G A signal x() with Fourier transform ( ) undergoes impulse-train sampling to generate 4 where For each of the following sets of constraints on x() or ( ), does the sampling guarantee that x() can be recovered from xxpp()? Justify your answer and show your work. Simple yes/no answers are not acceptable. a) ( )=0 for ||>5,000 b) ( )=0 for ||>15,000 c) {( )}=0 for ||>5,000 d) xx() is real and ( )=0 for ||>15,000 [hint: if xx() is real-valued then ( )=∗(− )] e) ( )∗( )=0 for ||>15,000 f) |( )|=0 for >5,000 Problem

(a) The Nyquist rate for the given signal is 2 × 5000π = 10000π. Therefore, in order to be able

to recover x(t) from xp(t), the sampling period must at most be Tmax =

2π

10000π = 2 × 10−4

sec. Since the sampling period used is T = 10−4 < Tmax, x(t) can be recovered from xp(t).

(b) The Nyquist rate for the given signal is 2 × 15000π = 30000π. Therefore, in order to be able

to recover x(t) from xp(t), the sampling period must at most be Tmax =

2π

30000π = 0.66×10−4

sec. Since the sampling period used is T = 10−4 > Tmax, x(t) cannot be recovered from

xp(t).

(c) Here, Im{X(jω)} is not specified. Therefore, the Nyquist rate for the signal x(t) is indeterminate. This implies that one cannot guarantee that x(t) would be recoverable from xp(t).

(d) Since x(t) is real, we may conclude that X(jω) = 0 for |ω| > 5000. Therefore, the answer to

this part is identical to that of part (a).

(e) Since x(t) is real, X(jω) = 0 for |ω| > 15000π. Therefore, the answer to this part is identical

to that of part (b).

(f) If X(jω) = 0 for |ω| > ω1, then X(jω) ∗ X(jω) = 0 for |ω| > 2ω1. Therefore, in this

part, X(jω) = 0 for |ω| > 7500π. The Nyquist rate for this signal is 2 × 7500π = 15000π.

Therefore, in order to be able to recover x(t) from xp(t), the sampling period must at most

be Tmax =

2π

15000π = 1.33 × 10−4

sec. Since the sampling period used is T = 10−4 < Tmax,

x(t) can be recovered from xp(t).

(g) If |X(jω)| = 0 for |ω| > 5000π, then X(jω) = 0 for |ω| > 5000π. Therefore, the answer to

this part is identical to the answer of part (a).

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step-by-step explanation:

in the math equation 8 x 7 = 56, the 8 and the 7 are both factors of 56, since in this lesson we will use square tiles to figure out the factors of a whole find as many different arrays as you can, but follow one important rule: all the rows must have the i can make two rows of nine, so 2 and 9 are both factors of 18.

step-by-step explanation: