All transformations assume a lowpass filter of length 2N+1.

#### Lowpass to Lowpass

Consider an ideal lowpass brickwall filter with normalized cutoff
frequency ω_{c1}. By taking the inverse
Discrete Fourier Transform of the ideal frequency response, and clipping
the resulting sequence to length 2N+1, the impulse response is:

where w(n) is the window
vector. The lowpass filter coefficients are tuned to a new cutoff
frequency ω_{c2} as follows:

There is no need to
recompute the window every time you tune the cutoff frequency.

#### Lowpass to Highpass

Assuming a lowpass filter with normalized 6–dB cutoff
frequency ω_{c}, a highpass filter with
the same cutoff frequency can be obtained by taking the complementary
of the lowpass frequency response: H_{HP}(e^{jω})
= 1 — H_{LP}(e^{jω})

Taking the inverse discrete Fourier transform of the above response,
we get the following highpass filter coefficients:

#### Lowpass to Bandpass

A bandpass filtered centered at frequency ω_{0} can
be obtained by shifting the lowpass response:

H_{BP}(e^{jω})
= H_{LP}(e^{j(ω–ω0}))
+ H_{LP}(e^{j(ω–ω0)})

The bandwidth of the resulting bandpass filter is 2ω_{c},
as measured between the two cutoff frequencies of the bandpass filter.
The equivalent bandpass filter coefficients are then:

#### Lowpass to Bandstop

We can transform a lowpass filter to a bandstop filter by combining
the highpass and bandpass transformations. That is, first make the
filter bandpass by shifting the lowpass response, and then invert
in to get a bandstop response centered at ω_{0}.

H_{BS}(e^{jω})
= 1 – (H_{LP}(e^{j(ω–ω0)})
+ H_{LP}(e^{j(ω+ω0)}))

This yields the following coefficients:

#### Generalized Transformation

The transformations highlighted above can be combined to transform
a lowpass filter to a lowpass, highpass, bandpass or bandstop filter
with arbitrary cutoffs.

For example, to transform a lowpass filter with cutoff ω_{c1} to
a highpass with cutoff ω_{c2}, you first
apply the lowpass-to-lowpass transformation to get a lowpass filter
with cutoff ω_{c2}, and then apply the lowpass-to-highpass
transformation to get the highpass with cutoff ω_{c2}.

To get a bandpass filter with center frequency ω_{0} and
bandwidth β, we first apply the lowpass-to-lowpass transformation
to go from a lowpass with cutoff ω_{c} to
a lowpass with cutoff β/2, and then apply the lowpass-to-bandpass
transformation to get the desired bandpass filter. The same approach
can be applied to a bandstop filter.