EE 233 Laboratory Exercise 01

A leaky integrator equation is a specific differential equation, 
used to describe a component or system that takes the integral of an input, 
but gradually leaks a small amount of input over time. 
It appears commonly in electronics, and hydraulics.

The continuous-time representation is the differential equation of the form

dx/dt + Ax = C

In discrete-time,
a leaky integrator is defined by the difference equation 

y[k] = ay[k - 1] + T x[k], 

where is typically 0 < a < 1.  An integrator that does not leak
has a = 1.  The system is stable for -1 < a < 1.

1. Plot the step response of the discrete-time version of the integrator 
for values a from 0 to 1 in increments of 0.2.  
Plot the step response of the system for values of a from -1 to 0 
in increments of 0.5.  Use T = 0.1 sec.

2. Plot the response of the system with the input being 
a sine wave and a = 0.9. Use a sine wave of frequency
1 rad/sec. Use sampling periods T = 0.1, 0.5, 1 sec.