Final answer:
The student must first calculate the impulse response of an ideal low-pass filter and then apply a frequency transformation to convert it into a high-pass filter considering the order N=8 to create a causal and time-centered filter. The resulting FIR high-pass filter's impulse response is derived by subtracting the low-pass prototype from a delta function representing an infinite impulse response approximation.
Step-by-step explanation:
The student is asking to calculate the impulse response h[n] of an ideal high-pass FIR filter with a cut-off frequency of 1kHz, for a system sampled at 8 kHz. to design this filter, a frequency transformation is to be applied to the impulse response of a correctly calculated low-pass filter. Given the filter's order, N=8, the low-pass prototype filter must first be defined before being transformed into a high-pass filter. an ideal low-pass FIR filter has an impulse response given by:
hlp[n] = 2Bs*sinc(2Bsn), where Bs is the normalized cut-off frequency and sinc(x) = sin(πx)/(πx). Now, applying the frequency transformation involves replacing n by n - (N-1)/2 to ensure the filter is causal and centered in time. Next, to convert it to a high-pass filter the impulse response is modified using hhp[n] = δ[n] - hlp[n], where δ[n] is the Kronecker delta function.
Therefore, the ideal impulse response of the high-pass filter can be obtained. However, it's important to note that in practice, the ideal high-pass FIR filter will be approximated since the true ideal filter requires an infinite impulse response.