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**Unformatted text preview: **x1 (t) 1
−1
1 t 1 0 2 −1 x2 (t) 1 −3 −2 −1 0 t 3 x3 (t) 1
−3 −2 −1 5
1 2 3 4 6
t −1 Determine which, if any, of the real signals depicted have Fourier transforms that satisfy
each of the following conditions:
(1) ≥e{X (j� )} = 0
Before testing this and other conditions, let’s review a useful property of the Fourier transform of a real signal x(t): F F E v {x(t)} �� ≥e{X (j� )}, O d{x(t)} �� j ∞m{X (j� )} (O & W, Section 4.3.3, p. 303). 17 Now to test for this condition, note that X (j� ) will have no real part only if x(t) is an
o dd function, i.e x(t) has only an odd component � x(−t) = −x(t).
By inspection, it is easy to see that only x1 (t) is odd. Therefore this condition is true for
x1 (t) and false for x2 (t) and x3 (t). Note that x3 (t) can not be described as either odd
or even, which means that it has both even and odd components, and hence its Fourier
transform would have b oth real and imaginary parts.
(2) ∞m{X (j� )} = 0
S...

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