Figure shows the textbook method of achieving a notch filter by simply taking the difference of lowpass- and highpass-filter outputs. However, a real notch need not actually use a highpass filter.
For example, Figure 1b shows how you can create a highpass filter using only a lowpass filter. This scheme involves subtracting the lowpass filter’s output from the input data, and the total frequency response of the system is highpass.
This scheme ensures that the frequency responses of the highpass and lowpass filters are mirror-symmetric. However, this scheme works only if the lowpass filter introduces no phase shift, that is, if the phase of the lowpass filter is zero. When the phase of the lowpass filter is not zero, you must compensate for the phase shift that this filter introduces into the datapath. You can perform the compensation by putting an allpass filter in the other path before subtracting the signals. You can design digital FIR filters to have exact linear-phase characteristics, and you can mathematically represent the frequency response of a FIR filter.
Because the phase characteristic of the FIR filter is linear (uphi(lom)=((N–1)/2)lom), you can easily design an allpass filter or phase shifter to compensate for the phase shift the filter introduces. This design yields a fixed delay of only N–1/2, where N is the filter order.
Now, you can create the notch filter as Figure 1c shows. Both of the lowpass filters have the same characteristics, so you can further simplify the circuit to use a single lowpass filter (Figure 1d). The resulting notch filter has infinite attenuation at the center frequency, and you can meet all other filter specifications, such as ripple in the passband and bandwidth by designing the lowpass filter accordingly.
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