Example 1

A somewhat different system has no "a" coefficients. Consider the difference equation

Because this system's output depends only on current and previous input values, we need not be concerned with initial conditions. When the input is a unit-sample, the output equals 1q 1 q for n∈0…q-1 n 0 … q 1 , then equals zero thereafter. Such systems are said to be FIR (Finite Impulse Response) because their unit sample responses have finite duration. Plotting this response (Figure 1) shows that the unit-sample response is a pulse of width qq and height 1q 1 q . This waveform is also known as a boxcar, hence the name boxcar filter given to this system. (We'll derive its frequency response and develop its filtering interpretation in the next section.) For now, note that the difference equation says that each output value equals the average of the input's current and previous values. Thus, the output equals the running average of input's previous qq values. Such a system could be used to produce the average weekly temperature ( q=7 q 7 ) that could be updated daily.