A sinusoidal signal of 300Hz frequency,3V amplitude is contaminated with line frequency (50Hz,1V). Design a digital filter including poles and zeros to remove the interference component. Consider the sampling rate at 8KHz.
I guess it's for a homework.
1 - You want to attenuate the 50-Hz component. Zeroes placed at 50 Hz will cancel the 50-Hz component, thus you need to place your zeros at 50 Hz.
2 - You want your 300-Hz signal to be unaffected by the zeros, i.e. you want it to keep the same amplitude and the same phase. If you only added the complex pair of zeroes at 50 Hz, it would increase the amplitude of your 300-Hz signal and affect the phase at 300 Hz also. Therefore, you need to "cancel" the effect of your complex zeroes pair. If you add a complex pole-pair at 49 Hz, then the combined effect of the pole pair at 49 Hz and the zero pair at 50 Hz will almost cancel each other at 300 Hz.
Intuitively, poles and zeros are opposites. At 300 Hz, the distance to the zero pair and the pole pair will be approximately the same, thus they will almost cancel each other out.
3 - I told you at which frequency to place your pair of poles and pair of zeroes, but now you need to select a damping factor for both pairs. This issue is a bit more complicated, but you can safely put the pair of zeroes with a damping factor of 0, i.e. on the complex circle. However, you need to put your pair of poles inside the unit circle, but close to the unit circle since poles outside the unit circle are unstable. Zeroes outside the unit circle are stable, but are not desirable.
4 - If you use this method, you will get a gain of approximately 1 at 300 Hz, but not exactly 1... Since I think this is a homework, this is good enough. In real-life, you would do best to work with the transfer function directly, whether in the s-domain or z-domain. https://en.wikipedia.org/wiki/Band-stop_filter