Link title

Problem Statement

6(a) Show $ℱ [\int_{- \infty}^{t} s (λ) d λ] = \frac{S (f)}{j 2 π f} if S (0) = 0$ .

6(b) If $S (0) \neq 0$ can you find $ℱ [\int_{- \infty}^{t} s (λ) d λ]$ in terms of $S (0)$ ?

Answer

a)

Remember that dummy variable $λ$ was used as a substitution such that $λ = t - t_{0}$

See then that $s (λ) = s (t - t_{0}) = ℱ [S (f) - S (f_{0})] < m a t h > f_{0} = 0$ where $S (0) = S (f) |_{f = 0} = \int_{- \infty}^{\infty} s (t) e^{- j 2 π f t} d t = \int_{- \infty}^{\infty} s (t) d t$

$if S (0) = 0 \int_{- \infty}^{\infty} s (t) d t = 0$

</math> and $\int_{- \infty}^{t} s (λ) d λ = \int_{- \infty}^{t} ℱ [S (f) - S (f_{0})] d λ$

$\int_{- \infty}^{t} s (λ) d λ = \int_{- \infty}^{t} ℱ [S (f)] d t$

$ℱ^{- 1} [S (f) - S (f_{0})] = \int_{- \infty}^{t} e^{j 2 π f t} d t \int_{- \infty}^{\infty} S (f) d f = \frac{e^{j 2 π f t}}{j 2 π f} \int_{- \infty}^{\infty} S (f) d f =$

$\int_{- \infty}^{t} s (λ) d λ = \int_{\infty}^{\infty} S (f) \frac{e^{j 2 π f t}}{j 2 π f} d f = ℱ^{- 1} [\frac{S (f)}{j 2 π f}]$

Therefore $ℱ [\int_{- \infty}^{t} s (λ) d λ] = \frac{S (f)}{j 2 π f}$

Link title

Navigation menu

Search