Problem 4: Let\'s assume we were given an x(n) discrete-time sequence, whose sam
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Question
Problem 4: Let's assume we were given an x(n) discrete-time sequence, whose sample rate is 20KHz, and its X(f) spectral magnitude is shown below. We were asked to design a linear-phase low-pass FIR filter to attenuate the undesired high-frequency noise, as seen in the spectral magnitude plot. Assume we already designed the filter and the frequency magnitude response of our filter is also shown below 10 To be "filtered out -10 (a) -20 10 (t/2) 8 Freq (kHz) dB -40 10 (t/2) 2 Freq (kHz) Sometime later, we unfortunately learned that our original sequence x(n) had actually been obtained at 40KHz sampling rate and not at 20KHz! So do we need to do anything with our low-pass filter coefficients h(k)'s which were originally obtained based on the assumption of 20KHz sampling rate so that our filter still attenuates the high-frequency noise when the sample rate is actually 40KHz? If yes, what? If no, why not? Please discussExplanation / Answer
The relation between continuous and discrete time filters bring about the relation between the sampling frequency and the magnitude of filter coefficients as given below.
Discrete filter h[n] = h(nTs) (continuous filter sampled at every Ts time interval: the frequency sampling at fs .
It is given that the frequency of sampling, later found to be twice of what was thought the signal was sampled at.
From the above relation given h[n] = h(n/fs). If fs is doubled, then the number of coefficients of the FIR filter would be doubled for the same specifications of the cutoff-frequency etc.
i.e. the FIR filter coefficients need to extrapolated by a factor of 2 in order for the filter to have same cutoff frequency and functions as desired. An alternative method is to reduce the sampling rate of the signal by factor of 2 in order for the filter coefficients to remain unchanged and ensure the filter functions as desired.