System types; output as convolution

Types of systems

• Linear system: obeys superposition, $a x_1 + b x_2 \to a y_1 + b y_2$. Example: an ideal amplifier $y = 5x(t)$, filters, cables.
• Non-linear system: does not obey superposition; creates new frequencies. Example: square-law device $y = x^2(t)$ used in modulators and detectors, a diode, a saturated amplifier.
• Causal system: output depends only on present and past inputs; $h(t) = 0$ for $t < 0$. All real-time systems are causal, e.g. $y(t) = x(t-1)$. A system such as $y(t) = x(t+1)$ is non-causal (ideal filters are non-causal).
• Time-invariant system: a delay of the input causes the same delay of the output, $x(t - t_0) \to y(t - t_0)$; its parameters do not change with time, e.g. a fixed RC filter. A system like $y(t) = t\,x(t)$ or a time-varying fading channel is time-variant.

Communication channels and filters are usually modelled as linear time-invariant (LTI); modulators and detectors are deliberately non-linear or time-varying.

Proof: output = input convolved with impulse response

Let $h(t)$ be the response of an LTI system to $\delta(t)$.

Step 1: represent the input as a sum of impulses (sifting property):

$$x(t) = \int_{-\infty}^{\infty}x(\lambda)\,\delta(t - \lambda)\,d\lambda$$

i.e. $x(t)$ is a continuous sum of impulses located at $t = \lambda$ with strength $x(\lambda)d\lambda$.

Step 2: use time invariance: $\delta(t - \lambda) \to h(t - \lambda)$.

Step 3: use homogeneity: $x(\lambda)\delta(t-\lambda)d\lambda \to x(\lambda)h(t-\lambda)d\lambda$.

Step 4: use additivity (superposition) over all $\lambda$:

$$y(t) = \int_{-\infty}^{\infty}x(\lambda)\,h(t-\lambda)\,d\lambda = x(t) * h(t)$$

This is the convolution integral. With $\mu = t - \lambda$ it can also be written $y(t) = \int h(\mu)x(t-\mu)d\mu = h(t) * x(t)$.

For a causal system with input starting at $t = 0$, the limits reduce to $y(t) = \int_0^t x(\lambda)h(t-\lambda)d\lambda$. In the frequency domain the result becomes $Y(f) = X(f)H(f)$.