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So I understand that a type 3 filter is not suitable for a highpass filter design, but is there any reason why it isnt suitable for a lowpass filter?

So ultimately, can a type 3 linear-phase FIR filter be used to design a lowpass filter? Why or why not?

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    $\begingroup$ I know the definitions in Oppenheim and Schafer, but you should spell out what you mean by a "Type 3, linear-phase FIR". $\endgroup$ – robert bristow-johnson May 2 '17 at 6:02
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A type 3 FIR filter has odd symmetry and an odd number of taps. For this reason it has frequency response zeros at $\omega=0$ (DC) and $\omega=\pi$ (Nyquist), corresponding to transfer function zeros at $z=1$ and $z=-1$:

$$H(1)=\sum_{n=-M}^{M}h[n]=0\\ H(-1)=\sum_{n=-M}^{M}(-1)^nh[n]=0$$

where $N=2M+1$ is the filter length. So it can neither be used as a high pass nor as a low pass filter.

Apart from that, the phase shift of $\pi/2$ caused by the odd symmetry is usually undesirable for frequency-selective filters. You could use such a filter for implementing Hilbert transformers or differentiators.

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