Spider DSA User’s Manual
313
A digital filter can be understood by considering the equation which defines how
the input signal is related to the output signal:
Where x[n] is the current input signal sample, x[n-1] is the previous signal sample
and x[n-N] is the last sample in the series. The series multiplies the most recent
N+1 samples with the associated N+1 filter coefficients. y[n] is the current output
signal and b
i
are the filter coefficients. The number N is known as the filter order;
an N
th
-order filter has (N + 1) terms on the right-hand side. N+1 filter coefficients
are also referred to as “taps”.
This equation illustrates why a higher order filter has a slower response time. It
takes more samples and therefore more time for an event to work its way through
the series until the output is no longer affected by the event. A lower order filter
has fewer coefficients and therefore a faster response time.
The previous equation can also be expressed as a convolution of the filter
coefficients and the input signal, specifically:
The Impulse Response of the filter shows how historical data affects the current
filtered value. The longer the impulse response, the more the older data will affect
the current filtered value. To find the impulse response we set
Where δ[n] is the Kronecker delta impulse. The equation below shows that the
impulse response for an FIR filter is simply the set of coefficients b
n
, as follows
FIR filters are stable because the output is a sum of a finite number of finite
multiples of the input values that can be no greater than
times the largest
value appearing in the input.
Data Windows FIR Filters
In the academic world, hundreds of methods are available to design FIR filters to
meet various criteria. EDM includes the most popular filter design methods: Data
Window and Remez. Both methods are discussed below.
The Data Window FIR Filter Design method is the easiest to understand. The
name "Window" comes from the fact that these filters are created by scaling a
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