Signals and Systems

Transform-domain signal properties

A transform pair converts an operation in one domain into an algebraic rule in another, and the questions in this group test a small set of rules that must be applied exactly. For the z-transform, the basic pairs are a^n u[n] \leftrightarrow \frac{z}{z-a} and n a^n u[n] \leftrightarrow \frac{az}{(z-a)^2}. Adding them gives (n+1)a^n u[n] \leftrightarrow \frac{z^2}{(z-a)^2}, so a double pole at z=a in a causal transform points to a linear factor in n, while a pole at z=a^2 describes a different sequence. Read the pole locations exactly before matching a pair.

For the continuous-time Fourier transform, X(0)=\int x(t)\,dt is the area under the signal, and |X(\omega)| \le \int |x(t)|\,dt for every \omega, with equality at \omega=0 whenever x(t)\ge 0. The inverse transform at t=0 gives x(0)=\frac{1}{2\pi}\int X(\omega)\,d\omega, so a spectrum equal to e^{-\omega^2/2} yields x(0)=\frac{1}{\sqrt{2\pi}}\approx 0.399. An equation of the form dX/d\omega=-\omega X is separable, and fixing X(0) determines this Gaussian shape.

The derivative of the transform at the origin satisfies \frac{dX}{d\omega}\big|_{\omega=0}=-j\int t\,x(t)\,dt. A real even signal has an even transform, so this derivative vanishes at the origin.

A rectangular spectrum of unit height over |\omega|<W_0 inverts to \frac{\sin(W_0 t)}{\pi t}, whose value at the origin is W_0/\pi. Its area stays equal to X(0)=1 while its width shrinks, so it tends to an impulse as W_0\to\infty.

For discrete-time signals, the DTFT is the z-transform evaluated on |z|=1, so it exists only when the region of convergence contains the unit circle. A region such as 2/3<|z|<3 includes the unit circle, but |z|>3 does not. Sidedness alone does not decide convergence.

A signal that is bounded and vanishes outside a finite interval has a Laplace transform that converges for every complex s, so its region of convergence is the whole s-plane.

The DFS synthesis sum x[n]=\sum_k a_k e^{j2\pi kn/N} requires the coefficients to be periodic with period N. A signal built from a constant and one exponential has a_0 as its mean value and the exponential's coefficient as its single harmonic term.

Decimation y[n]=x[2n] keeps only the even-indexed samples. In the z-domain, Y(z)=\frac{1}{2}\left[X(z^{1/2})+X(-z^{1/2})\right], because the two terms cancel the odd-indexed samples. Replacing z by z^2 instead describes upsampling, so the two operations must not be confused.

LTI system properties and cascade

A system is linear if superposition and scaling hold, and time-invariant if a shift of the input produces an identical shift of the output. A memoryless operation such as y(t)=\max(0,x(t)) violates superposition, because the maximum of a sum is not the sum of the maxima, so it is nonlinear. Since the rule has no explicit dependence on t, it is time-invariant.

Time-invariance is checked by seeing whether the kernel depends only on t-\tau, and causality by seeing whether the output at time t uses only inputs with \tau\le t. The integral \int_{-\infty}^{t} e^{-(t-\tau)}x(\tau)\,d\tau satisfies both conditions, so it describes a causal LTI system, namely convolution with e^{-t}u(t).

The impulse response of a cascade of LTI systems is the convolution of the individual responses, which in the z-domain is the product of the transfer functions. Multiplying (z+z^{-1}) by (1+z^{-1}) gives z+1+z^{-1}+z^{-2}, so the cascade has four unit impulses at consecutive shifts.

A system whose impulse response is the unit step u(t) is an ideal integrator. It is not BIBO stable, because u(t) is not absolutely integrable. A bounded step input produces the ramp t\,u(t), which is unbounded.

Convolution of finite-duration signals

Convolution with a finite-duration kernel is a finite sum of shifted copies of the input. In discrete time, if the impulse response is nonzero only at n=0,1,2, the output is y[n]=x[n]+x[n-1]+x[n-2], a three-term moving sum.

When one signal is a unit-height rectangle on [a,b], the convolution with another signal y is z(T)=\int_{T-b}^{T-a} y(s)\,ds. The output is therefore an integral of y over a sliding window, and its derivative is z'(T)=y(T-a)-y(T-b). The maximum occurs where these two sampled values of y are equal, which is the point where the window covers the largest accumulated area.

The integral of a convolution equals the product of the two individual integrals, because setting \omega=0 in Y(\omega)=X_1(\omega)X_2(\omega) gives \int y\,dt=\left(\int x_1\,dt\right)\left(\int x_2\,dt\right). The answer can therefore be found from the two areas alone, without computing the full convolution.

Sinusoidal response of LTI system

For a stable LTI system with transfer function H(s), a constant input A produces the steady-state output A\,H(0), and a sinusoid A\cos(\omega_0 t) produces A\,|H(j\omega_0)|\cos\left(\omega_0 t+\angle H(j\omega_0)\right). A sum of a constant and a sinusoid is therefore scaled term by term: the DC gain multiplies the constant, and the magnitude response at the sinusoidal frequency multiplies the sinusoid.

Frequency selection follows the same principle. Multiplying a sinc of cutoff W by 2\cos(\Omega t) produces a band-pass response with unit gain on \Omega-W<|\omega|<\Omega+W. Components inside this band pass unchanged, and components outside it are removed.

Impulse response from differential equation

For a first-order system written as a\,y'(t)+b\,y(t)=c\,x(t), the transfer function is H(s)=\frac{c/a}{s+b/a}. The causal impulse response is therefore h(t)=\frac{c}{a}e^{-(b/a)t}u(t). The pole sits at s=-b/a, and the unit step u(t) enforces causality for a system at rest.