by sampling a continuous-time 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 continuous-time 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 three-dimensional 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 three-dimensional 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 continuous-time 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 discrete-time 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 continuous-time 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 two-element 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 two-element 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 (one-line) 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.

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