odefiltering - ECE 421 Sum 2010 Notes Set 2 Filter Design...

ECE 421 - Sum 2010 Notes Set 2: Filter Design Using ODEs
1
INTRODUCTION
There are many methods for creating digital filters, and we will discuss some of them
in this course. In this Notes Set we study how to derive a basic digital filter from the
differential equation describing an analog filter. Such a digital filter might not have the best
performance compared to more sophisticated metods, but its derivation is straightforward
and provides a good transition from the analog domain to the digital domain. Concerning
terminology, in this course we use the terms “discrete-time” and “digital” interchangeably.
The basic topics to be discussed in this Notes Set are the following:
Begin with an analog filter (a circuit) design which is known to fulfill a set of desired
analog specifications.
From the analog circuit determine the Ordinary Differential Equation (ODE) relating
filter analog input and analog output.
Substitute a discrete-time approximation for the continuous derivatives in the ODE.
Determine the difference equation which then relates digital filter input and digital
filter output.
Understanding this type of digital filter design will be good preparation for learning later
about other digital filter design techniques. One property of any digital filter is that it gives
information about the output only at discrete points in time. We will discuss some of the
considerations necessary to guarantee that the output of the digital filter we design is an
accurate approximation to the corresponding analog (continuous-time) filter output.

ECE 421 - Sum 2010 Notes Set 2: Filter Design Using ODEs
2
FIRST-ORDER ODE
Let’s first examine how to obtain the ODE relating the analog filter input and output. As an
example of the procedure, we will use the analog filter having the RL -circuit implementa-
tion shown below.
o
v (t)
i
v (t)
i(t)
-
+
-
+
L
R
The signal v
i
t is the input voltage to the filter and v
o
t is the output voltage of the fil-
ter. We now need to obtain the ODE relating v
i
t and v
o
t . One way to do this is use
Kirchoff’s Voltage Law around the loop to obtain the following equations:
v
i
t L
d
dt
i t i t R 0 v
o
t i t R 1 a 1 b
We can solve (1a) for i t and then find v
o
t using (1b). Equation (1a) can be rearranged
as the familiar first-order ordinary differential equation
d
dt
i t
R
L
i t
1
L
v
i
t 2
Define the constants a and b as
a
R
L
b
1
L
3 a 3 b
To develop the general approach also set x t = v
i
t and y t = i t . Using these substitu-
tions and equation (3) in (2) then gives
d
dt
y t ay t b x t 4
Equation (4) relates the input and output for the analog filter. In mathematical work, the
general terms excitation and response are frequently used. In equation (4), we might say
that x t = v
i
t is the excitation and y t = i t is the response.

ECE 421 - Sum 2010 Notes Set 2: Filter Design Using ODEs
3
ANALOG FILTER OUTPUT
An important goal of many digital filter designs is that the digital filter output be “close” to
the same output provided by the original analog filter. Therefore, let’s review one method
for finding analog filter outputs.