How to produce a high-pass filter from a low-pass one?

Question

I already have a low-pass filter and want to remove the DC offset of a signal with a high-pass one. I guess you can subtract a low-pass filtered signal from the original signal and the result is a high-pass filtered signal?

My low-pass filter:

x = input, raw signal
y = output, low-pass filtered signal
n = current sample
n - 1 = previous sample

y(n) = y(n - 1) + (x(n) - y(n - 1)) / alpha

High-pass filter:

x = input, raw signal
y = input, low-pass filtered signal
z = output, high-pass filtered signal
n = current sample

z(n) = x(n) - y(n)

Is this last equation correct?

Answer 1

The result will indeed be a high pass filter. From your difference equation, the transfer function of the low pass filter is

$$H_l(z)=\frac{\beta}{1-(1-\beta)z^{-1}}\tag{1}$$

with $\beta=1/\alpha$. Note that this is actually a leaky integrator, not a classic low pass filter, because its frequency response does not have a zero at Nyquist.

The high pass filter has the transfer function

$$H_h(z)=1-H_l(z)=(1-\beta)\frac{1-z^{-1}}{1-(1-\beta)z^{-1}}\tag{2}$$

which has a zero at DC (i.e. at $z=1$), as it should. However, it is scaled, i.e. its value at Nyquist ($z=-1$) is not $1$, but $2(1-\beta)/(2-\beta)$. So in order to have a 0 dB gain at Nyquist, you need to scale your high pass filter with the inverse of that factor:

$$H'_h(z)=\frac{2-\beta}{2(1-\beta)}H_h(z)=\left(1-\frac{\beta}{2}\right)\frac{1-z^{-1}}{1-(1-\beta)z^{-1}}\tag{3}$$

EDIT:

You can compute the 3dB cut-off frequencies of the low pass filter (1) and of the high pass filter (2) by solving

$$|H_l(e^{j\omega_c})|^2=\frac12\quad\text{and}\quad |H'_h(e^{j\omega_c})|^2=\frac12$$

After some algebra this gives for the low pass filter

$$\cos\omega_c=1-\frac{\beta^2}{2(1-\beta)}$$

and for the high pass filter

$$\cos\omega_c=\frac{1}{1+\frac{\beta^2}{2(1-\beta)}}$$

where $\omega$ is the normalized frequency in radians:

$$\omega=2\pi\frac{f}{f_s}$$

with the sampling frequency $f_s$.

Answer 2

For an LTI system another method to generate a highpass filter impulse response $h_{hp}[n]$, from an existing lowpass filter impulse reponse $h_{lp}[n]$ is the following: $$h_{high}[n]=(-1)^n h_{low}[n]$$

From the DTFT properties we can see that: $$h_{high}[n]=e^{j\pi n} h_{low}[n]$$ $$H_{high}(e^{j\omega})=H_{low}(e^{j(\omega - \pi)})$$

which would produce a highpass filter from a lowpass filter, as a result of the right shift by $\pi$ in its spectrum. Note that this highpass filter will have the exact spectral shape of the prototype lowpass filter, however it won't be a complementary highpass counterpart of the given lowpass filter. Which means their spectral sum $H_{lp}(e^{j\omega}) + H_{hp}(e^{j\omega})$ is not unity. In your example, you would get the complementary filter if you use the definition as: $H_{hp}(e^{j\omega})= 1 - H_{lp}(e^{j\omega})$ assuming both DTFTs to be real (zero phase), if they are not, you may define the highpass filter as $$H_{high}(e^{j\omega})= e^{j\omega d} - H_{low}(e^{j\omega})$$