Fourier Transform Properties: Duality & Multiplication

Duality

• Duality property: If $x(t) \xrightarrow{FT} X(f)$, then $X(t) \xrightarrow{FT} x(-f)$.
• Example: Since $rect(t) \xrightarrow{FT} sinc(f)$, then $sinc(t) \xrightarrow{FT} rect(f)$.
• Example: Since $\delta(t) \xrightarrow{FT} 1$, then $1 \xrightarrow{FT} \delta(f)$.

Fourier Transform of Periodic Signals

• For a periodic signal $\tilde{x}(t)$ with period $T_0$ and Fourier Series $\tilde{x}(t) = \sum_{n = -\infty}^{\infty} a_n e^{j 2 \pi \frac{n}{T_0} t}$, the Fourier Transform is $\tilde{X}(f) = \sum_{n = -\infty}^{\infty} a_n \delta \left( f - \frac{n}{T_0} \right)$.

Multiplication Property

• Multiplication Property: If $x(t) \xrightarrow{FT} X(f)$ and $y(t) \xrightarrow{FT} Y(f)$, then $x(t) y(t) \xrightarrow{FT} X(f) * Y(f)$.
• Multiplication in the time domain corresponds to convolution in the frequency domain.
• Example: For $y(t) = x(t) \cos (2 \pi f_0 t)$, $Y(f) = \frac{1}{2} \left[ X(f - f_0) + X(f + f_0) \right]$.
• Multiplication by a cosine shifts the spectrum in the frequency domain.

Parseval's Theorem

• Parseval's Theorem: $\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$.
• Energy in the time domain is equal to the energy in the frequency domain.
• The last step in the proof of Parseval's theorem is valid only when $x(t)$ is real-valued.

Fourier Transform of Derivatives

• If $x(t) \xrightarrow{FT} X(f)$, then $\frac{d}{dt} x(t) \xrightarrow{FT} j 2 \pi f X(f)$.
• For the nth derivative, $\frac{d^n}{dt^n} x(t) \xrightarrow{FT} (j 2 \pi f)^n X(f)$.
• Also, $t x(t) \xrightarrow{FT} \frac{1}{-j 2 \pi} \frac{d}{df} X(f)$.