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    • Finite Impulse Response Filter - Overall Idea

    Finite Impulse Response filters are designed to attenuate, pass, etc. a desired set of frequencies using a difference equation that has a finite impulse response. They are typically used to separate or restore signals.

    Filters in practice

    High level gameplan

    Our input signal is a digital signal x[n]. This signal has a known sampling time sr. The image is passed through a filter with a frequency response of our choice, and then produces an output y[n].

    Filtering is based on two fundamental ideas:

  • Convolution in the time domain is equal to multiplication in the frequency domain. - See LSI/LTI
  • The fourier transform of the impulse response is the frequency response - See Impulse Response

    • These ideas allow us to design a frequency response of interest (One that multiplies in the frequency domain to do what we want). Then, we can implement it in the time domain using convolution with the impulse response!

    Italian Trulli The power of FIR is that the Filter (in this case a low pass filter LPF) can be designed! This lets us control time/frequency characteristics. The FIR is "finite" because its impulse response has a determined length. This necessarily means that the output y[n] must be stable. (This is the case because computation of this operation in the time domain is done with Convolution which slides the filter impulse response like a window, multiplying by the input. Since the window is finite, the "sliding" operation will eventually not overlap the input, and the output will be zero)

    Practical Implementation

    Filter design - Window- Sinc

    We can start with a desired frequency response for our filter. The desired FR has good properties when multiplied by our input(E.G low, band, high pass). Here is an example LPF: Italian Trulli We can find our impulse response from the frequency response by taking the inverse fourier transform (equation below) $$h(t) = \frac{1}{2\pi}\int^{+inf}_{-inf}X(w)e^{jwt}dw$$ Where $$w=2\pi f$$ However, for a rectangular pulse, we will find that our given Impulse Response is a continuous, sinc function in the time domain. This is not possible to implement since sinc functions have infinite tails, look at negative time, and aren't discrete. We can fix this issue by shifting the response to be in positive time, discritizing, and cutting off tails. However, this introduces high frequency lobes in the frequency domain due to truncation discontinuities. This process is shown below: Italian Trulli We can fix these side lobes (to some extent) with windowing. Intuitively, this is smoothing out the truncation transition, getting rid of some of the high frequency characteristics that accompany it.

    Filtering

    Things to consider:
  • Sampling Frequency - complexity/accuracy tradeoff
  • Filter Length - complexity/accuracy tradeoff
  • Windowing function - Hamming, Hann, Blackman, Rectangular, etc.
  • Frequency properties - What frequencies are attenuated in the frequency domain
  • Time-domain properties - rise time, overshoot, settling time, etc.

    • When implementing, there are two main steps:

  • Design a filter online - T-Filter
  • Implement the Convolution in code. The shifting operation in convolution is very expensive. But, this can be fixed with a circular buffer! Keep a pointer to the oldest sample, when a new sample is measured, replace the oldest sample and the second oldest now becomes oldest!