Signals and LTI Systems

Impulse and Step

$$\delta[n] = \begin{cases} 1 & n = 0 \\ 0 & \text{otherwise} \end{cases}, \qquad u[n] = \begin{cases} 1 & n \ge 0 \\ 0 & n < 0 \end{cases}, \qquad \delta[n] = u[n] - u[n - 1], \qquad u[n] = \sum_{m=-\infty}^{n}\delta[m] = \sum_{k=0}^\infty \delta[n - k]$$ $$x[n]\delta[n - n_0] = x[n_0]\delta[n - n_0], \qquad \sum_{n=-\infty}^\infty x[n]\delta[n] = x[0]$$

In continuous time, $u(t) = \int_{-\infty}^t \delta(\tau)\, d\tau$, $\frac{du(t)}{dt} = \delta(t)$, $\int_{-\infty}^\infty x(t)\delta(at)\, dt = \frac{1}{|a|}x(0)$, $\delta(t) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{j\omega t}\, d\omega$, and the ramp and parabola are $r(t) = \int_{-\infty}^t u$ and $p(t) = \int_{-\infty}^t r = \frac{t^2}{2}$ for $t \ge 0$, 0 otherwise.

Energy and Power

$$E_\infty = \int_{-\infty}^\infty |x(t)|^2\, dt = \lim_{T \to \infty}\int_{-T}^T |x(t)|^2\, dt, \qquad P_\infty = \lim_{T \to \infty}\frac{1}{2T}\int_{-T}^T |x(t)|^2\, dt$$

and in discrete time $E_\infty = \sum_{n=-\infty}^\infty |x[n]|^2$, $P_\infty = \lim_{N \to \infty}\frac{1}{2N + 1}\sum_{n=-N}^N |x[n]|^2$. A finite energy signal has zero power, and a signal with finite nonzero power has infinite energy. Periodic bounded signals are power signals, with $P_T = \frac{1}{T}\int_{-T/2}^{T/2} |x(t)|^2\, dt$.

Even, Odd and Periodic Signals

$$x_{ev}(t) = \frac{x(t) + x(-t)}{2}, \qquad x_{odd}(t) = \frac{x(t) - x(-t)}{2}, \qquad x(t) = x_{ev}(t) + x_{odd}(t)$$

$x$ is bounded if $|x(t)| \le K$, and periodic if $x(t) = x(t + T)$ for all $t$, with fundamental period $T_0$ and $\omega_0 = \frac{2\pi}{T_0}$ (in discrete time the period is a positive integer). The phasor $|c|e^{j(\omega_0 t + \phi)}$ is periodic with $T_0 = \frac{2\pi}{|\omega_0|}$. A sum of phasors is periodic iff $\omega_1 = k\omega_0$, $\omega_2 = l\omega_0$ for integers $k, l$, i.e. $\frac{\omega_1}{\omega_2} = \frac{k}{l}$ is rational, and then $T_0 = \frac{2\pi}{\omega_0}$.

A discrete-time phasor $e^{j\omega_0 n}$ is periodic iff $\omega_0$ is a rational multiple of $2\pi$. For $\omega_1 = \frac{m_1}{N_1}2\pi$ and $\omega_2 = \frac{m_2}{N_2}2\pi$ the sum is periodic with period $N = N_1 N_2$.

System Properties

• Causal The output depends only on the past and present input.
• Memoryless The output depends only on the present input.
• Time-invariant $x(t - t_0) \to y(t - t_0)$. To check, let $v(t) = x(t - t_0)$ and compare $S[v(t)]$ with $y(t - t_0)$.

Feedback: $y(t) = S_1[x(t) + S_2[y(t)]]$.

Convolution

An LTI system is described by its impulse response $h$,

$$y[n] = x[n] * h[n] = \sum_{k=-\infty}^\infty x[k]h[n - k]$$

with $\delta[n] * h[n] = h[n]$, $x[n] * \delta[n - n_0] = x[n - n_0]$, $u[n] * h[n] = \sum_{k=-\infty}^n h[k]$, and $y[n - n_0] = x[n] * h[n - n_0] = x[n - n_0] * h[n]$. For systems in parallel the impulse responses add, and in series they convolve.

• Causal iff $h[n] = 0$ for $n < 0$ ($h$ right-sided).
• Memoryless iff $h[n] = c\,\delta[n]$.
• Stable iff $\sum_{n=-\infty}^\infty |h[n]| < \infty$ (absolutely summable).
• Invertible iff there is $h_I$ with $h[n] * h_I[n] = \delta[n]$.

Useful sums: $\sum_{n=0}^{N-1}a^n = \frac{1 - a^N}{1 - a}$, and $\sum_{n=0}^\infty a^n = \frac{1}{1 - a}$ for $|a| < 1$.

Eigenfunctions

Exponentials are eigenfunctions of LTI systems. E.g. for a causal, stable LTI system and $x[n] = e^{\sigma_0 n}$, $\sigma_0 \ge 0$, $y[n] = c\,e^{\sigma_0 n}$ with

$$c = H(\sigma_0) = \sum_{k=0}^\infty h[k]e^{-\sigma_0 k}$$

First-order systems $y[n] + ay[n - 1] = bx[n]$ is LTI with $h[n] = (-a)^n b\, u[n]$: causal, memoryless iff $a = 0$, stable iff $|a| < 1$. In continuous time $\dot{y}(t) + ay(t) = bx(t)$ has $h(t) = be^{-at}u(t)$.

Transforms

If $x(t)$ has Fourier transform $X(\omega)$, then $X(t)$ has Fourier transform $2\pi x(-\omega)$. A periodic signal with Fourier series coefficients $a_k$ has

$$X(\omega) = \sum_{k=-\infty}^\infty 2\pi a_k\, \delta(\omega - k\omega_0)$$

For Laplace transforms, a causal system has a right-sided ROC, and a system is stable iff the ROC contains the $j\omega$ axis.

Feedback With $H_1$ in the forward path and $H_2$ in negative feedback, the closed loop is $\frac{H_1}{1 + H_1 H_2}$, i.e. $G(s) = \frac{Y(s)}{X(s)} = \frac{H(s)}{1 + gH(s)}$ for a feedback gain $g$. E.g. for $H(s) = \frac{1}{s - 1}$,

$$G(s) = \frac{1/(s - 1)}{1 + g/(s - 1)} = \frac{1}{s - 1 + g}$$

which is stable if $g > 1$.