Signals basics formulas

Every signal splits this way, uniquely; the DT form is identical with $n$ in place of $t$. Example: $u(t)=\frac{1}{2}+\frac{1}{2}\mathrm{sgn}(t)$, so $u_e(t)=\frac{1}{2}$ and $u_o(t)=\frac{1}{2}\mathrm{sgn}(t)$ for $t\ne 0$.

$\mathrm{sgn}(0)=0$, so $\mathrm{sgn}(t)=2u(t)-1$ holds for $t\ne 0$. Derivative chain: $\frac{d}{dt}r(t)=u(t)$, $\frac{d}{dt}u(t)=\delta (t)$, $\frac{d}{dt}\mathrm{sgn}(t)=2\delta (t)$.

For $\tau >0$: rect has width $\tau$ and area $\tau$; tri has half-width $\tau$ (base $2\tau$), peak 1 and area $\tau$. Also $\mathrm{rect}(t)\ast \mathrm{rect}(t)=\mathrm{tri}(t)$. Some texts write the triangle with total base $\tau$; read the definition given.

Requires $x$ continuous at $t_0$. When $t_0$ sits exactly on a limit, the value depends on a convention, so that case is left out. The impulse has unit area, ${\int }_{-\infty }^{\infty }\delta (t)dt=1$, and $u(t)={\int }_{-\infty }^t\delta (\tau )d\tau$.

With $a=-1$: $\delta (-t)=\delta (t)$, so the impulse is even. Hence ${\int }_{-\infty }^{\infty }x(t)\delta (at+b)dt=\frac{1}{|a|}x\left(-\frac{b}{a}\right)$.

The sum runs over the roots $t_k$ of $f(t)=0$, and every root must be simple ($f^{\prime }(t_k)\ne 0$); at a repeated root $\delta (f(t))$ is not defined. Example: $\delta (t^2-9)=\frac{1}{6}\left[\delta (t-3)+\delta (t+3)\right]$.

Requires $x^{(n)}$ continuous at $t_0$. For $n=1$ (the doublet): ${\int }_{-\infty }^{\infty }x(t){\delta }^{\prime }(t-t_0)dt=-x^{\prime }(t_0)$. The doublet ${\delta }^{\prime }(t)$ is odd, and $x(t){\delta }^{\prime }(t)=x(0){\delta }^{\prime }(t)-x^{\prime }(0)\delta (t)$, not just $x(0){\delta }^{\prime }(t)$.

$\delta [0]=1$ is an ordinary finite value, unlike $\delta (t)$. $\delta [an]=\delta [n]$ for every nonzero integer $a$: there is no $1/|a|$ factor in DT.

The DT form holds sample by sample, since only the $k=n$ term survives. The CT form is the sifting property with $\delta (t-\tau )=\delta (\tau -t)$, valid wherever $x$ is continuous. Both are the starting point for convolution.

Map each breakpoint of $x$ this way. A support $[t_1,t_2]$ becomes $\left[\frac{t_1-b}{a},\frac{t_2-b}{a}\right]$, with the ends swapped when $a<0$. This check does not depend on any order of operations.

Shifts change none of these. Time scaling divides area and energy by $|a|$ but leaves the power, average and RMS value of a periodic signal unchanged (its period becomes $T_0/|a|$). Amplitude scaling by $k$ multiplies $E$ and $P$ by $|k{|}^2$.

Take $p/q$ in lowest terms. For non-constant $x_1$, $x_2$ with fundamental periods $T_1$, $T_2$ the converse also holds: if $T_1/T_2$ is irrational, $x_1+x_2$ is not periodic. The LCM is always a period of the sum, and it is the fundamental period unless components cancel: $x_1=\cos t+\cos 2t$ and $x_2=-\cos t$ both have period $2\pi$, but their sum $\cos 2t$ has period $\pi$. For a sum of sinusoids with nonzero amplitudes at distinct positive frequencies, the fundamental angular frequency is the GCD of the component frequencies, ${\omega }_0=\gcd ({\omega }_1,{\omega }_2,\dots )$, and $T_0=2\pi /{\omega }_0$; a DC term does not change this.

