# Parallel Cascaded FIR Filters

I have a relatively simple question. I know if you connect multiple LTI systems in parallel, overall system's frequency response is the sum of each LTI system's frequency response. Therefore, what would be the frequency response of the overall system if two FIR Low-Pass filters are connected in parallel? Let's say one has a cut-off frequency of 50 Hz and the other one has 100 Hz. I'm guessing it would have a frequency response of 1 between 50-100 Hz and 2 below 50 Hz.

Similarly, how would you construct a band-pass filter from a pair of low-pass filters? I am totally stumped with this one.

## 1 Answer

As you've mentioned, you can simply add the impulse responses, and, consequently, the frequency responses of LTI systems connected in parallel. In general, frequency responses are complex-valued, so you can't just add their magnitudes. However, if their phase responses are identical, you can add their magnitudes. A special case are two linear-phase FIR filters of the same length, which have the same (linear) phase response, so their magnitudes can be added.

If both filters approximate a magnitude of $1$ in their passbands, and a magnitude of zero in their stopbands, then the resulting filter will approximate a magnitude of $2$ where their passbands overlap, a magnitude of $1$ where the passband of one filter overlaps with the stopband of the other, and a magnitude of zero where both stopbands overlap. Note that FIR filters cannot have a constant magnitude (unless they have only $1$ filter tap), so all constant values are just approximated.

Now that you've figured out the resulting response when the outputs of the two filters are added, maybe you can come up with a creative way to connect the outputs of two linear phase FIR low pass filters (of the same length) with different cut-off frequencies to implement a bandpass filter.