Common Fourier Transform Properties

Common Fourier Transform Properties#

The Fourier Transform has several fundamental properties that make it a powerful tool for signal analysis. These properties allow us to analyze and manipulate signals in the frequency domain and provide a deeper understanding of how different transformations affect a signal’s spectral content.

1. Linearity Property#

The linearity property states that if $x(t)$ and $y(t)$ are two signals with corresponding Fourier Transforms $X(f)$ and $Y(f)$, then the Fourier Transform of a linear combination of these signals is given by:

\[ \boxed{ \mathcal{F} \{ a x(t) + b y(t) \} = a X(f) + b Y(f) } \]

where $a$ and $b$ are constants. This property is useful for analyzing systems that are linear, such as electrical circuits and control systems.

2. Time Shifting Property#

If a signal $x(t)$ is shifted in time by $t_0$, its Fourier Transform is given by:

\[ \boxed{ \mathcal{F} \{ x(t - t_0) \} = X(f) e^{-j2\pi ft_0} } \]

This property indicates that a time shift in the time domain results in a phase shift in the frequency domain.

3. Frequency Shifting Property#

If a signal $x(t)$ is multiplied by a complex exponential $e^{j2\pi f_0 t}$, its Fourier Transform is shifted in the frequency domain:

\[ \boxed{ \mathcal{F} \{ x(t) e^{j2\pi f_0 t} \} = X(f - f_0) } \]

This property is commonly used in modulation and demodulation of signals in communication systems.

4. Time Scaling Property#

If a signal $x(t)$ is scaled in time by a factor of $a$, its Fourier Transform is given by:

\[ \boxed{ \mathcal{F} \{ x(at) \} = \frac{1}{|a|} X\left( \frac{f}{a} \right) } \]

Time scaling results in an inverse scaling in the frequency domain. This means that compressing a signal in the time domain (making it shorter) results in expanding it in the frequency domain.

5. Conjugate Symmetry Property#

If $x(t)$ is a real-valued signal, then its Fourier Transform $X(f)$ satisfies the conjugate symmetry property:

\[ \boxed{ X(-f) = X^*(f) } \]

This means that the Fourier Transform of a real signal has symmetrical magnitudes and opposite phases at positive and negative frequencies.

6. Parseval’s Theorem#

The Parseval’s theorem relates the total energy of a signal in the time domain to its energy in the frequency domain:

\[ \boxed{ \int_{-\infty}^{\infty} |x(t)|^2 \, dt = \int_{-\infty}^{\infty} |X(f)|^2 \, df } \]

This property is important in signal processing for understanding energy conservation and power spectral density.

7. Duality Property#

If $x(t)$ has a Fourier Transform $X(f)$, then the duality property states that:

\[ \boxed{ \mathcal{F} \{ X(t) \} = x(-f) } \]

This property swaps the roles of the time and frequency domains, allowing us to use the time-domain properties in the frequency domain.

Summary Table of Fourier Transform Properties#

Property	Time Domain	Frequency Domain
Linearity	$a x(t) + b y(t)$	$a X(f) + b Y(f)$
Time Shifting	$x(t - t_0)$	$X(f) e^{-j2\pi f t_0}$
Time Scaling	$x(at)$	$\frac{1}{
Time Reversal	$x(-t)$	$X(-f)$
Duality	$X(t)$	$x(-f)$
Frequency Shifting	$x(t) e^{j2\pi f_0 t}$	$X(f - f_0)$
Convolution	$x(t) * y(t)$	$X(f) \cdot Y(f)$
Differentiation	$\frac{d^n}{dt^n} x(t)$	$(j 2\pi f)^n X(f)$
Integration	$\int_{-\infty}^{t} x(\tau) \, d\tau$	$\frac{1}{j2\pi f} X(f) + \frac{1}{2} \delta(f)$

Property	Time Domain	Frequency Domain
Linearity	\(a x(t) + b y(t)\)	\(a X(f) + b Y(f)\)
Time Shifting	\(x(t - t_0)\)	\(X(f) e^{-j2\pi f t_0}\)
Time Scaling	\(x(at)\)	$\frac{1}{
Time Reversal	\(x(-t)\)	\(X(-f)\)
Duality	\(X(t)\)	\(x(-f)\)
Frequency Shifting	\(x(t) e^{j2\pi f_0 t}\)	\(X(f - f_0)\)
Convolution	\(x(t) * y(t)\)	\(X(f) \cdot Y(f)\)
Differentiation	\(\frac{d^n}{dt^n} x(t)\)	\((j 2\pi f)^n X(f)\)
Integration	\(\int_{-\infty}^{t} x(\tau) \, d\tau\)	\(\frac{1}{j2\pi f} X(f) + \frac{1}{2} \delta(f)\)