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A low-pass single pole infinite impulse response (IIR) filter is transformed into a band pass filter using simple mathematics!
A previous blog described the frequency response of the single pole IIR filter and how it can be a substitute for a moving average filter.
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Reviewing the Low Pass Filter
A previous blog described how a specific implementation of a single pole IIR filter can be used to approximate a low pass moving average filter. The impulse response of this filter is defined as
which may also be referred to as an exponential moving averager or a leaky integrator. The impulse response in (1)
is simplified by setting
The frequency response for the low pass filter (LPF) (3) is
Transformation into Band Pass Filter
The LPF single pole IIR filter can be transformed into a band pass filter (BPF) through complex frequency shifting. The center frequency of the band pass filter is defined to be and the frequency is shifted according to the substitution
The frequency response for the BPF filter is therefore
The LPF frequency response (4) is substituted into (6),
The frequency response is represented using the output and the input
which is rearranged into
and simplified as
The impulse response is obtained by applying the inverse Z-transform
which is more commonly written as
The feedback filter coefficient for the LPF is however the bandpass version includes a complex weight .
The following plots show the frequency response for the band pass single pole IIR with . Figure 1 gives the magnitude of the frequency response and Figure 2 gives the phase of the frequency response. Compare the plots against the LPF in the previous blog post.
A low-pass single pole IIR filter can be transformed into a band pass through complex frequency shifting. The impulse response for the IIR filter is modified by using a complex exponential as the feedback filter coefficient.