A Menagery of Linear Operations

• The following operations are all linear.
• The list indicates devices that performs each of the operations.
  

  Operation		Device
  Scaling	sout = c vin	resistor circuit, (linear) amplifier
  integration	$s_{out}(t) = \int_{-\infty}^t v_{in}(\tau) d\tau$	inductor (coil)
  differentiation	$s_{out}(t) = \frac{dv_{in}(t)}{dt}$	capacitor
  convolution	$s_{out}(t) = \int_{-\infty}^{\infty} v_{in}(\tau) h(t-\tau) d\tau$	general linear system

• Any combination of such operations yields a linear system.
• The convolution operation is the most general operation performed by a linear system.
• The input-output relationship of any linear, time-invariant system can be written as a convolution.
• The function h(t) is called the impulse response of the linear system.
• It characterizes the linear system completely.
• An example of a non-linear system is a squaring-device: s_out(t) = v_in²(t).