FILTERING
GG313 Lecture 27
December 1, 2005
A FILTER is a device or function that allows certain material
to pass through it while not others. In electronics and signal
analysis, a filter is a device or function that changes the
shape of the phase and/or amplitude spectrum in the
frequency domain, i.e. it attenuates energy at some
frequencies relative to others, and it may also make
considerable changes in phase, causing delays or advances
in the time domain.
A LINEAR TIME-INVARIENT (LTI) filter always behaves the
same way regardless of the input or the time. For example a
filter that attenuates 60 Hz energy by 48 dB, ALWAYS
attenuates energy at 60 Hz by 48 dB.
A LTI filter’s response is completely specified by its IMPULSE
RESPONSE - the output when the input is an impulse.
A filter is CAUSAL if the output does not anticipate the input.
A zero-phase filter (one that does not cause any delay)
cannot be physically reliasable (except for a unit impulse).
For example, consider a filter with the following impulse
response:
h=[.2 .2 .2 .2 .2]
This filter is symmetric about its center (zero lag) and it will
spread any signal at time t(i) over times t(i-2) to t(i+2). It
thus sees AHEAD, and it can not be built with physical
components. So it is noncausal. We can make this filter
causal by adding zeros at the front, adding a 2-step delay:
h=[0 0 .2 .2 .2 .2 .2]
Now it spreads out the signal at t(i) from t(i) to t(i+4), and it
does not see ahead.
Smoothing filters. Often you will have a power spectrum that is
very noisy with large changes from one spectral estimate to
another. In the case below it just looks like spikes.
First, we take the log to see the “real” spectrum, and note
that there is an underlying structure:
IF we want the underlying structure, we can convolve
the power spectrum with a 25-point boxcar (each point
=1/25) to obtain the heavy line below:
and then exponentiate back to linear amplitude:
This type of boxcar filter is a moving average smoother. It is
the same as taking a 25-point mean, moving one point, and
taking another.
Which domain you are in - time or frequency - is not
important, smoothing filters work in both domains.
Note that the transfer function of this smoothing filter is a sinc
function - not a particularly good filter for separating signals
with different frequencies. For that, we want a boxcar in the
FREQUENCY domain, so the impulse response in the TIME
domain will be a sinc function:
This will be a low-pass filter, with infinite length, corner d.
Unless we want processing to take an infinite amount of
time, we need to truncate the sinc function at some
reasonable length in the time domain. What does
truncation imply? - multiplying the impulse by a boxcar of
some length. What does this do in the frequency domain?
- convolves the boxcar with a sinc function:
The cost of truncating the sinc function in the time domain
is some jitter in the “pass and stop bands”, and a lesssteep slope between. The longer the sinc function kernel,
the better.
Note that this is a non-causal filter, but it is excellent for
frequency separation.
When is it important to use a causal filter?
Consider the following seismogram:
No trouble determining when the
signal starts - an important
feature of a good seismogram.
What happens when we use it for
input to a sinc filter:
The filtered seismogram looks
fine, with a sharp onset, but look
what happened to the start time:
The red curve is the filtered one.
Because the filter “sees” ahead
of the signal, the start time is not
correct.
What does a causal filter kernal look like? A causal lowpass filter, such as might result from an electronic filter, is an
exponential:
Causal R-C type low-pass
filter impulse response
Since this filter starts at zero time, it does not see ahead.
The red curve is filtered with the
causal filter. Note that the start
time is preserved.
FIR Filters. The filters we have been discussing are all
finite impulse response filters. These filters are defined
by a kernel that is a fixed length time series that gets
convolved with another time series to obtain a filtered
output. Alternatively, the Fourier Transforms of the filter
kernel can be multiplied by the Fourier Transform of the
data series and inverse transformed to obtain the filtered
signal. Let’s try one.
GENERATE a random ~Gaussian time series by the
following:
y=(rand(1,512)-.5).*(rand(1,512)-.5).*(rand(1,512)-.5);
subplot(1,2,1);
plot(y)
subplot(1,2,2);
periodogram(y);
Generate a 7-point FIR smoothing filter and apply it.
Also change subplot to “subplot( 2,2,-)”
h=[1/7 1/7 1/7 1/7 1/7 1/7 1/7]; smoothing filter
out=conv(y,h); convolution
subplot(2,2,3);
plot(out)
subplot(2,2,4);
periodogram(out);
Run it
Generate a new filter by changing the MIDDLE value of h to a
negative impulse equal in size to the sum of the other points:
change all the subplot calls to subplot(3,2,-);
hh=[1/7 1/7 1/7 -6/7 1/7 1/7 1/7];
outh=conv(y,hh);
subplot(3,2,5);
plot(outh);
subplot(3,2,6);
periodogram(outh);
run it
Edit the plots to get reasonable y axis limits.
There are many other ways to make high pass filters from
low pass, but this one works, and it generates a high pass
filter that has the same corner frequency as the low pass
filter.
IIR filters. Another type of filter has INFINITE IMPULSE
RESPONSE, aslo called a RECURSIVE filter. It is designed
using something called the z-transform, which we don’t have
time to get into, and it is applied differently from a FIR filter.
IIR filters are applied in a way more analogous to how filters
are applied in the analog world than FIR filters. The earth
cannot “convolve”.

Consider the function:
out(n)  a0 x(n)b1out(n 1)
This is not a simple convolution, because the output depends
on previous OUTPUTS.
Program this:
n=128;
x=zeros(1,n);
x(1)=1;
a0=1.
b1=.5;
out(1)=a0*x(1);
for ii=2:n
out(ii)=a0*x(ii)+b1*out(ii-1);
end
plot(out)
Notice that the input is an impulse. The output looks just like
our causal filter from before, but we’re only doing 2
multiplications per output point, and the impulse resplonse
goes on forever.
What happens if the sign of b1 is negative?
Note that a general IIR filter will be of the form:
out(n)  a0 x(n) a1 x(n 1) a2 x(n  2)....
b1out(n 1) b2 out(n 1)...
IIR filters are very fast relative to FIR filters, but they are
somewhat noisier.