Signal Reconstruction with Ideal LPF After Sampling

When a continuous-time signal cos(10000πt) is sampled at a frequency of 8kHz and then reconstructed using an ideal low-pass filter (LPF) with a cutoff frequency of 5kHz, the output signal will be a reconstructed version of the original signal, but with a limited frequency range.

The Nyquist-Shannon sampling theorem states that the maximum frequency that can be accurately represented after sampling is half of the sampling frequency. In this case, the maximum representable frequency is 4kHz (half of 8kHz).

Although the ideal LPF has a cutoff frequency of 5kHz, the output signal will only contain frequencies up to 4kHz due to the limitations imposed by the sampling process. Any frequency components above 4kHz in the original signal will be aliased or folded back into the lower frequency range.

Therefore, the output of the ideal LPF will be the reconstructed cos(10000πt) signal with frequencies above 4kHz removed. This means the output signal will have a frequency range from 0Hz to 4kHz. Within this range, the reconstructed signal will retain the same amplitude and phase characteristics as the original continuous-time signal. Frequencies above 4kHz will be completely attenuated or removed by the LPF.