Page 42 - Analog and Digital Filter Design

Introduction 39

so any sudden change at the input only affects the output for the duration of its
passage through the delay line. A nonrecursive filter is said to have a Finite
Impulse Response (FIR). The FIR filter is inherently stable, but truncating
the coefficients used as multiplying factors can lead to a nonideal frequency
response, such as a passband that is not flat (Le., one that has high levels of
ripple).

The Path to Digital Filter Design

To design a Finite Impulse Response (FIR) filter, the required number of delay
elements (or filter taps) must be calculated. This is determined by a number of
factors: the window function to be used, the sampling clock frequency, and
the ratio of passband frequency to stopband frequency. Once the number of
taps is known, the multiplying coefficients are found using the sin(x)/.v envelope.
Each coetficient is then multiplied by the chosen window function. If a high-
pass, bandpass, or bandstop filter is required. frequency scaling equations are
also used to convert the response from the lowpass prototype.

The design of Infinite Impulse Response (IIR) filters are based on analog
designs. The signal paths are arranged so that the output depends on both the
input signal and the output signal. Input signals are fed into a delay line. and
multiplying factors are used in the same way as in FIR filters to provide a feed-
forward source. In addition, output signals are fed into a second delay line, and
multiplying factors are used to provide a feedback source. These two sources are
then combined to produce the output. The required analog frequency response
is transformed using simple equations to give the feed-forward and feedback
filter coefficients.

Exercises

1.1 The signal power into a filter at a particular frequency is 6mW and the
output power is 0.3mW. What is the attenuation of the filter at this fre-
quency? If the input voltage is 2V, what will be the output voltage?
I .2 A second-order filter with a cutoff frequency of 1 MHz gives a signal
attenuation of 12dB at 2MHz. What will be the attenuation at 4MHz?
1.3 If the filter described in Exercise 1.2 has an input signal level of
IOmW, what will be the output level at 2MHz and 4MHz?
1.4 A simple RC lowpass filter has an input voltage of 10V What will be
the voltage across (a) the resistor and (b) the capacitor at the -3dB
point?