In the system shown below, two continuous time fnctions (t) and r2(t) where X1(S) anod X2(S2) represent their corresponding Fourier transforms are added together

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In the system shown below, two continuous time fnctions (t) and r2(t) where X1(S) anod X2(S2) represent their corresponding Fourier transforms are added together, and the result y(t) is sampled by a periodic impulse train e., ideal sampling. Then the sampled signal is passed gain G-T through the reconstruction filter, ideal low-pass filter with cut-off frequency Fe Fy2 and Tdeal ReionstNc Her ( LPF) Po H(n) ?c a) If the sampling frequency F-500 Hz., sketch both the sampled, y.(t), and reconstructed, yr(t), signals in the frequency domain (approximately) and comment on aliasing. b) If the sampling frequency F, 600 Hz., repeat part (a) c) Now, if the reconstruction filter is replaced by a more realistic (practical) low pass filter, whose Fourier transform shown below, repeat part (b) and comment on the reason why the Nyquist sampling frequency F 2F rather than simply F-2Fma H(L) 2. 2. d) Repeat part (c) for F 1000 Hz

The Correct Answer and Explanation is :

To solve this problem, let’s break it down step by step. Based on the given problem, we need to analyze the frequency domain behavior of the signals before and after sampling and reconstruction. Here’s how to approach each part:

a) Sampling Frequency Fs=500 HzF_s = 500 \, \text{Hz}

1. Signal Description:

  • Two continuous-time signals x1(t)x_1(t) and x2(t)x_2(t) are given in the time domain.
  • Their corresponding Fourier transforms X1(s)X_1(s) and X2(s)X_2(s) represent the frequency content of each signal.
  • The signals are added together, creating the composite signal y(t)=x1(t)+x2(t)y(t) = x_1(t) + x_2(t).
  • The signal y(t)y(t) is then sampled by a periodic impulse train with frequency FsF_s, leading to discrete-time signal ys(t)y_s(t).
  • The sampled signal is passed through an ideal reconstruction filter with cutoff frequency FrF_r, which ideally reconstructs the continuous signal from the discrete samples.

2. Frequency Domain Sketch for Sampling:

  • The sampling process introduces replicas (aliases) of the frequency spectrum of the continuous signal at multiples of the sampling frequency.
  • The original frequency components X1(s)X_1(s) and X2(s)X_2(s) will replicate at Fs=500 HzF_s = 500 \, \text{Hz}.
  • If the highest frequency component of X1(s)X_1(s) and X2(s)X_2(s) exceeds half of the sampling frequency, aliasing will occur. This is because frequencies above Fs/2F_s / 2 fold back onto lower frequencies.

3. Aliasing Observation:

  • If the highest frequency of the signal exceeds Fs/2=250 HzF_s / 2 = 250 \, \text{Hz}, aliasing occurs. The spectra of x1(t)x_1(t) and x2(t)x_2(t) will overlap, causing distortion in the reconstructed signal.
  • If the signals are well below 250 Hz250 \, \text{Hz}, there will be no aliasing.

b) Sampling Frequency Fs=600 HzF_s = 600 \, \text{Hz}

1. New Frequency Domain Behavior:

  • With the sampling frequency increased to 600 Hz, the aliasing behavior improves. The Nyquist frequency is now Fs/2=300 HzF_s / 2 = 300 \, \text{Hz}, so if the signals have frequencies lower than 300 Hz, there will be no aliasing.
  • The frequency spectra of the signal will still replicate at multiples of Fs=600 HzF_s = 600 \, \text{Hz}, but now there is a greater separation between the original spectrum and the first aliasing copy, so the chances of aliasing decrease.

c) Replacing the Ideal Low-pass Filter with a Practical One

1. Practical Reconstruction Filter:

  • A more realistic reconstruction filter has a frequency response that does not have a perfect brick-wall cutoff. This means that it may not fully attenuate high frequencies beyond the Nyquist frequency.
  • The practical filter would have a frequency response that rolls off gradually instead of abruptly, and it might allow some aliasing components to pass through.
  • Why Fs=2FmaxF_s = 2F_{\text{max}} instead of Fs=2FmaxF_s = 2F_{\text{max}}?
    • The ideal Nyquist rate assumes perfect reconstruction and no signal distortion. However, in practice, some non-idealities (like the finite roll-off of filters) require a higher sampling rate to prevent aliasing. This ensures that all frequency components can be captured and properly reconstructed.

d) Sampling Frequency Fs=1000 HzF_s = 1000 \, \text{Hz}

1. Higher Sampling Frequency:

  • With a sampling frequency of 1000 Hz1000 \, \text{Hz}, the Nyquist frequency is 500 Hz500 \, \text{Hz}, which is much higher than the previous cases.
  • The spectral replicas will be further apart, minimizing aliasing even more, and ensuring accurate signal reconstruction after filtering through both the ideal or practical low-pass filter.

Explanation Summary:

  • The key to preventing aliasing is ensuring that the sampling frequency is greater than twice the highest frequency of the signal.
  • In practical scenarios, filters have non-ideal characteristics that make it necessary to sample at a higher rate than the theoretical Nyquist rate to ensure accurate reconstruction.
  • For higher sampling frequencies (like 1000 Hz), aliasing becomes less of an issue because the spectral copies of the signal are spaced far apart, reducing overlap and distortion during reconstruction.

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