Fourier Transforms

Fourier Series

consider a periodic signal \(x(t)\) with period \(T_0\)

\[\begin{align} a_n &= \frac{1}{T_0} \int_{T_0} x(t) e^{-jn\omega_{0}t}dt\\ x(t) &= \sum_{k=-\infty}^{\infty} a_k e^{jk\omega_0 t} \end{align}\]

Parseval’s Theorem

\[\begin{align} \frac{1}{T_0} \int_{T_0} |x(t)|^2 dt = \sum_{k=-\infty}^{\infty} |a_k|^2 \end{align}\]

Fourier Transform

For a signal \(x(t)\), if the integration \(\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt\) exists for all \(-\infty < \omega < \infty\).

\[\begin{align} X(\omega) &= \int_{-\infty}^{\infty} x(t) e^{-j \omega t} dt\\ x(t) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} d\omega \end{align}\]

•   \(x(t) \text{ is real function } \quad \iff \quad X(\omega) = \overline{X(-\omega)}\)

\[\begin{align} \overline{X(-\omega)} &= \overline{\int_{-\infty}^{\infty} x(t) e^{j\omega t} dt}\\ &= \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt\\ &= X(\omega) \end{align}\]

• Since Fourier transform and inverse Fourier transform is dual, we also have

\[X(\omega) \text{ is real function } \iff x(t) = \overline{x(-t)}\]

• for example, if \(x(t)\) is real and even function, then we know \(X(\omega)\) is also real and even function.

Parseval’s Theorem

\[\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2 d\omega\]

Discrete-Time Fourier Series

for a discrete-time signal \(x[n]\) with period \(N\), let \(\Omega_0 = 2\pi/N\)

\[\begin{align} a_k &= \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-jk\Omega_0 n}\\ x[n] &= \sum_{k=0}^{N-1} a_k e^{jk\Omega_0 n} \end{align}\]

Parseval’s Theorem

\[\frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2 = \sum_{k=0}^{N-1} |a_k|^2\]

Sampling Theorem

• If \(x[n]\) is a sampling (with sampling frequency \(f_s = 1/T_s\)) of \(x(t)\), such that \(N \cdot T_s\) is the period of \(x(t)\), may or may not be the fundamental period.
• If \(x(t)\) does not includes any tones with frequency higher or equal than \(f_s/2\)
• Then DT FS gives exactly the same coefficients as \(x(t)\) for frequencies \(0, f_s/N, \dots, (N/2 - 1) f_s/N\).

Discrete Fourier Transform

\[X[k] = \sum_{n=0}^{N-1} x[n] e^{-jk \frac{2\pi}{N} n}\]

function psd = power_spectrum(signal, Fs)
    N = length(signal);
    spectrum = fft(signal)/N;
    psd = abs(spectrum).^2;
    psd = psd(1:N/2+1);
    psd(2:N/2) = 2*psd(2:N/2);
    f = (0:N/2)*Fs/N;
    semilogy(f, psd);
    title("SSB Power Spectrum Density");
    xlabel("f (Hz)");
end

Fourier Transform Table

Non-Causal Signal

\(f(t)\) with \(\, t \in R\)	\(F(\omega)\)	ROC?
\(f(t)\)	\(\int_{-\infty}^{\infty} f(t)e^{-j\omega t}\)
\(e^{-a\vert t \vert}, \quad a > 0\)	\(\dfrac{2a}{a^2+\omega^2}\)
\(\delta(t)\)	\(1\)
\(1\)	\(2\pi \delta(\omega)\)