Question

Problem 1 (10 pts). A time-domain signal x(t) is given in the figure below: - - - - - - - 9 X(t) 0 2 t, msec (a) Compute the simplest real-valued math function for the spectrum X($). (b) Plot X(f) over a Hz frequency range large enough to show all the important properties.

Answer 1

Answer:-

Given That:-

A time-domain signal x(t) is given in the figure.

dll) ---- 9 o 2 t, (m sec)

(a)

we will use the differentiate property do find the fourier transform of x(t)

differentiation property:-

d"z(t) LT (JW)" X (JW) dt

ก YIP t, lm sec) - - (ก 13 t2 E (1) Set)

ZIP

=

--9(0) + (t+ ਰ +(t - Bo)

Taking fourier transform:-

$(J\omega)^{2}x(J\omega) = -9 + \frac{9}{2}[e^{+J\frac{2\omega}{1000}} +e^{-J\frac{2\omega}{1000}} ]$

$-\omega^{2}x(J\omega) = -9 + 9 cos\frac{2\omega}{1000}$

$x(J\omega) = \frac{-9}{-\omega^{2}}(1 - cos\frac{2\omega}{1000})$

$x(J\omega) = \frac{+9}{\omega^{2}}(1 - (1 - 2sin^{2}(\frac{\omega}{1000})))$

$x(J\omega) = 9\times 2\frac{sin^{2}\frac{\omega}{1000}}{\omega^{2}}$

$x(J\omega) = 18\frac{sin^{2}\frac{\omega}{1000}}{\omega^{2}}$

$x(f) = 18\left ( \frac{sin 2\pi f/1000}{2\pi f\times \frac{1000}{1000}} \right )^{2}$

$x(f) = 18\times sinC^{2}\left ( \frac{2f}{1000} \right )\times 10^{6}$

(b)

$x(f) = 18 sinC^{2}(2f)\times 10^{6}$

at

$18sin C^{2}(\frac{2f}{1000}) = 0$

$18\frac{sin C^{2}(\frac{2\pi f}{1000})}{(\frac{2\pi f}{1000})^{2}} = 0$

$sin\left ( \frac{2\pi f}{10^{3}} \right ) = 0$

$2\pi f = n\pi \times 10^{3}$

$f = \frac{n}{2}\times 10^{3}$

f = 500 n

for $f\rightarrow 0\, \, \, x(f) = \lim_{f\rightarrow 0}18\left ( \frac{sin 2\pi f}{2\pi f} \right )^{2}\times 10^{6}$

= $18\times (1)^{2}\times 10^{6}$

$x(0) = (18)\times 10^{6}$

4 - f) là tổ Sinºt su ) 18x10 B.Sixco 0.29X106 -5DO -1000 - 500 SOU lovo 1500 f(H2)

The function spectrum is decaying oscillatory at $f\rightarrow \infty$ It will become 0 in magnitude.