Low-Pass Filters - Digital Filter Design Solutions - page 61

For example, the identifier LPF99F0.3 means a low-pass filter of length 99 with
normalized cut-off frequency at 0.3. The cut-off frequency of a digital filter is the
frequency at which the magnitude of the frequency response is 0.5. Thus, in this
example, the frequency response exhibits a value of 0.5 at F = 0.3.

3.3.2 Impulse Response

The impulse response is given in the second graph (first row, second column).
This shows a graph of the unnormalized filter coefficients or impulse response
amplitude against index k . The index, k is very commonly used to represent the
k th sample in time, but in general, could represent the k th sample in any
measurement space.

3.3.3 Step Response

The step response is the response of the filter to a step input function. This is
shown in the third plot (second row, first column) with amplitude plotted against
index k . Note that the step response is normalized, whereas the impulse response
is not.

3.3.4 Pass-Band Ripple

The pass-band ripple is shown in the last figure where the magnitude of the
frequency response is usually magnified by a factor of 10 3 . This gives the nature
of the filter in the pass band.

3.3.5 Filter Coefficients Table

The filter coefficients are listed for all low-pass filters except for those of lengths
255 and 511; this has been done chiefly for space reasons. All filters are archived
in the accompanying compact disc (CD) to full precision in ASCII format.
The coefficients listed on each page are to 4 decimal places and are closely
linked to 10-bit processing. In fact, the 10-bit quantized filter is easily created by
rounding the full-precision filter coefficients as given, however; there will be a
loss in filter order because many coefficients will be rounded to zero, with
consequential losses in attenuation and pass-band characteristics as noted above.
Note that application of the filter in any precision requires that the filtered result
be normalized with K = 2 9 to avoid undue amplification of the output signal.
Moreover, only the first L /2 + 1 coefficients are given as the filter is symmetric
about the L /2 + 1 coefficient. Note further that T in the performance table is the

sampling period and f N is the Nyquist frequency (= f s /2). The square brackets [

∆

∆ T,

f N ] in the units column of the performance features table mean the units of ∆ T or

f N .