${\Omega }_0$ is in rad/sample. Equivalently, $N_0$ is the smallest positive integer $N$ for which ${\Omega }_{0}N$ is a multiple of $2\pi$; for ${\Omega }_0\ne 0$ that is $N_0=2\pi m/{\Omega }_0$ with the smallest positive integer $m$ that makes it an integer. The same test applies to $e^{j{\Omega }_{0}n}$. The one exception is a sequence that is identically zero, such as $\sin (\pi n)$, which trivially has period 1. A sum of periodic sequences is always periodic: $\mathrm{LCM}(N_1,N_2)$ is a period, and it is the fundamental period unless components cancel.

For integers $k$ and $n$. Distinct DT sinusoids need only $\Omega \in (-\pi ,\pi ]$ (or $[0,\pi ]$ for real cosines). The fastest oscillation is at $\Omega =\pi$: $\cos (\pi n)=(-1{)}^n$.

Normalised energy, as if $x$ were a voltage across $1\Omega$. Use $|x{|}^2$, not $x^2$, for complex signals.

The window is symmetric about zero, which is why $Au(t)$ has power $A^2/2$. If $E<\infty$ then $P=0$; if $0<P<\infty$ then $E=\infty$.

Any one full period, from any starting point. For a periodic signal, $P=X_{\text{rms}}^2$.

For a finite sum with real amplitudes, whether or not $x$ is periodic. Terms at the same frequency must first be merged into one sinusoid; a negative frequency is the same frequency, since $\cos (-\omega t+\theta )=\cos (\omega t-\theta )$. For ${\sum }_k{C}_k{e}^{j{\omega }_{k}t}$ with distinct ${\omega }_k$ (here $\omega$ and $-\omega$ do count as different), $P={\sum }_k|C_k{|}^2$.

This is phasor addition, $A\angle \varphi =A_1\angle {\theta }_1+A_2\angle {\theta }_2$. The powers add directly only when $\cos ({\theta }_1-{\theta }_2)=0$; for example $a\cos \omega t+b\sin \omega t$ has amplitude $\sqrt{a^2+b^2}$.

Over any one full period of a real waveform $v(t)$. $V_{\text{rms}}^2$ is the average power in $1\Omega$: the RMS value is the DC level that would dissipate the same power in a resistor.

The AC part $v(t)-V_{\text{avg}}$ has zero mean, so it is orthogonal to the DC part. For $v=V_0+{\sum }_k{V}_k\cos ({\omega }_{k}t+{\theta }_k)$ with distinct positive ${\omega }_k$: $V_{\text{rms}}=\sqrt{V_0^2+{\sum }_k{V}_k^2/2}$. For a waveform with nonzero mean, the ratio $V_{\text{ac,rms}}/V_{\text{avg}}=\sqrt{{\text{FF}}^2-1}$ (FF as defined below, with the full-period average) is the ripple factor used for rectifier outputs.

Valid for any straight-line piece, whatever its slope or duration. Weight each segment's mean (or mean square) by its fraction of the period and add. A flat piece at level $c$ is the case $a=b=c$.

For a waveform with nonzero mean, $V_{\text{avg}}$ is the full-period average. For a zero-mean AC wave that average is zero, so $V_{\text{avg}}$ in FF means the average of $|v(t)|$ over a period, which for a sine, square or symmetric triangle wave equals the average over a positive half-cycle. Sine: FF $=\frac{\pi }{2\sqrt{2}}\approx 1.11$, CF $=\sqrt{2}\approx 1.414$. Square: FF $=$ CF $=1$. Triangle: FF $=\frac{2}{\sqrt{3}}\approx 1.155$, CF $=\sqrt{3}\approx 1.732$. Half-wave rectified sine: FF $=\frac{\pi }{2}\approx 1.571$, CF $=2$. Full-wave rectified sine: the same as the sine, FF $\approx 1.11$, CF $=\sqrt{2}$.