Consoles 96325.21 DSP Filtering and Equalization
25.21.1 Transversal, Blumlein, or FIR Filters
Yes, Alan Blumlein invented these, too. The train of
samples concept becomes very valuable in DSP. This
type of filter can produce a wide variety of time effects
and frequency response shapes, particularly bandpass
and cutoffs. While the determination of coefficients for
the various filter types is beyond the scope and intent of
this section, the underlying principle is shown in
Fig. 25-130. For each sample period (i.e., every 20μs) a
fresh input sample is inserted at the head of the train; all
the samples move along the train and the oldest one
falls out of the other end and is lost. Each sample is
multiplied by a coefficient specific to that position and
summed in the accumulator with other results from the
other multiplied samples. Each pickoff is subject to a
different coefficient and sum sense (normal or inverse).
The accumulator value is the new output word for that
particular sample time; 20μs later the whole routine
starts over again.
This passing of one set of data (in this case audio)
through another set of data (coefficients) is also called
convolution.
25.21.1.1 Impulse Response
As familiar as we are with using the tool of frequency
response measurement to analyze or describe the
transfer function of a device or circuit, there is an
equally powerful descriptor: the impulse response.
Embracing the impulse response concept aids gaining a
mental picture of how digital filters work. Fig. 25-131A
is what the waveform of a large bell excited by an
impulse could look like: a damped sine wave at the tone
of the bell. (Hardly dissimilar to that from a damped
oscillator, or bandpass filter. Hold that thought in mind.
Actually, looking at the response, it would probably
sound more like the dung of a lamp post, but please
suspend disbelief for now.)
Each of the vertical lines represents the instanta-
neous amplitude of the signal at each sampling period;
this, if you like, is a sample-by-sample digital recording
of the bell’s sound. If we were to play back the samples
at the rate they were recorded, we would hear the bell
thunk again. We now use the bell’s samples’ numeric
values as coefficients in a transversal filter, Fig.
25-131B, and send an impulse (one sample of full posi-
tive amplitude, the rest zero) into the filter; the effect is
exactly the same. As the impulse passes each coeffi-
cient, the bell sound will be reconstructed once more.
There is nothing to stop us from putting real audio
samples into the front of the transversal filter—the
effect will be as if the audio is being played through a
damped bandpass filter at the frequency of the bell. The
bell’s impulse response is impinging itself directly on
the audio passing through the transversal stages. It will
sound as though you’re listening to the audio with your
head stuck up inside that bell. Yes, it’s a filter! In short,
if we can describe a desired filter’s impulse response
and use its samples as coefficients in a transversal filter,
any signal passing through the transversal stages will be
filtered accordingly.
This kind of processing is commonly called FIR
(finite impulse response) filtering. If a transient
(impulse) were encoded and applied to such a filter, the
samples describing it would enter the train of stages.
Output summation contributions occur until they reach
the end. When the last relevant sample has fallen out the
end of the train, no further output samples that have
anything to do with the originally applied transient are
possible. The duration of the transient within the filter is
limited to the lifetime of its samples in the train; they
eventually all leave. The impulse’s existence is finite.
The filter’s length is finite—hence, finite impulse
response.
Intellectually, FIRs are very appealing through their
very simplicity. Unfortunately, this genre of filtering is
rather taxing in current DSP terms since it demands a
lot of processor time for any useful audio filters. As a
rough rule of thumb, to do anything meaningful at a
given frequency the filter must be able to contain a full
cycle of that frequency; to operate at 50 Hz an FIR
would need to be at least 20 ms long, which (assuming a
48 kHz sampling rate) would be about 1000 filter points
long. As mentioned earlier, a 200 MHz part only has
about 4000 cycles of processing available per
sample—this one filter has just eaten about a quarter ofFigure 25-130. Transversal or finite impulse response filter
(FIR).