HW 2 solution - HW 2 sol PDF

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EE348 Homework #2 Solution 1. (10 pts.) Determine whether or not each of the following signals is periodic. If the signal is periodic, determine its fundamental period. 𝜋

(a) sin (4𝑡 + 2 )

(b) cos[4𝑛 − 𝜋]

(c) 𝑒 𝑗

3𝜋 4

𝑛

− 𝑒𝑗

6𝜋 5

𝜋

𝑛+ 3

𝜋

(d) 𝑠𝑖𝑛 2 ( 3 𝑛)

3𝜋

(e) (𝑒 𝑗 2 𝑛 )3

2. (5 pts.) Determine if the following statement is true or not. Write T for Ture, and F for False. (a) Harmonically related signals 𝛷𝑘 (𝑡) = 𝑒𝑗𝑘𝑡 have a common period 2π. ( T )

(b) 𝑒 𝑗𝜔0 𝑡 is not periodic if

2𝜋

𝜔0

(c) 𝑒 𝑗𝜔0 𝑛 = 𝑒 𝑗𝜔0 (𝑛+𝑁) only if (d) 𝑒 𝑗(𝜔0 +2𝜋)𝑡 = 𝑒 𝑗𝜔0 𝑡 . ( F )

is not a rational number. ( F ) 2𝜋

𝜔0

is a rational number. ( T )

(e) 𝑒 𝑗(𝜔0 +2𝜋)𝑛 = 𝑒 𝑗𝜔0 𝑛 . ( T )

3. (15 pts.) Sketch each of the following signals and for each case, specify if the signal is causal/non-causal, periodic/non-periodic, odd/even. If the signal is periodic specify its period. (a)

𝑥1 (𝑡) = 4cos(4𝜋𝑡)

(b)

𝑥2 (𝑡) = {

2𝑒 −2𝑡 ,𝑡 ≥ 0 0,𝑡 < 0

(c)

1 𝑛

𝑥3 [𝑛] = − (− ) 2

4. (10 pts.) Let x(t) = cos(𝜔𝑥 (t + 𝜏𝑥 ) + 𝜃𝑥 ). Determine the frequency in hertz and the period of x(t) for each of the following three cases:

5. (10 pts.) Let x[n] = cos(Ω𝑥 (n + 𝑃𝑥 ) + 𝜃𝑥 ). Determine the period of x[n] for each of the following three cases:

2𝜋 2𝜋 = =6⇒𝑁=6 Ω 𝜋/3 2𝜋 2𝜋 8 (ii) = = ⇒𝑁=8 Ω 3𝜋/4 3 2𝜋 8𝜋 2𝜋 = = (iii) isnotarationalnumber, so𝑥 [𝑛]isaperiodic. Ω 3/4 3 (i)

6. (14 pts.) A discrete-time signal x[n] is shown in following figure:

(a) Sketch and carefully label each of the following signals: (i) 𝑥[𝑛 − 2]

(ii)

𝑥[4 − 𝑛]

(iii)

𝑥[2𝑛]

(b) What difficulty arises when we try to define a signal as 𝑥[𝑛/2]? The difficulty arises when we try to evaluate 𝑥[𝑛/2] at 𝑛 = 1, for example (or 1

generally for 𝑛 an odd integer). Since 𝑥[ 2] is not defined, the signal 𝑥[𝑛/2] does not

exist.

7. (6 pts.) For each of the following signals, determine whether it is even, odd, or neither.

Even

Neither even nor odd

odd

Neither even nor odd

Even

odd

8. (10 pts.) Consider the signal y[n] in the following figure

(a) Find the signal x[n] such that 𝐸𝑣{𝑥[𝑛]} = 𝑦[𝑛] for 𝑛 ≥ 0, and 𝑂𝑑(𝑥[𝑛]} = 𝑦[𝑛] for 𝑛 < 0.

(b) Suppose that 𝐸𝑣{𝑤[𝑛]} = 𝑦[𝑛] for all 𝑛. Also assume that 𝑤[𝑛] = 0 for 𝑛 < 0. Find 𝑤[𝑛].

9. (10 pts.) (a) Sketch 𝑥[𝑛] = 𝑎𝑛 for a typical a in the range −1 < 𝑎 < 0. 1

For 𝑎 = − 2

(b) Assume that 𝑎 = −𝑒 −1 and define 𝑦(𝑡) as 𝑦(𝑡) = 𝑒 𝛽𝑡 . Find a complex number 𝛽 such that 𝑦(𝑡), when evaluated at 𝑡 equal to an integer 𝑛, is described by (−𝑒 −1)𝑛 .

(c) For 𝑦(𝑡) found in part (b), find an expression for 𝑅𝑒{𝑦(𝑡)} and 𝐼𝑚{𝑦(𝑡)}. Plot 𝑅𝑒{𝑦(𝑡)} and 𝐼𝑚{𝑦(𝑡)} for 𝑡 equal to an integer.

10. (10 pts.) Let 𝑥(𝑡) = √2(1 + 𝑗)𝑒 𝑗𝜋/4 𝑒 (−1+𝑗2𝜋)𝑡 . Sketch and label the following: (a) 𝑅𝑒{𝑥(𝑡)} (b) 𝐼𝑚{𝑥(𝑡)} (c) 𝑥(𝑡 + 2) + 𝑥 ∗(𝑡 + 2)...