# FIR filter design sampling rate

Let's assume we have an $x(n)$ time sequence, whose $f_s$ sample rate is 20 kHz. We are required to design a linear-phase lowpass FIR filter that will attenuate the undesired high-frequency noise beyond 4kHz analog frequency. So we design a lowpass FIR filter and come out with an equation for the unit impulse response $h_{low}(n)$,and assume that our filter design exercise is complete. Sometime later, unfortunately, we learn that the original $x(n)$ sequence's sample rate was not 20kHz, but 40 kHz. What must we do to our lowpass filter's $h_{low}(n)$ coefficients, originally designed based on a 20kHz sample rate, so that they will attenuate $x(n)$'s undesired high-frequency noise when the $f_s$ sample rate is actually 40kHz?

Hint-typical low pass filter's frequency response makes a pair with its unit impulse response which is a sinc function

• do you have any ideas on what you need to do and how to achieve it? – Fat32 Dec 6 '16 at 23:39
• If you don't mind the subtle passband droop, insert zeros between each coefficient and convolve with [1 1] and divide by two, or for more high frequency attenuation of the image, convolve with [1 2 1] and divide by 4. Also could design a half band filter to reject the image that is created by the zero insert and convolve with that: Inserting zeros will replicate the current filter exactly without distortion in it's response from 0 to 10KHz but will create an image of the response in the 10KHz to 20Khz portion of the first Nyquist zone that needs to be filtered out. – Dan Boschen Dec 7 '16 at 0:36
• Please see the hint as well – ROHAN PAUL Dec 7 '16 at 0:57
• @ROHANPAUL If it is only once, how difficult would it be to re-design for the new cutoff frequency? – msm Dec 7 '16 at 2:44
• @ROHAN PAUL. You posted a jpeg image of a multi-part IIR filter problem recently. The problem had a designator of 'Q3', a block diagram of an IIR filter, and mentioned the idea of "120 Hz flicker noise from fluorescent lights contaminating a photodiode signal." ROHAN, I need to find the source of that problem: Did it come from a textbook, college lecture notes, a web site, or just where? If you help me find the origin of that IIR filter problem I will reward you. Don't reply to me here on Sig. Proc. StackExchange, but rather, please send me an e-mail at R_dot_Lyons_@_ieee_dot_org. – Richard Lyons Dec 8 '16 at 9:42