東京大学 情報理工学系研究科 電子情報学専攻 2016年度 専門 第5問

Author

Josuke

Description

Let us denote the Fourier transform of the signal \(f(t)\) as \(F(\omega)\), where \(t\) and \(\omega\) represent a time variable and an angle frequency, respectively.

(1) Show the definition of the Fourier transform \(F(\omega)\) of the signal \(f(t)\). Also, explain the difference between the Fourier transform and the Fourier series expansion.

(2) Explain why \(|F(\omega)|^2\) represents the power spectrum, i.e., the power at a certain angle frequency of \(\omega\).

(3) Derive the following Parseval's theorem in the Fourier transform and determine \(k\).

\[ \int_{-\infty}^{\infty}|f(t)|^2\text{d}t = k\int_{-\infty}^{\infty}|F(\omega)|^2\text{d}\omega, k\text{ is a real constant.} \]

You may use \(|f(t)|^2 = f(t)\overline{f(t)}\), \((\overline{f(t)} \text{ is the complex conjugate of }f(t))\). You may also use the Fourier transform of the following convolution integrals

\[ \int_{-\infty}^{\infty}f(t - \tau)g(\tau)\text{d}r = k'\int_{-\infty}^{\infty}F(\omega)G(\omega)e^{j\omega t}\text{d}\omega , \]

\(j\) is the imaginary unit and \(k'\) is a real constant.

You may include \(k'\) when answering \(k\).

(4) Explain the physical meaning of the Parseval's theorem in the Fourier transform.

Kai

(1)

\[ F(\omega) = \int_{-\infty}^{\infty}f(t)e^{-j\omega t}\text{d}t \]

Fourier series can be applied to periodic signal. Fourier transform can be applied to non-periodic signal.

(2)

\[ |F(\omega)|^2 = F(\omega)\overline{F(\omega)} = (\text{Real}\{F(\omega)\})^2 + (\text{Image}\{F(\omega)\})^2 \]

Thus \(|F(\omega)|^2\) represents the power of certain angle frequency \(\omega\).

(3)

\[ \begin{aligned} f(t) * g(t) &= \int_{-\infty}^{+\infty}f(t - \tau)g(\tau)\text{d}\tau = \int_{-\infty}^{+\infty}g(t - \tau)f(\tau)\text{d}\tau \\ f(0) * g(0) &= \int_{-\infty}^{+\infty}f(-\tau)g(\tau)\text{d}\tau = \int_{-\infty}^{+\infty}g(- \tau)f(\tau)\text{d}\tau \\ \end{aligned} \]

if \(g(\tau) = \overline{f(-\tau)}\)

\[ \begin{aligned} G(\omega) &= \int_{-\infty}^{+\infty}g(t)e^{-j\omega t}\text{d}t \\ &= \int_{-\infty}^{+\infty}\overline{f(-t)}e^{-j\omega t}\text{d}t \\ \overline{G(\omega)} &= \int_{-\infty}^{+\infty}f(-t)e^{-j\omega t} \\ &= \int_{-\infty}^{+\infty}f(t)e^{-j\omega t}\text{d}t = F(\omega) \qquad G(\omega) = \overline{F(\omega)} \end{aligned} \]

So

\[ \int_{-\infty}^{\infty}|f(t)|^2 = k'\int_{-\infty}^{\infty}|F(\omega)|^2 \text{d}\omega \]

(4)

The energy in time domain equals to \(k'\) times energy in frequency domain.