A.15  Fourier Analysis: Properties

In the sequel, it is assumed that $X(f)$, $Y\; (f)$ and $Z(f)$ are the Fourier transforms of $x(t)$, $y(t)$ and $z(t)$, respectively. A pair (time / frequency) is denoted by $\iff$. The following discussion assumes the Fourier transform, but the properties are valid for all four Fourier tools with subtle distinctions.

Linearity: if a signal $x(t)\; =\; \alpha y(t)\; +\; z(t)$ is obtained by multiplying $y(t)$ by a constant $\alpha$ and summing the result to $z(t)$, its transform is $X(f)\; =\; \alpha Y\; (f)\; +\; Z(f)$. Linearity can be stated as:

Linearity can be decomposed into two properties: a) homogeneity and b) additivity, which correspond to the properties $\alpha x(t)\; \iff \; \alpha X(f)$ and $x(t)\; +\; y(t)\; \iff \; X(f)\; +\; Y\; (f)$, respectively.

Time-shift:

Scaling:

Time-reversal (scaling with $a\; =\; -1$):

Complex-conjugate:

Combined time-reversal and complex-conjugate:

Multiplication:

Frequency-shift:

Convolution:

Duality:

Example: $\mathrm{\text{rect}}(t)\; \iff \mathrm{\text{sinc}}(f)$, then by duality $\mathrm{\text{sinc}}(t)\; \iff \mathrm{\text{rect}}(-f)\; =\mathrm{\text{rect}}(f)$ (because $\mathrm{\text{rect}}(\cdot )$ is an even function).

Energy and power conservation (Plancherel / Parseval theorem).
For energy signals:

For periodic (power) signals with fundamental period $T_0$:

where $c_k$ are the coefficients of the Fourier series of $x(t)$.

Autocorrelation (Wiener-Khinchin theorem):