Page 39 - Analog and Digital Filter Design

36 Analog and Digital Filter Design

processing on the signal passing through such that it causes a sudden reduction
in the output signal level relative to the input signal level as the frequency is
changed.

The reason why the “brick wall” filter cannot be built is because of the rela-
tionship between the time and frequency domains. Just as a voltage step func-
tion (a sudden change in the time domain) has frequency components that
extend across a wide band, a step function in the frequency domain has voltage
components that extend across a wide period of time. The frequency domain
can be considered to cover both positive and negative frequencies, so a 1 kHz
sine wave can be represented by a pair of spectral lines at +I kHz and -1 kHz.
The step frequency response will, by reciprocity, have time domain components
at positive and negative time, relative to the event. Since a response cannot occur
before an event has taken place (i.e., negative time), the step frequency response
cannot exist.

Digital FIR filters make use of the impulse-response relationship by taking
samples of the analog input signal and passing these through a multistep delay
line. At each step in the delay line the signal is used as the input to a multiplier:
the other input to the multiplier is a fixed value. The fixed values for each
multiplier are arranged so that the array overall has the equivalent of a sampled
sin(s)l.u envelope. The output of every multiplier is then summed to produce the
filter’s output. A single input pulse will produce a sin(x)lx envelope at the
output. A single pulse has energy at all frequencies, and the sin(x)lx envelope
has the spectral energy of the filter’s frequency response. Thus a sampled analog
signal fed into the FIR filter will be filtered in the frequency domain response
due to pulse shaping in the time domain.

The impulse response can be shortened (truncated) by making extreme values
equal to zero, symmetrically on either side of the response peak. The frequency
response is degraded by truncating the impulse response, particularly due to the
sudden change to zero values. However, modifying the values to give a smoother
response by shaping using a window results in a frequency response that is closer
to the desired “brick wall.” Windowing a truncated sin(x)/x envelope is illus-
trated in Figure 1.16.