How to convert a low-pass filter to a band-pass filter

Question

I have a a low pass filter described by the following transfer function:

h[n] = (w_c/Pi) * sinc( n * w_c / Pi ), where is w_c is the cutoff frequency

I have to convert this low-pass filter to a band-pass filter.

Answer 1

You h[n] transforms into a rect in frequency domain. To make it band pass you need to move its central frequency higher.

To do this, multiply h[n] by exp(j*w_offset*n), where w_offset is the amount to shift. If w_offset is positive, then you shift towards higher frequencies.

Multiplication in time domain is convolution in frequency domain. Since exp(j*w_offset*n) turns into impulse function centred on w_offset, the multiplication shifts the H(w) by w_offset.

See Discrete Time Fourier Transform for more details.

Note: such a filter will not be symmetric about 0, which means it will have complex values. To make it symmetric, you need to add h[n] multiplied by exp(-j*w_offset*n):

h_bandpass[n] = h[n](exp(j*w_offset*n)+exp(-j*w_offset*n))

Since cos(w*n) = (exp(j*w*n)+exp(-j*w*n))/2 we get:

h_bandpass[n] = h[n]cos(w_offset*n)

This filter then has purely real values.